Data Mining Approach, Data Mining Cycle • Algorithm and Problem Solving • Algorithm and Matrix Modeling
- Part A - Part A
Algorithm of Statistics - Part A Algorithm of Curricular Process - Part A Algorithm of Option Pricing and Hedging - Part A Applications in Statistical Investigation - Part B Applications in Higher Education Process - Part C Applications in Financial Derivatives - Part D
ISBN 978-80-8166-000-9
Educational & Didactic Communication 2013 Vol.1 – Algorithms as Significant Result of Data Mining Approach Monolingual English Version
DIDAKTIS 2014. First edition. No part of the present monograph may be reproduced and distributed in any way and in any form without express permission of the author and of the Publishing House Didaktis The publisher and author will appreciate possible comments concerning the work. They may be forwarded to the addresses of the publisher and author presented below.
The publisher: Publishing House DIDAKTIS Hýrošova 4, 811 04 Bratislava, Slovakia www.didaktis.sk
Global Author of Monograph and Its Conception: Assoc. Prof. RNDr. Premysl Zaskodny, CSc., Emy Destinove 17, CZ-370 01 Ceske Budejovice, Czech Republic e-mail:
[email protected] Co-Authors of Individual Chapters: Premysl Zaskodny, Dang Thi Thu Hien, Vilem Fasura, Dominika Masna, Adam Vlcek, Barbora Vesela, Miroslava Bartonova, Vladislav Pavlat, Ivan Havlicek, Martin Pasta, Vladimir Risky, Martin Soucek, Miroslav Sebest
Affiliation of Global Author: The University of South Bohemia, Ceske Budejovice, Czech Republic The University of Finance and Administration, Prague, Czech Republic
Reviewers: Mgr. Veronika Adamcikova, Slovakia RNDr. Jan Tarabek, Ph.D., Slovakia
On line presentation: http://csrggroup.org, www.didaktis.sk ISBN 978-80-8166-000-9
Foreword The monograph “Educational & Didactic Communication 2013 (Vol.1: Algorithms as Significant Result of Data Mining Approach)” is a follow-up to earlier monographs presenting the results of data mining approach in the field of educational communication of science. The theory of educational communication is based on the Doyle´s content pedagogy and the Brockmeyer´s communicative conception. Our intention in the monograph “Educational & Didactic Communication 2013 (Vol.1: Algorithms as Significant Result of Data Mining Approach)” is to carry on the tradition created by monographs Educational & Didactic Communication 2010, 2011, 2012. The monograph “Educational & Didactic Communication 2010” specified the complex and partial data mining tools in statistics, physics and psychology education. The monograph “Educational & Didactic Communication 2011” specified the curricular process and its association with educational data mining. The monograph “Educational & Didactic Communication 2012 (Vol.1: Educational Data Mining and Its Application)” dealt with the applications of educational data mining in statistics and physics. The monographs “Educational & Didactic Communication 2012” and “Educational & Didactic Communication 2013” are also the result of the papers from the “1st and 2nd International e-Conference on Optimization, Education and Data Mining in Science, Engineering and Risk Management (OEDM-SERM 2011, OEDM-SERM 2012)”. The monograph “Educational & Didactic Communication 2013 (Vol.1: Algorithms as Significant Result of Data Mining Approach)” is consisted of four parts: Part A (The Algorithm of Problem Solving in Data Mining Approach) was processed by P.Zaskodny. Part B (The Algorithm of Problem Solving in Statistics) was processed by Dang Thi Thu Hien, V.Fasura, D.Masna, A.Vlcek. Part C (The Algorithm of Problem Solving in Curricular Process) was elaborated by B.Vesela and M.Bartonova. Part D (The Algorithm of Problem Solving in Option Pricing and Hedging) was elaborated by P.Zaskodny, V.Pavlat, I.Havlicek, M.Pasta, V.Risky, M.Soucek, M.Sebest. The monograph “Educational & Didactic Communication 2013 (Vol.1: Algorithms as Significant Result of Data Mining Approach)” is operating not only with educational data mining approach (see Part C) but also with general data mining approach (see Part B and Part D). The conception of this monograph represents an attempt to find the joint base of educational and general data mining approach (see Part A). Such joint base can be given by the detection of algorithms enabling problem solving. The monographs “Educational & Didactic Communication” are the expression of our cooperation with significant Slovakian scientist Pavol Tarabek who has co-instituted the Curriculum Studies Research Group and who has been in the course of many years the intellectual co-promoter of scientific activities, above all in the area of educational data mining. Unfortunately, our thanks to Pavol Tarabek can be pronounced only in memoriam. Premysl Zaskodny, Global author of monograph
CONTENTS of Monograph Parts and Monograph Chapters
Foreword Author of Foreword: P.Zaskodny
Part A: The Algorithm of Problem Solving in Data Mining Approach…….5 Author of Part A: P.Zaskodny
Part B: The Algorithm of Problem Solving in Statistics……………………49 Authors of Part B: Dang Thi Thu Hien V.Fasura D.Masna A.Vlcek
Part C: The Algorithm of Problem Solving in Curricular Process………135 Authors of Part C: B.Vesela M.Bartonova
Part D: The Algorithm of Problem Solving in Option Pricing and Hedging……………………………………180 Authors of Part D: P.Zaskodny V.Pavlat I.Havlicek M.Pasta V.Risky M.Soucek M.Sebest
Epilogue………………………………………………………………………229 Author of Epilogue: P.Zaskodny
Contents of Part A: The Algorithm of Problem Solving in Data Mining Approach……………..5 (Author of Part A: P.Zaskodny) A1. Data Mining as Problem Solving and Its Structure……………………………………7 A1.1. Meaning of Data Mining....…………………………………………………………….………..7 A1.2. Basic Structural Elements………………………………………………………………………8 A1.3. Detailed Structural Description………………………………………………………………...9
A2. Data Mining Approach as Realization of Data Mining Cycle………………………..10 A2.1. Data Mining Approach and Structure of Data Mining Cycle..………………………….…10 A2.2. Data Mining Cycle in Literature..……………………………………………………………11
A3. Tools of Data Mining…………………………………………………………………...12 A3.1. Definition and Classification of Tools in the light of Algorithms.……………………….…12 A3.2. Illustration and Substantiation of Data Mining Tools.……………………………………..12
A4. Complex Tool of Educational Data Mining – Curricular Process…………………..14 A4.1. Procedure of Complex Tool Seeking in area of Science Education……………..………...14 A4.2. Data Preprocessing in Science Education……………..…………………………………….14 A4.3. Data Processing in Science Education……………..………………………………………...15 A4.4. Complex Tool of Data Mining in Science Education…………….…………………………17 A4.5. Application of Complex Tool (Curricular Process) in Science Education……………..…17
A5. Significant Partial Tool of Data Mining – Analytical Synthetic Modeling………….19 A5.1. Universal Analytical Synthetic Modeling…………………………………………...……....19 A5.2. Illustration of Analytical Synthetic Modeling…………………………………………...…20
A6. Significant Partial Tool of Data Mining – Matrix Modeling and Main Diagonal of Matrix………………………………………………………………………………...24 A6.1. Representation of Matrix Model of Complex Problem Solving.………………………..…24 A6.2. Description of Matrix Model and Its Main Diagonal.………………………..…………….24
A7. Algorithm of Statistics………………………………………………………………….27 A7.1. Analytical Synthetic Model of Statistics…………………………………………………….27 A7.2. Groups of Definition Line Elements of Matrix Model of Statistics……………………….31 A7.3. Individual Stages of Algorithm of Statistics, Recommended Literature…………………32
A8. Algorithm of Curricular Process………………………………………………………33 A8.1. Possibility of Collaboration of European and Anglo-American Conception…………..…33 A8.2. Analytical Synthetic Model of Science Education………….………………………………33 A8.3. Groups of Definition Line Elements of Matrix Model of Science Education………….…36 A8.4. Individual Stages of Algorithm of Curricular Process………….………………………….37 A8.5. Recommended Literature……………………………………………………………………38
A9. Algorithm of Option Pricing and Hedging……………………………………………41 A9.1. Basic Concepts – Financial Derivatives, Pricing, Hedging………………………………....41 A9.2. Analytical Synthetic Model of Option Pricing and Hedging……………………………….42 A9.3. Groups of Definition Line Elements of Matrix Model of Option Pricing and Hedging…44 A9.4. Recommended Literature…………………………………………………………………….45
A10. Role of Algorithms in data Mining Approach……………………………………….46 A10.1. Brief Description of Role…………………………………….……………………………...46 A10.2. Illustration of Data Mining Approach Algorithms…………………………………….…47
Contents of Part B: The Algorithm of Problem Solving in Statistics…………………………….49 B1. Investigation of Comparison between Prices of Natural 95 and Diesel……………...50 (Author of Chapter B1: Dang Thi Thu Hien) B1.1. Introduction…………………….…….…………………………….……………………51 B1.2. Descriptive Part of Investigation of Statistical Sign SS1……………………..…….…52 B1.3. Non-parametric Testing for Statistical Sign SS1…………………….…….………….56 B1.4. Descriptive Part of Investigation of Statistical Sign SS2…………………….…….…59 B1.5. Non-parametric Testing for Statistical Sign SS2.…………………….……………….62 B1.6. Measurement of Statistical Dependences…………….………………………………..63 B1.7. Conclusion…………………….…………………………………………………………67 B1.8. Literature…………………….………………………………………………………….67
B2. Investigation of Price Movements of Selected Stocks in The S&P500 within One Month………………………………………………………………………………68 (Author of Chapter B2: V.Fasura) B2.1. Introduction…………………….……………………………………………………….70 B2.2. Descriptive Part of Investigation of Statistical Sign SS1…………………….…….…71 B2.3. Descriptive Part of Investigation of Statistical Sign SS2…………………….…….…78 B2.4. Non-parametric Testing for Statistical Sign SS1…………………….……………….81 B2.5. Non-parametric Testing for Statistical Sign SS2…………………….……………….83 B2.6. Measurement of Statistical Dependences……………………………………………..84 B2.7. Conclusion………………………………………………………………………………89 B2.8. Literature………………………....…………………………………………………….89
B3. Comparison between Percentage Price Movements of Coca Cola Company and PepsiCoCompany within Relevant Financial Market…………………………...90 (Author of Chapter B3: D.Masna) B3.1. Introduction……………………..……………………………………………………...91 B3.2. Descriptive Part of Investigation of Statistical Sign SS1…………….………..……..92 B3.3. Descriptive Part of Investigation of Statistical Sign SS2…………….………..……..99 B3.4. Non-parametric Testing for Statistical Signs SS1 and SS2.…………….……….…103 B3.5. Measurement of Statistical Dependences…………….……………………………..107 B3.6. Conclusion…………….………………………………………………………………112 B3.7. Literature…………….……………………………………………………………….112
B4. Measurement of Different Luminous Intensity of Star: WASP-39b……………….113 (Author of Chapter B4: A.Vlcek) B4.1. Introduction…………………….…….……………………………………………...115 B4.2. Formulation of Statistical Investigation……………………..…….……………….116 B4.3. Scaling…………………….…….……………………………………………………126 B4.4. Measurement – Description of Observation Process…………….……….…….…126 B4.5. Elementary Statistical Processing.…………………….…………………………....128 B4.6. Non-parametric Testing.…………………….………………………………………130 B4.7. Conclusion.…………………….…………………………………………………......134 B4.8. Literature…………….……….……………………………………….……………..134
Contents of Part C: The Algorithm of Problem Solving in Curricular Process………………..135 C1. Application of Curricular Process in Explanation of Physics Bases of Classical Circular Accelerators……………………………………………………136 (Author of Chapter C1: B.Vesela) C1.1. Algorithm of Curricular Process.……………………………………………………………….......137 C1.2. Theoretical Formulation of Application of Curricular Process Algorithm……………………...138 C1.3. Application of Conceptual Curriculum…………………………………………………………….140 C1.3.1. Application of Lagrangian Formalism………………………………………………………...140 C1.3.2. Solution of Lagrange Equation………………………………………………………………...143 C1.3.3. Interpretation of Solution………………………………………………………………………145 C1.4. Application of Intended and Projected Curriculum……………………………………………….145 C1.4.1. Synthesis of Intended and Projected Curriculum…………………………………………......145 C1.4.2. Presentation of Education Text “The Physics Base of Cyclotron for Radiology Students…146 C1.5. Application of Implemented Curriculum-2………………………………………………….……..152 C1.5.1. Education Test…………………………………………………………………………………...152 C1.5.2. Statistical Assessment of Education Test………………………………………………………154 C1.6. Conclusion…………………………………………………………………………………………….162 C1.7. Literature……………………………………………………………………………………………..163
C2. Curricular Process of Radiological Physics within Higher Education Level……...164 (Author of Chapter C2: M.Bartonova) C2.1. Universal Curricular Process of Physics – Universal Algorithm…………..…………………….164 C2.2. Application of Radiological Physics Curricular Process to Higher Education Level – Specific Algorithm of Curricular Process………….……………………………………………..166 C2.3. Conceptual Curriculum of Radiological Physics for Higher Education Level………….……..167 C2.4. Intended Curriculum of Radiological Physics for Higher Education Level…………………...170 C2.5. Projected Curriculum of Radiological Physics for Higher Education Level…………………..171 C2.6. Implemented Curriculum-1 of Radiological Physics for Higher Education Level……………174 C2.7. Implemented Curriculum-2 of Radiological Physics for Higher Education Level……………175 C2.8. Attained Curriculum of Radiological Physics for Higher Education Level……….......……….177 C2.9. Conclusion…………...……………………………………………………………………………...178 C2.10. Literature……….......……………………………………………………………………………..179
Contents of Part D: The Algorithm of Problem Solving in Option Pricing and Hedging……..180 D. Initial Orientation………………………………………………………………………182 (Author of Chapter D. Initial Orientation: P.Zaskodny) D1. Research, Historical and Economical Substantiations……………………………...183 (Author of Chapter D1: V.Pavlat) D1.1. Summary of Chapter……………….………………………………………………...183 D1.2. Main Characteristics of Economical and Historical Dimension………………..…183 D1.3. Main Characteristics of Research Dimension……………….……………………...186 D1.4. Conclusion……………….……………………………………………………………186 D1.5. Selected Literature...…………….…………………………………………………...187
D2. Algorithms Identified in Option Portfolio Hedging…………………………………188 (Authors of Chapter D2: P.Zaskodny, I.Havlicek) D2.1. Summary of Chapter………………………..………………………………………..188 D2.2. Introduction………………………..………………………………………………….188 D2.3. Structure of Conceptual Algorithm...……………………..………………………...189 D2.4. structure of Theoretical Algorithm...……………………...………………………...189 D2.5. Structure of Practical Algorithm...……………………..…………………………...190 D2.6. Structure of Model Algorithm..……………………..……………………………....191 D2.7. Description of Potential Way of Algorithms Application..……………………..…191 D2.8. Selected Literature...…………………….…………………………………………...192
D3. From Algorithm to Programming and Way of Verification………………………..197 (Author of Chapter D3: M.Pasta) D3.1. Summary of Chapter..……………………………………………………………….197 D3.2. Basic Concepts..………………………………………………………………………197 D3.3. Algorithms, Programming, Future Verification..……………………………….…198 D3.4. Selected Literature..………………………………………………………………….199
D4. From Algorithm to Software for Trinomial Option Pricing………………………..200 (Author of Chapter D4: V.Risky) D4.1. Summary of Chapter..………………………………………………………………..200 D4.2. Introduction..………………………………………………………………………….200 D4.3. Validation of Hypothesis H1.………………………………………………………...201 D4.3.1. Comparison of Binomial and Trinomial Model.………………………………….201 D4.3.2. Validation of Hypothesis in Practice and Brief Description of Application.…...204 D4.4. Validation of Hypothesis H2…………………………………………………………206 D4.5. Conclusion…………………………………………………………………………….206 D4.6. Selected Literature…………………………………………………………………....207
D5. From Algorithm to Software for Quadrinomial Option Pricing…………………...208 (Author of Chapter D5: M.Soucek) D5.1. Summary of Chapter…………………………………………………………………208 D5.2. Introduction…………………………………………………………………………...208 D5.3. Algorithm of Quadrinomial Model……………………………………………….…209 D5.4. Graphics Interface……………………………………………………………………212 D5.5. Conclusion……………………………………………...……………………………..213 D5.6. Selected Literature……………………………………………………………………214
D6. From Algorithm to Software in Binomial Delta Hedging Based on Call Option….216 (Author of Chapter D6: M.Sebest) D6.1. Summary of Chapter…..…………………………………………………………..216 D6.2. Basic Terms…..…………………………………………………………………….217 D6.3. Creation of Dynamic Hedging Schemes Delta….………………………………..218 D6.4. Delta Rebalancing….……………………………………………………………...219 D6.5. Selection of Algorithm for Programming Relevant Option Hedging.....………220 D6.6. Software Processing of Practical Algorithm of Relevant Option Hedging...…221 D6.7. Conclusion...……………………………………………………………………….226 D6.8. Selected Literature...……………………………………………………………...226
D. Closing Orientation……………………………………………………………………..227 (Author of Chapter D. Closing Orientation: P.Zaskodny)
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PART A The Algorithm of Problem Solving in Data Mining Approach With support of grant 316MVVG20132015122, Czech Republic Author Premysl Zaskodny University of South Bohemia, University of Finance and Administration
[email protected]
Abstract The significant result of data mining approach can be seen in the delimitation of role of the algorithms in problem solving. The problem solving is expressing very often the essence of data mining and the algorithm of problem solving is showing the way how to reach the concrete conclusions. It is showing not only how to substantiate the concrete conclusions but also how to continue by an expression in the form of model representation or in the form of programming language application. In the framework of monograph “Educational & Didactic Communication 2013”, Volume 1 “Algorithms as Significant Result of Data Mining Approach” the role of algorithms within data mining approach will be described generally in the PART A. The concrete substantiation of importance of the algorithms in problem solving will be realized by means of PART B, PART C, and PART D. In the framework of PART A “The Algorithm of Problem Solving in Data Mining Approach” the fundamental basics of algorithm applications in statistics, curricular process and option pricing and hedging will be presented, cumulatively with the general principles of data mining approach. In the framework of PART B “The Algorithm of Problem Solving in Statistics” it will be shown the concrete applications of problem solution by means of the algorithm of statistics and probability. In the framework of PART C “The Algorithm of Problem Solving in Curricular Process” it will be shown the concrete applications of problem solution by means of the algorithm of curricular process as complex tool of educational data mining. In the framework of PART D “The Algorithm of Problem Solving in Option Pricing and Hedging” it will be shown the concrete applications of problem solution above all by means of the practical algorithm delimited in the area of option hedging.
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The structure of delimitation of the role of algorithms in problem solving within data mining approach will be described through following succession of steps: 1. Data Mining as Problem Solving and Its Structure 2. Data Mining Approach as Realization of Data Mining Cycle 3. Tools of Data Mining 4. Complex Tool of Educational Data Mining – Curricular Process 5. Significant Partial Tool of Data Mining – Analytical Synthetic Modeling 6. Significant Partial Tool of Data Mining – Matrix Modeling and Main Diagonal of Matrix 7. Algorithm of Statistics 8. Algorithm of Curricular Process 9. Algorithm of Option Pricing and Hedging 10. Role of Algorithms in Data Mining Approach
Key Words Data mining, Data mining cycle, Data mining approach, Tools of data mining, Analytical synthetic modeling and problem solving, Matrix modeling and algorithm, Algorithm of problem solving, Algorithm of statistics, Algorithm of curricular process, Algorithm of option pricing and hedging
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A1. Data Mining as Problem Solving and Its Structure A1.1. Meaning of Data Mining An imperative of data mining and a need of cooperation of the human with today´s computers are emphasized by D.A.Keim: “The progress made in hardware technology allows today´s computer systems to store very large amounts of data. Researchers from the University of Berkeley estimate that every year 1 Exabyte (= 1 Million Terabyte) of data are generated, of which a large portion is available in digital form. This means that in the next three years more data will be generated than in all of human history before”. “If the data is presented textually, the amount of data which can be displayed is in range one hundred data items, but this is like a drop in the ocean when dealing with data sets containing millions of data items”. “For data mining to be effective, it is important to include the human in the data exploration process and combine the flexibility, creativity, and general knowledge of the human with the enormous storage capacity and the computational power of today´s computers.” Modeling as a partial tool of data mining – quotation according to J.K.Gilbert: “In a nightmare world, we would perceive the world around us being continuous and without structure. However, our survival as a species has been possible because we have evolved the ability do “cut up” that world mentally into chunks about which we can think and hence give meaning to”. “This process of chunking, a part of all cognition, is modeling and the products of the mental actions that have taken place are models. Science, being concerned with the provision of explanations about the natural world, places an especial reliance on the generation and testing of models”.
- Keim,D.A. (2002) Information Visualization and Visual Data Mining IEEE Transactions on Visualization and Computer Graphics. Vol.7, No.1, January-March - Gilbert,J.K. (2008) Visualization: An Emergent Field of Practice and Enquiry In: Visualization: Theory and Practice in Science (Models and Modeling in Science Education) New York: Springer Science + Business Media
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A1.2. Basic Structural Elements Data Mining – an analytical synthetic way of extraction of hidden and potencially useful information from the large data files (continuum data-information-knowledge, knowledge discovery). Described analytical synthetic way can be taken as the important assumption for problems solving by relevant scientific branch Data Mining Techniques – system functions of the structure of formerly hidden relations and patterns (e.g. classification, association, clustering, prediction). The delimitation of system functions can be taken as the important assumption for structuring the relevant scientific branch Data Mining Tool – a concrete procedure how to reach the intended system functions. Data mining tools can be divided to the complex tools (the basic characteristics of relevant scientific branch) and to the partial tools (the essential procedures how to find the structures of problems solved) Complex Tool – a resolution of the complex problem of relevant scientific branch (the succession of phases in the course of solution of complex problem, cumulatively it is connected with methodology of relevant scientific branch) Partial Tool – a resolution of the partial problem of relevant scientific branch (the construction of global structure or partial structures of problems which are investigated by relevant scientific branch) Result of Data Mining – a result of the data mining tool application (the described structure of methodology of relevant scientific branch, the descriptions of structures of investigated partial problems The significant way of structure description – by means of algorithms of problem solving Representation of Data Mining Result – a model formalization of this what is expressed (usually the formalization of model construction of problems resolution) The significant content of model formalization – by means of algorithms of problem solving Visualization of Data Mining Result – an optical retrieval of the data mining result (usually the optical visualization of model expression of problems resolution) The significant optical visualization – by means of algorithms of problem solving
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A1.3. Detailed Structural Description The more detailed explanation can be found in following recommended literature: - Záškodný,P. (2011) Data Mining Tools in Science Education (312-318). In Proceedings of The 2nd International Multi-Conference on Complexity, Informatics and Cybernetics, Orlando, Florida, USA. Orlando, Florida, USA: International Institute of Informatics and Systemics. ISBN 978-1-936338-20-7 - Záškodný,P. (2012) Data Mining in Content Pedagogy (120-129). In Proceedings of The 1st International e-Conference on Optimization, Education and Data Mining in Science, Engineering and Risk Management 2011, Bratislava, Slovakia. Prague, Czech Republic: Curriculum ISBN 978-80-904948-1-7 - Záškodný,P. (2013) Educational & Didactic Communication 2012, Vol.1 Educational Data Mining and Its Applications (139 ps, Monograph). Bratislava, Slovakia: Didaktis ISBN 978-80-89160-97-6
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A2. Data Mining Approach as Realization of Data Mining Cycle A2.1. Data Mining Approach and Structure of Data Mining Cycle Data mining cycle is given by the succession of stages enabling to obtain the required results of data mining. Such succession is expressing the structure of data mining cycle. Data mining approach is connected with the sequential realization of individual stages, data mining approach is given by the global realization of data mining cycle structure.
The individual stages of data mining cycle will be described:
- Data Definition, Data Gathering (data accumulation data in the relation to an identified problem)
- Data Preprocessing, Data Processing (progressive processing data on the basis of identified problem analysis, delimitation of the partial problems)
- Data Mining Techniques and Data Mining Tools (delimitation of system functions of formerly hidden relations and patterns, application of concrete procedures in the course of problem solving, by means of abstraction the delimitation of partial problems essence)
- Discovering Knowledge or Patterns (transformation of data in the framework of a continuum data-information-knowledge) By means of synthesis the intellectual reconstruction of identified problem in the form of algorithm of problem solving
- Representation and Visualization of Data Mining Results (usually construction of structural model of worked problem out, optical retrieval of reached results) The structural model and optical retrieval in the light of problem solving algorithm
- Application (utilization of the resolved problems for next development of scientific theory or for substantiation of practical application) Utilization of problem solving algorithm
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A2.2. Data Mining Cycle in Literature U.M.Fayyad, G.Piatelsky-Shapiro, P.Smyth (1996) Cycle of Data Mining Data Mining can be viewed as a cycle that consists of several steps: - Identify a problem where analyzing data can provide value - Collect the data - Preprocess the data obtain a clean, mineable table - Build a model that summarizes patterns of interest in a particular representational form - Interpret/Evaluate the model - Deploy the results incorporating the model into another system for further action. J.Luan (2002) Steps for Data Mining Preparation (algorithm, building, visualization) a) Investigate the possibility of overlaying Data Mining algorithms directly on a data warehouse b) Select a solid querying tool to build Data Mining files. These files closely resemble multidimensional cubes c) Data Visualization and Validation. This means both examining frequency counts as well as generating scatter plots, histograms, and other graphics, including clustering models d) Mine your data Le Jun (2008) Main Processes of Data Mining - The main processes include data definition, data gathering, preprocessing, data processing and discovering knowledge or patterns (Data Mining techniques can be implemented rapidly on existing software and hardware) - Application of Data Mining tools: To solve the task of prediction, classification, explicit modeling and clustering. The application can help understand learners´learning behaviors.
- Fayyad,U.M., Piatelsky-Shapiro,G., Smyth,P. (1996) From Data Mining to Knowledge Discovery in Databases AI Magazine, 17, 37-54 - Luan,J. (2002) Data Mining and Knowledge Management in Higher Education – Potential Applications In Proceedings of The 42nd Annual Forum for the Association for Institutional Research, Toronto, Ontario, Canada Toronto, Ontario, Canada: Association for Institutional Research - Le Jun (2008) Knowledge Management in Distance Education: A Case Study of Curriculum Teaching and Learning In: Kwan,R., Fox.R., Chan,F.T.: Enhancing Learning Through Technology (Research on Emerging Technologies and Pedagogies Singapore, Hachensack, London: World Scientific Publishing ISBN 10 981-279-944-3, ISBN 13 978-981-279-944-9
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A3. Tools of Data Mining A3.1. Definition and Classification of Tools in the light of Algorithms Definition of Data Mining Tool – a concrete procedure how to reach the intended system functions. Data mining tools can be divided to the complex tools (the basic characteristics of relevant scientific branch) and to the partial tools (the essential procedures how to find the structures of problems solved) Dividing of Data Mining Tools can be carried out into Complex Tools of Data Mining and Partial Tools of Data Mining. The complex tool and partial tool can be delimited by means of following way: Complex Tool – a resolution of the complex problem of relevant scientific branch (the succession of phases in the course of solution of complex problem, cumulatively it is connected with methodology of relevant scientific branch) Partial Tool – a resolution of the partial problem of relevant scientific branch (the construction of global structure or partial structures of problems which are investigated by relevant scientific branch) The described structure of methodology of relevant scientific branch can be taken as the result of complex tool application, the descriptions of structures of investigated partial problems of relevant scientific branch can be taken as the results of partial tools applications. The significant way of structure description may be connected with the algorithms of problem solving. The role of algorithms in data mining approach could be explained in respective scientific branch by means of the complex and partial data mining tools – the result of data mining is given by the application of data mining tools.
A3.2. Illustration and Substantiation of Data Mining Tools The illustration of data mining tools and their algorithms will be shown through the mediation of several scientific branches. In the framework of PART B “The Algorithm of Problem Solving in Statistics” will be shown the concrete applications of problem solution by means of the algorithm of statistics and probability – by means of application of data mining tools which are specific for statistics and probability. In the framework of PART C “The Algorithm of Problem Solving in Curricular Process” will be shown the concrete applications of problem solution by means of the algorithm of curricular process as complex tool of educational data mining – by means of application of data mining tools which are specific for education and content pedagogy.
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In the framework of PART D “The Algorithm of Problem Solving in Option Pricing and Hedging” will be shown the concrete applications of problem solution above all by means of the practical algorithm delimited in the area of option hedging – by means of application of data mining tools which are specific for the theory of financial derivatives in the area of option pricing and hedging. The more detailed explanation of complex and partial data mining tools can be found in following recommended literature: - Záškodný,P., Pavlát,V. (2010) Data Mining Process – Brief Recherche (54-76) In: Educational & Didactic Communication 2009 Bratislava, Slovakia: Didaktis ISBN 978-80-89160-69-3 - Záškodný,P., Novák,V. (2010) Data Mining Process – Brief Summary (76-81) In: Educational & Didactic Communication 2009 Bratislava, Slovakia: Didaktis ISBN 978-80-89160-69-3 - Záškodný,P. (2011) Data Mining Tools in Science Education (312-318). In: Proceedings of The 2nd International Multi-Conference on Complexity, Informatics and Cybernetics, Orlando, Florida, USA. Orlando, Florida, USA: International Institute of Informatics and Systemics. ISBN 978-1-936338-20-7 - Záškodný,P. (2012) Data Mining in Content Pedagogy (120-129). In: Proceedings of The 1st International e-Conference on Optimization, Education and Data Mining in Science, Engineering and Risk Management 2011, Bratislava, Slovakia. Prague, Czech Republic: Curriculum ISBN 978-80-904948-1-7 - Záškodný,P. (2013) Educational & Didactic Communication 2012, Vol.1 Educational Data Mining and Its Applications (139 ps, Monograph). Bratislava, Slovakia: Didaktis ISBN 978-80-89160-97-6
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A4. Complex Tool of Educational Data Mining–Curricular Process A4.1. Procedure of Complex Tool Seeking in area of Science Education The main principle is data mining approach in education from the point of view of content pedagogy (see works of W.Doyle, 1992, 2003) – the representation of education from the point of view of content pedagogy will be demonstrated by means of science education. Data mining approach is given by the realization of data mining cycle structure, data mining approach is taken as problem solving. The main goal is consisting in delimitation of complex data mining tool. The procedure is consisting of data preprocessing in science education, data processing in science education, description of curricular process as complex data mining tool in science education and finally application.
A4.2. Data Preprocessing in Science Education Result of Data Preprocessing – Educational Communication of Natural Science as a succession of transformations of education content forms: • The transformation T1 is transformation of scientific system of natural science to communicable scientific system of natural science (the first form of education content existence), • The transformation T2 is transformation of communicable scientific system of natural science to educational system of natural science (the second form of education content existence), • The transformation T3 is transformation of educational system of natural science to both instructional project of natural science and preparedness of educator to education (the third and fourth forms of education content existence), • The transformation T4 is transformation of both instructional project of natural science and preparedness of educator to results of education (the fifth form of education content existence), • The transformation T5 is transformation of results of natural science education to applicable results of natural science education (the sixth form of education content existence)
The more detailed explanation of the educational communication of natural science can be found in following recommended literature: - Brockmeyerová,J. (1982) Introduction into Theory and Methodology of Physics Education Prague, Czech Republic: SPN
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- Doyle,W. (1992a) Curriculum and Pedagogy (p.486-516). In: Handbook of Research on Curriculum New York: Macmillan - Doyle,W. (1992b) Constructing Curriculum in the Classroom (p.66-79). In: Effective and Responsible Teaching – The New Synthesis San Francisco: Jossey-Bass Publ. - Doyle,W., Carter,K. (2003) Narrative and Learning to Teach: Implications for Teachers – Education Curriculum Tucson: Taylor and Francis - Tarábek,P, Záškodný,P. (2008) Educational and Didactic Communication 2007, Vol.1 – Theory Bratislava, Slovak Republic: Didaktis ISBN 987-80-89160-56-3 - Tarábek,P, Záškodný,P. (2008) Educational and Didactic Communication 2007, Vol.2 – Methods Bratislava, Slovak Republic: Didaktis ISBN 987-80-89160-56-3 - Tarábek,P, Záškodný,P. (2008) Educational and Didactic Communication 2007, Vol.3 – Applications Bratislava, Slovak Republic: Didaktis ISBN 987-80-89160-56-3 - Tarábek,P, Záškodný,P. (2009) Educational and Didactic Communication 2008 Bratislava, Slovak Republic: Didaktis ISBN 987-80-89160-62-4
A4.3. Data Processing in Science Education Result of Data Processing – Curricular Process of Natural Science as a succession of transformations of algorithmized and formalized education content forms: i. The form of education content existence – “variant form of curriculum” ii. The curriculum – “education content” (see J.Prucha, 2005) iii. The variant forms of curriculum have got the universal structure (four structural elements – sense and interpretation, set of objectives, conceptual knowledge system, factor of following transformation) iv. The variant forms of curriculum were selected on the basis of fusion of AngloAmerican curricular tradition and European didactic tradition
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v. The curricular process is defined as the succession of transformations T1-T5 of curriculum variant forms: “conceptual curriculum” (output of T1, the first variant form of curriculum) - the communicable scientific system “intended curriculum” (output of T2, the second variant form of curriculum) - the educational system of natural science “projected curriculum” (output of T3, the third variant form of curriculum) - the instructional project of natural science “implemented curriculum-1” (output of T3, the fourth variant form of curriculum) - the preparedness of educator to education “implemented curriculum-2” (output of T4, the fifth variant form of curriculum) – the results of education “attained curriculum” (output of T5, the sixth variant form of curriculum) - applicable results of education
The more detailed explanation of the curricular process of natural science can be found in following recommended literature: - Záškodný,P., Procházka,P. (2010) Collective Scheme of Both Educational Communication and Curricular Process In: Educational and Didactic Communication 2009 Bratislava, Slovak Republic: Didaktis ISBN 978-80-89160- 69-3 - Záškodný,P. (2006) Survey of Principles of Theoretical Physics (with Application to Radiology) (in English) Lucerne, Switzerland, Ostrava, Czech Republic: Avenira, Algoritmus ISBN 80-902491-9-1 - Záškodný,P. (2009) Curicular Process of Physics (with Survey of Principles of Theoretical Physics) (in Czech) Lucerne, Switzerland, Ostrava, Czech Republic: Avenira, Algoritmus ISBN 978-80-902491-0-3 - Průcha,J. (2005) Moderní pedagogika (Modern Educational Science) Prague, Czech Republic: Portál ISBN 80-7367-045-X
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A4.4. Complex Tool of Data Mining in Science Education Complex tool of data mining in science education is given by curricular process of natural science. Curricular process delimits the correct education content via succession of transformations T1-T5: The first variant form of curriculum The communicable scientific system as “conceptual curriculum” Output of transformation T1 The second variant form of curriculum The educational system of natural science as “intended curriculum” Output of transformation T2 The third variant form of curriculum The instructional project of natural science “projected curriculum” Output of transformation T3 The fourth variant form of curriculum The preparedness of educator to education “implemented curriculum-1” Output of transformation T3 The fifth variant form of curriculum The results of education “implemented curriculum-2” Output of transformation T4 The sixth variant form of curriculum Applicable results of education “attained curriculum” Output of transformation T5
A4.5. Application of Complex Tool (Curricular Process) in Science Education The application of complex tool in science education was performed through physics education. The contents of book “Curricular Process of Physics” (Zaskodny,P., 2009, Lucerne, Switzerland, Ostrava, Czech Republic: Avenira, Algoritmus, ISBN 978-80-902491-0-3) is showing how to apply the curricular process in concrete natural science – in physics.
The brief description of contents of mentioned book can be described by means of following form: Introduction I: Delimitation of Curricular Process of Physics Introduction II: Structure of Physics
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Construction and Representation of Curriculum Variant Form: i) Conceptual Curriculum (Statistical and Non-Statistical Physics. Classical, Quantum and Relativistic Dimension of Statistical and Non-Statistical Physics. Scientific Structural Models) ii) Intended Curriculum (Adaptation of Scientific Structural Models of Physics to Possibilities of Addressees of Education – Creation of Cognitive Models) iii) Projected Curriculum (Projection of Cognitive Models within Creation of Textbook System for Classical, Quantum and Relativistic Dimension of Statistical and Non-Statistical Physics ) iv) Implemented Curriculum-1(Physics Teacher Preparedness for Education of Concrete Theme of Physics Adjusted to Addressees Possibilities – Mediated Solution of Problems) v) Implemented Curriculum-2 (Detection of Results Achieved by Physics Instruction with Utilization of Appropriate Test Techniques) vi) Attained Curriculum (Detection of which Achieved Results of Classical, Quantum and Relativistic Dimension of Statistical and Non-Statistical Physics Can Be Applied in Practice) Workout of Curricular Process in Physics Workout of Relevant Parts of Physics
The more detailed explanation of the curricular process of physics can be found in recommended literature: - Záškodný,P. (2006) Survey of Principles of Theoretical Physics (with Application to Radiology) (in English) Lucerne, Switzerland, Ostrava, Czech Republic: Avenira, Algoritmus ISBN 80-902491-9-1 - Záškodný,P. (2009) Curicular Process of Physics (with Survey of Principles of Theoretical Physics) (in Czech) Lucerne, Switzerland, Ostrava, Czech Republic: Avenira, Algoritmus ISBN 978-80-902491-0-3
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A5. Significant Partial Tool of Data Mining – Analytical Synthetic Modeling A5.1. Universal Analytical Synthetic Modeling Significant partial tool of data mining is given by analytical synthetic modeling. Analytical synthetic modeling describes the mediated or real problem solving within the inputs and outputs of individual transformations T1-T5. The description of analytical synthetic modeling is realized by means of both visualization Vis.1 and Legend to Vis.1.
Legend to Vis.1
a
(Identified Complex Problem) –Investigated area of reality, investigated phenomenon
Bk (Analysis) – Analytical segmentation of complex problem to partial problems bk (Partial problems PP-k) – Result of analysis: essential attributes and features of investigated phenomenon Ck (Abstraction) – Delimitation of partial problems essences by abstraction with goal to acquire the partial solutions ck (Partial solutions PS-k) – Result of abstraction: partial concepts, partial pieces of knowledge, various relations, etc. Dk (Synthesis) – Synthetic finding dependences among results of abstraction dk (Partial conclusions PC-k) – Result of synthesis: principle, law, dependence, continuity Ek (Intellectual reconstruction) – Intellectual reconstruction of investigated phenomenon/ investigated area of reality e (Total solution of complex problem “a”) – Result of intellectual reconstruction: analytical synthetic structure of final knowledge (conceptual knowledge system)
Vis.1: Universal Analytical Synthetic Model of Problem Solving
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a - Identified Complex Problem
B1
B2
b1 - Partial Problem No. 1 (PP-1) C1
C2
c1-Partial Solution No.1(PS-1)
c2-Partial Solution No.2(PS-2)
D1
ANALYSIS
b2 - Partial Problem No. 2 (PP-2) C3
bk - Partial Problem No. k (PP-k)
C4
c3-Partial Solution No.3(PS-3)
ABSTRACTION
c4-Partial Solution No.4(PS-4)
D2
d1 - Partial Conclusion No. 1 (PC-1)
E1
Ck
ck-Partial Solution No.k(PS-k)
SYNTHESIS
d2 - Partial Conclusion No. 2 (PC-2)
E2
Bk
Dk
dk - Partial Conclusion No. k (PC-k)
RECONSTRUCTION
Ek
e - Total Solution of Complex Problem "a" formed by means of PC-1, PC-2, .., PC-k
A5.2. Illustration of Analytical Synthetic Modeling The application of analytical synthetic modeling is the visualization Vis.2 from the area of physics education. The visualization Vis.2 is analytical synthetic model of classical nonstatistical physics (classical mechanics, mechanics of continuum, free electromagnetic field). This visualization constitutes a part of physics conceptual curriculum as a part of communicable scientific system of physics (a part of output of transformation T1). Vis.2: The analytical synthetic model of classical non-statistical physics (a part of conceptual curriculum of physics – a part of communicable scientific system of physics – output of transformation T1)
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System of particles, continuum, free electromagnetic field – motional states and their changes
Non-stationary and quasi-stationary motional states and their changes caused by forces
Stationary and static motional states and conditions of their duration
D´Alembert principle, Hamilton principle: Lagrange (Hamilton) function L (H) is dependent on time
D´Alembert principle, Hamilton principle: Lagrange (Hamilton) function L (H) is independent of time
Motional equation and motional law: Description of quasi-stationary and nonstationary states and of their changes
Motional equation and motional law: Description of stationary and static states and of their changes
General description of states and of their changes for the system of particles (second type Lagrange equations, Hamilton canonic equations, the principle of virtual works), for the continuum (general motional equation and the general equation of continuum equilibrium), and for the free monochromatic electromagnetic field (the motional equation of the free electromagnetic field)
Particular description of the system of particles
Particular description of the continuum
Particular description of the free electromagnetic field
Motional states and their changes in models of particle systems (free and bound system, solid body), in models of the continuum (Pascalian perfect fluid, Newtonian viscous fluid, Euclidean solid, Hooke´s elastic continuum), in the free magnetic field with a given frequency (an enormous number of lowfrequency photons – electromagnetic wave motion, small number of high-frequency photons – an ordered flux of particles)
Simple applications of classical mechanics: Switch to Newtonian formalism
Simple applications of electromagnetic field theory: Field sources and vortices (Maxwell equations, scalar and vector potential) Common application of classical mechanics and electromagnetic field theory: the Lorentz force and the motion of the classical electron in the constant electromagnetic field
The further application of analytical synthetic modeling is the visualization Vis.3 from the area of content pedagogy. The content pedagogy is presented as result of curriculum research and development.
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Vis.3: The analytical synthetic model of curriculum research and development Theory and Practice of Curriculum as Curriculum Research and Development Implementation of primary analysis and synthesis
Definition of curriculum (DC)
Philosophical line in theory of curriculum (PL)
Empirical and practical line in theory of curriculum (EPL)
Survey of DC
Survey of PL
Survey of EPL
Curriculum- Content of Education
Existence of Various Curricular Variant Forms
Curriculum Development-Construction and Formation of Curriculum Variant Forms for Purposes of Education Practice Curriculum Development-Description of Curriculum Variant Forms Succession as Curricular Process of Relevant Science
Curriculum Research-Delimitation of Succession of Curriculum Variant Forms and Transformations of Education Contents from Relevant Science
Primary synthesis: - Useful approach for modern educational science within curriculum research – curriculum is taken as content of education, curriculum has got several different forms of its existence and it means an approach to curriculum as to variant phenomenon (J.Průcha) - Useful approach for modern educational science within curriculum development – development of curriculum is taken as conversion of content of human knowledge to addressees of education, construction of anybody curriculum is derived from possibilities of addressees of education and it means to construct especially adequate knowledge and concepts by acceptable methods (W.Doyle, H.Spada, H.Mandl) - Implementation of secondary analysis and synthesis
Solution of curriculum research – systems of forms and transformations of curriculum existence (SFTC)
Solution of curriculum development – theories of conversion of content of human knowledge, structure and communicability of transformed knowledge (SCTK)
Survey of SFTC-Survey of investigated variant forms of curriculum and investigated transformations among them
Selection of theories - Theory of educational communication. Theory of contect pedagogy
Secondary synthesis: - Hypothetical conceptions about variability of curriculum forms are in no way abstract. They are elaborated both by theoretical way and by empirical way. For example, F.Achtenhagen (1992), W.Doyle (1992b, Constructing Curriculum in Classroom). - Implementation of terciary analysis and synthesis with objective to classify terms “Curricular Process” and “Variant Form of Curriculum” in the frame of Curriculum Research and Development (on the basis of theories educational communication and content pedagogy and on the basis of approach to curriculum as variant phenomenon – J.Brockmeyer, W.Doyle, J.Průcha)
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The more detailed explanation of the analytical synthetic modeling in physics and content pedagogy can be found in recommended literature: - Záškodný,P. (2006) Survey of Principles of Theoretical Physics (with Application to Radiology) (in English) Lucerne, Switzerland, Ostrava, Czech Republic: Avenira, Algoritmus ISBN 80-902491-9-1 - Záškodný,P. (2009) Curicular Process of Physics (with Survey of Principles of Theoretical Physics) (in Czech) Lucerne, Switzerland, Ostrava, Czech Republic: Avenira, Algoritmus ISBN 978-80-902491-0-3
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A6. Significant Partial Tool of Data Mining – Matrix Modeling and Main Diagonal of Matrix A6.1. Representation of Matrix Model of Complex Problem Solving Vis.4: The matrix model of complex problem solving
1 = = =
= 2 = =
= = 3 =
= = = 4 = = = =
I = 5 + + +
= + 6 + +
= + + 7 =
= + + = 8 =
II = 9 = = = = =
= 10 + = = =
= + 11 = = =
= = = 12 + =
= = = + 13 =
= = = = = 14 = = = =
III
= 15 + + =
= + 16 + = V
= + + 17 =
= = = = 18 = = =
IV = 19 + +
= + 20 +
= + + 21
A6.2. Description of Matrix Model and Its Main Diagonal Within the framework of creation of matrix model of scientific or cognitive structure it is first of all necessary to delimit the survey of concurring elements of matrix main diagonal by linear way. The brief description of their contents is necessary to perform. The elements of main diagonal form a definition line of matrix and, in the case of cognitive structure, they are often called by subject matter units. The delimitation of definition line of matrix is starting from linear arrangement of the analytical synthetic model of scientific or cognitive structure. Often it is possible to take the important segments of definition line as the partial conclusions dk of analytical synthetic model as partial scientific or conceptual knowledge systems (see Vis.1). Afterwards it is possible these segments to complete by the found essences ck, eventually by the partial problems bk (again see Vis.1).
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The projection of definition line into creation of textbook should enable to carry out the intellectual reconstruction “e” of problem solved in the cooperation of addressees of education with teacher or in the cooperation with relevant scientific community. The projection of definition line into solution of scientific task should enable to carry out the intellectual reconstruction “e” of identified problem “a” (see Vis.1). The definition line of matrix (see visualization Vis.4) contains 21 elements (21 subject matter units or 21 scientific pieces of knowledge). The individual subject matter units or scientific pieces of knowledge will be marked by numbers from 1 to 21 and they will be to occupy the main diagonal of matrix. Thereafter it is possible the matrix model of investigated scientific or cognitive structure to represent by Vis.4. The construction of matrix model of scientific or cognitive structure with 21 subject matter units or scientific pieces of knowledge will be described in the following way. Resulted matrix will be squared with 21 matrix rows and matrix columns. The numeral succession of sequential numbers will be written down into main diagonal – definition line. The being relations (associations and discriminations) among 21 elements of definition line (among 21 subject matter units or scientific pieces of knowledge) will be finally plotted in matrix (associations will be marked by =, discriminations by +). In the course of developing the same conceptual knowledge system on the basis of collective elements or in the course of direct sequence in light of the arrangement into definition line, the elements (subject matter units or scientific pieces of knowledge) are associated. In the course of developing the same conceptual knowledge system on the basis of diversity, the relation of discrimination will be among the elements of definition line (among the subject matter units or scientific pieces of knowledge). The matrix element given by i-th row and j-th column bears the usual indication aij. Element aij remains without indication in that case – none of delimited relations (neither relation of association, nor relation of discrimination) is between elements aii and ajj of definition line. In the course of matrix filling the elements a11 and a22 of definition line will be at the earliest investigated. The relation between them is given by association = or discrimination +, the element a12 will be marked = or +. All the elements aij for j greater than i will be gradually explored by this way. Thus the elements above definition line will be marked or unmarked. The verification of analysis correctness of relations between the elements of definition line is given by the execution of this analysis in contrary order – at the earliest the relation of elements a21,21 and a20,20 will be determined, then a20,20 and a19,19 etc. Subsequently the elements aij for i greater than j will be marked or unmarked. The both halves of filled matrix should be axially symmetrical according to definition line. In the case of symmetry the analysis of relations among elements of definition line was carried out correctly. The matrix capable of interpreting should mark all the elements aij for j=i+1 and i=j+1 (so called ideal matrix). Groups of marked elements which are going from follow-up elements of definition line in the various directions are important. It is showing to the close associations among corresponding elements of definition line – the definition line elements of such group contribute to delimitation of the same conceptual knowledge system. Mistaken construction of matrix (caused, for example, by replacement of definition line elements sequence) shows itself by disturbance of matrix ideality.
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From the matrix model the matrix shape of curriculum variant forms or partial scientific conceptual knowledge systems is evident. The matrix connected with Vis.4 should contain five curriculum variant forms or conceptual knowledge systems I, II, III, IV, and V. The main diagonal of matrix, so called definition line, can be by the carrier of algorithm describing the result of data mining approach or the way of application of data mining tools. The role of algorithms in data mining approach is connected with the existence of ideal matrix. The disturbance of matrix ideality indicates the errors in delimited algorithm. Correctly described definition line of scientific or cognitive matrix can be identified with sought algorithm.
The more detailed explanation of the matrix modeling can be found in recommended literature: - Záškodný,P., Škrabánková,J. Modeling and Visualization of Problem Solving In: Educational and Didactic Communication 2009 Bratislava, Slovak Republic: Didaktis ISBN 978-80-89160- 69-3 - Záškodný,P. (2009) Curicular Process of Physics (with Survey of Principles of Theoretical Physics) (in Czech) Lucerne, Switzerland, Ostrava, Czech Republic: Avenira, Algoritmus ISBN 978-80-902491-0-3
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A7. Algorithm of Statistics A7.1. Analytical Synthetic Model of Statistics The applications of descriptive and mathematical statistics and probability theory in an investigation of the collective random phenomena are the subject of probability and statistics. To describe these applications it is necessary to first be concerned in descriptive and mathematical statistics and probability theory. In view of the fact that primarily the algorithm of statistics will be sought (as definition line of matrix model of scientific structure of statistics) the analytical synthetic model of statistics structure should be created. From this reason it will be effective to acquaint ourselves above all with main statistical methods, continuously to describe them by the survey of substantial concepts, marginally to touch of some concepts of probability theory and finally to form the analytical synthetic model. The investigation of so structured orientation is although accessible for the construction of analytical synthetic model and definition line of matrix, it cannot, however, be confused with a continuous and coherent study of statistics and probability theory as a separate scientific disciplines. The derivation of matrix definition line (algorithm of statistics) will be introduced by analytical-synthetic model of the structure of statistics as a whole. This model can be used for the immediate classification of statistical method and for the immediate location of previous and follow-up methods. The model also has a significant cognitive dimension – it is showing which the operations of analysis, abstraction and synthesis are to be carried out to be complete the adoption of relevant statistical method. The presented model in visualization Vis.5 contains the four partial analytical-synthetic structures. The model in visualization Vis.5, the legend to visualization Vis.5 and the description of component structural parts will be presented now. The utilization of analytical synthetic model of statistics represents data mining approach to the exploration of the principles of statistics and several needful concepts of probability. The data mining approach enables to work with the integral concepts and knowledge pieces in their system shape (see analytical-synthetic model). The data mining approach can be taken as the realization of data mining cycle, the utilization of analytical synthetic model can be taken as the realization of significant partial data mining tool. The immediate structural orientation, showing which part of the statistics and its probability applications is just applied in the course of the problem solving, isn´t useless. It is always good to know whether the selective statistical set (SSS) is “only” determined (the first partial structure from element a-1 up to element e-1), whether the empirical picture of set SSS is already created (the second partial structure from element a-2 up to element e-2) or whether the probability picture of set SSS is already even explored (the third partial structure from element a-3 up to element e-3) or whether it was already entered to the process of creation of the associative picture of set SSS (the fourth partial structure from element a-4 up to element e-4). In addition, the linear arrangement of analytical synthetic model enables to characterize the algorithm of statistics as the definition line of matrix model. Vis.5: The analytical synthetic model of statistics
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Collective random phenomenon and reason of its investigation a-1
Statistical unit
Variants (values) of statistical sign
Statistical sign
Choice of statistical units
Selective statistical set (SSS) as a part of basic statistical set, Goals of statistical examination e-1=a-2 Statistical probability
Creating of scale Measurement
Frequencies tables (Empirical distribution)
Graphical expression
Empirical parameters
Empirical picture of selective statistical set, Necessity of probable investigation e-2=a-3 Probability distributions
Comparison of theoretical and empirical parameters
Choice of acceptable theoretical distribution
Quantification of theoretical parameters
Testing non-parametric hypotheses
Point & interval estimation (e.g. confidence interval)
Empirical & probable picture of selective statistical set,
Statistical dependence (causal, non-causal)
Testing parametric hypotheses
Necessity of association investigation e-3=a-4
Regression analysis
Correlation analysis
Empirical & probable & association picture of selective statistical set Interpretation and conclusions as the statistical & probable dimension e-4 of investigation collective random phenomenon
Applied probability and statistics (see Part B of monograph)
Vis.5 Analytical synthetic model of statistics and needful probability concepts
formed by four partial models a1-e1, a2-e2, a3-e3, a4-e4
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LEGEND to whole visualization Vis.5 a-1
e-1
,
a-2
e-2
,
a-3
e-3
,
a-4
e-4
One – Sample Analysis, Two / Multiple – Sample Analysis
LEGEND to partial models of visualization Vis.5 a-1
e-1
Formulation of statistical examination
a-2
e-2
Relative & Cumulative Frequencies (Empirical distribution) Plotting functions: e.g. Plot Frequency Polygon (Graphical expression) Average-Means (Arithmetic Mean), Variance-Standard (Determinative) Deviation, Obliqueness (Skewness), Pointedness (Kurtosis) – (Empirical parameters)
a-3
e-3
Theoretical Distribution (partial survey in alphabetical order): Bernoulli, Beta, Binomial, Chi-square, Discrete Uniform, Erlang, Exponential, F, Gamma, Geometric, Lognormal, Negative binomial, Normal, Poisson, Student´s, Triangular, Trinomial, Uniform, Weibull Testing Non-parametric Hypotheses (Hypothesis test for H0 – receive or reject H0): e.g. computed Wilcoxon´s test, Kolmogorov-Smirnov test, Chi-square test e.g. at alpha = 0,05 Point & Interval Estimation: e.g. confidence interval for Mean, confidence interval for Standard Deviation Testing Parametric Hypotheses (Hypothesis test for H0 – receive or reject H0): e.g. computed u-statistic, t-statistic, F-statistic, Chi-square statistic, Cochran´s test, Barlett´s test, Hartley´s test e.g. at alpha = 0,05
a-4
e-4
Statistical dependence: e.g. confidence interval for difference in Means (Equal variances, Unequal variances) e.g. confidence interval for Ratio of Variances Regression analysis: simple – multiple, linear – non-linear Correlation analysis: e.g. Rank correlation coefficient, Pearson´s correlation coefficient
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Description of four partial analytical synthetic structures The example of applicability of analytical synthetic modeling presented via Vis.5 is introduced by means of description of statistics as a whole. In the framework of this description it is possible to indicate four partial analytical-synthetic structures of statistical dimension of investigated problem. Now, these four partial analytical synthetic structures will be presented. Within this presentation let us compare general model of analytical synthetic structure of investigated problem (from investigated phenomenon to the result of solution given by intellectual reconstruction) with visualization Vis.5 "Analytical synthetic model of statistics formed by four partial models". First structure
a-1
e-1
(see Vis.5)
From investigated phenomenon (marked a-1) "Collective random phenomenon and reason of its investigation" to the result of intellectual reconstruction (marked e-1) "Selective statistical set as a part of basic statistical set"
Second structure
a-2
e-2
(see Vis.5)
From investigated phenomenon (marked a-2) "Selective statistical set as a part of basic statistical set" to the result of intellectual reconstruction (marked e-2) "Empirical picture of selective statistical set"
Third structure
a-3
e-3
(see Vis.5)
From investigated phenomenon (marked a-3) "Empirical picture of selective statistical set" to the result of intellectual reconstruction (marked e-3) "Probable picture of selective statistical set"
Fourth structure
a-4
e-4
(see Vis.5)
From investigated phenomenon (marked a-4) "Probable picture of selective statistical set" to the result of intellectual reconstruction (marked e-4) "Association picture of selective statistical set" Applied statistics
a5
(see Part B of monograph)
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A7.2. Groups of Definition Line Elements of Matrix Model of Statistics The creation of matrix model definition line will reflect the model represented by visualization Vis.5. Therefore, the interpretation of individual elements of definition line can be described by means of the structural elements a-1 up to a-5 and e-1 up to e-4. The application of matrix model definition line as the algorithm of statistics should be fulfilled for persons interested in deeper understanding by both some basic concepts of probability theory and the survey of basic statistical tables. The groups of definition line elements forming the main methods of statistics (the main conceptual knowledge systems as stages of statistics algorithm) should keep the order determined by definition line of matrix model. The succession of main statistics methods can be considered as algorithm of statistics. Survey of arranged main methods of statistics: The main methods of descriptive statistics, Statistical probability i)
Formulation of statistical investigation (from element a-1 to element e-1) ii) Creation of scale (from element a-2 to element e-2) iii) Measurement, Probability (from element a-2 to element e-2) iv) Elementary statistical processing (from element a-2 to element e-2) The main methods of mathematical statistics, Probability distribution v)
Assignment of theoretical distribution to empirical distribution – testing non-parametric hypotheses, Probability – theoretical distributions (from element a-3 to element e-3) vi), vii) Comparison of empirical and theoretical parameters – estimations of theoretical parameters, – testing parametric hypotheses (from element a-3 to element e-3) viii) Measurement of statistical dependences – some fundaments of regression and correlation analysis (from element a-4 to element e-4) Applications (see Part B of monograph) Statistical tables (see Part B of monograph)
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A7.3. Individual Stages of Algorithm of Statistics, Recommended Literature The algorithm of statistics is created by the succession of following stages: i) Formulation of Statistical Investigation (Collective Random Phenomenon, Statistical Unit, Statistical Sign, Selective Statistical Set, Random Selection, Basic Statistical Set – Population) ii) Creation of Scale (Classification of Scales, Parameters of Selective Type of Scale) iii) Measurement in Descriptive Statistics (Absolute Frequency, Relative Frequency, Cumulative Frequency) iv) Elementary Statistical Processing (Frequencies Tables, Empirical Distribution, Graphical Expression, Plotting Function – Graphical Expression of Empirical Distribution, Frequency Polygon, Empirical Parameters, General Moments, Central Moments, Standardized Moments) v) Non-parametric Testing (Theoretical Distribution, Testing Nonparametric Hypotheses, Receiving or Rejecting of Zero Hypothesis, Level of Statistical Significance) vi) Theory of Estimation (Point Estimation, Interval Estimation, Confidence Interval, Confidence Interval for Mean Value, Confidence Interval for Standard Deviation) vii) Parametric Testing (Testing Parametric Hypotheses, Computed u-Statistic, Computed t-Statistic, Computed F-Statistic, Computed chi-Square Statistic) viii) Regression and Correlation Analysis (Simple and Multiple Selective Statistical Set, Statistical Dependence, Simple and Multiple Regression Dependence, Linear and Nonlinear Regression Dependence, Regression Analysis, Regression Function, Simple and Multiple Correlation, Correlation Analysis, Correlation Coefficient) The more detailed explanation of the matrix modeling can be found in recommended literature: - Zaskodny,P. (2013) The Principles of Probability and Statistics Prague, Czech Republic: Curriculum (Bilingual Czech-English Version) ISBN 978-80-904948-5-5 - Zaskodny,P. (2013) The Principles of Probability and Statistics Prague, Czech Republic: Curriculum (Monolingual English Version) ISBN 978-80-904948-6-2
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A8. Algorithm of Curricular Process A8.1. Possibility of Collaboration of European and Anglo-American Conception The representation of education from the point of view of content pedagogy was demonstrated by means of science education in Chapter A4. This representation was realized in Chapter A4 by data preprocessing in science education, by data processing in science education, by description of curricular process as complex data mining tool in science education and finally by application. The basic result of data mining approach in science education is the comparison of transformations of relevant natural science knowledge with variant forms of curriculum. Organized sequence of respective natural science knowledge transformations is the expression of educational communication of respective natural science, organized sequence of curriculum variant forms is the expression of curricular process - it is essence of knowledge discovery described. The communicative conception of science education as an order of relevant natural science knowledge transformations was defined in the Czech-Slovak conception (and may be also in conjunction with continental Europe). Also the interdisciplinary cooperation with relevant natural science was pointed out. On the other hand, several forms of education content were described in Anglo-American conception. Thus the interdisciplinary collaboration with educational science was pointed out. The “assimilation” of science education with natural science in European conception on one side and the “assimilation” of science education with educational science in AngloAmerican conception on the second side led to the special phenomena: Anglo-American research quitted to use the concept “natural science didactics” and European research quitted to use the concept “science education”. The algorithm of curricular process expresses the hopeful way for collaboration of European communicative conception and Anglo-American education content conception. Such hopeful way would enable to work with the idea of Carl Wieman (recipient of the Nobel Prize in 2001) – “Why not try a scientific approach to science education?” (by Carl Wieman, www.cwsei.ubc.ca).
A8.2. Analytical Synthetic Model of Science Education According to J.Brockmeyer (1982), P.Tarabek, P.Zaskodny (2009, 2010, 2011, 2012, 2013) the subject of science education is a whole continuous process of forwarding and negotiation of results and methods of natural science knowledge to the sense of individuals, who are not directly bounded with the knowledge creation. This process is leading to the transfer of natural science knowledge to the sense of whole society. This process is done by
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various participants with educational intention and includes not only the teaching and education in all levels of educational system, but also lifelong studies carried out institutionally and information transfer from natural science to society. From this reason it will be effective to acquaint ourselves above all with main structural elements of science education, continuously to describe them by the survey of substantial concepts connected with respective natural science curricular process, and finally to form the analytical synthetic model. The investigation of so structured orientation is although accessible for the construction of analytical synthetic model and definition line of matrix, it cannot, however, be confused with a continuous and coherent study of relevant natural science theory as a separate scientific disciplines. The derivation of matrix definition line (algorithm of curricular process) will be introduced by analytical-synthetic model of the structure of science education as a whole. This model can be used for the immediate classification of curriculum variant forms and for the immediate location of previous and follow-up consequences. The model also has a significant cognitive dimension – it is showing which the operations of analysis, abstraction and synthesis are to be carried out to be complete the adoption of relevant curriculum variant forms. The presented model (see visualization Vis.6 and visualization Vis.7) contains three partial analytical-synthetic structures. The model in visualization Vis.6 and in visualization Vis.7 will be presented now. The utilization of analytical synthetic model of science education represents data mining approach to the exploration of the principles of science education. The data mining approach enables to work with the integral concepts and knowledge pieces in their system shape (see analytical-synthetic model). It is necessary to remind the starting point for construction of analytical synthetic model of science education – the theory of content pedagogy expressed by analytical synthetic model of curriculum research and development (see visualization Vis.3 in Chapter A4). The data mining approach can be taken as the realization of data mining cycle, the utilization of analytical synthetic model can be taken as the realization of significant partial data mining tool. The immediate structural orientation, showing which part of the science education and curricular process of respective natural science is just applied in the course of the problem solving, isn´t useless. It is always good to know whether the first partial structure and the second partial structure are used in the framework of problem solving (see visualization Vis.6) or whether the third partial structure is used for the final resolution of investigated problem (see visualization Vis.7). In addition, the linear arrangement of analytical synthetic model enables to characterize the algorithm of curricular process as the definition line of matrix model
Vis.6: Analytical synthetic model of science education – 1. and 2. partial structures
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The conception of science education as a combination of European tradition and Anglo-American tradition with the application takeover of findings of education science and relevant natural science
European tradition:
Anglo-American tradition:
Communicative Conception – transformations of relevant natural science knowledge
Curriculum Conceptions – phenomenon of several forms of education content
Communication of relevant natural science
Curricular process of relevant natural science
Application Takeover of Findings of Educational Science: Endogenic and exogenous side of educational process
Application Takeover of Findings of Respective Natural Science: Scientific system of relevant natural science from the point of education
Possibilities of educants. Methods, forms and means in natural science. Education management.
Scientific system of natural science from the point of its communication. Principle of scientific character
a) Educational communication of respective natural science as curricular process of respective natural science b) Sequence of natural science knowledge transformations, Specification of endogenic side and exigenous side of natural science education (Brockmeyer, Kotasek, Prucha, Tarabek, Zaskodny) c) Cathegory of content of natural science education gives reasons for existence of science education as scientific branch
Structure of transfer of natural science cognition (Bloom, Carroll, Sochor, Thomas, Tollinger, Fajkus)
Structure of transfer forms of natural science cognition (Bruner, Pulpan)
Specification of endogenic side of natural science education (Piaget, Pulpan)
Specification of exogenous side of natural science education (Prucha)
Modeling transfer of Modeling concept and of cognitive Methods of science Obr.2 Analyticko-syntetický model kognitivníHierarchy struktury – 2.část natural science conceptual knowledge levels didaktiky fyzikyeducation cognition systems (Tarabek) (Many authors) (Zaskodny) (Tarabek) a) Aplikace kognitivně strukturních metod v didaktické komunikaci fyziky a kurikulárním procesu fyziky (vymezování obsahové stránky fyzikální edukace) a specifické podoby endogenní a exogenní stránky fyzikální edukace:
Kategorie edukacein zdůvodňuje existenci didaktiky fyziky jakoof a) b) Application of obsahu cognitivefyzikální structural methods curricular process of natural science (delimitation vědního content side ofoboru natural science education) and specific forms of endogenic and exogenous side of natural science education a vede k potřebě zkoumat formy existence kurikula v rámci transformací fyzikálního b) Category of content of natural science education leads to the need to investigate the forms of poznatku curriculum existence within the transformations of natural science finding
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Vis.7: Analytical synthetic model of science education – 3. partial structure a) Application of cognitive structural methods in curricular process of natural science (delimitation of content side of natural science education) and specific forms of endogenic and exogenous side of natural science education b) Category of content of natural science education leads to the need to investigate the forms of curriculum existence within the transformations of natural science finding
Transformation T1 – Conceptual Curriculum (communicative scientific system of natural science)
Transformation T2 – Intended Curriculum (expression of endogenic side)
Transformation T3 – Projected Curriculum and Implemented Curriculum-1 (expression of exogenous side)
Transformation T4 – Implemented Curriculum-2 (the results of natural science education)
Transformation T5 – Attained Curriculum (usable results of natural science education)
Construction and expression of variant forms of curriculum as investigation of individual problem areas of science education
A8.3. Groups of Definition Line Elements of Matrix Model of Science Education The creation of matrix model definition line will reflect the model represented by visualization Vis.3 and visualization Vis.4. Therefore, the interpretation of individual elements of definition line can be described by means of the structural elements of the first partial structure (see Vis.6), the second partial structure (see Vis.6) and the third partial structure (see Vis.7). The application of matrix model definition line as the algorithm of relevant natural science curricular process should be fulfilled for persons interested in deeper understanding methodology of science education. The groups of definition line elements forming the main problem areas of science education (the main conceptual knowledge systems as stages of natural science curricular process algorithm) should keep the order determined by definition line of matrix model. The succession of main problem areas of science education can be considered as algorithm of natural science curricular process. Survey of arranged main problem areas of science education (see also Chapter A4.4.): The first problem area of science education – The first variant form of curriculum The communicable scientific system as “conceptual curriculum” Output of transformation T1
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The second problem area of science education – The second variant form of curriculum The educational system of natural science as “intended curriculum” Output of transformation T2 The third problem area of science education – The third variant form of curriculum The instructional project of natural science “projected curriculum” Output of transformation T3 The fourth problem area of science education – The fourth variant form of curriculum The preparedness of educator to education “implemented curriculum-1” Output of transformation T3 The fifth problem area of science education – The fifth variant form of curriculum The results of education “implemented curriculum-2” Output of transformation T4 The sixth problem area of science education – The sixth variant form of curriculum Applicable results of education “attained curriculum” Output of transformation T5
A8.4. Individual Stages of Algorithm of Curricular Process The algorithm of natural science curricular process is created by the succession of following stages: i) Conceptual curriculum Variant form of curriculum in literature: Conceptual form (the conception of the education content in schools). Curriculum conception: The conception which is focused to the structure of scientific knowledge (structured and communicable set of knowledge of particular sciences). Transformation of natural science finding: This form can be connected with transformation T1. Conceptual curriculum can be identified with the result of transformation T1. ii) Intended curriculum Variant form of curriculum in literature: Intended curriculum – planned goals and content of education with explicit definition in curriculum documents (curriculum, textbooks). There are three categories of content: content of education itself, its operational level (the actions of students and teachers e.g. during solving suitable types of tasks), prospects level (planned changes of student’s attitudes, interests and motivation). Curriculum conception: Conception is focused on structure of knowledge (curriculum – content – as a structured set of knowledge of particular sciences optimized for abilities of students). This conception is also based on development of cognitive processes (the ability of thinking is more than the list of facts!). Transformation of natural science finding: This form can be connected with transformation T2. Intended curriculum can be identified with the result of transformation T2.
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iii) Projected curriculum and Implemented curriculum-1 Variant form of curriculum in literature: Project and realization form (concrete planned projects of content of education, the content of education presented to the subjects of education). Curriculum conception: This conception is focused on the technologies of education (the method of forwarding is in the centre of knowledge). Transformation of natural science finding: These forms can be identified with the result of transformation T3. The expecting result of transformation T3, “Projected curriculum”, can be extended to a new variant form of curriculum bounded with preparing of teacher for education. This has not been mentioned in literature yet. This new variant form of curriculum can be named “Implemented curriculum-1” and thus it can be divided from variant form of implemented curriculum connected with subject of matter took over by educants (Projected curriculum and Implemented curriculum-1). iv) Implemented curriculum-2 Variant form of curriculum in literature: Resulting form (content of education accepted with the subjects of education). Curriculum conception: Conception is based on self-realization of educant (to give the educant the space to investigate the world with his own action, to start with his interests). Transformation ofnatural science finding: This form can be identified with transformation T4. We can identify implemented curriculum as a content of education accepted with the subjects of education with the result of transformation T4. Implemented curriculum is in terms of this analysis of variant forms of curriculum divided into Implemented curriculum 1 (identified with transformation T3) and Implemented curriculum 2 (identified with transformation T4).
v) Attained curriculum Variant form of curriculum in literature: Effect form as an achieved curriculum (the content of education operating on the side of subjects of education), the form of acquired knowledge modified by educants in the term of their own experiences and interests. Curriculum conception: The conception of society reparation (to solve the problems of society by education). Transformation of natural science finding: This form can be connected with transformation T5. Attained curriculum can be identified with the result of transformation T5 as a permanent component of education.
A8.5. Recommended Literature The more detailed explanation of the science education methodology (the main problem areas as individual curriculum variant forms of natural science curricular process) can be found in recommended literature. Recommended literature is given by survey of monographs Educational & Didactic Communication (see also www.csrggroup.org):
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Tarabek,P., Zaskodny,P. (2001) Analytical-Synthetic Modelling of Cognitive Structures (volume 1: New structural methods and their application). Educational Publisher Didaktis Ltd., Bratislava, London ISBN 80-85456-77-X Tarabek,P., Zaskodny,P. (2002) Analytical-Synthetic Modelling of Cognitive Structures (volume 2: Didactic communication and educational sciences). Educational Publisher Didaktis Ltd., Bratislava, New York ISBN 80-85456-77-X Tarabek,P., Zaskodny,P. (2003) Structure, Formation and Design of Textbook (volume 1: Theoretical basis). Educational Publisher Didaktis Ltd., Bratislava, London ISBN 80-85456-09-5 Tarabek,P., Zaskodny,P. (2004) Structure, Formation and Design of Textbook (volume 2: Theory and practice). Educational Publisher Didaktis Ltd., Bratislava, London ISBN 80-85456-09-5 Tarabek,P., Zaskodny,P. (2005) Modern Science and Textbook Creation (volume 1: Projection of scientific systems). Educational Publisher Didaktis Ltd., Bratislava, Frankfurt a.M. ISBN 80-85456-12-3 Tarabek,P., Zaskodny,P. (2006) Modern Science and Textbook Creation (volume 2: Modern tendencies in textbook creation). Educational Publisher Didaktis Ltd., Bratislava, Frankfurt a.M. ISBN 80-85456-12-3 Tarabek,P., Zaskodny,P. (2008) Educational and Didactic Communication 2007 Vol. 1 – Theory Educational Publisher Didaktis Ltd., Bratislava, Frankfurt a.M. ISBN 987-80-89160-56-3 Tarabek,P., Zaskodny,P. (2008) Educational and Didactic Communication 2007 Vol. 2 – Methods Educational Publisher Didaktis Ltd., Bratislava, Frankfurt a.M. ISBN 987-80-89160-56-3 Tarabek,P., Zaskodny,P. (2008) Educational and Didactic Communication 2007 Vol. 3 – Aplications Educational Publisher Didaktis Ltd., Bratislava, Frankfurt a.M. ISBN 987-80-89160-56-3
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Tarabek,P., Zaskodny,P. (2009) Educational and Didactic Communication 2008 Educational Publisher Didaktis Ltd., Bratislava, Frankfurt a.M. ISBN 987-80-89160-62-4 Tarabek,P., Zaskodny,P. (2010) Educational and Didactic Communication 2009 Educational Publisher Didaktis Ltd., Bratislava, Slovakia ISBN 978-80-89160-69-3 Tarabek,P., Zaskodny,P. (2011) Educational and Didactic Communication 2010 Educational Publisher Didaktis Ltd., Bratislava, Slovakia ISBN 978-80-89160-78-5 Tarabek,P., Zaskodny,P. (2012) Educational and Didactic Communication 2011 Educational Publisher Didaktis Ltd., Bratislava, Slovakia ISBN 978-80-89160-93-8 Zaskodny,P. (2013) Educational and Didactic Communication 2012 (volume 1: Educational Data Mining and Its Application) Educational Publisher Didaktis Ltd., Bratislava, Slovakia ISBN 978-80-89160-97-6
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A9. Algorithm of Option Pricing and Hedging A9.1. Basic Concepts – Financial Options, Pricing, Hedging The problem of option pricing and option hedging is connected primarily with theoretical studies of financial markets. The reached results of theoretical studies are often given by continuous and discrete models (Black-Scholes model and its innovation, Figlewski model, multinomial models). Secondarily, from the point of view of data mining approach, the reached results should be described by applicable algorithms. Tertiarily, the finalization of given problem solved should be expressed by functional software. Financial options are such type of derivative contracts in which the underlying securities are financial instruments such as stocks, bonds or an interest rate. The options on financial instruments provide a buyer with the right to either buy or sell the underlying financial instruments at a specified price on a specified future date. Although the buyer gets the rights to buy or sell the underlying options, there is no obligation to apply this option. However, the seller of the contract is under an obligation to buy or sell the underlying instruments if the option is applied. Two types of financial options exist, namely call options and put options. Under a call option, the buyer of the contract gets the right to buy the financial instrument at the specified price at a future date, whereas a put option gives the buyer the right to sell the same at the specified price at the specified future date. The price that is paid by the buyer to the seller for exercising this level of flexibility is called the premium (the fair price). The prescribed future price is called the strike price. The theoretical calculation of premium is connected namely with both the Black-Scholes model (continuous statistical model based on normal distribution) and the multinomial model (discrete statistical models based e.g. on binomial or trinomial distribution). Financial options are either traded in an organized stock exchange or over-the-counter. The options traded through exchange are known as standardized options. The options exchange is responsible for this standardization. This is done by specifying the quantity of the underlying financial instrument, its price and the future date of expiration. The details of these specifications may very vary from exchange to exchange. However, the broad outlines are similar. Financial options are used either to hedge against risks by buying contracts that will pay out if something with negative financial consequences happens, or because it allows traders to magnify gains in spite of limiting risks are existing. Financial options involve the risk of losing some or all of the contract prices, if the market moves against the trend expected, and counterparty risk, such as broker insolvency or contractors who do not fulfill their contractual obligations.
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A9.2. Analytical Synthetic Model of Option Pricing and Hedging The deciding idea is to develop financial option pricing structural levels and algorithms of a form that can be connected to the relevant discrete option hedging structural levels and algorithms. Applying the algorithms will enable to create the software with a suitable programming language, considering the reality of the financial markets. The identification of all the types of interconnected option pricing and hedging structural levels (according to V.Pavlat, P.Zaskodny, 2012) is the way leading to the differentiation of analytical synthetic model and matrix model. The analytical synthetic modeling and matrix modeling are belonging to the very important partial data mining tools. The way how to solve the differentiation the analytical synthetic and matrix model is proposed as follows: i) It is necessary to identify option pricing structural levels and related option hedging structural levels. ii) The four option hedging structural levels connected with relevant option pricing structural levels can be described as follows: - Model structural level based on links between related models of the successive application of various option pricing and hedging types. - Conceptual structural level based on fixed and proportional hedging of an option portfolio. - Theoretical structural level based on option pricing and hedging theoretical continuities. - Practical structural level based on financial risk management using a specific data mining matrix modeling iii) The structural levels should incorporate a detailed series of steps to allow us to choose, and apply, a suitable programming language (e.g., the high-level object oriented programming language C#, with the software’s graphics developed in Windows Forms and data stored with ADO.NET technology). The model structural level, conceptual structural level and theoretical structural level can be connected with the description of analytical synthetic model of option pricing and hedging. The analytical synthetic model of above all option hedging (with input given by acceptable model of option pricing) is created by three structural levels (see visualizations Vis.8, Vis.9, Vis.10): i) The first structural level of analytical synthetic model – model structural level ii) The second structural level of analytical synthetic model – conceptual structural level iii) The third structural level of analytical synthetic model – theoretical structural level
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Vis.8: Analytical synthetic model of option hedging – The first structural level i) The first structural level of analytical synthetic model – model structural level associated with the selection of becoming sequences of models. Implied sequence: Black-Scholes model and its innovations; Implied binomial model; Implied trinomial model; Implied quadrinomial model; Alternatively implied pentanomial model. Autonomously implied sequence: Autonomous binomial model; Autonomously implied trinomial model; Autonomously implied quadrinomial model; Alternatively autonomously implied pentanomial model. Autonomous sequence: Autonomous binomial model; Autonomous trinomial model; Autonomous quadrinomial model; Alternatively autonomous pentanomial model. Vis.9: Analytical synthetic model of option hedging – The second structural level ii) The second structural level of analytical synthetic model – conceptual structural level, based on fixed and proportional hedging of option portfolios, is defined by the following series of concepts which are transformed gradually to each other: Hedged option portfolio: A fixed-value portfolio resulting from option hedging. Fixed and proportional hedge: Fixed hedge relating to a single hedged option portfolio, and a proportional hedge that relates to a set of hedged portfolios. Four input positions to apply option hedging to: Call option buyer, put option buyer, call option seller and put option seller. Constructing needful position: Constructing “call option seller” and “put option seller” position to finance a hedged option portfolio during option hedging implementation. Relation between continuous and discrete option hedging: Transition from continuous option hedging to discrete option hedging, i.e., a link between continuous and discrete option pricing models. The role of discrete option pricing models: The role of discrete option pricing models as a means to go from a fixed to a proportional hedge. Implementing a discrete option hedge in two steps: Developing dynamic hedging strategies and rebalancing. Developing a dynamic hedging strategy: Forming a lattice that represents a chosen type of random walk, putting relevant option hedge factor values at the nodes. Rebalancing the hedge: Transforming the original hedged option portfolio into new hedged portfolios within a predetermined random walk along the time series of consecutive nodes.
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Vis.10: Analytical synthetic model of option hedging – The third structural level iii) The third structural level of analytical synthetic model – theoretical structural level, based on option pricing and hedging theoretical continuities, is defined by the following series of steps which are transformed gradually to each other: Selecting a random walk type: The selection is based on the choice of an s-fold multinomial model. Developing a lattice with nodes and time layers, i.e., taking the selected type of random walk: E.g., developing a binomial lattice (binomial tree), a trinomial lattice (trinomial tree) or a quadrinomial lattice (quadrinomial tree). Developing a dynamic hedging strategy: E.g., developing a dynamic delta hedge and developing a dynamic gamma hedge. Selecting a specific random walk, i.e., a time series of consecutive nodes on each time layer: E.g., selecting a specific binomial random walk on a binomial tree, selecting a specific trinomial random walk on a trinomial tree and selecting a specific quadrinomial random walk on a quadrinomial tree. Rebalancing, and developing a set of hedged option portfolios at specific nodes on consecutive time layers of the lattice: E.g., developing a set of binomial hedged option portfolios, developing a set of trinomial hedged blocks of trinomial option portfolios and developing a set of quadrinomial hedged blocks of quadrinomial option portfolios.
A9.3. Groups of Definition Line Elements of Matrix Model of Option Pricing and Hedging The four option hedging structural levels (connected with relevant option pricing structural levels) were found in A9.2.: - Model structural level based on links between related models of the successive application of various option pricing and hedging types. - Conceptual structural level based on fixed and proportional hedging of an option portfolio. - Theoretical structural level based on option pricing and hedging theoretical continuities. - Practical structural level based on financial risk management using a specific data mining matrix modeling. The analytical synthetic model of option pricing and hedging was formed using model structural level, conceptual structural level, theoretical structural level. The remaining structural level, practical structural level, enables to describe the line elements groups of option hedging matrix model. This option hedging matrix model is emanating from relevant option pricing models. The sequence of line elements groups is expressing the algorithm of option pricing and hedging.
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The algorithm of option pricing and hedging is created by the succession of following stages (a) to (i): (a) Selecting a suitable type of a discrete option pricing model, defining the parameters of the model selected, determining the values of the input parameters (growth and decline indices, in particular) and describing the selected type of random walk. (b) Defining the origins of the growth and decline indices in terms of their links to the implied model or the autonomously implied model or the autonomous model. (c) Calculating underlying stock prices and theoretically correct option prices at the nodes of a lattice resulting from the implementation of a selected type of random walk. (d) Selecting the appropriate equations and formulae to prepare the implementation of a delta hedge. (e) Developing a dynamic delta hedge strategy. (f) Delta rebalancing. (g) Selecting the appropriate equations and formulae to prepare the implementation of a gamma hedge. (h) Developing a dynamic gamma hedge strategy. (i) Gamma rebalancing. (j) Preparing and implementing other option hedging types that have been selected (e.g., speed hedging and vega hedging).
A9.4. Recommended Literature The more detailed explanation of the option pricing & hedging algorithm can be found in recommended literature. Tarabek,P., Zaskodny,P. (2010) Educational and Didactic Communication 2009 Bratislava, Slovakia: Educational Publisher Didaktis Ltd. ISBN 978-80-89160-78-5 Pavlat,V., Zaskodny,P. (2012) From Financial Derivatives to Option Hedging (343 ps) Prague, Czech Republic: Curriculum ISBN 978-80-904948-3-1 Pavlát, V., Záškodný,P Budík,J. (2007) Financial Derivatives and Their Evaluation (161 ps) Prague, Czech Republic: VSFS, ISBN 978-80-86754-73-4.
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A10. Role of Algorithms in Data Mining Approach A10.1. Brief Description of Role The role of algorithms in data mining approach can be described by means of three conclusions gained in the framework of previous chapters. The first conclusion: Data mining approach is given by the global realization of data mining cycle structure (see Chapter A2). Data mining approach is closely associated with problem solving. And the resolution of concrete identified problem is usually given by the location of algorithm how to reach intellectual reconstruction of identified problem. The second conclusion: Data mining tool is defined as a concrete procedure how to reach the results of data mining approach (see Chapter A1 and Chapter A3). Data mining tools can be divided to the complex tools (e.g. structured methodology of relevant scientific branch) and to the partial tools (the essential procedures how to find the structures of problems solved). The described structure of methodology of relevant scientific branch can be taken as the result of complex tool applications, the descriptions of structures of investigated partial problems of relevant scientific branch can be taken as the results of partial tools applications. The significant way of structure description may be connected with the algorithms of relevant problem solving. The role of algorithms in data mining approach could be explained in respective scientific branch by means of the complex and partial data mining tools. The third conclusion: The significant data mining tools are analytical synthetic modeling and matrix modeling (see Chapter A5 and Chapter A6). Within the framework of creation of matrix model of scientific or cognitive structure it is first of all necessary to delimit the survey of follow-up elements of matrix main diagonal by linear way. The brief description of their contents is necessary to perform. The delimitation of definition line of matrix is starting from linear arrangement of the analytical synthetic model of scientific or cognitive structure. Often it is possible to take the important segments of definition line as the partial conclusions dk of analytical synthetic model (see Vis.1). The main diagonal of matrix, so called definition line, can be by the carrier of algorithm describing the result of data mining approach or the way of application of data mining tools. The role of algorithms in data mining approach is connected with the discovery of definition line of matrix model.
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A10.2. Illustration of Data Mining Approach Algorithms For realization of PART B “The Algorithm of Problem Solving in Statistics” the concrete algorithm of statistics and probability was delimited (see Chapter A7): i) Formulation of Statistical Investigation ii) Creation of Scale iii) Measurement in Descriptive Statistics, Statistical Probability iv) Elementary Statistical Processing v) Non-parametric Testing, Continuous and Discrete Probability Distribution vi) Theory of Estimation vii) Parametric Testing viii) Regression and Correlation Analysis For realization of PART C “The Algorithm of Problem Solving in Curricular Process” the concrete algorithm of curricular process was delimited (see Chapter A4 and Chapter A8): i) Conceptual curriculum ii) Intended curriculum iii) Projected curriculum and Implemented curriculum-1 iv) Implemented curriculum-2 v) Attained curriculum For realization of PART D “The Algorithm of Problem Solving in Option Pricing and Hedging” the practical algorithm of option hedging (in cooperation with corresponding algorithm of option pricing) was delimited (see Chapter A9): i) Selecting a suitable type of a discrete option pricing model, defining the parameters of the model selected, determining the values of the input parameters (growth and decline indices, in particular) and describing the selected type of random walk. ii) Defining the origins of the growth and decline indices in terms of their links to the implied model or the autonomously implied model or the autonomous model.
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iii) Calculating underlying stock prices and theoretically correct option prices at the nodes of a lattice resulting from the implementation of a selected type of random walk. iv) Selecting the appropriate equations and formulae to prepare the implementation of a delta hedge. v) Developing a dynamic delta hedge strategy. vi) Delta rebalancing. vii) Selecting the appropriate equations and formulae to prepare the implementation of a gamma hedge. viii) Developing a dynamic gamma hedge strategy. ix) Gamma rebalancing. x) Preparing and implementing other option hedging types that have been selected (e.g., speed hedging and vega hedging).
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PART B The Algorithm of Problem Solving in Statistics
Authors of Chapters (according to order of the single chapters) Dang Thi Thu Hien University of Finance and Administration
[email protected] Vilem Fasura University of Finance and Administration
[email protected] Dominika Masna University of Finance and Administration
[email protected] Adam Vlcek University of South Bohemia
[email protected]
The goal of Part B is to validate the hypothesis that it is possible to apply the algorithm of statistics to the solution of concrete problems of statistical investigation. The goal of Part B was completed using by the follow-up chapters: B1. Investigation of Comparison between Prices of Natural 95 and Diesel (according to Dang Thi Thu Hien) B2. Investigation of Price Movement of Selected Stocks in The S&P500 within One Month (according toV.Fasura) B3. Comparison between Percentage Price Movements of Coca Cola Company and PepsiCoCompany within Relevant Financial Market (according to D.Masna) B4. Measurement of Different Luminous Intensity of Star: WASP-39b (according to A.Vlcek)
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B1. Investigation of Comparison between Prices of Natural 95 and Diesel Author Dang Thi Thu Hien University of Finance and Administration
[email protected]
Abstract This chapter investigates the correlation between the prices of Natural 95 and Diesel, which are among the most used fuels in the automotive market. The methodology for this study includes descriptive and mathematical statistics, statistical probability and probability distribution. Prices were gathered from the metropolitan area of Prague, Czech Republic. The results may be or may be not surprising, that the prices of these two types of fuel correlate positively.
Key Words Algorithm of Statistics and Probability, Fuel, Non-parametric Testing, Regression Analysis, Correlation Analysis.
Contents B1.1. Introduction B1.2. Descriptive Part for Statistical Sign SS1 B1.3. Non-parametric Testing for Statistical Sign SS1 B1.4. Descriptive Part for Statistical Sign SS2 B1.5. Non-parametric Testing for Statistical Sign SS2 B1.6. Measurement of Statistical Dependences B1.6.1. Linear regression analysis B1.6.2. Linear correlation analysis B1.7. Conclusion B1.8. Literature
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B1.1. Introduction The prices of petrol and diesel are changing daily based on world conditions. The post crisis economic situation also causes fuel prices to fluctuate. Basically, the usage and pricing of fuel results myriad factors such as crude oil prices processing and distribution costs, local demand, strength of local currencies, taxation, and the availability of local supply. Fuels are traded worldwide so the prices are similar; however customers often pay extra due to national pricing policies. For example, in Europe there are relatively high taxes on fuel, and profit and margin demands of station owner operators can influence the final price per liter. There have been some studies on the correlation between prices of petrol and diesel, but not that many in the Czech Republic, specifically in Prague. The application of descriptive and mathematical statistics and probability theory in investigation of the above mentioned correlation is according to P.Záškodný (2013) connected with utilization of the algorithm of probability and statistics. This algorithm is created by the succession of following steps: - Formulation of Statistical Investigation - Creation of Scale - Measurement in Descriptive Statistics, Statistical Probability - Elementary Statistical Processing - Non-parametric Testing - Theory of Estimation - Parametric Testing - Regression and Correlation Analysis Presented work is seeking the correlation between prices of petrol and diesel. The assumptions of work are suggesting - the empirical distribution of prices is possible to substitute by normal distribution - the correlation between these two types of fuel will be strongly positive. The verification of submitted assumptions will be associated with an application of following steps from presented algorithm – formulation of statistical investigation, scaling, measurement, elementary statistical processing, non-parametric testing and regression and correlation analysis. The formulation of statistical investigation is given by utilization of the definition of basic concepts: - Collective Random Phenomenon (CRP – Correlation between two types of fuel) - Statistical Unit (SU as Carrier of CRP – Petrol station in Prague) - Statistical Signs (SS1 and SS2 as investigated properties of SU: SS1- Prices of Natural 95, SS2- Prices of Diesel) - Values of Statistical Signs (the set of prices of two types of fuel – VSS1, VSS2) - Basic Statistical Set (BSS as the set of all the statistical units – all the petrol stations in Prague) - Random Selection (RS as the choice of fifty petrol stations) - Selective Statistical Set (SSS is given by fifty petrol stations randomly selected)
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B1.2. Descriptive Part of Investigation for Statistical Sign SS1 CRP: correlation between the prices of two types of fuel in Prague in March 2013 Statistical sign SS1: price of Natural 95 in Prague (CZK/l) SU: petrol station BSS: 126 units-126 petrol stations in Prague Petrol station Tank Ono Praha - Kutnohorská DB9 Praha 10 – Kutnohorská MV Plus Praha - Uhříněves - Podleská AMODO s.r.o. Praha 10 - Dřevčická Auto Jarov s.r.o. Praha 3 - Osiková 2 LukOil Praha - Dálnice D8 Prim Praha – Uhříněves 1 Praha 9/Vysočany TUKAS AUTO Praha 10 - Štěrboholská TUKAS AUTO Praha 10 - Radiová LukOil Praha 3 - U nákladového nádraţí Kont.cz Praha 4 – Opatovská LukOil Praha 5 - K Barrandovu Praţské sluţby Praha 9 - Pod Šancemi Makro Praha 13 – Jeremiášova Globus Praha 9 - Čakovice - Kostelecká Shell Praha 5 – Geologická Prim Praha 10 - Na Hroudě 1955/51
SSS: 50 units-50 petrol stations randomly selected
Price(Kč/l) 34,50 34,50 34,50 34,90 34,90 34,90 34,90 34,90 35,10 35,10 35,20 35,30 35,30 35,30 35,40 35,40 35,50 35,50
Petrol station Slovnaft Jana Ţelivského Globus Praha 5 - Sárská 133/5 Shell Praha U staré cihelny Globus Centrum Černý most Way24 Praha Skandinávská MEDOS Dolní Počernice TEXACO Praha 10 - Průběţná PETR CAR Českobrodská Gaspoint, a.s.Vestec Vídeňská LukOil Nad Vršovskou horou LukOil Praha Horní Počernice Shell Praha 10 – Bohdalecká Avanti Praha 8 – prosecká ČSAO Praha Černokostelecká EuroOil Praha 9 - Poděbradská Auto Leon s.r.o. Průmyslová Benzina Novovysočanská Makro Praha Chlumecká 2424
Price(Kč/l) 35,50 35,60 35,80 35,90 35,90 35,90 35,90 35,90 35,90 36,00 36,10 36,20 36,20 36,20 36,20 36,20 36,20 36,30
VSS1: 34,50 - 36,30 Kč/l Scaling for SS1: Quantitative metric scale Scale elements: Scale elements marked by x1 = 1, x2 = 2, x3 = 3, x4 = 4, x5 = 5 Price (Kč/l) Scale
under 34,86 1
34,86-35,22 2
35,23 – 35,58 3
35,59 – 35,94 4
above 35,94 5
Measurement for SS1 (table, graphs, empirical parameters): One from k scale elements x1, x2, x3, x4, x5, (k = 5) is assigned to statistical unit of selective statistical set SSS (extent n of SSS is given by equality n = 36) The measurement results are given by determination: Scale element xi (x1, x2, x3, x4, x5) was measured by ni times. Absolute frequencies ni – summation of all the values ni ( i = 1, 2, …, k) is equal to the extent n selective statistical set.
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Relative frequencies ni/n (statistical probability) – summation of ni/n must be equal to 1. Cumulative frequencies ∑ni/n (cumulative probability) – it can be determined only within of quantitative metric or absolute metric scales. Measurement has fulfilled the conditions of validity (the price was measured), reliability (measurement reducibility) and objectivity (various researchers will measure by the same way).
Table: xi
ni
ni/n
∑ ni/n
xini
xi2ni
xi3ni
xi4ni
1 2 3 4 5
3 8 8 8 9 ∑ 36
0,08 0,22 0,22 0,22 0,25 ∑ 1,00
0,08 0,30 0,52 0,74 1
3 16 24 32 45 ∑ 120
3 32 72 128 225 ∑ 460
3 64 216 384 1125 ∑ 1792
3 128 648 2048 5625 ∑ 8452
The first four columns - creation of graphical representation of frequency empirical distributions The second four columns - computation of empirical parameters
Graphs: GEDAF – graph of empirical distribution of absolute frequencies
GEDAF 10 8 6 4 2 0 0
1
2
3
4
5
6
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GEDRF – graph of empirical distribution of relative frequencies
GEDRF 0,3 0,25 0,2 0,15 0,1 0,05 0 0
1
2
3
4
5
6
GEDCF – graph of empirical distribution of cumulative frequencies
GEDCF 1,2 1 0,8 0,6 0,4 0,2 0 0
1
2
3
4
5
6
Empirical parameters for SS1: General moment of r-th order:
1
𝑂𝑟 𝑥 = 𝑛
𝑛𝑖 ∙ 𝑥𝑖 𝑟
1
𝑂1 𝑥 = 36 ∙ 120 = 3,33 𝑂2 𝑥 =
1 36
∙ 460 = 12,77
𝑂3 𝑥 =
1 36
∙ 1792 = 49,77
𝑂1 - „arithmetic mean“ It shows the placement of frequencies empirical distribution on the horizontal axis.
1
𝑂4 𝑥 = 36 ∙ 8452 = 234,77 Central moment r-th order: Central moment of 2. order: 1
𝐶2 𝑥 = 36 ∙ 60 = 1,6667
𝐶𝑟 𝑥 = 𝑛
1
𝑛𝑖 𝑥𝑖 − 𝑂1
𝑟
1
𝑛𝑖 𝑥𝑖 − 𝑂1
2
𝐶2 𝑥 = 𝑛
𝐶2 – „empirical variance“
Educational & Didactic Communication 2013, Vol.1, Part B Standard deviation: 𝑆𝑥 = 𝑆𝑥 = 1,6667 = 1,291
𝐶2
55
Sx shows what the information value is given to arithmetic mean. If it is large, the arithmetic value of arithmetic mean is small and vice versa.
𝑆
Variation coefficient: V = 𝑂𝑥
1
𝑆𝑥 𝑂1
=
1,291 3,33
= 0,38 V = 38 %
It shows how many percent from arithmetic mean is created by standard deviation.
𝐶2 = 𝑂2 − 𝑂1 2 𝐶2 = 12,77 − 3,332 = 1,66 𝐶3 = 𝑂3 − 3𝑂2 𝑂1 + 2𝑂1 3 𝐶3 = 49,77 − 3.12,77.3,33 + 2.3,333 = -3,95 𝐶4 = 𝑂4 − 4𝑂3 𝑂1 + 6𝑂2 𝑂1 2 − 3𝑂1 4 𝐶4 = 234,77 − 4.49,77.3,33 + 6.12,77.3,332 − 3. 3,334 = 52,57 Standardized moments of 3. and 4. order for SS1 𝑁3 = 𝑁3 =
𝐶3
𝑁3 – „coefficient of skewness“ If the skewness
𝐶2 𝐶2 −3,95 1,66 1,66
= -1,8469
𝐶
𝑁4 = 𝐶 42
𝑁4 – „coefficient of kurtosis“ The greater value of
2
𝑁4 =
52,57 1,66 2
coefficient is positive, then the scale elements lying to the left of the arithmetic mean have greater frequencies and vice versa.
= 19,0775
Quantity excess = N4 – 3 excess = 19,0775 – 3 = 16,0775
kurtosis coefficient corresponds to more pointed distribution of frequencies for a given variance. The excess compares the kurtosis of empirical distribution with the kurtosis of known standardized normal distribution. If the excess is positive, the empirical distribution is more pointed than this distribution.
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B1.3. Non-parametric Testing for Statistical Sign SS1 The following table of basic properties of normal distribution and of standardized normal distribution will be used for realization of non-parametric testing. For completeness the comparison of continuous distributions (normal distribution and standardized normal distribution) and chosen discrete distributions (binomial distribution, Poisson distribution) is presented.
Normal distribution
Standardized ND
Binomial distrib.
Poisson distrib.
Mathematical Probability Descriptive Probability
N(μ,σ)=> r=2
N(0,1)=> r=2
Bi(n,p)
Po(λ)
2 theor.parameters
2 theor. parameters
2 theoretical
1 theoretical
μ~O1, σ~Sx
μ~0, σ~1
parameters n,p
parameter λ
ni/n
Probab. density
Probable function
Probable function
relative frequency
𝛒 𝒙 =
∑ni/n cumulative frequency
∙𝐞
𝟏
𝛒 𝒙 =
𝟐𝛑
∙𝐞
𝛍𝟐 −𝟐
Distrib.function
Distrib.function
𝑥 −∞
F(u)=
𝑢 −∞
F(∞)=
∞ −∞
F(x)=
∑ni/n standardized
𝟏 𝛔∙ 𝟐 𝛑
Probab. density
(𝐱−𝛍)𝟐 − 𝟐 ∙ 𝛔𝟐
F(∞)=
𝜌 𝑥 𝑑𝑥
∞ −∞
𝜌 𝑥 𝑑𝑥 =1
Pi=
O1=
∞ −∞
𝑥𝜌 𝑥 𝑑𝑥 =
=μ=E ∞ −∞
O1=
∞ −∞
𝜌 𝑢 𝑑𝑢
Fj=
𝜌 𝑢 𝑑𝑢 =1
𝑢𝜌 𝑢 𝑑𝑢 =
=0=E ∞ −∞
i
p (1-p)
n--i
i Pi =e-λ λ i!
Distrib.function
condition
O1=∑(ni/n) xi
𝑛 𝑖
j i=0 Pi
Distrib.function Fj=
j i=0 Pi
i=0,1,2..n
i=0,1,2..∞
Standardized
Standardized
condition
condition
n i=0 Pi=1
∞ i=0 Pi=1
O1=np=E
O1=λ=E
Expected value
Expected value
C2=np(1-p)=D
C2=λ=D
C2=∑ni/n(xi-O1)2
C2=
Emp. dispersion
C2=D=𝝈2
C2=1=D
Dispersion value
Dispersion value
Sx=√C2
Continuous
Continuous
Discrete
Discrete
Stand.deviation
distribution
distribution
distribution
distribution
(x-μ)2ρ(x)dx
C2=
u2ρ(u)du
The basic goal: To determine if the empirical distribution of prices (empirical distribution associated with statistical sign SS1) can be substituted by normal distribution.
xi
interval
ni
ni/n
∑ ni/n
xini
xi2ni
xi3ni
xi4ni
1 2 3 4 5
(-∞; 1,5>
3 8 8 8 9 ∑ 36
0,08 0,22 0,22 0,22 0,25 ∑ 1,00
0,08 0,30 0,52 0,74 1
3 16 24 32 45 ∑ 120
3 32 72 128 225 ∑ 460
3 64 216 384 1125 ∑ 1792
3 128 648 2048 5625 ∑ 8452
(1,5; 2,5> (2,5; 3,5> (3,5; 4,5> (4,5; ∞)
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Transformation of normal distribution N(μ,σ) to standardized normal distribution N(0,1). Computation of values of new variable u: u
x O1 Sx
u1 = -1,4175 u2 = -0,6429 u3 = 0,1316 u4 = 0,9062 u5 = ∞
Computation of surfaces under Gauss curve through the values of Laplace function (distribution function of standardized normal distribution):
p1 = F(–1,42) = 1– F(1,42) = 1– 0,922 = 0,078 p2 = F(–0,64) – F(–1,42) = 1– F(0,64) – 0,078 = 1- 0,738 – 0,078= 0,262 – 0,078 = 0,184 p3 = F(0,1316) – F(–0,64) = 0,551– 0,262 = 0,289 p4 = F(0,9062) – F(0,1316) = 0,815 – 0,551 = 0,264 p5= F(∞) – F(0,9062) = 1 – 0,815 = 0,185
Comparison of 5 segments of line and 5 surfaces under Gauss curve:
n1 = 0,08 ≈ p1 = 0,078 n
n2 = 0,22 ≈ p2 = 0,184 n
n n4 = 0,22 ≈ p4 = 0,264 5 = 0,25 ≈ p5 = 0,185 n n
n3 = 0,22 ≈ p3 = 0,289 n
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Real difference between 5 segments of line and 5 surfaces determined by empirical way: Computation of experimental value χ2 Application of χ2- test
(ni npi )2 npi i 1 k
2 EXP
n1 np1 2 np1
n2 np2 2 np2
n3 np3 2 np3
n4 np4 2 np4
n5 np5 2 np5
=
(3−36𝑥0,078)2
=
=
=
=
36𝑥0,078
= 0,013
(8−36𝑥0,184)2 36𝑥0,184 (8−36𝑥0,289)2 36𝑥0,289 (8−36𝑥0,264)2 36𝑥0,264 (9−36𝑥0,185)2 36𝑥0,185
= 0,286
= 0,555
= 0,238
= 0,822
2 𝜒𝐸𝑋𝑃 = 1,914
The highest allowed difference between 5 segments of line and 5 surfaces determined by means of statistical tables: Computation of theoretical (critical) value χ2 and of freedom degrees number ν for significance level (statistical error of 1.type) α = 0,05 2 TH k2r 1 0,05 (k – number of scale elements, r – number of theoretical parameters)
k r 1
ν = 5 – 2 – 1=2
2 𝜒𝑇𝐻 = 5,99 2 2 TEOR EXP → it behaves normally
For the significance level α = 0,05 the investigated empirical distribution for statistical sign SS1 may be substituted by normal distribution, the empirical graph may be substituted by Gauss curve.
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B1.4. Descriptive Part of Investigation for Statistical Sign SS2 CRP: correlation between the prices of two types of fuel in Prague in March 2013 Statistical sign SS2: price of Diesel in Prague (CZK/l) Petrol station Tank Ono Praha - Kutnohorská DB9 Praha 10 - Kutnohorská MV Plus Praha - Uhříněves - Podleská AMODO s.r.o. Praha 10 - Dřevčická Auto Jarov s.r.o. Praha 3 - Osiková 2 LukOil Praha - Dálnice D8 Prim Praha - Uhříněves 1 Praha 9/Vysočany TUKAS AUTO Praha 10 - Štěrboholská TUKAS AUTO Praha 10 - Radiová LukOil Praha 3 - U nákladového nádraţí Kont.cz Praha 4 - Opatovská LukOil Praha 5 - K Barrandovu Praţské sluţby Praha 9 - Pod Šancemi Makro Praha 13 - Jeremiášova Globus Praha 9 - Čakovice - Kostelecká Shell Praha 5 - Geologická Prim Praha 10 - Na Hroudě 1955/51
Price(Kč/l) 33,90 34,50 34,50 34,90 34,70 34,30 34,90 34,50 35,20 35,20 34,20 34,90 34,30 34,70 34,80 34,90 34,50 34,90
Petrol station Slovnaft Jana Ţelivského Globus Praha 5 - Sárská 133/5 Shell Praha U staré cihelny Globus Centrum Černý most Way24 Praha Skandinávská MEDOS Dolní Počernice TEXACO Praha 10 - Průběţná PETR CAR Českobrodská Gaspoint, a.s.Vestec Vídeňská LukOil Nad Vršovskou horou LukOil Praha Horní Počernice Shell Praha 10 – Bohdalecká Avanti Praha 8 – prosecká ČSAO Praha Černokostelecká EuroOil Praha 9 - Poděbradská Auto Leon s.r.o. Průmyslová Benzina Novovysočanská Makro Praha Chlumecká 2424
Price(Kč/l) 34,50 35,00 35,70 34,90 35,20 34,90 35,50 35,80 35,90 34,20 34,90 35,40 35,50 35,80 36,50 36,20 35,40 34,90
VSS2: 33,90 - 36,50 Kč/l Scaling for SS2: the quantitative metric scale Price (Kč/l) Scale
under 34,42 1
34,42-34,94 2
34,95 – 35,46 3
35,47 – 35,98 4
above 35,98 5
Measurement for SS2 (table, graphs, empirical parameters):
Table: xi
ni
ni/n
∑ ni/n
xini
xi2ni
xi3ni
xi4ni
1 2 3 4 5
5 18 5 6 2 ∑ 36
0,14 0,50 0,14 0,17 0,05 ∑ 1,00
0,14 0,64 0,78 0,95 1
5 36 15 24 10 ∑ 90
5 72 45 96 50 ∑ 268
5 144 135 384 250 ∑ 918
5 80 405 1536 1250 ∑ 3276
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Graphs: GEDAF – graph of empirical distribution of absolute frequencies
GEDAF 20 15 10 5 0 0
1
2
3
4
5
6
GEDRF – graph of empirical distribution of relative frequencies
GEDRF 0,6 0,5 0,4 0,3 0,2 0,1 0 0
1
2
3
4
5
6
GEDCF – graph of empirical distribution of cumulative frequencies
GEDCF 1,2 1 0,8 0,6 0,4 0,2 0 0
1
2
3
4
5
6
60
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Empirical parameters for SS2: 1
𝑂1 𝑥 = 36 ∙ 90 = 2,50 1
𝑂2 𝑥 = 36 ∙ 268 = 7,44 𝑂3 𝑥 =
1 36
∙ 918 = 25,50
𝑂4 𝑥 =
1 36
∙ 3276 = 91,00
𝐶2 𝑥 =
1 36
∙ 43 = 1,19444
𝑆𝑥 = 1,19444 = 1,0929 𝑆𝑥 𝑂1
=
1,0929 2,5
= 0,44 V = 44 %
𝐶2 = 7,44 − 2,52 = 1,19 𝐶3 = 25,5 − 3.7,44.2,5 + 2.2,53 = -14,675 𝐶4 = 91 − 4.25,5.2,5 + 6.7,44.2,52 − 3. 2,54 = 2,19 𝑁3 =
−14,675 1,19 1,19
= -11,3049
2,19
𝑁4 = 1,192 = 1,5465 excess = 1,5465 – 3 = - 1,4535
61
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B1.5. Non-parametric Testing for Statistical Sign SS2 From the point of view the probability distribution significance the table comparing the properties of continuous distributions (normal distribution, standardized normal distribution) and discrete distributions (binomial distribution, Poisson distribution) is reminded. Without cooperation between theory of probability and mathematical statistics the non-parametric testing is not possible to perform. Normal distribution
Standardized ND Binomial distrib.
Poisson distrib.
Mathematical Probability Descriptive Probability
N(μ,σ)=> r=2
N(0,1)=> r=2
Bi(n,p)
Po(λ)
2 theor.parameters
2 theor. parameters
2 theoretical
1 theoretical
μ~O1, σ~Sx
μ~0, σ~1
parameters n,p
parameter λ
ni/n
relative
Probab. density
Probab. density
Probable function
Probable function
frequency
𝛒 𝒙 =
∑ni/n cumulative frequency
∑ni/n=1
𝟏 𝛔∙ 𝟐 𝛑
∙𝐞
−
(𝐱−𝛍)𝟐 𝟐 ∙ 𝛔𝟐
𝛍𝟐
𝟏
𝛒 𝒙 =
𝟐𝛑
∙ 𝐞− 𝟐
Distrib.function
Distrib.function
𝑥 −∞
F(u)=
𝑢 −∞
F(∞)=
∞ −∞
F(x)=
F(∞)=
𝜌 𝑥 𝑑𝑥
∞ −∞
𝜌 𝑥 𝑑𝑥 = 1
𝜌 𝑢 𝑑𝑢
𝜌 𝑢 𝑑𝑢 =1
Standardized condition O1=∑(ni/n) xi
O1=
∞ −∞
𝑥𝜌 𝑥 𝑑𝑥 =
=μ=E ∞ −∞
O1=
∞ −∞
𝑢𝜌 𝑢 𝑑𝑢 =
=0=E ∞ −∞
Pi=
𝑛 𝑖
pi(1-p)n--i
Distrib.function Fj=
j i=0 Pi
i Pi =e-λ λ i!
Distrib.function Fj=
j i=0 Pi
i=0,1,2..n
i=0,1,2..∞
Standardized
Standardized
condition
condition
n i=0 Pi=1
∞ i=0 Pi=1
O1=np=E
O1=λ=E expected
expected value
value
C2=np(1-p)=D
C2=λ=D
C2=∑ni/n(xi-O1)2
C2=
Emp. dispersion
C2=D=𝝈2
C2=1=D
Dispersion value
Dispersion value
Sx=√C2
Continuous
Continuous
Discrete
Discrete
Stand.deviation
distribution
distribution
distribution
distribution
(x-μ)2ρ(x)dx
C2=
u2ρ(u)du
The basic goal: To determine if the empirical distribution of prices (empirical distribution associated with statistical sign SS2) can be substituted by normal distribution. By means of the same procedure, which was used for statistical sign SS1, it is possible to prove: For the significance level α = 0,05 the investigated empirical distribution for statistical sign SS2 may be substituted by normal distribution, the empirical graph may be substituted by Gauss curve.
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B1.6. Measurement of Statistical Dependences The two statistical signs SS1 and SS2 (Natural 95, Diesel) are examining from the point of bonds between them. The bonds between these statistical signs have two basic features: a) Character of bond - is given by the type of regression curve (Regression curve – the curve which is interposed by “probable cloud”. The probable cloud is created by graph the statistical sign SS1 is applied to the x-axis, it will be marked SS-x, the statistical sign SS2 is applied to the y-axis, it will be marked SS-s). To find the suitable regression curve is basic assignment for regression analysis. In this preparatory part of correlation analysis the line as regression curve will be searched. It is assignment for linear regression analysis. b) Tightness of bond - in this final part the linear correlation analysis will be applied (on the basis of above mentioned linear regression analysis). The result of linear correlation analysis evaluates the summation of distances of probable cloud points from regression line. To smaller summation, to tighter bond between statistical signs SS-x (SS1), SS-s (SS2) Quantification of bond tightness is characterized by value of Pearson' correlation coefficient 𝑘𝑥𝑠 . The values of 𝑘𝑥𝑠 ∈ < −1; 1 >. The bond is tighter and tighter with approaching the 𝑘𝑥𝑠 to values –1 and 1. positive correlation = the values of both statistical signs increase or decrease at the same time negative correlation = the values of one statistical sign increase, the values of the second sign decrease the values around 0 indicate the signs don´t correlate B1.6.1. Linear regression analysis Normal equations of linear regression analysis: Σ𝑠𝑖 = 𝑘 𝑏0 + 𝑏1 Σ𝑥𝑖 Σ𝑠𝑖 𝑥𝑖 = 𝑏0 Σ𝑥𝑖 + 𝑏1 Σ𝑥𝑖2
• firstly the reformulation of statistical investigation must be carried out • the number k of scale elements will be renamed to n Σ𝑠𝑖 = 𝑛 𝑏0 + 𝑏1 Σ𝑥𝑖
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Reformulation of statistical investigation xi
si
ni
ni / n
xi s i
xi2
3
5
1
0,2
15
9
8
18
1
0,2
144
64
8
5
1
0,2
40
64
8
6
1
0,2
48
64
9
2
1
0,2
18
81
Σ 36
Σ 36
Σ5
Σ 265
Σ 282
Installment to the normal equations of linear regression analysis
36 = 5𝑏0 + 36𝑏1 265 = 36𝑏0 + 282𝑏1 −1296 = −180𝑏0 − 1296𝑏1 1325 = 180𝑏0 + 1410𝑏1 114𝑏1 = 29 𝑏1 = 0,25
36 = 5𝑏0 + 36 ∙ 0,25 36 = 5𝑏0 + 9 𝑏0 = 5,4
Application of computations to the equation of line • analytical expression of line equation 𝑦 = 𝑏0 + 𝑏1 𝑥
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• illustration (applied value 7 to the variable x and computed value of variable y) 𝑦1 = 5,4 + 0,25 ∙ 3 𝑦1 = 6,15
It shows - 3 petrol stations with the lowest price of Natural 95 are approximately relating to 6 petrol stations with the lowest price for Diesel B1.6.2. Linear correlation analysis • the linear Pearson correlation coefficient 𝑘𝑥𝑠 will be used • the formula for Pearson correlation coefficient 𝑘𝑥𝑠 =
𝑆𝑥𝑠 𝑆𝑥 𝑆𝑠
𝑆𝑥𝑠 – the mixed central moment of 2. order
S xs
ni xi O1x si O1s n
O1x= 1/36 . 36 =1
O1s= 1/36 . 36 =1
𝑆𝑥 , 𝑆𝑠 – the standard deviations of reformulated statistical signs 𝑆𝑥 =
𝐶2𝑥 , 𝑆𝑠 =
𝐶2𝑠
Computation of mixed central moment of 2. order
S xs
𝑆𝑥𝑠 =
1 5
ni xi O1x si O1s n
3 − 1 5 − 1 + 8 − 1 18 − 1 + 8 − 1 5 − 1 + 8 − 1 6 − 1 + 9− 1 2−1 𝑆𝑥𝑠 =
1 8 + 119 + 28 + 35 + 8 5 𝑆𝑥𝑠 =
1 163 5
𝑆𝑥𝑠 = 32,6
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Computation of standard deviations of single statistical signs 𝐶2𝑥 = Σ
𝐶2𝑥 =
1 5
3− 1
2
+ 8− 1
𝐶2𝑥 =
2
𝐶2𝑥 =
𝐶2𝑠 = Σ
1 5
5− 1
2
+ 18 − 1
𝐶2𝑠 =
+ 8− 1
2
+ 8− 1
2
+ 9− 1
2
1 (4 + 49 + 49 + 49 + 64) 5 1 𝐶2𝑥 = (215) 5 𝐶2𝑥 = 43
𝑆𝑥 =
𝐶2𝑠 =
𝑛𝑖 (𝑥 − 𝑂1𝑥 )2 𝑛 𝑖
43 = 6,56
𝑛𝑖 (𝑠 − 𝑂1𝑠 )2 n 𝑖
2
+ 5− 1
2
+ 6− 1
2
+ 2− 1
2
1 (16 + 289 + 16 + 25 + 1) 5 1 𝐶2𝑠 = (347) 5 𝐶2𝑠 = 69,4
𝑆𝑠 =
𝐶2𝑠 =
69,4 = 8,33
Computation of Pearson correlation coefficient 𝑘𝑥𝑠 =
𝑘𝑥𝑠 =
𝑆𝑥𝑠 𝑆𝑥 ∙ 𝑆𝑠
32,6 6,56 ∙ 8,33
𝑘𝑥𝑠 = 0,596
The statistical sign SS-x (SS1 - price of Natural 95) and statistical sign SS-s (SS2 - price of Diesel) are correlating positively.
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B1.7. Conclusion On the basis of application of the algorithm of statistics and probability the basic assumptions were verified: - the empirical distribution of prices is possible to substitute by normal distribution - the correlation between these two types of fuel will be strongly positive The first assumption was confirmed by means of non-parametric testing (see the second chapter and the fourth chapter). The prices of Natural 95 and Diesel within Prague petrol stations are behaving normally, their empirical distributions may be substituted by normal distribution. The second assumption was not confirmed in the complete extent. The statistical sign SS-x (SS1 - price of Natural 95) and statistical sign SS-s (SS2 - price of Diesel) are although correlating positively but this correlation cannot be taken like a strong correlation.
B1.8. Literature Záškodný, P. (2013) The principles of probability and statistics (data mining approach) (monolingual English version) Prague, Czech Republic: Curriculum ISBN 978-80-904948-6-2 Záškodný, P. (2013) Základy pravděpodobnosti a statistiky (data miningový přístup) (bilingual Czech-English version) Prague, Czech Republic: Curriculum ISBN 978-80-904948-5-5
retrieved March 2013 retrieved April 2014
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B2. Investigation of Price Movements of Selected Stocks in The S&P500 within One Month
Author Vilem Fasura University of Finance and Administration [email protected]
Abstract The subject, or collective random phenomenon of this chapter is investigation of price movements in a stock market. I was analyzing randomly selected stocks by means of both descriptive and mathematical version of statistics and probability. For my project, I have chosen the widely-followed American stock index S&P 500, which represents 500 leading publicly-traded companies in the U.S. I was analyzing price movements of selected stocks in the course of one month, specifically from January 7 till February 7, 2013, to find out whether there is a correlation between them. I used drawing as a tool for random selection of 50 companies in the market index. After choosing given companies, I took their initial prices from January 7, all the way till February 7. The same was conducted for closing prices. These data were collected from finance.yahoo.com. I analyzed the price movements of selected stocks by using data mining approach. I found out that price movements of initial prices are positively correlating with price movements of closing prices. What is more, this correlation is very strong, almost 1. It implies that with an increase in the initial prices of given stocks, their closing prices also increase. Similarly, with a decrease in the initial prices, the closing prices follow this downward direction too. That can be attributed to the fact that the intraday movement of a stock is not so significant. Yet there are certain discrepancies, for instance with Amazon stock. Its difference of initial price was 0,43%, but its difference of closing price was -3,07%. To my conclusion, I would say the finding of a strong positive correlation between initial and closing prices is not surprising. This is not to say, however, that it does not bring any worth points. Given the result found, it can be used as a sort of hedging. It is very likely that the difference in the initial price of a stock in a concrete month will be of same direction as the difference of the closing price, or either both prices will simultaneously increase or decrease.
Key Words Financial markets, S&P500, Price movements of initial prices, Price movements of closing prices, Correlation, Statistical investigation, Descriptive version, Mathematical version
Educational & Didactic Communication 2013, Vol.1, Part B
Contents B2.1. Introduction B2.2. Descriptive Part for Statistical Sign SS1 B2.2.1. Formulation of statistical investigation B2.2.2. Scaling B2.2.3. Measurement for statistical sign SS1 B2.2.4. Elementary statistical processing B2.2.4.1. Empirical distribution of frequencies for statistical sign SS1 B2.2.4.2. Empirical parameters for statistical sign SS1 B2.3. Descriptive Part for Statistical Sign SS2 B2.4. Non-parametric Testing for Statistical Sign SS1 B2.5. Non-parametric Testing for Statistical Sign SS2 B2.6. Measurement of Statistical Dependences B2.6.1. Linear regression analysis B2.6.2. Linear correlation analysis B2.7. Conclusion B2.8. Literature
69
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B2.1. Introduction Financial markets constitute a perfect mechanism where to look for nuances and “noises”. There is a vast amount of data for analyzing and trying to find a certain pattern in the behavior of these data. Vast amount of data is vital for statistics and probability. First it is necessary to have many results, and second to have different probabilities of these results. The most trustworthy picture of the US economy, the S&P 500, best fulfills these two prerequisites. This chapter aims to find certain correlation among stock indices in their initial and closing prices in the course of one month. Correlation in the financial markets is represented by Greek letter Beta (ß). Each stock has its designated Beta, which implies to what extent does the stock moves with the overall stock market. Positive correlation indicates that the two variables move in tandem and vice versa with negative correlation. However, in my case I took 50 stocks as a whole and analyzed them, therefore the Beta cannot be assigned to individual stocks per se. Yet what I found out from the investigation is that the price movements of initial prices move in tandem with price movements of closing prices, or they have strong positive correlation. Descriptive statistics and probability was used for conducting the first 2 chapters, and subsequently the mathematical statistics and probability was used for the second 3 chapters. It was completed in accordance with P.Záškodný (2013). Specifically the result of correlation has been attained through elaboration of the chapter 1 to the chapter 5. The application of descriptive and mathematical statistics and probability theory in investigation of the above mentioned correlation is connected with utilization of the algorithm of probability and statistics. This algorithm is created by the succession of following steps: - Formulation of Statistical Investigation, - Creation of Scale- Measurement in Descriptive Statistics, - Elementary Statistical Processing, - Non-parametric Testing, - Theory of Estimation, - Parametric Testing, - Regression and Correlation Analysis Presented work is seeking the correlation between price movements of initial and closing prices of 50 stocks from the S&P 500. The verified hypotheses of work are following - the empirical distribution of initial and closing prices is possible to substitute by normal distribution - the correlation between these two types of prices will be strongly positive. The verification of submitted hypotheses will be associated with an application of following steps from presented algorithm – formulation of statistical investigation, scaling, measurement, elementary statistical processing, non-parametric testing and regression and correlation analysis. The formulation of statistical investigation is given by utilization of the definition of basic concepts:
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- Collective Random Phenomenon (CRP – Correlation between initial and closing prices) - Statistical Unit (SU as Carrier of CRP – Stock of the S&P 500) - Statistical Signs (SS1 and SS2 as investigated properties of SU: SS1- Initial prices of stocks, SS2- Closing prices of stocks) - Values of Statistical Signs (the set of price movements of two types of prices – VSS1, VSS2) - Basic Statistical Set (BSS as the set of all the statistical units – all the stocks in the S&P 500) - Random Selection (RS as the choice of fifty stocks) - Selective Statistical Set (SSS is given by fifty stocks randomly selected)
B2.2. Descriptive Part for Statistical Sign SS1 B2.2.1. Formulation of statistical investigation Collective Random Phenomenon (CRP): Investigation of price movements of selected stocks in the S&P500 within 1 month Statistical Unit (SU): Stock Statistical Set (SS1): Price movements of initial prices Statistical Set (SS2): Price movements of closing prices Values of Statistical Set (VSS): Percentage change Basic Statistical Set (BSS): 500 stocks Random Selection (RS): Drawing Selective Statistical Set (SSS): 50 stocks
Data taken in the period from 7.1.2013 to 7.2.2013
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B2.2.2. Scaling Statistical sign SS1: Price movements of Initial prices Stock Ticker GAS ANF ACT A MMM ALL AMZN AMT AON BAC BCR BEAM BRCM CELG CERN SCHW CMG CTXS CCE CMCSA DTE FLIR BEN EXPD FTR
Difference (%) -1,81 7,83 2,44 5,4 8,03 8,56 0,43 -0,77 -2,4 -1,48 0,83 1,56 -5,34 19,85 5,25 8,76 7,17 8,07 5,54 2,17 4,07 0,25 7,15 4,14 3,44
Stock Ticker GPS HES HPQ HSP JBL L MAT MDT MRK TAP MS NFLX NSC PDCO PEP RF SHW SPG TER TSO TRIP VTR WMT WDC ZION
Difference (%) 4,49 19,56 9,55 6,87 1,4 4,96 9,74 10,18 -2,24 3,21 16,68 92,71 6,23 7,67 4,73 4,87 2,86 2,39 -0,93 21,07 6,79 2,8 3,79 11,92 6,66
Table No.1: Percentage differences of initial prices
Values of statistical sign (VSS1):
-5,34 to 92,71%
Scaling for SS1:
Robust analysis (Quantitative metric scale)
Scale elements:
x1 = 1, x2 = 2, x3 = 3, x4 = 4, x5 = 5, x6 = 6, x7 = 7
difference %
under -2
-1,99 – 0,00
0,01 – 3,00
3,01 – 7,00
7,01 – 10,00
10,01 – 20,00
above 20
scale
1
2
3
4
5
6
7
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B2.2.3. Measurement for statistical sign SS1 One from k scale elements x1, x2, x3, x4, x5, x6, x7 (k = 7) is assigned to statistical unit of selective statistical set SSS (extent n of SSS is given by equality n = 50) The number of scale elements can be computed by Sturges rule k = 1 + 3,3 log10n.
The calculation of scale elements number gives the value k = 7: k = 1 + 3,3 log1050, k = 6,6 →7
Absolute frequencies ni as the first results of measurement for statistical sign SS1 Summation of all the values ni ( i = 1, 2, 3,…, k) is equal to the extent n selective statistical set.
Relative frequencies ni/n (statistical probability) as the second results of measurement for statistical sign SS1 Summation of ni/n must be equal to 1.
Cumulative frequencies ∑ni/n (cumulative probability) as the third results of measurement for statistical sign SS1 It can be determined only within the quantitative metric or absolute metric scales.
Data for the finance.yahoo.com.
measurement
of
statistical
sign
SS1
were
collected
from
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B2.2.4. Elementary statistical processing
xi
ni
ni/n
Σ ni/n
xini
xi2ni
xi3ni
xi4ni
1 2 3 4 5 6 7
3 4 10 16 10 5 2
0,06 0,08 0,2 0,32 0,2 0,1 0,04 ∑1
0,06 0,14 0,34 0,66 0,86 0,96 1
3 8 30 64 50 30 14 ∑199
3 16 90 256 250 180 98 ∑893
3 32 270 1024 1250 1080 686 ∑4345
3 64 810 4096 6250 6480 4802 ∑22505
∑50 Table No. 2: Elaboration of 50 values for SS1 – percentage differences of initial prices
The first four columns: Creation of graphical representation of frequency empirical distributions
The second four columns: Computation of empirical parameters
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B2.2.5. Empirical distribution of frequencies for statistical sign SS1 20 15 10 5 0 1
2
3
4
5
6
7
Graph 1: Polygon of absolute frequencies for SS1 From the first graph it is possible to determine that scale element 4 is the largest → the largest number of stocks moved in the range 3,01 – 7,00%. The shape of polygon resembles Gauss curve. 0,4 0,3 0,2 0,1
0 1
2
3
4
5
6
7
Graph 2: Polygon of relative frequencies for SS1 The summation is equal to 1, which means that every price movement has to be within one of these 7 categories of scale. It is 100% probability. 1,2 1 0,8 0,6 0,4 0,2 0 1
2
3
4
5
6
7
Graph 3: Polygon cumulative frequencies for SS1 The graph is increasing. For example in point 4 there is 66% probability, meaning that the chosen stock is an element of scale 4 or less (3, 2 and 1), its price movement will be from -2 to 7 %. Remaining 34% of stocks moved more than 7%, however the maximum rise was 92,71%.
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B2.2.6. Empirical parameters for statistical sign SS1
General moment of r-th order:
𝟏
𝐧𝐢 ∙ 𝐱 𝐢 𝐫
𝐎𝐫 𝒙 = 𝒏
1
O1 x = n
General moment of 1.order:
ni ∙ x i 1
1
O1 x = 50 ∙ 199 = 3,98 It determines the parameter of location, “arithmetic mean“. It shows the placement of frequencies of empirical distribution within horizontal axis – the arithmetic mean of price movement through scale elements is 3,98. 1
O2 x = 50 ∙ 893 = 17,86 O3 x =
1 50
∙ 4 345 = 86,9
1
O4 x = 50 ∙ 22 505 = 450,1
𝟏
𝐂𝐫 𝐱 = 𝐧
Central moment r-th order:
𝐧𝐢 𝐱 𝐢 − 𝐎𝟏
𝐫
Central moment of 2.order:
C2 = O2 – O12
C2 = 2,0196
It determines the parameter of variability, “empirical dispersion”.
𝐒𝐱 =
Standard deviation:
𝐂𝟐
Sx = 2,0196 = 1,4211 It shows the attesting value of arithmetic mean – if this one is great, the attesting value of arithmetic mean is small.
𝐒
V = 𝐎𝐱
Variation coefficient: Sx O1
=
1,4211 3,98
= 0,357
𝟏
V = 35,7 %
It shows how many percent from arithmetic mean is created by standard deviation (thus 35,7%).
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Computation of central moments for SS1: C3 = O3 − 3O2 O1 + 2O1 3 C3 = 86,9 − 3 ∗ 17,86 ∗ 3,98 + 2 ∗ 3,983 = - 0,259 C4 = O4 − 4O3 O1 + 6O2 O1 2 − 3O1 4 C4 = 450,1 − 4 ∗ 86,9 ∗ 3,98 + 6 ∗ 17,86 ∗ 3,982 − 3 ∗ 3,984 = 11,354 Standardized moments of 3.and 4.order for SS1
N3 =
C3 C2 C2
−0,259
N3 = 2,0196
2,0196
= -0,09
It determines the parameter of skewness, “coefficient of skewness“. If it is positive, the scale elements lying to the left from arithmetic mean have higher frequencies (positively skewed frequency distribution – the larger concentration of the smaller scale elements, of the smaller values of statistical sign SS1). In my case the parameter is negative, hence the left tail is longer; the mass of the distribution is concentrated on the right of the figure.
C
N4 = C 42 2
11,354
N4 = 2,0196 2 = 2,78
It determines the parameter of kurtosis (of pointedness), “coefficient of kurtosis”. Ideal coefficient of kurtosis is equal to 3. Quantity excess = N4 – 3
excess = 2,78 – 3 = - 0,22
The value of kurtosis coefficient N4 is approaching to the ideal value 3. This depicts the comparison with the pointedness of standardized normal distribution.
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B2.3. Descriptive Part for Statistical Sign SS2 Statistical sign SS2: Price movements of Closing prices Stock Ticker GAS ANF ACT A MMM ALL AMZN AMT AON BAC BCR BEAM BRCM CELG CERN SCHW CMG CTXS CCE CMCSA DTE FLIR BEN EXPD FTR
Difference (%) -1,53 7,7 1,72 5,78 7,05 7,95 -3,07 -1,39 -1,39 -2,07 -0,76 1,22 -5,32 14,13 4,26 12,05 6,3 5,46 6,24 1,9 5,27 10,67 6,97 4,13 2,24
Stock Ticker GPS HES HPQ HSP JBL L MAT MDT MRK TAP MS NFLX NSC PDCO PEP RF SHW SPG TER TSO TRIP VTR WMT WDC ZION
Difference (%) 3,3 19,22 8,37 5,29 -0,87 3,87 10,13 8,34 -2,66 5,6 16,67 83,43 6,09 6,03 4,84 6,7 2,78 0,67 -1,17 30,66 3,83 2,39 4,14 10,96 6,3
Table No.3: Percentage differences of closing prices Values of statistical sign (VSS2):
-5,32 to 83,43%
Scaling for SS2:
Robust analysis (Quantitative metric scale)
Scale elements:
x1 = 1, x2 = 2, x3 = 3, x4 = 4, x5 = 5, x6 = 6, x7 = 7
difference %
under -2
-1,99 – 0,00
0,01 – 3,00
3,01 – 7,00
7,01 – 12,00
12,01 – 30,00
above 30
scale
1
2
3
4
5
6
7
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xi
ni
ni/n
Σ ni/n
xini
xi2ni
xi3ni
xi4ni
1 2 3 4 5 6 7
4 6 7 19 7 5 2
0,08 0,12 0,14 0,38 0,14 0,1 0,04 ∑1
0,08 0,2 0,34 0,72 0,86 0,96 1
4 12 21 76 35 30 14 ∑192
4 24 63 304 175 180 98 ∑848
4 48 189 1216 875 1080 686 ∑4098
4 96 567 4864 4375 6480 4802 ∑21188
∑50 Table No. 4: Elaboration of 50 values for SS2 – percentage differences of closing prices
20 15 10 5 0 1
2
3
4
5
6
7
1
2
3
4
5
6
7
Graph 4: Polygon of absolute frequencies for SS2
0,4 0,3 0,2 0,1 0
Graph 5: Polygon of relative frequencies for SS2
1,2 1 0,8 0,6 0,4 0,2 0 1
2
3
4
5
6
7
Graph 6: Polygon of cumulative frequencies for SS2
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Empirical parameters for SS2 𝟏
𝐧𝐢 ∙ 𝐱 𝐢 𝐫
𝟏
𝐧𝐢 𝐱 𝐢 − 𝐎𝟏
𝐎𝐫 𝐱 = 𝐧
General moment: 1
O1 x = 50 ∙ 192 = 3,84 1
O2 x = 50 ∙ 848 = 16,96 1
O3 x = 50 ∙ 4098 = 81,96 1
O4 x = 50 ∙ 21188 = 423,76
𝐂𝐫 𝐱 = 𝐧
Central moment:
𝐫
C2 = 2,2144 C3 = 81,96 − 3 ∗ 16,96 ∗ 3,84 + 2 ∗ 3,843 = - 0,173 C4 = 423,76 − 4 ∗ 81,96 ∗ 3,84 + 6 ∗ 16,96 ∗ 3,842 − 3 ∗ 3,844 = 13,068
𝐒𝐱 =
Standard deviation:
𝐂𝟐
Sx = 2,2144 = 1,488
𝐒
V = 𝐎𝐱
Variation coefficient: 1,488 3,84
= 0,3875
𝟏
V = 38,75 %
Standardized moments: 0,173
N3 = 2,2144
2,2144
N3 =
𝐂𝟑 𝐂𝟐 𝐂𝟐
= 0,0525 𝐍𝟒 =
𝐂𝟒
𝐂𝟐 𝟐
13,068
N4 = 2,2144 2 = 2,665 Quantity excess = N4 - 3
excess = 2,665 – 3 = - 0,345
80
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B2.4. Non-parametric Testing for Statistical Sign SS1 The aim is to determine whether the empirical distribution of initial prices (empirical distribution associated with statistical sign SS1) can be substituted by normal distribution.
xi
interval
ni
u1
F(ui)
pi
npi
(ni-npi)2/npi
1
(-∞;1,5>
3
-1,75
0,04006
0,04006
2,003
0,49626011
2 3 4 5 6 7
(1,5;2,5> (2,5;3,5> (3,5;4,5> (4,5;5,5> (5,5;6,5> (6,5;∞)
4 10 16 10 5 2 Σ 50
-1,04 -0,34 0,37 1,07 1,77 ∞
0,14917 0,36693 0,64431 0,85769 0,96164 1
0,10911 0,21776 0,27738 0,21338 0,10395 0,03836 Σ1
5,4555 10,888 13,869 10,669 5,1975 1,918
0,38832009 0,072423218 0,327432475 0,041949667 0,00750481 0,003505735 Σ 1,337396106
Table No. 5: Interval division of frequencies
(ni-npi)2/npi
xi
ni
npi
1+2
7
7,4585
0,028185594
3 4 5 6+7
10 16 10 7 Σ 50
10,888 13,869 10,669 7,1155 Σ 50
0,072423218 0,327432475 0,041949667 0,001874816 Σ0,47186577
2 Table No. 6: The adjustment of interval numbers, the calculation of χ exp
Given that there are 7 measurement results, the table above reacts to the requirement that at least 5 or more measurement results must be in each interval in the course of normality test. The neighboring intervals come together to reach 5 or more measurement results. At the same time the additional calculations, enabling to establish the experimental value of statistical criterion, are carried out in this table.
Difference between 5 segments of line and 5 surfaces determined by empirical way:
2 χ exp
(n i -npi )2 npi i=1 k
χ2exp = 0,47
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The highest allowed difference determined by means of statistical tables: Computation of theoretical (critical) value χ2 and of freedom degrees number ν for significance level (statistical error of 1.type) α = 0,05
χ 2th =χ k-r-1 α=0,05 (k – number of scale elements, r – number of theoretical parameters) ν=k–r–1 ν=5–2–1 ν=2
χ 2th (0,05)=5,99
2 χ 2th >χ exp
→ the zero hypothesis H0 can be reached
For the significance level α = 0,05 the investigated empirical distribution may be substituted by normal distribution, the empirical graph may be substituted by Gauss curve.
The price movements of initial prices of selected stocks behave normally.
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B2.5. Non-parametric Testing for Statistical Sign SS2 On the basis of the probability distribution significance the table comparing the properties of continuous distributions (normal distribution, standardized normal distribution) and discrete distributions (binomial distribution, Poisson distribution) is presented. Without association between theory of probability and mathematical statistics the non-parametric testing is not possible to carry out. Normal distribution ND Standardized ND
Binomial distribution Poisson distrib.
Mathematical Probability Descriptive Probability
N(μ,σ)=> r=2
N(0,1)=> r=2
Bi(n,p)
Po(λ)
2 theor.parameters
2 theor. parameters
2 theoretical
1 theoretical
μ~O1, σ~Sx
μ~0, σ~1
parameters n,p
parameter λ
ni/n
Probability density
Probability density
Probable function
Probable function
Relative
𝛒 𝒙 =
𝟏 𝛔∙ 𝟐 𝛑
∙𝐞
−
(𝐱−𝛍)𝟐 𝟐 ∙ 𝛔𝟐
𝛍𝟐
𝟏
𝛒 𝒙 =
𝟐𝛑
∙ 𝐞− 𝟐
Pi=
𝑛 𝑖
pi(1-p)n--i
frequency ∑ni/n
Distrib.function
Cumulative
F(x)=
𝑥 −∞
𝜌 𝑥 𝑑𝑥
Distrib.function F(u)=
𝑢 −∞
F(∞)=
∞ −∞
𝜌 𝑢 𝑑𝑢
frequency ∑ni/n
F(∞)=
∞ −∞
𝜌 𝑥 𝑑𝑥 = 1
𝜌 𝑢 𝑑𝑢 =1
Standardized condition O1=∑(ni/n) xi
O1=
∞ −∞
𝑥𝜌 𝑥 𝑑𝑥 =
=μ=E ∞ −∞
O1=
∞ −∞
𝑢𝜌 𝑢 𝑑𝑢 =
=0=E ∞ −∞
Distrib.function Fj=
j i=0 Pi
i Pi =e-λ λ i!
Distrib.function Fj=
j i=0 Pi
i=0,1,2..n
i=0,1,2..∞
Standardized
Standardized
condition
condition
n i=0 Pi=1
∞ i=0 Pi=1
O1=np=E
O1=λ=E
Expected value
Expected value
C2=np(1-p)=D
C2=λ=D
C2=∑ni/n(xi-O1)2
C2=
Emp. dispersion
C2=D=σ2
C2=1=D
Dispersion value
Dispersion value
Sx=√C2
Continuous
Continuous
Discrete
Discrete
Standard deviation
distribution
distribution
distribution
distribution
(x-μ)2ρ(x)dx
C2=
u2ρ(u)du
The basic goal for statistical sign SS2: To determine if the empirical distribution of price movements of closing prices (empirical distribution associated with statistical sign SS2) can be substituted by normal distribution. On the basis of the same procedure, which was applied to statistical sign SS1, it is possible to state: For the significance level α = 0,05 the investigated empirical distribution for statistical sign SS2 may be substituted by normal distribution, the empirical graph may be substituted by Gauss curve.
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B2.6. Measurement of Statistical Dependences The two statistical signs SS1 and SS2 (price movements of initial prices and closing prices) are examined from the point of bonds between them. The bonds between these statistical signs have two basic features: 1) Character of bond -it is given by the type of regression curve. Finding the suitable regression curve is the basic assignment for regression analysis. - in this statistical project the line as regression curve will be searched. It is an assignment for the linear regression analysis. -the other types of regression analysis exist as well (e.g. quadratic, trigonometric, exponential) 2) Tightness of bond - quantification of bond tightness is characterized by the value of correlation coefficient k xs . The values of k xs ∈ < −1; 1 > . The bond is tighter and tighter with approaching the k xs to values –1 and 1.
The potential typology of bond tightness: • 5 potential cases 1. k xs ∈ (0,6; 1)
= strong positive correlation
2. k xs ∈ (0,3; 0,6 >
= weak positive correlation
3. k xs ∈ < −0,3; 0,3 >
= uncorrelated
4. k xs ∈ < −0,6; −0,3) = weak negative correlation 5. k xs ∈ < −1; −0,6)
= strong negative correlation
Positive correlation = both statistical signs simultaneously increase or decrease Negative correlation = one from ones increases, the remaining from ones decreases Statistical sign SS1 will be marked SSx (its values will be applied to x-axis) Statistical sign SS2 will be marked SSs (its values will be although applied to y-axis but letter y will be used for the equation of regression line)
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B2.6.1. Linear regression analysis Normal equations of linear regression analysis: Σsi = k 𝑏0 + 𝑏1 Σxi Σsi xi = 𝑏0 Σxi + 𝑏1 Σxi2
The application of respective values to these equations can be realized by: 1) method A - the single statistical unit can be identified 2) method B - the single statistical unit cannot be identified
In my statistical project the statistical units were not possible to identify. The method B has to be selected. Method B • first, the reformulation of statistical investigation must be carried out • method B is not utilized for time series (opposite method A) • the number k of scale elements will be renamed to n Σsi = n 𝑏0 + 𝑏1 Σxi
Reformulation of statistical investigation
xi 3 4 10 16 10 5 2 Σ 50 ( O1x= 7,14)
si 4 6 7 19 7 5 2 Σ 50 ( O1s = 7,14)
ni 1 1 1 1 1 1 1
Σ7
ni/n 0,02 0,02 0,02 0,02 0,02 0,02 0,02
xisi 12 24 70 304 70 25 4
xi2 9 16 100 256 100 25 4
Σ 509
Σ 510
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Installment to the normal equations of linear regression analysis on the basis of Method B
50 = 7𝑏0 + 50𝑏1 /. (−10,2) 509 = 50𝑏0 + 510𝑏1 −510 = −71,4𝑏0 − 510𝑏1 509 = 50𝑏0 + 510 𝑏1 21,4𝑏0 = 1 𝑏0 = 0,05
50 = 7 ∙ 0,05 + 50𝑏1 50 = 0,35 + 50𝑏1 𝑏1 = 1
Application of computations to the equation of line Analytical expression of line equation: y = 𝑏0 + 𝑏1 x
Illustration (applied value 9 to the variable x and computed value of variable y) y1 = 0,05 + 1 ∙ 9 y1 = 9,05
Interpretation: To 9 initial prices (their price movements) approximately 9 closing prices (their price movements) is responding.
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B2.6.2. Linear correlation analysis The linear Pearson correlation coefficient k xs will be used The formula is as follows: kxs =
Sxs Sx Ss
Sxs – mixed central moment of 2.order Sxs = Σ
ni (x − O1x )(si − O1s ) n i
Sx , Ss – standard deviation of reformulated statistical signs Sx =
C2x , Ss =
C2s
Computation of mixed central moment of 2. order Sxs = Σ
Sxs =
1 7
ni (x − O1x )(si − O1s ) n i
3 − 7,14 4 − 7,14 + 4 − 7,14 6 − 7,14 + 10 − 7,14 7 − 7,14 + 16 − 7,14 19 − 7,14 + 10 − 7,14 7 − 7,14 + 5 − 7,14 5 − 7,14 + (2 − 7,14)(2 − 7,14) Sxs =
1 13 + 3,58 + −0,4 + 105.1 + −0,4 + 4,58 + 26,42 7 Sxs =
1 151,88 7
Sxs = 21,7 Computation of standard deviations of single statistical signs C2x = Σ
C2x =
1 7
3 − 7,14
2
+ 4 − 7,14
+ 5 − 7,14 C2x =
2
2
ni (xi − O1x )2 n
+ 10 − 7,14
+ 2 − 7,14
2
+ 16 − 7,14
2
+ 10 − 7,14
2
1 (17,14 + 9,86 + 8,18 + 78,5 + 8,18 + 4,58 + 26,42) 7
2
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C2x =
88
1 (152,86) 7
C2x = 21,83
Sx =
C2x =
C2s = Σ
21,83 = 4,67
ni (s − O1s )2 n i
C2s= 17 4-7,142+6-7,142+7-7,142+19-7,142+7-7,142+5-7,142+2-7,142 C2s =
1 (9,86 + 1,3 + 0,02 + 140,66 + 0,02 + 4,58 + 26,42) 7 C2s =
1 (182,86) 7
C2s = 26,12
Ss= C2s=26,12=5,11
Computation of Pearson correlation coefficient kxs =
Sxs Sx ∙ Ss
kxs= 21,74,67∙5,11 kxs= 0,91
Interpretation: The statistical sign SS1 (price movements SSx of initial prices) has a strong positive correlation with statistical sign SS2 (price movements SSs of closing prices). That might come as no surprise, for the price movement of a certain stock in the course of one single day is rather low.
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B2.7. Conclusion By means of application of the algorithm of statistics and probability the two hypotheses were investigated: - the empirical distributions of initial prices (their price movements) and closing prices (their price movements) of 50 stocks selected from the S&P500 can be substituted by normal distribution - the correlation between these two types of prices (their price movements) will be strongly positive The first hypothesis was confirmed by means of non-parametric testing (see the third chapter and the fourth chapter). The price movements of initial prices and closing prices are behaving normally, their empirical distributions may be substituted by normal distribution. It is showing the interesting possibility – the S&P500 can be investigated by means of a random selection of stocks. This possibility as the first conclusion of presented work should be verified by means of the potential following random selections. The second hypothesis was confirmed in the complete extent. The statistical sign SS-x (SS1 – price movements of initial prices) and statistical sign SS-s (SS2 – price movements of closing prices) are correlating strongly positively. This second conclusion was expected, in spite of confirmed expectation such result required a deeper analysis.
B2.8. Literature Záškodný, P. (2013), The principles of probability and statistics (data mining approach) (monolingual English version) Prague, Czech Republic: Curriculum ISBN 978-80-904948-6-2 Záškodný, P. (2013), Základy pravděpodobnosti a statistiky (data miningový přístup) (bilingual Czech-English version) Prague, Czech Republic: Curriculum ISBN 978-80-904948-5-5
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B3. Comparison between Percentage Price Movements of Coca Cola Company and PepsiCoCompany within Relevant Financial Market Author Dominika Masna University of Finance and Administration [email protected]
Abstract In this chapter probability and statistic is provided by investigation of percentage price movements of two selected companies in fifty days of relevant financial market. The result of this investigation is achieved by different measuring, scaling and evaluating processes. The collective random phenomena of the investigation of dependence between the percentage differences of Coca cola Company and PepsiCo Company are very interesting. With the help of Pearson correlation coefficient the result of this project has been achieved – strong positive correlation.
Key Words Companies of relevant financial market, Statistical unit, Statistical sign, Empirical parameters, Normal distribution, Normal equations, Linear regression, Linear correlation
Contents B3.1. Introduction B3.2. Descriptive Part for Statistical Sign SS1 B3.2.1. Formulation of statistical investigation B3.2.2. Scaling B3.2.3. Measurement B3.2.4. Elementary statistical processing B3.3. Descriptive Part for Statistical Sign SS2 B3.4. Non-parametric Testing for Statistical Signs SS1 and SS2 B3.5. Measurement of Statistical Dependences B3.5.1. Simple linear regression analysis B3.5.2. Simple linear correlation analysis B3.6. Conclusion B3.7. Literature
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B3.1. Introduction The applications of statistics and probability theory in an investigation of the collective random phenomena are the subject of probability and statistics. This chapter uses the collective random phenomena of the investigation of dependence between the percentage differences of Coca cola and PepsiCo Company. To be able to process the result of this investigation the project is supported by the theory of probability and statistics. As the main object of this project was the investigation of correlation of the Coca Cola and PepsiCo Company. These two companies are producing quite very similar project, therefore the correlation of these companies is interesting. The application of descriptive and mathematical statistics and probability theory in investigation of the above mentioned correlation is according to P.Zaskodny (2013) connected with utilization of the algorithm of probability and statistics. This algorithm is created by the succession of following steps: - Formulation of Statistical Investigation - Creation of Scale - Measurement in Descriptive Statistics, Statistical Probability - Elementary Statistical Processing - Non-parametric Testing - Theory of Estimation - Parametric Testing - Regression and Correlation Analysis Presented work is seeking the correlation between the percentage differences of Coca cola and PepsiCo Company. The hypotheses of work are assuming - the empirical distribution of percentage price movements is possible to substitute by normal distribution - the correlation between these two percentage differences will be strongly positive. The verification of submitted hypotheses will be connected with an application of following steps from presented algorithm – formulation of statistical investigation, scaling, measurement, elementary statistical processing, non-parametric testing and regression and correlation analysis. The formulation of statistical investigation is given by utilization of the definition of basic concepts: - Collective Random Phenomenon (CRP – Correlation between the percentage differences) - Statistical Unit (SU as Carrier of CRP – One day of relevant financial market) - Statistical Signs (SS1 and SS2 as investigated properties of SU: SS1- Percentage price movements of Coca cola Company, SS2- Percentage price movements of PepsiCoCompany) - Values of Statistical Signs (the set of percentage price movements prices of two companies– VSS1, VSS2) - Basic Statistical Set (BSS as the set of all the statistical units – 50 days of relevant financial market) - Random Selection (RS – the random selection was not applied) - Selective Statistical Set (SSS is given by 50 days of relevant financial market)
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B3.2. Descriptive Part of Investigation for Statistical Sign SS1 B3.2.1. Formulation of statistical and probability investigation Collective Random Phenomenon (CRP): The investigation of dependence between the percentage differences of Coca Cola and PepsiCoCompany. Statistical Unit (SU): One day of relevant financial market Statistical Set (SS): Price movements Values of Statistical Set (VSS): Percentage change Basic Statistical Set (BSS): 50 statistical units (50 days of relevant financial market) Random Selection (RS): The random selection will not be carried out Selective Statistical Set (SSS): 50 statistical units Statistical sign 1: Percentage difference of open and close price of Coca Cola Company Statistical sign 2: Percentage difference of open and close price of PepsiCo, Inc.
Data taken in the period from 1.2.2013 to 15.4.2013
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B3.2.2. Scaling
Statistical sign SS1: Percentage difference of open and close price of Coca Cola Company The date were collected in 50 days from 1.2.2013 to 15.4.2013 The numbers from 1- 50 were assigned to the dates. Day number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Difference (%) -1,81 7,83 2,44 5,4 8,03 8,56 0,43 -0,77 -2,4 -1,48 0,83 1,56 -5,34 19,85 5,25 8,76 7,17 8,07 5,54 2,17 4,07 0,25 7,15 4,14 3,44
Day number 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
Difference (%) 4,49 19,56 9,55 6,87 1,4 4,96 9,74 10,18 -2,24 3,21 16,68 92,71 6,23 7,67 4,73 4,87 2,86 2,39 -0,93 21,07 6,79 2,8 3,79 11,92 6,66
Table No. 1: Percentage differences of open and close price in Coca Cola Company
Values of statistical sign (VSS1):
-2, 23 % to 1,97%
Scaling for SS1:
Robust analysis (Quantitative metric scale)
Scale elements:
x1 = 1, x2 = 2, x3 = 3, x4 = 4, x5 = 5, x6 = 6, x7 = 7
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difference %
Under -1,74
- 1,73 - -0,77
-0,76 – 0,02
0,03 – 0,56
0,57 – 1,26
1,27 – 1,65
Above 1,65
scale
1
2
3
4
5
6
7
B3.2.3. Measurement for statistical sign SS1
One from k scale elements x1, x2, x3, x4, x5, x6, x7 (k = 7) is assigned to statistical unit of selective statistical set SSS (extent n of SSS is given by equality n = 50) The measurement results are given by Sturges rule: k = 1 + 3.3 log (50) k = 6.6 →7 Absolute frequencies ni – Summation of all the values ni ( i = 1, 2, 3,…, k) is equal to the extent n of selective statistical set. Relative frequencies ni/n (statistical probability) – Summation of ni/n must be equal to 1. Cumulative frequencies ∑ni/n (cumulative probability) – it can be determined only within of quantitative metric or absolute metric scales. Data for SS1 provided from www.finance.yahoo.com
B3.2.4. Elementary statistical processing (table, graphs, empirical parameters) Table: xi
ni
ni/n
Σ ni/n
xini
xi2ni
xi3ni
xi4ni
1 2 3 4 5 6 7
3 5 9 17 10 4 2
0,06 0,1 0,18 0,34 0,2 0,08 0,04 ∑1
0,06 0,16 0,34 0,68 0,88 0,96 1
3 10 27 68 50 24 14 ∑196
3 20 81 272 250 144 98 ∑868
3 40 243 1088 1250 864 686 ∑4174
3 80 729 4352 6250 5184 4802 ∑21400
∑50 Table No. 2: Elaboration of 50 values for SS1 – percentage differences of open and close prices of Coca Cola Company
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The first four columns – creation of graphical representation of frequency empirical distributions The second four columns - computation of empirical parameters
Graphs:
Empirical distribution of frequencies for SS1
20 15 10 5 0 1
2
3
4
5
6
7
Graph 1: Polygon of absolute frequencies for SS1 Comment: From the graph 1 it is possible to determine that scale element 4 is the largest→ the largest number of differences between open and close prices in the range 0,03 – 0,56 %.
The shape of polygon resembles Gauss curve
0,4
0,3 0,2 0,1 0 1
2
3
4
5
6
7
Graph 2: Polygon of relative frequencies for SS1 Comment: Graph 2 – statistical probability of result, summation is equal to 1. It means that every number of percentage difference of open and close price must be the element of one of the 7 categories of scale, it is 100% probability or certainty 1. The number of percentage difference, which would be the element of no categories, does not exist.
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1,2 1 0,8 0,6 0,4 0,2 0 1
2
3
4
5
6
7
Graph 3: Polygon cumulative frequencies for SS1 Comment: Graph 3 – the increasing graph of probabilities may be interpreted by following way : For example in the point 4 the 68 % probability is occurring, the chosen number of percentage difference in prices is an element of category 4 or less (3,2 and 1), it’s percentage difference in prices will be from -1,74% - 0,56%. Remaining 32% have the percentage difference in prices greater than 0, 56% but not more than 1,65.
Empirical parameters for SS1:
1
𝑛𝑖 ∙ 𝑥𝑖 𝑟
𝑂𝑟 𝑥 = 𝑛
General moment of r-th order:
1
General moment of 1st order: 𝑂1 𝑥 = 𝑛
𝑛𝑖 ∙ 𝑥𝑖 (1)
1
𝑂1 𝑥 = 50 ∙ 196 = 3,92 It determines the parameter of location – the arithmetic mean. It shows the placement of frequencies empirical distribution within horizontal axis –the arithmetic mean of percentage difference between open and close price is through scale elements 3, 92. Computation of general moments for SS1: 1
General moment of 2nd order:
𝑂2 𝑥 = 50 ∙ 862 = 17, 24
General moment of 3rd order:
𝑂3 𝑥 = 50 ∙ 4174 = 83, 48
General moment of 4th order:
𝑂4 𝑥 =
1
1 50
∙ 21400 = 428
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𝐶𝑟 𝑥 = 𝑛
Central moment of r-th order:
1
Central moment of 2ndorder: 𝐶2 𝑥 = 𝑛
𝑛𝑖 𝑥𝑖 − 𝑂1
𝑛𝑖 𝑥𝑖 − 𝑂1
97
𝑟
2
𝐶2 𝑥 = 1,8736 It determines the parameter of variability, “empirical dispersion”.
Determinative (Standard) deviation:
𝑆𝑥 =
𝐶2
𝑆𝑥 =
1,8736
𝑆𝑥 = 1.3688
Variation coefficient: 𝑆𝑥 𝑂1
=
1,3688 3,92
𝑆
V = 𝑂𝑥
1
= 0,35, V = 35 %
It shows how many percent from arithmetic mean is created by standard deviation (In this case it means 35%)
Computation of central moments for SS1:
𝐶2 = 𝑂2 − 𝑂1 2
𝐶2 = 17,24 − 3,922 = 1,8736
𝐶3 = 𝑂3 − 3𝑂2 𝑂1 + 2𝑂1 3 𝐶3 = 83,48 − 3(17,24.3,92) + 2(3,923) =1,2097
𝐶4 = 𝑂4 − 4𝑂3 𝑂1 + 6𝑂2 𝑂1 2 − 3𝑂1 4
𝐶4 = 428 − 4(83,48.3,92) + 6(17,28. (3,922 )) − – 3(3,924 )= 3,8433
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Standardized moments of 3. and 4. order
𝑁3 = 𝐶
𝐶3
2 𝐶2
1,2097
𝑁3 = 1,8736
1,8736
= 0,4717
It determines the parameter of skewness, “coefficient of skewness“. If the skewness coefficient is positive, then the scale elements lying to the left from arithmetic mean have the higher frequencies (positively skewed frequency distribution – the larger concentration of the lower scale elements, of the smaller values of statistical sign SS1). The used graph 1 is confirming this determination.
𝐶
𝑁4 = 𝐶 42 2
3,8433
𝑁4 = 1,8736 2 = 1,0949
It determines the pointedness parameter of kurtosis using standardized moment of 4th order and then it has the name “coefficient of kurtosis”. Ideal coefficient of kurtosis is equal to 3.
Quantity ”excess” = N4 – 3
excess = 1,0949 – 3 = – 1,9051
The value of kurtosis coefficient N4 is approaching to the ideal value 3. This one shows the comparison with the pointedness of standardized normal distribution.
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B3.3. Descriptive Part of Investigation for Statistical Sign SS2 Statistical sign SS2: Percentage difference of open and close price of PepsiCo, Inc. The date were collected in 50 days from 1.2.2013 to 15.4.2013 The numbers from 1- 50 were assigned to the dates. Day number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Difference (%) -0,77 1,33 -0,09 0,79 -0,49 1,17 -0,46 1,55 -1,50 0,49 0,43 0,77 -0,27 1,29 -1,00 2,42 -0,18 -0,84 -0,14 0,03 0,25 0,17 0,08 0,08 -0,35
Day number 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50
Difference (%) 0,05 0,49 -0,26 0,08 1,07 0,50 0,44 0,50 0,29 -0,65 0,20 0,12 0,17 2,32 1,62 0,22 -1,03 -0,35 -0,19 -0,22 -0,01 -0,19 0,74 -0,04 -0,56
Table No. 3: Percentage differences of open and close price in PepsiCo, Inc. Company
Values of statistical sign (VSS2):
-1,50 % to 2,42%
Scaling for SS2:
Robust analysis (Quantitative metric scale)
Scale elements:
x1 = 1, x2 = 2, x3 = 3, x4 = 4, x5 = 5, x6 = 6, x7 = 7
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difference %
Under -1,00
- 0.99 - -0,49
-0,48 - -0,19
-0,18 – 0,29
0,30 – 0,80
0,81 – 1,6
Above 1,61
scale
1
2
3
4
5
6
7
Table for SS2
xi
ni
ni/n
Σ ni/n
xini
xi2ni
xi3ni
xi4ni
1 2 3 4 5 6 7
3 5 8 17 9 5 3
0,06 0,1 0,16 0,34 0,18 0,1 0,06 ∑1
0,06 0,16 0,32 0,66 0,84 0,94 1
3 10 24 68 45 30 21 ∑201
3 20 72 272 225 180 147 ∑919
3 40 216 1088 1125 1080 1029 ∑4581
3 80 648 4352 5652 6480 7203 ∑24391
∑50 Table No. 4: Elaboration of 50 values for SS2 – percentage differences of open and close prices of PepsiCo, Inc. Company
The first four columns of this table show the creation of graphical representation of frequency empirical distribution. The other four columns show the computation of empirical parameters.
Empirical distribution of frequencies for SS2
20 15 10 5 0 1
2
3
4
5
6
7
Graph 1: Polygon of absolute frequencies for SS1 Comment: From the graph 1 it is possible to determine that scale element 4 is the largest → the largest number of differences between open and close prices in the range -0,18 – 0,29 %.
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0,4 0,3 0,2 0,1 0 1
2
3
4
5
6
7
Graph 2: Polygon of relative frequencies for SS1 Comment: Graph 2 – statistical probability of result, summation is equal to 1. It means that every number of percentage difference of open and close price must be the element of one of the 7 categories of scale, it is 100% probability or certainty 1.
1,2 1 0,8 0,6 0,4 0,2 0 1
2
3
4
5
6
7
Graph 3: Polygon cumulative frequencies for SS1 Comment: Graph 3 – the increasing graph of probabilities.
Empirical parameters for SS2:
General moments 1
𝑂1 𝑥 = 50 ∙ 201 = 4,02 1
𝑂2 𝑥 = 50 ∙ 919 = 18,38 1
𝑂3 𝑥 = 50 ∙ 4581 = 91,62 1
𝑂4 𝑥 = 50 ∙ 24391 = 487,82
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Central moments
𝐶2 = 𝑂2 − 𝑂1 2
𝐶2 = 2,2196
𝐶3 = 𝑂3 − 3𝑂2 𝑂1 + 2𝑂1 3
𝐶3 = -0,1132
𝐶4 = 𝑂4 − 4𝑂3 𝑂1 + 6𝑂2 𝑂1 2 − 3𝑂1 4
𝐶4 =13,4245
𝑆𝑥 =
Standard deviation:
𝐶2
𝑆𝑥 = 2,2196 = 1,0322 𝑆𝑥 𝑂1
𝑆
V = 𝑂𝑥
Variation coefficient: =
1,0322 4,02
1
= 0,257 V = 26 %
Standardized moments
𝑁3 = 𝐶
𝐶3
2
𝐶
𝑁4 = 𝐶 42 2
𝐶2
−0,1132
𝑁3 = 2,2196
𝑁4 =
13,4245 2,2196 2
2,2196
= -0,0342
= 2,7249
It determines the parameter of kurtosis, “coefficient of kurtosis “. Ideal value is equal to = 3. Quantity “excess” = N4 - 3 excess = 2,7249 – 3 = -0,2751
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B3.4. Non-parametric Testing for Statistical Signs SS1 and SS2 i) Non-parametric testing of normality for statistical sign SS1 (percentage difference of open and close price of Coca Cola Company) The basic goal: To determine if the empirical distribution of difference in open and close price of Coca-Cola Company (empirical distribution associated with statistical sign SS1) can be substituted by normal distribution (another theoretical distributions will not be explored because the normal distribution and the testing of normality are occurring the most frequently) xi
1 2 3 4 5 6 7
Interval (-∞; 1,5> (1,5; 2,5> (2,5; 3,5> (3,5; 4,5> (4,5; 5,5> (5,5; 6,5> (6,5; ∞)
ni
ni/n
Σ ni/n
xini
xi2ni
xi3ni
xi4ni
3 5 9 17 10 4 2
0,06 0,1 0,18 0,34 0,2 0,08 0,04 ∑1
0,06 0,16 0,34 0,68 0,88 0,96 1
3 10 27 68 50 24 14 ∑196
3 20 81 272 250 144 98 ∑868
3 40 243 1088 1250 864 686 ∑4174
3 80 729 4352 6250 5184 4802 ∑21400
∑50
Table No. 3: Interval division of frequencies
Transformation of normal distribution N(μ,σ) to standardized normal distribution N(0,1)
Computation of values of new variable u:
u
x O1 Sx
u1
1,5 3,92 1, 766 1,37
u2
2,5 3,92 1, 036 1,37
u3
3,5 3,92 0,306 1,37
u4
4,5 3,92 0, 423 1,37
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u5
5,5 3,92 1,153 1,37
u6
6,5 3,92 1,883 1,37
u7
104
3,92 1,37
Computation of surfaces pi under Gauss curve through the values of Laplace function (distribution function of standardized normal distribution):
p1
u1
u du F (u1 ) F 0, 039
p2 u du F (u2 ) F u1 0,112 u2
u1
p3 u du F (u3 ) F u2 0, 231 u3
u2
p4 u du F (u4 ) F u3 0, 037 u4
u3
p5 u du F (u5 ) F u4 0, 220 u5
u4
p6 u du F (u6 ) F u5 0, 094 u6
u5
p7 u du F () F u6 0, 031 u6
Comparison of 7 segments ni/n of line and 7 surfaces pi under Gauss curve:
n1 = 0,06 ≈ 𝑝1 = 0,03 n n2 = 0,10 ≈ 𝑝2 = 0,11 n
n3 = 0,18 ≈ 𝑝3 = 0,23 n n4 = 0,34 ≈ 𝑝4 = 0,03 n
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n5 = 0,2 ≈ 𝑝5 = 0,22 n n6 = 0,08 ≈ 𝑝6 = 0,09 n n7 = 0,04 ≈ 𝑝7 = 0,03 n
Application of χ2- test
- Real difference between 7 segments of line and 7 surfaces determined by empirical way: Computation of experimental value χ2
(ni npi )2 npi i 1 k
2 EXP
2 EXP
(𝑛1 − 𝑛𝑝1 )2 (3 − 50 × 0,039)2 = = = 0,565 𝑛𝑝1 50 × 0,039 (𝑛2 − 𝑛𝑝2 )2 (5 − 50 × 0,112)2 = = = 0,064 𝑛𝑝2 50 × 0,112 =
(𝑛3 − 𝑛𝑝3 )2 (9 − 50 × 0,231)2 = = 0,563 𝑛𝑝3 50 × 0,231
=
(𝑛4 − 𝑛𝑝4 )2 (17 − 50 × 0,037)2 = = 1,241 𝑛𝑝4 50 × 0,037
=
(𝑛5 − 𝑛𝑝5 )2 (10 − 50 × 0,22)2 = = 0,090 𝑛𝑝5 50 × 0,22
=
(𝑛6 − 𝑛𝑝6 )2 (4 − 50 × 0,094)2 = = 0,104 𝑛𝑝6 50 × 0,094
(𝑛7 − 𝑛𝑝7 )2 (2 − 50 × 0,031)2 = = = 0,131 𝑛𝑝7 50 × 0,031 𝜒2𝐸𝑋𝑃 = 2,758
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- The highest allowed difference between 7 segments of line and 7 surfaces determined by
means of statistical tables: Computation of theoretical (critical) value χ2 and of freedom degrees number ν for significance level (statistical error of 1.type) α = 0, 05 2 TH k2r 0, 05 (k – number of scale elements, r – number of theoretical parameters)
k r 1 7 2 1 4
42 (0,05) 9, 49
2 2 → We can receive the zero hypothesis H0 TH EXP
For the significance level α = 0,05 the empirical graph may be substituted by Gauss curve and the investigated empirical distribution may be substituted by normal distribution. The percentage difference of open and close price in Coca – Cola Company is behaving in order. We can also determine the percentage difference with the most probability (under the top of Gauss Curve) and the probabilities of smaller percentage differences and higher percentage differences are decreasing smoothly with the shape of Gauss curve. Remark: In the case of reduction from 7 intervals to 5 intervals (the connection of 1. interval and 2. interval, the connection of 6. interval and 7. interval – these connections should be carried out due to small numbers of the statistical units within mentioned intervals) the positive result of non-parametric testing will not be changed.
ii) Non-parametric testing of normality for statistical sign SS2 (percentage difference of open and close price of PepsiCoCompany)
By means of the same procedure, which was used for statistical sign SS1, it is possible to prove: For the significance level α = 0,05 the investigated empirical distribution for statistical sign SS2 may be substituted by normal distribution, the empirical graph may be substituted by Gauss curve.
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B3.5. Measurement of Statistical Dependences The statistical dependence between the signs SS1, SS2 is given by an instruction which assigns exactly one empirical distribution of the frequencies of statistical sign SS2 to measured or entered values of sign SS1. The concept correlation dependence is the narrower concept than regression dependence. The simple correlation can be understood as the mutual dependences of two random variables(statistical signs SS1, SS2) with a change of the arithmetic mean deduced from the exploration of the second statistical sign. The two chosen statistical signs in this chapter are: SS1: The percentage differences between open and close price of Coca Cola Company (Statistical sign SS1 will be marked SS-x, it will be applied to axis x) SS2: The percentage difference between open and close price of Pepsico, Inc. (Statistical sign SS2 will be marked SS-s, it will be applied to axis y) The two statistical signs SS1 (SS-x) and SS2 (SS-s) are examining from the point of popularity among customers and competitiveness between them. The popularity between these statistical signs has two basic features: The popularity among customers is characterized by value of correlation coefficient 𝑘𝑥𝑠 . The values of 𝑘𝑥𝑠 ∈ < −1; 1 >. The bond is tighter and tighter with approaching the 𝑘𝑥𝑠 to values –1 and 1.
The most used measure of simple linear correlation tightness is Pearson’s correlation coefficient𝑘𝑥𝑠 . The coefficient is given by: 𝑆𝑥𝑠 𝑘𝑥𝑠 = 𝑆𝑥𝑆𝑆
The potential typology of bond tightness: • 7 potential cases 𝑘𝑥𝑠 𝑘𝑥𝑠 𝑘𝑥𝑠 𝑘𝑥𝑠 𝑘𝑥𝑠 𝑘𝑥𝑠 𝑘𝑥𝑠
∈ (0,714; 1) ∈ (0,428; 0,714 > ∈ < 0,142; 0,428 > ∈ < −0,144; 0,142) ∈ < −0,43; −0,144) ∈ < −0,716; −0,43) ∈ < −1; −0,716)
= strong positive correlation = middle positive correlation = weak positive correlation = uncorrelated = weak negative correlation = middle negative correlation = strong negative correlation
Positive correlation means that both statistical signs simultaneously increase or decrease Negative correlation means that one from ones increases, the remaining from ones decreases
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B3.5.1. Simple linear regression analysis
The system of normal equations for the linear regression: Σ𝑠𝑖 = 𝑘 𝑏0 + 𝑏1 Σ𝑥𝑖 Σ𝑠𝑖 𝑥𝑖 = 𝑏0 Σ𝑥𝑖 + 𝑏1 Σ𝑥𝑖2
For computing in our project we will use method B - the single statistical unit cannot be identified.
the number k of scale elements will be renamed to n Σ𝑠𝑖 = 𝑛 𝑏0 + 𝑏1 Σ𝑥𝑖
The system of normal equations for the concrete case
xi
si
ni
ni/ n
xi si
x i2
3
3
1
0,14
9
9
5
5
1
0,14
10
25
9
8
1
0,14
72
81
17
17
1
0,14
289
289
10
9
1
0,14
90
100
4
5
1
0,14
20
16
2
3
1
0,14
6
4
Σ 50
Σ 50
Σ 498
Σ 524
(O1x = 7,14)
(O1s = 7,14)
Σ7
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Installment to the normal equations of linear regression analysis 50 = 7𝑏0 + 50𝑏1 /. (−50) 498 = 50𝑏0 + 524𝑏1 /. (7) −2500 = −350𝑏0 − 250𝑂𝑏1 3486 = 350𝑏0 + 524𝑏1 1168𝑏1 = 986 𝑏1 = 0,84
50 = 7𝑏0 + 50 ∙ 0,84 50 = 7𝑏0 + 42 𝑏0 = 1,14
Application of computations to the equation of line
• analytical expression of line equation
𝑦 = 𝑏0 + 𝑏1 𝑥
• illustration (applied value 3 to the variable x and computed value of variable y)
𝑦1 = 1,14 + 0,84 ∙ 3 𝑦1 = 3,66
It mean 3,66 approximately 4 lowest percentage differences between open and close prices of the companies.
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B3.5.2. Linear correlation analysis
The formula for Pearson correlation coefficient will be used
𝑘𝑥𝑠 =
𝑆𝑥𝑠 𝑆𝑥𝑆𝑆
Computation of mixed central moment of 2nd order
S xs 𝑆𝑥𝑠 =
ni xi O1x si O1s n
1 3 − 7,14 3 − 7,14 + 5 − 7,14 5 − 7,14 + 9 − 7,14 8 − 7,14 7 + 17 − 7,14 17 − 7,14 + 10 − 7,14 9 − 7,14 + 4 − 7,14 5 − 7,14 + 2 − 7,14 3 − 7,14 𝑆𝑥𝑠 =
1 17,13 + 4,57 + 1,59 + 97,31 + 5,31 + 6,71 + 21,27 5 𝑆𝑥𝑠 =
1 153,89 7
𝑆𝑥𝑠 = 21,98
Computation of standard deviations of single statistical signs
C2 x
𝐶2𝑥 =
ni xi O1x 2 n
1 3 − 7,14 2 + 5 − 7,14 2 + 9 − 7,14 7 + 4 − 7,14 2 + 2 − 7,14 2 𝐶2𝑥 =
2
+ 17 − 7,14
2
+ 10 − 7,14
1 (17,13 + 4,57 + 3,45 + 97,21 + 8,17 + 9,85 + 24,41) 7 𝐶2𝑥 =
1 (164,79) 7
𝐶2𝑥 = 23,54
2
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𝑆𝑥 =
𝐶2𝑥 =
C2 s
𝐶2𝑠 =
23,54 = 4,85
ni si O1s 2 n
1 3 − 7,14 2 + 5 − 7,14 2 + 8 − 7,14 7 + 5 − 7,14 2 + 3 − 7,14 2 𝐶2𝑠 =
111
2
+ 17 − 7,14
2
+ 9 − 7,14
2
1 (17,13 + 4,57 + 0,73 + 97,21 + 3,45 + 4.57 + 17,13) 7 𝐶2𝑠 =
1 (144,79) 7
𝐶2𝑠 = 20,68
𝑆𝑠 =
𝐶2𝑠 =
20,68 = 4,55
Computation of Pearson correlation coefficient 𝑘𝑥𝑠 =
𝑘𝑥𝑠 =
𝑆𝑥𝑠 𝑆𝑥 ∙ 𝑆𝑠
21,98 4,85 ∙ 4,55
𝑘𝑥𝑠 = 0,99
The statistical sign SS-x (the percentage difference of open and close price of Coca Cola Company) and statistical sign SS-s (the percentage difference of open and close price of PepsiCo Inc. Company) are correlating strongly positively because the coefficient is almost 1. This determination was expected – with increasing popularity and selling of Coca Cola also PepsiCo Inc. is growing
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B3.6. Conclusion On the basis of application the algorithm of statistics and probability the basic hypotheses were verified: - the empirical distribution of percentage price movements is possible to substitute by normal distribution - the correlation between these two percentage differences will be strongly positive The first assumption was confirmed by means of non-parametric testing (see the third chapter). The percentage price movements of Coca Cola Company and PepsiCoCompany within relevant financial market are behaving normally, their empirical distributions may be substituted by normal distribution. The second assumption was also confirmed in the complete extent. The statistical sign SS-x (SS1 - percentage price movements of Coca Cola Company) and statistical sign SS-s (SS2 - percentage price movements of PepsiCoCompany) are correlating positively and this correlation can be taken like the strong correlation.
B3.7. Literature Záškodný,P. (2013) The principles of probability and statistics (Monolingual English Version) Prague, Czech Republic: Curriculum ISBN 978-80-904 948-6-2 Záškodný,P. et al (2007) Principles of Economical Statistics (Partly on English) Prague, Czech Republic: Eupress. ISBN 80-86754-00-6 Yahoo Finance. (2013). The Coca-Cola Company (KO). Retreived April, 1, 2013, from http://finance.yahoo.com/q/hp?s=KO+Historical+Prices Yahoo Finance.(2013). Pepsico, Inc. (PEP).Retreived April, 1, 2013, fromhttp://finance.yahoo.com/q/hp?s=PEP+Historical+Prices
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B4. Measurement of Different Luminous Intensity of Star: WASP-39b
Author Adam Vlcek University of South Bohemia [email protected]
Abstract The work “Measurement of Different Luminous Intensity of Star: WASP-39b” presents the attempt to apply the algorithm of statistics to the investigation of variable light intensity of star WASP-39b. This variability should be caused by circulation of exoplanet. The algorithm of statistics was used as the sequence of stages i) Formulation of Statistical Investigation ii) Creation of Scale iii) Measurement in Descriptive Statistics iv) Elementary Statistical Processing v) Non-parametric Testing vi) Theory of Estimation vii) Parametric Testing viii) Regression and Correlation Analysis. For verification of formulated hypothesis “The measured values of light intensity (in proportion to the auxiliary constant “star C” caused by circulation of exoplanet) have got the normal distribution” were chosen the following stages of statistics algorithm: - Formulation of statistical investigation, - Scaling, - Measurement in descriptive statistics, - Elementary statistical processing, - Non-parametric testing. In the course of gradual application of individual algorithmic stages was performed the random selection of 50 measurement of light intensity from total number 456 measured values. The methods of descriptive statistics were applied by very broad range. From the method of mathematical statistics was used χ2-test for the verification of above mentioned hypothesis. The verification hypothesis was, although closely, confirmed – the statistical character of carried out measurement of variable light intensity of investigated star is given by normal
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distribution. The influence of circulating exoplanet for the changes of measured luminous intensity of maternal star was demonstrated in the form of normal distribution. The deeper astronomical interpretation of existence of exoplanet was not performed – the basic goal of presented work consisted in the application of statistics algorithm to concrete problem solving.
Key Words Statistics algorithm, Sequence of method of descriptive statistics, Sequence of method of mathematical statistics, Used stages of statistics algorithm for exploration of concrete problem, Variable light intensity of concrete star, Influence of circulating exoplanet, Normal distribution of realized measurements
Contents B4.1. Introduction B4.2. Formulation of Statistical Investigation B4.3. Scaling B4.4. Measurement – Description of Observation Process B4.5. Elementary Statistical Processing B4.6. Non-parametric Testing B4.7. Conclusion B4.8. Literature
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B4.1. Introduction As a subject of investigation was chosen star WASP-39b where variable light intensity was measured in proportion to the auxiliary constant “star C“, caused by circulation of exoplanet. Number of measurement was reduced from the original count of 456 values to 50, using a random number generator. The investigation was based on data provided by F. Lomoz, measured on the observatory of Josef Sadil in Sedlčany. The basic task of presented investigation has consisted in the verification of hypothesis “The measured values of light intensity (in proportion to the auxiliary constant “star C” caused by circulation of exoplanet) have got the normal distribution”. This assumption expressed by formulated hypothesis is in the harmony with astronomical science and the verification of mentioned assumption would be by the good substantiation of seeking of measurement statistical character. For verification of formulated hypothesis the algorithm of statistics (see P.Zaskodny, 2013a, 2013b) was used. The algorithm of statistics is created by the succession of following stages: i) Formulation of Statistical Investigation (For our investigation it was necessary to define mass random event, statistical unit, statistical code, selective statistical file, way of random selection, basic statistical set) ii) Creation of Scale (For our investigation it was necessary to introduce the quantitative metric scale) iii) Measurement in descriptive statistics (For our investigation it was necessary to introduce the absolute, relative, cumulative frequency) iv) Elementary statistical processing (For our investigation it was necessary to introduce the frequencies table, to calculate the empirical parameters, to draw the frequencies polygon) v) Non-parametric Testing (For our investigation it was necessary to demonstrate the normality of realized measurement of proportional light intensities) vi) Theory of Estimation (For our investigation this algorithmic stage was not used) vii) Parametric Testing (For our investigation this algorithmic stage was not used)
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viii) Regression and Correlation Analysis (For our investigation this algorithmic stage was not used) The first four stages are connected with the methods of descriptive statistics and probability theory, the second four stages are connected with the methods of mathematical statistics and probability theory. For our investigation of the normality demonstration of proportional light intensities realized measurement were applied the methods of descriptive statistics and non-parametric testing as the needful method of mathematical statistics. These used methods were applied with the various deepness of utilization. It means, for the verification of formulated hypothesis were applied of following stages from statistics algorithm: - Formulation of statistical investigation, - Scaling, - Measurement in descriptive statistics, - Elementary statistical processing, - Non-parametric testing.
B4.2. Formulation of Statistical Investigation As a subject of survey was chosen star WASP-39b where variable light intensity was measured in proportion to the auxiliary constant star„C“, caused by circulation of exoplanet. Number of measurement was reduced from the original count of 456 values to 50, using a random number generator. The survey was based on data provided by F. Lomoz, measured on the observatory of Josef Sadil in Sedlčany. The basic concepts of the first stage of statistics algorithm were defined by following way: Mass random event: Measurement of the intensity of illumination Statistical unit: Star WASP-39b Statistical Code: Change of luminous intensity caused by circulation of exoplanet Value of statistical code: Description of size ratio of the intensity of illumination [mag.] Basic statistical set: 456 measured values during the time period of one day Random selection: Selection of fifty values from the original 456 using a random number generator Selective statistical file: 50 measured values
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Statistical selective file as part of the basic statistical set (see the measurement marked by red color):
Exoplanet WASP-39b - Day of observation: 4.12.2012 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47.
JD geocentric 2456040,33598 2456040,33654 2456040,33707 2456040,33763 2456040,33817 2456040,33872 2456040,33926 2456040,3398 2456040,34034 2456040,34089 2456040,34142 2456040,34199 2456040,34253 2456040,34308 2456040,34362 2456040,34417 2456040,34471 2456040,34526 2456040,3458 2456040,34634 2456040,34689 2456040,34743 2456040,34797 2456040,34852 2456040,34909 2456040,34963 2456040,35016 2456040,35072 2456040,35126 2456040,35179 2456040,35235 2456040,35289 2456040,35343 2456040,35398 2456040,35452 2456040,35506 2456040,35562 2456040,35617 2456040,35671 2456040,35727 2456040,35781 2456040,35836 2456040,3589 2456040,35944 2456040,35999 2456040,36053 2456040,36108
V-C -1,1416 -1,1305 -1,1299 -1,1312 -1,142 -1,1385 -1,1377 -1,1408 -1,1685 -1,1365 -1,1501 -1,1259 -1,1367 -1,1199 -1,134 -1,144 -1,1295 -1,131 -1,1255 -1,1242 -1,1286 -1,1172 -1,1277 -1,1449 -1,1428 -1,1329 -1,1079 -1,1284 -1,1232 -1,1325 -1,1477
s1 0,0067 0,0064 0,0065 0,0065 0,0064 0,0063 0,0063 0,0065 0,0063 0,0062 0,0062 0,0064 0,0063 0,0066 0,0064 0,0064 0,0066 0,0065 0,0069 0,007 0,0065 0,0061 0,0063 0,0061 0,0061 0,0061 0,006 0,0059 0,0059 0,0061 0,0059
V-C1 -2,2709 -2,271 -2,2459 -2,2425 -2,2668 -2,2468 -2,2863 -2,2709 -2,2821 -2,2595 -2,2668 -2,2201 -2,2912 -2,2777 -2,2882 -2,2531 -2,2292 -2,2882 -2,2555 -2,2566 -2,2457 -2,2807 -2,2314 -2,2827 -2,2953 -2,2865 -2,2293 -2,2769 -2,279 -2,2245 -2,2977
s2 0,0163 0,0159 0,0158 0,0157 0,016 0,0156 0,0157 0,0156 0,0153 0,0155 0,0153 0,0149 0,016 0,017 0,0159 0,0157 0,0154 0,0161 0,0172 0,0171 0,0156 0,0155 0,0149 0,0153 0,0153 0,0151 0,0145 0,015 0,015 0,0144 0,0148
V-C3 -1,8425 -1,8289 -1,8275 -1,8339 -1,8396 -1,8575 -1,853 -1,8521 -1,8573 -1,8348 -1,8618 -1,8315 -1,8588 -1,8381 -1,8164 -1,8644 -1,8384 -1,8274 -1,8329 -1,811 -1,8336 -1,8157 -1,8385 -1,8201 -1,8557 -1,8464 -1,826 -1,8496 -1,8357 -1,8257 -1,86
s3 0,0113 0,0111 0,0112 0,0112 0,011 0,011 0,0111 0,0112 0,0107 0,0111 0,0108 0,0109 0,011 0,0114 0,0106 0,0116 0,0112 0,0108 0,0121 0,0121 0,011 0,0099 0,0109 0,0101 0,0104 0,0105 0,0104 0,0103 0,0102 0,0102 0,01
C-C1 -1,1293 -1,1405 -1,116 -1,1113 -1,1248 -1,1083 -1,1486 -1,1301 -1,1136 -1,123 -1,1167 -1,0942 -1,1545 -1,1578 -1,1542 -1,1091 -1,0997 -1,1572 -1,13 -1,1324 -1,1171 -1,1635 -1,1037 -1,1378 -1,1525 -1,1536 -1,1214 -1,1485 -1,1558 -1,092 -1,15
s4 0,0173 0,0168 0,0167 0,0166 0,0169 0,0165 0,0166 0,0165 0,0162 0,0164 0,0162 0,0158 0,0168 0,0179 0,0168 0,0166 0,0164 0,017 0,0182 0,0181 0,0165 0,0163 0,0158 0,0161 0,0161 0,0159 0,0153 0,0158 0,0158 0,0153 0,0156
-1,1409 -1,1164 -1,1302 -1,1362 -1,1191 -1,136 -1,1199 -1,1285 -1,1523 -1,1331 -1,1286 -1,1518 -1,1385 -1,1417 -1,1188
0,0059 0,0058 0,0058 0,0058 0,0058 0,0059 0,0057 0,0058 0,0058 0,0057 0,0057 0,0058 0,0057 0,0057 0,0057
-2,2763 -2,2448 -2,285 -2,2442 -2,2747 -2,2749 -2,2943 -2,2589 -2,2483 -2,2475 -2,277 -2,2696 -2,2862 -2,2716 -2,2599
0,0146 0,0144 0,015 0,014 0,0148 0,0145 0,0144 0,0143 0,0141 0,014 0,0144 0,0141 0,0141 0,0138 0,0139
-1,8373 -1,82 -1,8411 -1,8402 -1,8404 -1,818 -1,8462 -1,8423 -1,8247 -1,8324 -1,861 -1,8324 -1,8383 -1,837 -1,817
0,0099 0,01 0,01 0,0102 0,0101 0,0099 0,01 0,01 0,0098 0,0096 0,0098 0,0097 0,0097 0,0095 0,0096
-1,1354 -1,1284 -1,1548 -1,108 -1,1556 -1,1389 -1,1744 -1,1304 -1,096 -1,1144 -1,1484 -1,1178 -1,1477 -1,1299 -1,1411
0,0154 0,0152 0,0158 0,0148 0,0156 0,0153 0,0152 0,0151 0,0149 0,0148 0,0152 0,0149 0,0149 0,0146 0,0147
Educational & Didactic Communication 2013, Vol.1, Part B 48. 49. 50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101.
2456040,36163 2456040,36219 2456040,36273 2456040,36327 2456040,36381 2456040,36435 2456040,3649 2456040,36544 2456040,36598 2456040,36654 2456040,36708 2456040,36763 2456040,36817 2456040,3687 2456040,36925 2456040,36979 2456040,37034 2456040,37088 2456040,37141 2456040,37198 2456040,37252 2456040,37307 2456040,37361 2456040,37415 2456040,3747 2456040,37524 2456040,37578 2456040,37632 2456040,37686 2456040,37741 2456040,37795 2456040,37849 2456040,37905 2456040,37959 2456040,38014 2456040,38068 2456040,38123 2456040,38177 2456040,3823 2456040,38286 2456040,3834 2456040,38395 2456040,38449 2456040,38502 2456040,38559 2456040,38613 2456040,38668 2456040,38722 2456040,38777 2456040,38831 2456040,38885 2456040,3894 2456040,38994 2456040,39047
-1,1344 -1,1363 -1,1325 -1,1394 -1,1471 -1,1259 -1,1423 -1,1333 -1,1343 -1,1225 -1,1283 -1,1444 -1,144 -1,1104 -1,1711 -1,1371 -1,1558 -1,1481 -1,1476 -1,1567 -1,1201 -1,1323 -1,1346 -1,1585 -1,0969 -1,1581 -1,13 -1,1208 -1,1646 -1,1357 -1,1428 -1,1274 -1,1474 -1,1205 -1,134 -1,1484 -1,1512 -1,1386 -1,1395 -1,1419 -1,1305 -1,1274 -1,1495 -1,1478 -1,1368 -1,1304 -1,1244 -1,1438 -1,1334 -1,1257 -1,1481 -1,1246 -1,1207 -1,1458
0,0056 0,0056 0,0057 0,0056 0,0057 0,0056 0,0057 0,0056 0,0056 0,0056 0,0055 0,0056 0,0055 0,0054 0,0056 0,0054 0,0055 0,0055 0,0054 0,0055 0,0054 0,0055 0,0054 0,0056 0,0054 0,0059 0,0055 0,0055 0,0055 0,0054 0,0054 0,0053 0,0054 0,0055 0,0052 0,0054 0,0053 0,0052 0,0053 0,0053 0,0053 0,0052 0,0052 0,0053 0,0052 0,0051 0,0051 0,0052 0,0052 0,0052 0,0052 0,0052 0,0051 0,0052
-2,2678 -2,2386 -2,2311 -2,2603 -2,2768 -2,2465 -2,2579 -2,243 -2,2487 -2,2457 -2,2731 -2,2636 -2,2499 -2,2711 -2,2758 -2,2533 -2,2957 -2,2497 -2,2725 -2,295 -2,2543 -2,2695 -2,2697 -2,2811 -2,2325 -2,278 -2,2861 -2,2916 -2,2896 -2,2528 -2,2678 -2,2601 -2,2551 -2,2415 -2,2358 -2,2586 -2,2927 -2,296 -2,258 -2,2647 -2,2645 -2,2552 -2,2476 -2,2687 -2,2628 -2,2823 -2,2674 -2,2531 -2,2599 -2,2466 -2,2793 -2,2592 -2,2642 -2,2702
0,0138 0,0137 0,0136 0,0135 0,0138 0,0134 0,0136 0,0133 0,0133 0,0135 0,0137 0,0135 0,013 0,0136 0,0135 0,0133 0,0138 0,0129 0,0131 0,0135 0,0129 0,0134 0,0132 0,0136 0,0129 0,0144 0,014 0,014 0,0134 0,0128 0,0133 0,0131 0,013 0,0133 0,0125 0,0126 0,013 0,0128 0,0128 0,0129 0,0126 0,0126 0,0123 0,0124 0,0123 0,0126 0,0125 0,0122 0,0123 0,0123 0,0123 0,0125 0,0127 0,0123
-1,8137 -1,8326 -1,8566 -1,8434 -1,8297 -1,8186 -1,8398 -1,8478 -1,8418 -1,8248 -1,8316 -1,8279 -1,8349 -1,7907 -1,8551 -1,8276 -1,8456 -1,8476 -1,8484 -1,8293 -1,8121 -1,8417 -1,8147 -1,8663 -1,832 -1,843 -1,8245 -1,8528 -1,8356 -1,8512 -1,8219 -1,8413 -1,8443 -1,8257 -1,8282 -1,8259 -1,8378 -1,8368 -1,8165 -1,832 -1,8309 -1,8397 -1,841 -1,8486 -1,8333 -1,8348 -1,8361 -1,824 -1,8308 -1,8222 -1,8287 -1,8456 -1,8335 -1,8154
0,0095 0,0097 0,0097 0,0097 0,0094 0,0097 0,0095 0,0097 0,0094 0,0094 0,0092 0,0092 0,0093 0,009 0,0093 0,0094 0,0094 0,0093 0,0093 0,009 0,009 0,0093 0,0089 0,0096 0,0095 0,0099 0,0094 0,0098 0,0091 0,0092 0,0091 0,0091 0,0089 0,0095 0,0089 0,0088 0,0088 0,0088 0,0088 0,0087 0,0089 0,0087 0,0087 0,0088 0,0085 0,0085 0,0087 0,0084 0,0086 0,0086 0,0086 0,0088 0,0087 0,0085
118 -1,1334 -1,1023 -1,0986 -1,1209 -1,1297 -1,1206 -1,1156 -1,1097 -1,1144 -1,1232 -1,1448 -1,1192 -1,1059 -1,1607 -1,1047 -1,1162 -1,1399 -1,1016 -1,1249 -1,1383 -1,1342 -1,1372 -1,1351 -1,1226 -1,1356 -1,1199 -1,1561 -1,1708 -1,125 -1,1171 -1,125 -1,1327 -1,1077 -1,121 -1,1018 -1,1102 -1,1415 -1,1574 -1,1185 -1,1228 -1,134 -1,1278 -1,0981 -1,1209 -1,126 -1,1519 -1,143 -1,1093 -1,1265 -1,1209 -1,1312 -1,1346 -1,1435 -1,1244
0,0146 0,0145 0,0144 0,0143 0,0146 0,0142 0,0144 0,0141 0,0141 0,0142 0,0144 0,0143 0,0138 0,0143 0,0143 0,014 0,0145 0,0137 0,0138 0,0142 0,0136 0,0142 0,0139 0,0144 0,0136 0,0152 0,0147 0,0147 0,0142 0,0136 0,014 0,0138 0,0137 0,0141 0,0132 0,0134 0,0137 0,0135 0,0135 0,0136 0,0133 0,0133 0,013 0,0131 0,013 0,0133 0,0132 0,0129 0,013 0,013 0,013 0,0132 0,0134 0,013
Educational & Didactic Communication 2013, Vol.1, Part B 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113. 114. 115. 116. 117. 118. 119. 120. 121. 122. 123. 124. 125. 126. 127. 128. 129. 130. 131. 132. 133. 134. 135. 136. 137. 138. 139. 140. 141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155.
2456040,39102 2456040,39156 2456040,39211 2456040,39266 2456040,39321 2456040,39375 2456040,39429 2456040,39484 2456040,39538 2456040,39593 2456040,39646 2456040,39701 2456040,39756 2456040,3981 2456040,39866 2456040,3992 2456040,39975 2456040,40029 2456040,40082 2456040,40137 2456040,40191 2456040,40245 2456040,403 2456040,40354 2456040,40409 2456040,40462 2456040,40517 2456040,40571 2456040,40626 2456040,40681 2456040,40735 2456040,40789 2456040,40843 2456040,40898 2456040,40953 2456040,41007 2456040,41061 2456040,41115 2456040,4117 2456040,41226 2456040,4128 2456040,41334 2456040,41389 2456040,41443 2456040,41498 2456040,41552 2456040,41607 2456040,41661 2456040,41715 2456040,4177 2456040,41826 2456040,41881 2456040,41935 2456040,4199
119
-1,1273 -1,1363 -1,1301 -1,1293 -1,1332 -1,1506 -1,1385 -1,1334 -1,122 -1,1277 -1,1321 -1,1344 -1,1428 -1,1393 -1,1368 -1,1489 -1,1429 -1,1475 -1,1441 -1,1381 -1,1378 -1,1417 -1,1449 -1,1324 -1,1401 -1,1362 -1,1327 -1,1486 -1,1416 -1,1423 -1,1434 -1,1485 -1,1193 -1,137 -1,1451 -1,1372 -1,1328 -1,1318 -1,1413 -1,1405 -1,1348 -1,1411
0,005 0,0051 0,0051 0,0051 0,0051 0,0051 0,0051 0,0051 0,005 0,005 0,0051 0,005 0,005 0,0051 0,005 0,005 0,005 0,005 0,005 0,005 0,005 0,0049 0,005 0,005 0,0049 0,0049 0,0048 0,005 0,0048 0,0048 0,0049 0,0049 0,0048 0,0049 0,005 0,0049 0,0049 0,0049 0,0048 0,0049 0,0049 0,0049
-2,274 -2,2666 -2,2838 -2,2641 -2,2607 -2,2724 -2,2568 -2,2668 -2,2605 -2,2541 -2,2515 -2,2749 -2,2504 -2,2817 -2,28 -2,2783 -2,2788 -2,2795 -2,287 -2,2764 -2,2477 -2,2898 -2,2805 -2,2985 -2,2691 -2,2635 -2,2579 -2,2585 -2,2961 -2,2813 -2,2758 -2,2866 -2,2619 -2,2796 -2,2887 -2,2875 -2,2957 -2,2782 -2,2934 -2,2595 -2,2541 -2,2565
0,0125 0,0126 0,0123 0,0122 0,0122 0,0122 0,012 0,012 0,012 0,012 0,012 0,0119 0,012 0,012 0,0121 0,0119 0,0118 0,012 0,012 0,0121 0,0115 0,012 0,012 0,0122 0,0119 0,012 0,0118 0,0117 0,0119 0,0117 0,0117 0,0117 0,0114 0,0115 0,0117 0,0119 0,0121 0,0117 0,0119 0,0114 0,0114 0,0114
-1,8171 -1,8284 -1,8404 -1,832 -1,8251 -1,8496 -1,817 -1,8331 -1,8281 -1,8279 -1,8191 -1,8373 -1,8321 -1,841 -1,8375 -1,8532 -1,8342 -1,8181 -1,8484 -1,8271 -1,8281 -1,8389 -1,8353 -1,8272 -1,8529 -1,8065 -1,8527 -1,8399 -1,833 -1,8544 -1,8484 -1,8468 -1,8242 -1,8346 -1,8443 -1,8428 -1,8479 -1,8397 -1,8525 -1,8441 -1,842 -1,8452
0,0084 0,0085 0,0085 0,0085 0,0084 0,0085 0,0086 0,0086 0,0084 0,0085 0,0082 0,0084 0,0083 0,0083 0,0083 0,0084 0,0082 0,0081 0,0083 0,0081 0,0081 0,0083 0,0083 0,0083 0,0084 0,008 0,0082 0,0083 0,0081 0,0082 0,0082 0,0082 0,0079 0,008 0,0082 0,0081 0,0083 0,0081 0,0083 0,0082 0,0081 0,0082
-1,1467 -1,1303 -1,1537 -1,1348 -1,1275 -1,1218 -1,1183 -1,1334 -1,1385 -1,1264 -1,1194 -1,1405 -1,1076 -1,1424 -1,1432 -1,1294 -1,1359 -1,132 -1,1429 -1,1383 -1,1099 -1,1481 -1,1356 -1,1661 -1,129 -1,1273 -1,1252 -1,1099 -1,1545 -1,139 -1,1324 -1,1381 -1,1426 -1,1426 -1,1436 -1,1503 -1,1629 -1,1464 -1,1521 -1,119 -1,1193 -1,1154
0,0131 0,0133 0,013 0,0129 0,0129 0,0129 0,0127 0,0127 0,0127 0,0127 0,0127 0,0126 0,0127 0,0127 0,0129 0,0126 0,0125 0,0127 0,0127 0,0128 0,0122 0,0126 0,0127 0,0129 0,0125 0,0126 0,0124 0,0124 0,0125 0,0123 0,0123 0,0123 0,012 0,0122 0,0124 0,0125 0,0127 0,0123 0,0125 0,0121 0,0121 0,0121
-1,1459 -1,1406 -1,1378 -1,1322 -1,1422 -1,1264 -1,1216 -1,1324 -1,1318 -1,1294 -1,1388
0,0049 0,0049 0,0048 0,0048 0,0049 0,0049 0,0048 0,0049 0,0049 0,0049 0,0048
-2,2559 -2,2847 -2,2697 -2,2792 -2,2707 -2,2824 -2,2781 -2,2672 -2,252 -2,2646 -2,2765
0,0116 0,0119 0,0119 0,0118 0,0117 0,0117 0,0121 0,0117 0,0117 0,0117 0,0117
-1,8325 -1,8472 -1,8389 -1,8379 -1,8337 -1,8288 -1,8228 -1,836 -1,8337 -1,8427 -1,837
0,0081 0,008 0,0081 0,0081 0,008 0,008 0,008 0,0081 0,0081 0,0083 0,008
-1,11 -1,1441 -1,1319 -1,147 -1,1285 -1,156 -1,1565 -1,1348 -1,1202 -1,1352 -1,1377
0,0123 0,0125 0,0125 0,0124 0,0123 0,0123 0,0127 0,0123 0,0123 0,0123 0,0123
Educational & Didactic Communication 2013, Vol.1, Part B 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166. 167. 168. 169. 170. 171. 172. 173. 174. 175. 176. 177. 178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199. 200. 201. 202. 203. 204. 205. 206. 207. 208. 209.
2456040,42044 2456040,42099 2456040,42154 2456040,42208 2456040,42263 2456040,42317 2456040,42374 2456040,42428 2456040,42483 2456040,42537 2456040,42591 2456040,42646 2456040,427 2456040,42755 2456040,42809 2456040,42863 2456040,42919 2456040,42974 2456040,43029 2456040,43083 2456040,43138 2456040,43192 2456040,43248 2456040,43302 2456040,43355 2456040,43411 2456040,43465 2456040,43521 2456040,43576 2456040,43631 2456040,43685 2456040,43738 2456040,43793 2456040,43847 2456040,43902 2456040,43956 2456040,44009 2456040,44066 2456040,44119 2456040,44175 2456040,44229 2456040,44282 2456040,44337 2456040,44391 2456040,44446 2456040,445 2456040,44554 2456040,4461 2456040,44664 2456040,44719 2456040,44772 2456040,44827 2456040,44882 2456040,44935
120
-1,1221 -1,1247 -1,1236 -1,1154 -1,143 -1,1134 -1,1193 -1,1249 -1,1333 -1,1258 -1,1247 -1,1392 -1,117 -1,1118 -1,1149 -1,1078 -1,1136 -1,1267 -1,1226 -1,1271 -1,108 -1,1121 -1,1179 -1,1257 -1,1191 -1,1281 -1,114
0,0048 0,0048 0,0048 0,0048 0,0049 0,0047 0,0048 0,0048 0,0047 0,0048 0,0048 0,0048 0,0047 0,0048 0,0047 0,0047 0,0047 0,0048 0,0047 0,0047 0,0047 0,0047 0,0047 0,0047 0,0048 0,0047 0,0047
-2,2644 -2,2467 -2,2535 -2,2428 -2,2528 -2,2611 -2,2587 -2,2593 -2,2476 -2,2465 -2,2597 -2,2292 -2,2592 -2,2629 -2,283 -2,2432 -2,2321 -2,2356 -2,2642 -2,2459 -2,2716 -2,2538 -2,2599 -2,2431 -2,2614 -2,2683 -2,2543
0,0116 0,0115 0,0114 0,0114 0,0113 0,0116 0,0116 0,0115 0,0111 0,0111 0,0112 0,011 0,0111 0,0111 0,0115 0,0113 0,0111 0,011 0,0114 0,0109 0,0113 0,0113 0,0113 0,0111 0,0113 0,0114 0,0113
-1,822 -1,8405 -1,8264 -1,8332 -1,8412 -1,8199 -1,8245 -1,8147 -1,8278 -1,8323 -1,8301 -1,8315 -1,8358 -1,8292 -1,8192 -1,8246 -1,8046 -1,8276 -1,827 -1,8414 -1,8228 -1,8211 -1,8232 -1,8274 -1,8222 -1,8079 -1,8128
0,008 0,0082 0,008 0,0081 0,008 0,008 0,008 0,0079 0,0079 0,0078 0,0078 0,008 0,008 0,0079 0,0078 0,0079 0,0077 0,0078 0,0078 0,008 0,0078 0,0077 0,0078 0,0078 0,0079 0,0077 0,0079
-1,1423 -1,122 -1,1299 -1,1274 -1,1098 -1,1477 -1,1394 -1,1344 -1,1143 -1,1207 -1,135 -1,09 -1,1422 -1,1511 -1,1681 -1,1354 -1,1185 -1,1089 -1,1416 -1,1188 -1,1636 -1,1417 -1,142 -1,1174 -1,1423 -1,1402 -1,1403
0,0122 0,0121 0,012 0,012 0,012 0,0122 0,0122 0,0121 0,0117 0,0118 0,0119 0,0117 0,0117 0,0118 0,0121 0,0119 0,0117 0,0117 0,012 0,0116 0,012 0,0119 0,012 0,0117 0,0119 0,0121 0,0119
-1,1058 -1,106 -1,1104 -1,1155 -1,109 -1,1013 -1,1067 -1,0977 -1,116 -1,1016 -1,1119 -1,1175 -1,1107 -1,1115 -1,123 -1,1023 -1,106 -1,1072 -1,1061 -1,1194 -1,1057 -1,0947 -1,106 -1,0904 -1,1005 -1,1012
0,0047 0,0047 0,0047 0,0046 0,0047 0,0047 0,0046 0,0046 0,0047 0,0046 0,0046 0,0046 0,0046 0,0046 0,0046 0,0046 0,0046 0,0046 0,0047 0,0046 0,0046 0,0047 0,0046 0,0046 0,0046 0,0046
-2,2613 -2,2416 -2,2448 -2,2356 -2,2374 -2,2399 -2,2452 -2,2208 -2,234 -2,2239 -2,2505 -2,2627 -2,2726 -2,216 -2,2555 -2,2384 -2,2524 -2,2366 -2,2174 -2,2348 -2,249 -2,2304 -2,2455 -2,2273 -2,2527 -2,2416
0,0114 0,0111 0,0112 0,0109 0,0109 0,0109 0,011 0,0108 0,0109 0,0108 0,0109 0,011 0,0113 0,0107 0,011 0,0109 0,0109 0,011 0,0107 0,0109 0,011 0,0112 0,0109 0,0111 0,0111 0,0109
-1,8237 -1,8159 -1,8378 -1,8175 -1,8267 -1,8104 -1,8104 -1,8111 -1,8355 -1,8378 -1,8193 -1,8287 -1,812 -1,802 -1,8208 -1,8162 -1,8098 -1,799 -1,8106 -1,8177 -1,8418 -1,7936 -1,8139 -1,8124 -1,8204 -1,8068
0,0078 0,0077 0,0079 0,0078 0,0077 0,0076 0,0076 0,0077 0,0077 0,0078 0,0076 0,0076 0,0077 0,0075 0,0076 0,0076 0,0076 0,0076 0,0077 0,0077 0,0078 0,0075 0,0076 0,0077 0,0076 0,0076
-1,1555 -1,1356 -1,1344 -1,1201 -1,1284 -1,1386 -1,1385 -1,1231 -1,118 -1,1223 -1,1386 -1,1452 -1,1619 -1,1045 -1,1325 -1,1361 -1,1464 -1,1294 -1,1113 -1,1154 -1,1433 -1,1357 -1,1395 -1,1369 -1,1522 -1,1404
0,012 0,0117 0,0118 0,0115 0,0116 0,0115 0,0117 0,0114 0,0115 0,0114 0,0116 0,0117 0,012 0,0113 0,0117 0,0116 0,0116 0,0117 0,0113 0,0115 0,0116 0,0118 0,0116 0,0117 0,0117 0,0115
Educational & Didactic Communication 2013, Vol.1, Part B 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 235. 236. 237. 238. 239. 240. 241. 242. 243. 244. 245. 246. 247. 248. 249. 250. 251. 252. 253. 254. 255. 256. 257. 258. 259. 260. 261. 262. 263.
2456040,44991 2456040,45045 2456040,45098 2456040,45155 2456040,4521 2456040,45263 2456040,45318 2456040,45373 2456040,45427 2456040,45481 2456040,45535 2456040,45589 2456040,45645 2456040,457 2456040,45755 2456040,45809 2456040,45863 2456040,45918 2456040,45972 2456040,46027 2456040,46081 2456040,46135 2456040,4619 2456040,46246 2456040,46301 2456040,46355 2456040,4641 2456040,46464 2456040,46519 2456040,46573 2456040,46627 2456040,46682 2456040,46736 2456040,46792 2456040,46846 2456040,469 2456040,46955 2456040,47009 2456040,47064 2456040,47117 2456040,47173 2456040,47226 2456040,47281 2456040,47337 2456040,47391 2456040,47446 2456040,475 2456040,47554 2456040,47609 2456040,47663 2456040,47718 2456040,47772 2456040,47826 2456040,47882
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0,0047 0,0047 0,0046 0,0046 0,0047 0,0047 0,0047 0,0046 0,0047 0,0047 0,0047 0,0047 0,0047 0,0046 0,0046 0,0047 0,0047 0,0046 0,0047 0,0047 0,0047 0,0046 0,0047 0,0048 0,0048 0,0048 0,0047 0,0047 0,0047 0,0048 0,0048 0,0047 0,0047 0,0048 0,0047 0,0048 0,0048 0,0048 0,0047 0,0047 0,0048 0,0048 0,0047 0,0047 0,0048 0,0047 0,0049 0,0048 0,0047 0,0047 0,0048 0,0048 0,0047 0,0047
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121 -1,1431 -1,1194 -1,1629 -1,1312 -1,1036 -1,1414 -1,1396 -1,1488 -1,1341 -1,1284 -1,1391 -1,1428 -1,152 -1,127 -1,1647 -1,1638 -1,1306 -1,1512 -1,1754 -1,1319 -1,1454 -1,1428 -1,1737 -1,1225 -1,1478 -1,1513 -1,1395 -1,1264 -1,1321 -1,1261 -1,1504 -1,1522 -1,1299 -1,1564 -1,1447 -1,1396 -1,1293 -1,1507 -1,1261 -1,1254 -1,1518 -1,1294 -1,1442 -1,1404 -1,1489 -1,1448 -1,1355 -1,1166 -1,1562 -1,157 -1,1243 -1,1091 -1,1452 -1,1634
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Educational & Didactic Communication 2013, Vol.1, Part B 264. 265. 266. 267. 268. 269. 270. 271. 272. 273. 274. 275. 276. 277. 278. 279. 280. 281. 282. 283. 284. 285. 286. 287. 288. 289. 290. 291. 292. 293. 294. 295. 296. 297. 298. 299. 300. 301. 302. 303. 304. 305. 306. 307. 308. 309. 310. 311. 312. 313. 314. 315. 316. 317.
2456040,47938 2456040,47991 2456040,48045 2456040,48101 2456040,48155 2456040,48209 2456040,48264 2456040,48318 2456040,48374 2456040,48428 2456040,48483 2456040,48536 2456040,4859 2456040,48645 2456040,48698 2456040,48752 2456040,48807 2456040,48861 2456040,48917 2456040,48971 2456040,49026 2456040,4908 2456040,49133 2456040,49187 2456040,49242 2456040,49295 2456040,4935 2456040,49404 2456040,4946 2456040,49514 2456040,49568 2456040,49623 2456040,49677 2456040,49731 2456040,49786 2456040,4984 2456040,49895 2456040,49948 2456040,50002 2456040,50057 2456040,50111 2456040,50165 2456040,5022 2456040,50274 2456040,50329 2456040,50383 2456040,50438 2456040,50493 2456040,50548 2456040,50602 2456040,50656 2456040,50711 2456040,50765 2456040,50819
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-2,2872 -2,2738 -2,2294 -2,2584 -2,2444 -2,24 -2,2691 -2,2366 -2,2508 -2,2389 -2,2382 -2,2596 -2,2486 -2,2635 -2,2668 -2,2348 -2,233 -2,2402 -2,2424 -2,2384 -2,255 -2,2467 -2,2449 -2,2461 -2,2475 -2,2204 -2,2288 -2,2469 -2,2283 -2,2438 -2,2448 -2,2478 -2,2333 -2,2446 -2,242 -2,2581 -2,2517 -2,2449 -2,2723 -2,2633 -2,2665 -2,2709 -2,2426 -2,2619 -2,2365 -2,2587 -2,2349 -2,2377 -2,2518 -2,2427 -2,2701 -2,2489 -2,2519 -2,2471
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0,0079 0,0078 0,0079 0,0079 0,0079 0,0079 0,0084 0,008 0,0078 0,0078 0,008 0,0078 0,0078 0,0078 0,0079 0,0079 0,0078 0,0077 0,0078 0,008 0,0078 0,0078 0,0077 0,0079 0,0077 0,0078 0,0078 0,0077 0,008 0,0076 0,0078 0,0078 0,0078 0,0079 0,0078 0,0078 0,0078 0,0079 0,0078 0,0078 0,008 0,0079 0,0076 0,0078 0,0078 0,008 0,0078 0,0078 0,0078 0,008 0,0079 0,0077 0,0078 0,0079
122 -1,1692 -1,1756 -1,1237 -1,1446 -1,1334 -1,1291 -1,1638 -1,1316 -1,1365 -1,1377 -1,1241 -1,1584 -1,1313 -1,1422 -1,1469 -1,1119 -1,1194 -1,1345 -1,1269 -1,1234 -1,1332 -1,1381 -1,1244 -1,1362 -1,1346 -1,11 -1,1132 -1,1372 -1,1301 -1,1392 -1,1513 -1,1388 -1,1168 -1,1431 -1,1111 -1,142 -1,1479 -1,1372 -1,1584 -1,1428 -1,1445 -1,1706 -1,1391 -1,154 -1,1231 -1,1498 -1,1337 -1,1237 -1,1467 -1,1268 -1,1647 -1,1398 -1,1361 -1,1462
0,0123 0,012 0,0118 0,0122 0,012 0,0121 0,0132 0,0124 0,0119 0,0117 0,0117 0,0121 0,0118 0,0121 0,0121 0,0119 0,0118 0,0117 0,0119 0,012 0,012 0,0121 0,0118 0,0118 0,0116 0,0117 0,0117 0,0118 0,0117 0,0119 0,0119 0,0119 0,0116 0,0118 0,0116 0,0118 0,012 0,0119 0,0122 0,0122 0,0121 0,0121 0,0118 0,0119 0,0118 0,012 0,0118 0,0117 0,0119 0,0119 0,0122 0,012 0,0117 0,0118
Educational & Didactic Communication 2013, Vol.1, Part B 318. 319. 320. 321. 322. 323. 324. 325. 326. 327. 328. 329. 330. 331. 332. 333. 334. 335. 336. 337. 338. 339. 340. 341. 342. 343. 344. 345. 346. 347. 348. 349. 350. 351. 352. 353. 354. 355. 356. 357. 358. 359. 360. 361. 362. 363. 364. 365. 366. 367. 368. 369. 370. 371.
2456040,50874 2456040,50928 2456040,50982 2456040,51038 2456040,51093 2456040,51146 2456040,512 2456040,51255 2456040,51309 2456040,51363 2456040,51417 2456040,51471 2456040,51526 2456040,51581 2456040,51635 2456040,51689 2456040,51743 2456040,51798 2456040,51852 2456040,51905 2456040,5196 2456040,52014 2456040,52068 2456040,52124 2456040,52178 2456040,52233 2456040,52287 2456040,52341 2456040,52396 2456040,5245 2456040,52505 2456040,52559 2456040,52613 2456040,52669 2456040,52723 2456040,52777 2456040,52831 2456040,52885 2456040,5294 2456040,52993 2456040,53047 2456040,53102 2456040,53156 2456040,53211 2456040,53266 2456040,53321 2456040,53375 2456040,53429 2456040,53484 2456040,53538 2456040,53593 2456040,53647 2456040,53701 2456040,53756
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0,0047 0,0047 0,0047 0,0047 0,0046 0,0046 0,0047 0,0048 0,0047 0,0048 0,0047 0,0047 0,0047 0,0047 0,0047 0,0048 0,0047 0,0047 0,0047 0,0047 0,0047 0,0047 0,0047 0,0047 0,0048 0,0047 0,0047 0,0047 0,0046 0,0048 0,0048 0,0048 0,0048 0,0048 0,0047 0,0047 0,0048 0,0047 0,0048 0,0048 0,0048 0,0048 0,0048 0,0049 0,0048 0,0048 0,0049 0,0048 0,0048 0,0048 0,0049 0,0048 0,0049 0,0049
-2,2864 -2,2365 -2,2308 -2,239 -2,2621 -2,2739 -2,2778 -2,258 -2,2449 -2,2826 -2,2594 -2,2608 -2,2657 -2,271 -2,2509 -2,2619 -2,2548 -2,2629 -2,2773 -2,2723 -2,2873 -2,2664 -2,2721 -2,2596 -2,2722 -2,2522 -2,2804 -2,2609 -2,2601 -2,2635 -2,2584 -2,26 -2,2646 -2,268 -2,2679 -2,2686 -2,2828 -2,2735 -2,2838 -2,285 -2,2755 -2,2905 -2,2802 -2,2706 -2,2759 -2,2458 -2,2813 -2,2849 -2,2713 -2,2591 -2,2905 -2,2912 -2,2788 -2,2674
0,0116 0,011 0,0113 0,011 0,0116 0,0116 0,0116 0,0113 0,0113 0,0113 0,0115 0,0111 0,0115 0,0113 0,0109 0,0113 0,0115 0,0111 0,0116 0,0111 0,0115 0,0112 0,0114 0,011 0,0113 0,0112 0,0113 0,0112 0,0112 0,011 0,0112 0,0114 0,0115 0,0114 0,0113 0,0113 0,0114 0,0112 0,0113 0,0115 0,0113 0,0115 0,0117 0,0115 0,0117 0,0113 0,0116 0,0117 0,0116 0,0116 0,0116 0,0118 0,0119 0,0117
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0,0078 0,008 0,0077 0,0077 0,0079 0,0076 0,0079 0,0079 0,0079 0,0078 0,0079 0,0077 0,0078 0,0078 0,0075 0,0078 0,0078 0,0079 0,0078 0,008 0,0078 0,0078 0,0077 0,0078 0,0077 0,0079 0,0078 0,0078 0,0077 0,0078 0,0077 0,0079 0,0079 0,008 0,0078 0,0078 0,0078 0,0076 0,0079 0,0078 0,0077 0,0078 0,0079 0,0081 0,008 0,0081 0,0081 0,0081 0,0078 0,0082 0,008 0,0079 0,0081 0,0082
123 -1,166 -1,1287 -1,1312 -1,1362 -1,1593 -1,1707 -1,1807 -1,1467 -1,1242 -1,156 -1,1422 -1,1434 -1,1538 -1,142 -1,1322 -1,1374 -1,1347 -1,1409 -1,1637 -1,144 -1,1615 -1,1365 -1,1493 -1,1178 -1,1336 -1,1313 -1,1445 -1,1385 -1,1248 -1,1183 -1,1308 -1,1306 -1,1424 -1,1337 -1,1492 -1,1411 -1,1442 -1,1453 -1,1394 -1,1514 -1,1354 -1,136 -1,1433 -1,1273 -1,1452 -1,1162 -1,1265 -1,1344 -1,1319 -1,1381 -1,1522 -1,1602 -1,1308 -1,1336
0,0122 0,0116 0,0119 0,0116 0,0121 0,0121 0,0122 0,0119 0,0119 0,0119 0,0121 0,0117 0,0121 0,0119 0,0115 0,0119 0,0121 0,0118 0,0122 0,0118 0,0121 0,0118 0,012 0,0117 0,012 0,0118 0,012 0,0118 0,0119 0,0118 0,0118 0,012 0,0121 0,012 0,0119 0,0119 0,0121 0,0118 0,012 0,0121 0,012 0,0122 0,0123 0,0122 0,0123 0,0119 0,0123 0,0123 0,0122 0,0122 0,0123 0,0124 0,0125 0,0123
Educational & Didactic Communication 2013, Vol.1, Part B 372. 373. 374. 375. 376. 377. 378. 379. 380. 381. 382. 383. 384. 385. 386. 387. 388. 389. 390. 391. 392. 393. 394. 395. 396. 397. 398. 399. 400. 401. 402. 403. 404. 405. 406. 407. 408. 409. 410. 411. 412. 413. 414. 415. 416. 417. 418. 419. 420. 421. 422. 423. 424. 425.
2456040,53811 2456040,53866 2456040,53919 2456040,53975 2456040,54029 2456040,54083 2456040,54137 2456040,54192 2456040,54245 2456040,54301 2456040,54356 2456040,54411 2456040,54465 2456040,5452 2456040,54574 2456040,54629 2456040,54683 2456040,54737 2456040,54792 2456040,54846 2456040,54902 2456040,54956 2456040,5501 2456040,55065 2456040,55119 2456040,55174 2456040,55228 2456040,55338 2456040,55394 2456040,55449 2456040,55502 2456040,55611 2456040,55666 2456040,55719 2456040,55773 2456040,55828 2456040,55883 2456040,55938 2456040,55992 2456040,56045 2456040,56099 2456040,56154 2456040,56208 2456040,56263 2456040,56316 2456040,5637 2456040,56427 2456040,56482 2456040,56536 2456040,5659 2456040,56645 2456040,56752 2456040,56807 2456040,56861
-1,1518 -1,1316 -1,1452 -1,1432 -1,133 -1,1273 -1,1411 -1,1275 -1,1342 -1,1457 -1,1485 -1,1423 -1,1483 -1,1471 -1,1384 -1,148 -1,1424 -1,1329 -1,1378 -1,1195 -1,1542 -1,1388 -1,1366 -1,1368 -1,1357 -1,1482 -1,1413 -1,1482 -1,1358 -1,1103 -1,1373 -1,1513 -1,1241 -1,147 -1,1132 -1,1483 -1,1456 -1,1308 -1,1422 -1,1648 -1,1418 -1,1328 -1,1516 -1,1386 -1,13 -1,1389 -1,1546 -1,1283 -1,1443 -1,136 -1,1321 -1,1429 -1,1611 -1,1434
0,0049 0,0049 0,0049 0,0049 0,0048 0,0048 0,0049 0,0049 0,0048 0,0049 0,0048 0,0048 0,0048 0,0049 0,0048 0,0049 0,0049 0,0049 0,0049 0,0048 0,0052 0,0049 0,0048 0,0048 0,0049 0,0083 0,0049 0,005 0,0049 0,0067 0,0049 0,005 0,005 0,0063 0,0052 0,0053 0,005 0,005 0,0057 0,0052 0,0051 0,0051 0,0052 0,0056 0,0053 0,0055 0,0055 0,0051 0,0051 0,0051 0,005 0,005 0,0051 0,0055
-2,2597 -2,2617 -2,249 -2,2691 -2,2736 -2,2858 -2,2835 -2,2782 -2,2634 -2,2768 -2,2836 -2,2676 -2,2921 -2,2541 -2,2665 -2,2737 -2,291 -2,2797 -2,2708 -2,2523 -2,269 -2,2539 -2,2715 -2,2542 -2,2759 -2,2846 -2,2543 -2,2608 -2,2556 -2,263 -2,2586 -2,2759 -2,2737 -2,2837 -2,2642 -2,2569 -2,2582 -2,2633 -2,2787 -2,2484 -2,246 -2,2397 -2,2552 -2,2875 -2,2518 -2,2696 -2,2504 -2,2638 -2,2717 -2,2669 -2,2569 -2,2616 -2,265 -2,2693
0,0117 0,0114 0,0114 0,0116 0,0116 0,0117 0,0118 0,0119 0,0115 0,0115 0,0118 0,0114 0,0114 0,0113 0,0116 0,0114 0,0118 0,0119 0,0117 0,0115 0,012 0,0115 0,0116 0,0113 0,0118 0,0208 0,0117 0,0117 0,0118 0,0166 0,0119 0,0119 0,0121 0,0154 0,0125 0,0127 0,0116 0,012 0,0136 0,0119 0,0119 0,0117 0,0121 0,0139 0,0127 0,0137 0,0129 0,0125 0,0126 0,0119 0,0121 0,0121 0,0122 0,0128
-1,8408 -1,8437 -1,8294 -1,8435 -1,8658 -1,8467 -1,8481 -1,8275 -1,8342 -1,8456 -1,8347 -1,8379 -1,864 -1,8522 -1,8237 -1,8452 -1,8437 -1,8353 -1,8416 -1,823 -1,8436 -1,8432 -1,8457 -1,8194 -1,8334 -1,846 -1,8554 -1,8493 -1,8409 -1,8329 -1,8596 -1,8594 -1,8238 -1,825 -1,8038 -1,8555 -1,8338 -1,8494 -1,8563 -1,8465 -1,8244 -1,8383 -1,8541 -1,8445 -1,8382 -1,842 -1,8272 -1,8254 -1,8297 -1,842 -1,8436 -1,8405 -1,8378 -1,8505
0,0082 0,0082 0,0079 0,008 0,0081 0,0081 0,0081 0,008 0,008 0,0082 0,0079 0,008 0,0082 0,0081 0,0079 0,0081 0,0081 0,008 0,0082 0,0081 0,0086 0,0081 0,008 0,008 0,0081 0,014 0,0083 0,0082 0,0082 0,0118 0,0086 0,0084 0,0084 0,0104 0,0087 0,0092 0,0082 0,0083 0,0096 0,0082 0,0081 0,0083 0,0086 0,0095 0,0089 0,0092 0,0089 0,0084 0,0086 0,0084 0,0085 0,0084 0,0085 0,009
124 -1,1079 -1,1301 -1,1038 -1,1259 -1,1406 -1,1585 -1,1424 -1,1507 -1,1292 -1,1311 -1,1351 -1,1253 -1,1438 -1,107 -1,1281 -1,1257 -1,1486 -1,1468 -1,133 -1,1328 -1,1148 -1,1151 -1,1349 -1,1174 -1,1402 -1,1364 -1,113 -1,1126 -1,1198 -1,1527 -1,1213 -1,1246 -1,1496 -1,1367 -1,151 -1,1086 -1,1126 -1,1325 -1,1365 -1,0836 -1,1042 -1,1069 -1,1036 -1,1489 -1,1218 -1,1307 -1,0958 -1,1355 -1,1274 -1,1309 -1,1248 -1,1187 -1,1039 -1,1259
0,0123 0,0121 0,0121 0,0123 0,0122 0,0123 0,0124 0,0125 0,0121 0,0122 0,0124 0,012 0,012 0,012 0,0122 0,0121 0,0124 0,0125 0,0123 0,0121 0,0127 0,0122 0,0122 0,0119 0,0124 0,022 0,0123 0,0124 0,0124 0,0175 0,0125 0,0126 0,0128 0,0163 0,0132 0,0134 0,0123 0,0127 0,0144 0,0126 0,0126 0,0124 0,0128 0,0147 0,0134 0,0144 0,0137 0,0132 0,0133 0,0126 0,0128 0,0128 0,013 0,0136
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2456040,56917 2456040,56971 2456040,57026 2456040,5708 2456040,57134 2456040,57189 2456040,57297 2456040,57352 2456040,57462 2456040,57571 2456040,57625 2456040,57679 2456040,57734 2456040,57788 2456040,57897 2456040,57953 2456040,58007 2456040,58061 2456040,58116 2456040,5817 2456040,58225 2456040,58279 2456040,58333 2456040,58388 2456040,58442 2456040,58498 2456040,58552 2456040,58606 2456040,58661 2456040,58715 2456040,5877
-1,1316 -1,1267 -1,1419 -1,1355 -1,1391 -1,1328 -1,1331 -1,1483 -1,1319 -1,1527 -1,1328 -1,1577 -1,1327 -1,135 -1,1392 -1,1365 -1,1484 -1,1426 -1,1533 -1,1468 -1,1319 -1,1504 -1,1424 -1,1483 -1,1336 -1,12 -1,1385 -1,1485 -1,1264 -1,1565 -1,16
0,0051 0,0059 0,0051 0,0069 0,0055 0,0052 0,0052 0,0053 0,0052 0,0052 0,0067 0,0061 0,0052 0,0051 0,0058 0,0073 0,0052 0,0053 0,0053 0,0055 0,0053 0,0054 0,0053 0,0053 0,0053 0,0054 0,0054 0,0055 0,0055 0,0055 0,0057
-2,2578 -2,2711 -2,269 -2,2911 -2,266 -2,274 -2,2819 -2,2879 -2,2876 -2,259 -2,2544 -2,2605 -2,2534 -2,26 -2,2738 -2,2648 -2,2561 -2,2677 -2,2727 -2,259 -2,2703 -2,2549 -2,2746 -2,2611 -2,2603 -2,2965 -2,2494 -2,3014 -2,2858 -2,2677 -2,2556
0,0119 0,0141 0,0126 0,0176 0,0137 0,0123 0,0125 0,0129 0,0126 0,0126 0,0159 0,0146 0,0121 0,0123 0,0141 0,0174 0,0123 0,0126 0,0129 0,0129 0,0127 0,0128 0,0129 0,0128 0,013 0,0132 0,0127 0,0137 0,0138 0,0133 0,0136
-1,8358 -1,8525 -1,8467 -1,8285 -1,8482 -1,843 -1,8456 -1,8306 -1,8257 -1,8692 -1,8298 -1,8569 -1,8503 -1,8636 -1,8653 -1,8675 -1,8595 -1,8375 -1,8641 -1,857 -1,8709 -1,8311 -1,8644 -1,8615 -1,8542 -1,8455 -1,8285 -1,8462 -1,8197 -1,8653 -1,8416
0,0085 0,0098 0,0086 0,0118 0,0094 0,0086 0,0088 0,0085 0,0085 0,0089 0,0113 0,0106 0,0088 0,0087 0,0099 0,0124 0,0087 0,0088 0,009 0,0089 0,009 0,009 0,0091 0,0091 0,0091 0,0092 0,0091 0,0092 0,009 0,0095 0,0098
125 -1,1262 -1,1444 -1,1271 -1,1556 -1,1269 -1,1412 -1,1488 -1,1396 -1,1557 -1,1063 -1,1216 -1,1028 -1,1207 -1,125 -1,1346 -1,1283 -1,1077 -1,1251 -1,1194 -1,1122 -1,1384 -1,1045 -1,1322 -1,1128 -1,1267 -1,1765 -1,1109 -1,1529 -1,1594 -1,1112 -1,0956
0,0126 0,0149 0,0133 0,0185 0,0144 0,013 0,0132 0,0136 0,0133 0,0133 0,0169 0,0155 0,0128 0,013 0,0149 0,0185 0,013 0,0133 0,0136 0,0137 0,0134 0,0136 0,0136 0,0135 0,0137 0,0139 0,0135 0,0144 0,0145 0,0141 0,0144
B4.3. Scaling From the measurements marked by red color it was possible to determine the spread of measured value: Minimum Value of Light Intensity: –1,1711 Maximum value of Light Intensity: –1,0944 For construction of scale the quantitative metric scale was selected. By means of application of Sturges rule the number of scale elements xi is equal to 6. From the point of creation of quantitative metric scale it is possible to determine the scale elements xi by following way:
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x1=1: From –1,18 to –1,165 x2=2: From –1,165 to –1,150 x3=3: From –1,150 to –1,135 x4=4: From –1,135 to –1,120 x5=5: From –1,120 to –1,105 x6=6: From –1,105 to –1,090
B4.4. Measurement – Description of Observation Process The values were measured by F. Lomoz on the Observatory of Josef Sadil in Sedlčany by mirror telescope Newton 254/1016+G2-8300 during one day (night): 4. 12. 2012. Determination of the values was done by the ratio of intensity due to a nearby “star C“ (see figure) and two other stars, which for this measurement are not used.
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B4.5. Elementary Statistical Processing B4.5.1. Table xi – scale elements, ni – absolute frequencies, ni/n – relative frequencies, Σni/n – cumulative frequencies, n – extent of selective statistical file
ni
ni/n
∑ ni/n
xini
xi ni
2
xi ni
3
xi ni
1
0,02
0,02
1
1
1
1
1 16 18 8 6 50
0,02 0,32 0,36 0,16 0,12 1
0,04 0,36 0,72 0,88 1
2 48 72 40 36 199
4 144 288 200 216 853
8 432 1152 1000 1296 3889
16 1296 4608 5000 7776 18697
B4.5.2. Empirical parametres General moment of r-th order: Or (x) = 1/n Σ ni . (xi )r
general moment of 1. order: general moment of 2. order: general moment of 3. order: general moment of 4. order:
O1 (x) = 3,98 O2 (x) = 17,06 O3 (x) = 77,78 O4 (x) = 373,94
Central moment of r-th order: Cr (x) = 1/n Σ ni .(xi – O1)r Standard deviation: Sx =
C2 = 1,104355015
Central moment of 2. order: C2(x) = 1,2196 (empirical variance) Central moment of 3. order: C3(x) = 0,173184 Central moment of 4. order: C4(x) = 4,35092752 Standardized moments: N3(x), N4(x) N3(x) = 0,1285794642 (skewness) N4(x) = 2,925142429 (kurtosis) Excess Ex = -0,074857571
4
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B4.5.3. Charts
Absolute frequency polygon is graphical expression of discrete function ni = f (xi).
Absolute frequency polygon: 20 18 16 14 12 10 8 6 4 2 0 1
2
3
4
5
6
7
Cumulative frequency polygon is graphical expression of discrete function Σni/n = f (xi)
Cumulative frequency polygon: 1,2 1 0,8 0,6 0,4 0,2 0 1
2
3
4
5
6
7
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B4.6. Non-parametric Testing i) Interval division of frequencies, substitution of scale elements
In the table the scale elements are substituted by the midpoints of individual intervals. Instead of scale elements xi it will be worked with the midpoints xi – the designation xi of scale elements will be transferred to the designation of midpoints.
xi -1,1725 -1,1575 -1,1425 -1,1275 -1,1175 -1,0975
Interval (-∞; -1,165> (-1,165; -1,15> (-1,15; -1,135> (-1,135; -1,12> (-1,12 ; -1,105> (-1,105; ∞)
ni 1 1 16 18 8 6 50
ni/n 0,02 0,02 0,32 0,36 0,16 0,12 1
∑ ni/n 0,02 0,04 0,36 0,72 0,88 1
xini -1,1725 -1,1575 -18,28 -20,295 -8,94 -6,585 -56,43
2
xi ni 1,37476 1,33981 20,8849 22,8826 9,99045 7,22704 63,6996
3
xi ni -1,6119 -1,5508 -23,861 -25,8 -11,164 -7,9317 -71,92
4
xi ni 1,88995 1,79508 27,2612 29,0897 12,4761 8,70501 81,217
The scale elements were substituted by the midpoints of individual intervals. Such change enables to work with the real values of varying light intensity instead of the auxiliary values of scale elements. From this reason the reduction of intervals number (the robust analysis) was not applied. From the point of this change the new value of empirical parameters must be computed.
O1 O2 O3 O4
-1,1286 1,27399125 -1,4383974563 1,6243407686
Sx
0,0159150872
C2 C3 C4
0,00025329 7,94688E-007 2,14869E-007
N3 N4
0,1971376563 3,3491702541
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ii) The brief theory of non parametric testing – substitution of empirical distribution by normal distribution The normal distribution is graphically expressed by Gauss curve. The normal distribution is belonging among the continuous distributions. The continuous distribution is described by probability density. For normal distribution the shape of probability density and distribution function are given by formulas x 2 x exp 2 2 2 1
F x
x
x dx
- theoretical general moment O1 of 1.order 2 - theoretical central moment C 2 of 2.order From the point of application of statistical table it is necessary to transform the normal distribution to the standardized normal distribution. The standardized normal distribution is again belonging between continuous distributions. The graphical expression of standardized normal distribution is given by standardized Gauss curve. The shape of probability density and the distribution function for standardized normal distribution are given by formulas u2 exp 2 2 1
u u
x
F u
u
u du
The transformation consists in computation of the values u = U. On the basis of point estimation of theoretical parameters μ, σ2 by general moment of 1.order O1 and by central moment of 2.order C2 it is possible within process of standardization to apply μ = – 1,1286, σ = 0,0159. In the formula x u the upper limits of individual intervals will be applied. Instead of value – ∞ the value – 1,18 will be used. Instead of value + ∞ the value – 1,09 will be used. The surfaces p under standardized Gauss curve are given by the differences between relevant values of distribution function F(u).
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iii) Standardization
limits
variable u
-1,18 -1,165 -1,15 -1,135 -1,12 -1,105 -1,09
-3,229639864 -2,287137958 -1,344636052 -0,402134146 0,540367759 1,482869665 2,425371571
F(u) 0,0006 0,0113 0,0901 0,3446 0,7054 0,9306 0,9922
ss
0,0107 0,0788 0,2545 0,3608 0,2252 0,0616
n. ss abs.fr. test 0,535 1 0,4042 3,94 1 2,1938 12,725 16 0,8429 18,04 18 9E-05 11,26 8 0,9438 3,08 6 2,7683
0,9916
μ σ
7,1531 χ square
-1,1286 0,015915087
F(u) – the distribution function of standardized normal distribution (Laplace function) ss – the surfaces pi under Gauss curve n.ss – the surfaces ss multiplied by extent n of selective statistical file (n.pj) limits – the limits determined on the basis of interval division of frequencies (limit – ∞ substituted by limit – 1,18 due to reality of astronomical observation) (limit ∞ substituted by limit – 1,09 due to reality of astronomical observation) abs.fr. – absolute frequencies nj 2 7,15 ) test ( χ square) – the calculation of experimental value of χ2-test ( exp
iv) Application of χ2-test The χ2-test is connected with the computation of experimental and theoretical value 2 2 2 exp and theor . The value χ exp was computed within process of standardization by means of
formula
n j np j np j j 1 The result of computation is given by result 2 exp
6
2
.
2 exp 7,15 (see the above situated table). 2 The value theor will be determined by means of statistical tables. The value χ 2theor will
be sought on the basis of selection of the usual significance level α = 0,05 and on the basis of determination of freedom degrees number ν = k – r – 1. The value k is given by number of the intervals within interval division (k = 6). The value r is given by number of theoretical parameters of used theoretical distribution. The used theoretical distribution was given by normal distribution with formula
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.
This formula contains two theoretical parameters (r = 2) - theoretical general moment of 1.order,
2 - theoretical central moment of 2.order. The theoretical value can be described in the shape
t2heor k2-r -1 0,05 . After application of the values for significance level and freedom degrees number this shape can be written in the form 2 theor k2-r -1 0, 05 32 0, 05 .
By means of utilization of statistical tables it is possible to find 2 theor 7,81.
2 2 The experimental value exp (7,15) is less then theoretical value theor (7,81). It means, the
empirical distribution of light intensity distribution can be substituted by normal distribution. The measured values of light intensity have got the normal distribution. It can be graphically expressed by means of following figure:
Graphical representation of the probability density: 20 18 16 14 12 10 8 6 4 2 0 -4
-3
-2
-1
0
1
2
3
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B4.7. Conclusion The investigation of variable light intensity of star WASP-39b (caused by circulation of exoplanet – see Chapter B4.1.) was realized by means of the application of statistics algorithm. This algorithm was used in the form of sequence of algorithm stages - Formulation of statistical investigation (see Chapter B4.2), - Scaling (see Chapter B4.3.), - Measurement in descriptive statistics (see Chapter B4.4.), - Elementary statistical processing (see Chapter B4.5.), - Non-parametric testing (see Chapter B4.6.).
The application of statistics algorithm has enabled to confirm the hypothesis “The measured values of light intensity (in proportion to the auxiliary constant “star C” caused by circulation of exoplanet) have got the normal distribution”. This verification can be interpreted as the confirmation of found statistical character of realized measurement by F. Lomoz on the Observatory of Josef Sadil in Sedlčany by mirror telescope Newton 254/1016+G2-8300 during one day (night): 4. 12. 2012. The main goal of presented work was fulfilled – the applicability of statistics algorithm for the seeking of statistical character of astronomical measurement in concrete area was confirmed. The deeper astronomical interpretation (from the point of existence of circulating exoplanet) was not carried out.
B4.8. Literature Zaskodny,P. (2013a) The Principles of Probability and Statistics Prague, Czech Republic: Curriculum (Bilingual Czech-English Version) ISBN 978-80-904948-5-5 Zaskodny,P. (2013b) The Principles of Probability and Statistics Prague, Czech Republic: Curriculum (Monolingual English Version) ISBN 978-80-904948-6-2 Zaskodny,P., Havrankova,R., Havranek,J., Vurm,V. (2012) The Principles of Statistics (with application to health care) Prague, Czech Republic: Curriculum (Monolingual Czech Version) ISBN 978-80-904948-2-4
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PART C The Algorithm of Problem Solving in Curricular Process
Authors of Chapters (according to order of the single chapters) Barbora Vesela University of South Bohemia [email protected] Miroslava Bartonova University of South Bohemia [email protected]
The goal of Part C is to validate the hypothesis that it is possible to apply the algorithm of curricular process to the solution if concrete educational problems.
The goal of Part C was completed using by the follow-up chapters: C1. Application of Curricular Process in Explanation of Physics Bases of Classical Circular Accelerators (according to B.Vesela) C2. Curricular Process of Radiological Physics within Higher Education Level (according to M.Bartonova)
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C1. Application of Curricular Process in Explanation of Physics Bases of Classical Circular Accelerators
Author Barbora Vesela University of South Bohemia [email protected]
Abstract As the selected theory of transfer of knowledge from the scientific radiological physics (in the sphere of physical bases of circular accelerators) to students of radiological branches the theory of curricular process was used.This theory was formulated in the world by M.Pasch, T.G.Gardner, M.Certon, M.Gayl, in the Czech and Slovak Republics by J.Průcha, J.Brockmeyerová, P.Tarábek, P.Záškodný (see B.Vesela, 2013a). In this work the transformations between variant forms of curricular process were used. In the sphere of physical bases of circular accelerators the transformations among the conceptual curriculum, intended curriculum, projected curriculum, implemented curriculum-1 and implemented curriculum-2 were applied to the explanation of this part of radiological physics. These transformations were carried out by means of the bonds among scientific system of radiological physics, educating text, preparation of experimental teaching, realization of experimental teaching and application of educational test to students of radiological branches to find out the results of experimental teaching in the sphere of physical bases of circular accelerators.
Key Words Curricular proces, Conceptual curriculum, Intended curriculum,Projected curriculum, Implemented curriculum-1, Implemented curriculum-2, Attained curriculum, Radiological physics, Classical circular accelerators, Educational text, Educational test, Experimental teaching, Methods of descriptive and mathematical statistics
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C1.1. Algorithm of Curricular Process On the base of curriculum process theory, in the sphere of radiological physics curriculum process, the sequence of transformations T1-T5 of curriculum variant forms can be defined (see B.Vesela, 2013a): Conceptual curriculum as a communicable scientific system of radiological physics (the first variant form of curriculum as the transformation T1 output from a scientific system to a communicable scientific system) Intended curriculum as an educational system of radiological physics (the second variant form of curriculum as the transformation T2 output from a communicable scientific system to an educational system) Projected curriculum as an instructional project of radiological physics (the third variant form of curriculum as the transformation T3 output from an educational system to a teaching project) Implemented curriculum-1 as the preparedness of educator to education in radiological physics (the fourth variant form of curriculum as again the transformation T3 output from an educational project to implementation of teaching) Implemented curriculum-2 as the results of education in radiological physics (the fifth variant form of curriculum as the transformation T4 output from an educational project to a reached result of educational process) Attained curriculum as the applicable results of education in radiological physics (the sixth variant form of curriculum as the transformation T5 output from implementation of teaching to application of the results of teaching). The algorithm of curricular process is given by the arranged succession of curriculum variant forms in the shape i) Conceptual curriculum ii) Intended curriculum iii) Projected curriculum iv) Implemented curriculum-1 v) Implemented curriculum-2 vi) Attained curriculum
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C1.2. Theoretical Foundation of Application of Curricular Process Algorithm Within the investigation of physics principles of classical circular accelerators for radiology students the methods of mathematical statistics (apart from methods of descriptive statistics) were used non-parametric tests and two sample t-test. Non-parametric test was used to test the normality of distribution knowledge. This knowledge was acquired by experimental teaching. Two-sample t-test was used to compare the knowledge of fulltime students and part-time students.
The investigation of physics principles of classical circular accelerators for radiology students was based on the verification of two hypotheses: H1 An adequate educational text in the sphere of physical bases of circular accelerators can be created by application of curriculum process H2 Students´ knowledge in the sphere of physical bases of circular accelerators acquired on the base of worked out educational text will have normal distribution. Both of these hypotheses were confirmed by statistical processing of the results which were obtained by experimental teaching and by application of educational test to students. Partial hypothesis was also confirmed – the knowledge between full-time students and parttime students was not statistically different on the level of statistical significance 0.05 (see B.Vesela, 2013). The verification of the adequacy of educational text and the appropriate construction of educational test was carried out by the application of suitable statistical methods. Statistical methods used in the research were as follows:
1. Formulation of statistical investigation 2. Scaling 3. Measurement - construction of the test 4. Elementary statistical processing - tables, graphs, empirical parameters 5. Non-parametric testing - test of normality 6. Theory of estimation 7. Parametric testing - resolution full-time and part-time studies
Brief description according to B.Vesela, 2013a:
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Formulation of statistical investigation The formulation of statistical investigation must be accurately characterized all subsequent terms. It is the investigated random mass phenomenon, the definition of the statistical unit for the analysis of statistical character, the characteristic value of a statistical character, the precise definition of basic statistical file and the following procedures for delimitation of random selection. Scaling The scaling transforms a real phenomenon on a numerical scale and classifies value of a statistical character into groups. There are known various types of scales, as these four: ordinal, nominal, absolute metric and quantitative metric. It is a quantitative metric scale which was used within theoretical foundation of application of curricular process algorithm. The elements of the scale are expressed by numerical size which was used to determine the distance between two consecutive statistical units. Measurement – construction of the test This step includes primarily the test that would fully reflect the structure of the educational text, and secondarily the presentation of test for the students of study programme Radiological Assistant. The presented test was then statistically processed to verify the normal distribution of students' knowledge and educational functions of the educational text. The measurement of each statistical unit of selection statistical file is connected with the scale elements x1, x2, ..., xk. The result of the measurement is the finding that the element scale xi was measured ni times. Values ni are called absolute frequencies and their sum is equal to the range of statistical file. Among the results of measurements xi the assessment of probability of their occurrence is also belonging. The statistical probability p(xi) of outcome xi is based on the relative frequency ni/n. When adding up all the relative frequencies thein summation must be equal to 1. Another result of the measurement is cumulative frequency. Cumulative frequency Σ ni/n denotes the probability that the measurement result is less than or equal to the result xi. Elementary statistical processing - tables, graphs, empirical parameters This step is processed through a table, creation of a graph of empirical frequency distribution and a calculation of empirical parameters of empirical distribution. These parameters include above all the arithmetic mean and standard deviation. Non-parametric testing - test of normality It consists in assigning theoretical distribution to the empirical distribution. It also states the term "non-parametric testing of hypotheses". Testing nonparametric hypothesis lies in the fact that it is advantageous to replace the empirical distribution to the theoretical distribution.
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Thanks theoretical division it is possible to obtain information otherwise unavailable and using by simple mathematical tools. Theory of estimation The next step in statistical survey is to estimate the theoretical parameters. Using point estimatiun we can quantify the theoretical parameters and using interval estimation we can construct the confidence interval. The point estimations of unknown parameters (parametric functions) are actually appropriately selected statistics. The estimation is preferable, depending on how close is the actual value of the estimated parameter (parametric function). The point estimation can be done either by the moment method where certain conditions can be considered (as an empirical parameter estimates corresponding theoretical parameter) or the maximum likelihood method which is mathematically more difficult. Parametric testing - resolution full-time and part-time studies Parametric testing is based on apparatus of two main hypotheses - the null hypothesis H0 (the population parameter) and the alternative hypothesis Ha (if H0 does not pay then Ha defines the situation). This apparatus is usually accompanied by an apparatus for a critical field W. Parametric test of theoretical parameters of a normal distribution is belonging among very important parametric test. Normal distribution N(μ,σ) has two theoretical parameters that are theoretically by general moment of the first order O1 = μ = E(x) and theoretically by central moment of second order C2 = σ2 = D(x). These theoretical parameters can be tested e.g. by means of one-selective t-test and one-selective χ2-test. Also the two-selective parametric testing is useful to apply.
C1.3. Application of Conceptual Curriculum In the course of gradual application of the curricular process theory the several publications were created in the framework of international conferences OEDM-SERM'11 and OEDM-SERM'12. Publications have dealt with the different variant forms of curriculum. The application of conceptual curriculum is connected with description of scientific system of radiological physics in the area of classical circular accelerators. The explanation of physics base of cyklotron on the scientific level is realized by means of B.Vesela, 2012, 2013a and P.Zaskodny, 2006, 2009 C1.3.1. Application of Lagrangian Formalism The part of an electromagnetic field is the magnetic field. Stationary magnetic field is a magnetic field that does not change over time, the quantities characterizing the field are constant.
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Homogeneous and constant electromagnetic field has magnetic induction B 0,0, B and intensity of electric field E 0,0,0 . The initial conditions of charge are: v 0, v0 ,0 , r 0,0,0 . By substitution of the Lagrangian function Q 1 L mv 2 QEr B r v 2 2 we will get its form.
To calculate this equation we will need these mathematic operations: Vector multiplication:
v r yz zy i zx xz j xy yx k
Velocity vector
v x , y , z
Motion vector
r x, y, z , r xi y j zk
Scalar multiplication:
Er E1r1 E2r2 E3r3 0 xi 0 y j 0zk
Application of vector and scalar multiplication:
Er 0 B r v 0 yz zy i 0 zx xz j B xy yx k B r v B xy yx
From the general form of Lagrange function we get concrete function:
1 Q L m x 2 y 2 z 2 B xy yx 2 2 After a substitution of Lagrange function into Lagrange equations of second type
d L L 0 dt q j q j we can get motion equations. We will be to substitute coordinates x 0, y 0, z B
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d L L 0 dt x x d 1 2 1 d 2 1 mx m x 2mx mx dx 2 2 dx 2 d Q Q d Q B xy By B xy dx 2 2 dx 2 d Q Q d Q Bxy B xy By dx 2 2 dx 2 d Q Q mx By By 0 dt 2 2 d 1 2 1 d 2 1 my m y 2my my dy 2 2 dy 2 d Q Q d Q B xy B xy Bx dy 2 2 dy 2 d Q Q d Q B yx B yx Bx dy 2 2 dy 2 d Q Q my Bx Bx 0 dt 2 2 d 1 2 1 d 1 mz mz 2 2mz mz dz 2 2 dz 2 mz 0 mz 0 z 0 Now we implement the second derivative and we will be to get the Lagrange equations:
Q Q By By 0 2 2 mx QBy 0 Q Q my Bx Bx 0 2 2 my QBx 0 mz 0 mx
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C1.3.2. Solution of Lagrange Equations We will be gradually to formulate x, y, z from the equations and we will be to use the QB constant : m
QB y m x y x
QB x m y x y
The outcome is the shape of movement equations:
x y y x z0 We will be to multiple the equation y x by imaginary unit i for easier numeration: i x iy
We will be to add up the equations y i x and x y : x iy y i x x iy ( y ix )
The fraction will be to extend by imaginary unit:
x iy
i
( x iy )
We will be to establish the relation x iy : i and the equation will be prepared for integration:
d
i dt ln t ln C We will be to apply the inverse function eln eln C eln it with result Ceit and we use the initial conditions:
0 iv0 ln iv0 0 ln C ln C iv0 C iv0
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Now we will be to apply Euler´s relation:
exp(ik ) cos k i sin k e it cos t i sin t x iy iv0 (cos t sin t ) x iy iv0 sin t iv0 cos t x v0 sin t y v0 cos t We use MacLaurin´s and Taylor´s expansion for the proof of Euler´s relation:
x1 x 2 .... 1! 2! x2 x4 cos x 1 .... 2! 4! x3 x5 sin x x .... 3! 5! cos x i sin x exp( ix) ex 1
We will be to integrate right and left side:
x y
v0
v0
cos t C1
sin t C2
We will be to discover the constants C1 and C2 according to the initial conditions:
0 0
v0
cos 0 C1 C1
v0
0 C2 C2 0
x y
v0
v0
v0
cos t
v0
sin t
The result of Lagarange equations solution can be described with the help of motion vector:
v v v r ( 0 cos t 0 )i ( 0 sin t ) j 0k
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C1.3.3. Interpretation of Solution The result is given by the circle (the shape of cyclotron) with usual equation
( x m)2 ( y n)2 r 2 The centre of cyclotron v S 0, 0
and the radius of cyclotron
v0
r
are leading to the equation
(x
v0
)2 y 2
v0
(x
v0
) y
v0 2
2
2
(cos2 t sin 2 t )
2
The resonance frequency of the circulation of this circle is
QB QB and v . m 2 m
This frequency must be realized for the correct function of a cyclotron.
The explanation of cyclotron physics principles will be given as an essential part of conceptual curriculum for higher education. The presented explanation is needful for understanding cyclotron operation in radiology.
C1.4. Application of Intended and Projected Curriculum C1.4.1. Synthesis of Intended and Projected Curriculum In the course of gradual application of the curricular process theory the several publications were created in the framework of international conferences OEDM-SERM'11 and OEDM-SERM'12. Publications have dealt with the different variant forms of curriculum. The application of intended and projected curriculum is connected with creation of education text „The physics Base of Cycloctron for Radiology Students“. This education text will be presented by means of B.Vesela, 2013b.
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The education text „The physics Base of Cycloctron for Radiology Students“ expresses the synthesis of intended curriculum and projected curriculum. Presented text could be written on the basis of adjustment of conceptual curriculum (see Chapter C1.3.) to the possibilities of radiology students, i.e. on the basis of delimitation of intended curriculum. The education text was presented to students of University of South Bohemia in Czech language. This education text was successful and fulfilled the educational function of projected curriculum. C1.4.2. Presentation of Education Text “The Physics Base of Cyclotron for Radiology Students” Physics in the thirties of the 20th century knew the proton, electron, neutron, positron, neutrino and meson and explained with these particals the structure of the atoms. These particles were called „elementary“. This means that they have been considered as the basic building blocks of atoms that do not consist of many smaller particles. Over time they were found other particles, including collisions of protons, electrons and other particles accelerated to high energies in accelerators. It is also recognized that the elementary particles are considered to decay into other particles. Born has articulated the question: "Which of them are actually elementary thus?" Nowadays we know about 280 particles, fundamental (basic) are particles with no internal structure, but even if they are composed of smaller particles, collisions can transform into other particles. Elementary particles are then other microparticles (proton, electron, deuteron, α particles). Accelerated microparticles in accelerators create friendly beam of particles which are precipitated, and collide with the hard target. Within these collisions can occur other particles which are recorded by the detectors. By this way we can find and study new particles. The principle of the accelerator can be simply described as the microparticles with an electric charge Q flies between locations with a potential difference U and thus increases their kinetic energy by QU. To obtain high energy particles, we let the particle fly through this place many times. Accelerators are classified according to the particle trajectory. We have linear accelerators with the trajectory as a straight line and circular accelerators with the trajectory as a spiral or a circle. The important type of circular accelerators is the cyclotron. Cyclotron (ie cyclic high-frequency accelerator) is used for acceleration of heavy charged particles using a high frequency electric field. Unlike linear accelerators are not as robust, but are likely to be more complicated construction. Accelerator has three main parts: the electromagnet (very strong), the accelerating chamber “duants” (i.e. pairs of hollow accelerating electrodes), the source of high frequency. The particles are moving inside hollow accelerating electrodes with their inertia track the curving magnetic field is oriented perpendicular to the tracks. Hollow electrode acts as a Faraday cage. It is mean that the trajectory of particles inside of electrodes doesn´t influence
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electric field but only magnetic field. The acceleration takes place only in the space between holow electrodes. The particles are powered by alternating high-frequency current of appropriate frequency. The electric field between electrodes always works in such a way to increase the speed of particles. The kinetic energy of the accelerated particles after many orbits can reach up to 50 MeV. In the cyclotron magnetic field is used only for keeping beam of particles, but not used for the acceleration. The field is oriented perpendicular to the trajectory of particles. The field causes folding trajectory into a circle, and as the particle velocity increases, the radius of the trajectory (circle) increases, it means that the particles move in a spiral trajectory. Transit time of individual loops spiral is but constant. Cyclotron frequency (f) is based on size of electrically charged particles (Q), size of magnetic induction (B) and mass of particle (m): f = QB/2πm. Simply we can describe the principle of a cyclotron, so that the particles move between the poles of the large magnet, and the magnetic field is maintained in a circular trajectory. Microparticles are accelerated by the electric field between the semicircular electrodes („duants“) to which is connected a high voltage. Constant frequency is chosen so that the voltage polarity is changed over time which requires the particles in flown semicircular trajectory from one slot to another, and because the velocity of the particle increases, the trajectory is a spiral. Cyclotron is used in radiotherapy to manufacture the artificial radioisotopes (for manufacturing of neutron radiation) and to the accelerating of particles with subsequent use of radiation to treat the malignant tumors. It is also used for the production of radioisotopes in nuclear medicine. For clarity, three pictures of cyclotron will be introduced: The first picture (the description of picture in Czech) was taken from http://fyzika.jreichl.com/main.article/view/859-cyklotron
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The second picture was taken from http://www.fkg-wuerzburg.de/schule/faecher/physik/lk/referate/r12/zyklotr.php
The third picture (the description of picture in Czech) was taken from http://www.pet-spect.fbmi.cvut.cz/pet/index.php/Fotogalerie
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This text also deals with the physical principles of the cyclotron and the aim is to find an equation that describes the motion of particles in a cyclotron. Now it is important to clarify concepts and terms used in this text even before we begin to solve this issue: Magnetic induction B is a vector physical quantity and it describes the magnetic fiel and expresses strength effects of magnetic field on a moving electric charges. Electric charge is used in two senses. Firstly this one can express the state of electrically charged particles or bodies which have electric charge, this charge can be transferred from one body to another. We can briefly say "charge". Second meaning of "electric charge" is physical quantity that characterizes the degree of state electrically charged particles or bodies. Because it is a quantity, electric charge has the unit called coulomb C. We consider an electron in a uniform magnetic field with initial conditions
B 0,0, B , v0 0, v0 ,0 , r0 0,0,0 . Following sketch is a reflection of Fleming's left hand rule. The charge Q operates in axis y and magnetic force Fmg operates in the axis z . If it is valid for electron – its movement is in the opposite direction than the charge Q. The presented sketch
typifies the situation for derivation of radius r by means of steps i) - v):
i) The formula of magnetic force F mg Q v B will be applied for electron in the shape
Fm e v B , Fm e B v .
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ii) After substituting B Bx , By , Bz , v x , y , z Fm e By z Bz y , Bz x Bx z, Bx y By x , Fm e 0, Bz, By , we express the components of the motion equation
0 mx eBz my eBy mz m 2 x 2 z 2 e 2 B 2 y 2 z 2 iii) The equations will be raised to the power of two and added up (the component in axis x can be omitted)
m2 y 2 z 2 e2 B 2 y 2 z 2 and
x
2
z 2 presents acceleration, y 2 z 2 presents speed.
iv) Known formulas will be adjusted and radius r will be sought
v4 2 v4 m r e B v , r v, r 2 , m 2 e2 B 2v r r r 2 2
4
2
2 2
2
v) We calculate the radius using the formula
F
mv 2 , Fm 2 m2an 2 , r
mv0 2 eBv0 where an is centripetal acceleration: r 2
We gain form m2an 2 e2 B2v02 i.e. man eBv0
mv0 2 mv eBv0 and we express the expression for radius r : r 0 r eB The derivation of radius r by means of steps i) - v) was finalized.
Maybe it would be good to explain the concept of centripetal acceleration ad: Centripetal force caused by motion of a body along a curve such as a circle, which is directed to the center of the circle. This force is causing centripetal acceleration. According to the second Newton's law, the following relations can be written:
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v2 Fd mad , ad 2 r r
If the electron was thrown with angles other than 900, it would be a oblique litter with angle α:
v0 v0 cos , v0 sin ,0
m2an 2 e2 B2v02 sin2 , it implicates man eBv0 sin , r
mv0 sin eB
We itemize into individual components x, y, z :
x v0 cos t y r sin t z r cos t v r an 2 r
v0 sin r
It would be a spiral (rather than a circle) and such particle has an unsuitable trajectory. I tried to sketch for an oblique litter:
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C1.5. Application of Implemented Curriculum-2 In the course of gradual application of the curricular process theory the several publications were created in the framework of international conferences OEDM-SERM'11 and OEDM-SERM'12. Publications have dealt with the different variant forms of curriculum. The application of implemented curriculum-2 is connected with creation of education test and with its statistical assessment. The test should reflex the structure of education text „The Physics Base of Cycloctron for Radiology Students“. This education test and its statistical assessment will be presented by means of B.Vesela, 2013b, 2013c. C1.5.1. Education Test After my lecture and the study of students at home my education test was presented to students. This test was reworked in relation to the abilities of addressees. The test looked as follows: 1) What is the difference between linear accelerator and circular accelerator? a) Linear accelerator has a trajectory as a circle, circular accelerator has a trajectory as a line. b) Linear accelerator has a trajectory as a line, circular accelerator has a trajectory as a circle or a spiral but the cyclotron is not circular accelerator. c) Linear accelerator has a trajectory as a line, circular accelerator has a trajectory as a circle or a spiral and the cyclotron is circular accelerator. 2) What kind of accelerator is cyclotron? a) Linear accelerator. b) Cyclic high-frequency accelerator. c) It may be circle and linear accelerator too. 3) Explain the concept of „stationary magnetic field “. a) Stacionary magnetic field is the magnetic field which does not change over time, quantities characterizing the field are constant. b) Stacionary magnetic field is the magnetic field which does not change over time, quantities characterizing the field are not constant. c) Stacionary magnetic field is the magnetic field which changes over time, quantities characterizing the field are not constant. 4) What are „duants“? a) Duants are couples of hollow acceleration electrodes. b) It is particle accelerated in the accelerator. c) Duants are not component of accelerator. 5) Explain the concept of „fundamental particles“. a) Fundamental particles can not transfer to other particles. b) Fundamental particles are elementary particles. c) Fundamental particles are particles withoutinternal structure but particles can transfer to other particles during the collisions.
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6) Why is the trajectory of particles in cyclotron as a spiral? a) Speed of particles decreases. b) Speed of particles does not change. c) Speed of particles increases. 7) What determines the frequency of cyclotron? a) The frequency does not depend on no other quantity. b) Cyclotron frequency (f) is based on size of electrically charged particles (Q), size of magnetic induction (B) and mass of particle (m): f = QB/2πm. c) The frequency does depend only on size of electrically charged particles (Q). 8) How is significant magnetic field for cyclotron? a) Magnetic field curves the trajectory of particles as a circle. b) It accelerates particles. c) Magnetic field is not significant fo cyclotron. 9) Why we use the Fleming´s rule of left arm? a) We use Fleming´s rule to determine the direction of gravitational force. b) We use it to determine the direction of induction lines. c) We use it to determine the direction of magnetic force. 10) How is significant electric field for cyclotron? a) Electric field curves the trajectory of particles. b) It accelerates particles. c) Electric field is not significant fo cyclotron. 11) Where we use cyclotron in the medicine? a) There is cyclotron uses toradiation and treatment of malignant tumors in the radiotherapy. b) There is cyclotron uses not in the medicine. c) There is cyclotron uses in all areas of the medicine. 12) How 3 main parts has cyclotron? a) Waveguide, cavity resonator, acceleration chamber. b) Acceleration chambre, electromagnet, modulator. c) Electromagnet, acceleration chambre, a source of high frequency. 13) Whereaccelerated particles move in cyclotron? a) There is the movement of particles inside of electrodes. b) There is the movement of particles above the electrodes. 14) What powers electrodes? a) Electrodes are not powered. b) Electrodes are powered by high-frequency alternating current. c) Electrodes are powered by low-frequencydirect current. 15) Where exactly is the acceleration of particles in a cyclotron? a) The gap between electrodes. b) Inside of electrodes. c) Throughout the cyclotron.
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16) The formula for centripetal force is: a) Fd m ad ccording to the second Newton's law for a body moving in a circle
v2 then apply Fd m m 2 r r 2 b) Fd m d c) Fd r a d2 17) In the area of circular accelerators is centripetal force expressed: a) F Q v B b) F Q B c) F B v 18) ) In the area of circular accelerators equation has the form: a) Fel mr b) r xi yj zk c) Fmg mi 19) The radius of circular accelerators can be calculated according to the formula: a) r mv 2QB b) QvB
mv 2 mv 2 r r QvB
c) r v 2 Bm 20) For a given type of particle velocity depends on the particle: a) Q b) v c) B C1.5.2. Statistical Assessment of Education Test i) Formulation of statistical investigation random mass phenomenon statistical unit
measuring students' knowledge 1. and 2. the year 2011/2012
student
Value of range of maximum and minimum range of knowledge: statistical 7points - 20 points code statistical code
Range of knowledge Selective statistical file = basic statistical file
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ii) Scaling 57 measurements were made in the form of a test on a scale of 1 to 5 (1 - 11 and less points; 2 - 12, 13; 3 - 14,15; 4 - 16,17; 5 - 18 and more points). Quantitative metric scale, i.e. range is the same. Scaling – see the first column shows it in the table. iii) Measurement Measurement – see the first and the second columns in the table. The table I.:
∑ ni/n
xi.ni
xi2ni xi3ni xi4ni
xi
ni ni/n
1
6
0,105263 0,105263 6
6
6
6
2
9
0,157895 0,263158 18
36
72
144
3
26 0,45614
0,719298 78
234
702
2106
4
11 0,192982 0,912281 44
176
704
2816
5
5
25
125
625
3125
∑
57 1
171
577
2109 8197
0,087719 1,00
The measured values: 1. year – the full-time study (FTS) and the combined study (CS) The measured values: 2. year – the full-time study (FTS) and the combined study (CS)
Number of students Range – 1. year 1 2 3 4 5
FTS
CS
0 1 4 6 0
5 5 6 3 0
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Number of students
Range – 2. year
FTS 1 2 8 1 2
1 2 3 4 5
CS 0 1 8 1 3
iv) Elementary statistical processing Students of 1. and 2. year together a) The table – see the table I. b) Empirical distribution of frequency
graph of distribution of absolute frequency 30 20 10 0 1
2
3
4
5
elements of range(xi)
graph of distribution of relative frequency 0,5 0,4 0,3 0,2 0,1 0 1
2
3 4 5 elements of range(xi)
6
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graph of distribution of cumulative frequency 1,5 1 0,5 0 1
2
3
4
5
elements of range(xi)
-
The parameter of position (general moment of 1. order O1): O1 = 171/57 = 3,00
-
The parameter of variability (calculation of central moment of 2. order C2): C2 = O2 - O12 O2 = ∑ni/n.xi2
O2 = 577/57
O2 = 10,122807 C2 = 10,122807 - 32 C2 = 1,122807 Sx=√C2 Sx=1,06 (standard deviation) -
The parameter of skewness (calculation of normal moment of 3. order N3): O3 = ni/n.xi3 O3 = 2109/57 O3 = 37 C3 = O3 - 3O2O1 + 2O13 C3 = 37 - 3.10,122807.3 + 2.33 C3 = -0,11 N3 = -0,11/1,122807.(O2 - O12) 1/2 N3 = -0,11/1,1897554
-
N3 = -0,11/ 1,122807.1,0596259
N3 = -0,09
The parameter of kurtosis (calculation of normal moment of 4. order N4): N4 = C4/C22 O4 = ni/n.xi4
O4 = 8197/57
O4 = 143,80702 C4 = O4 - 4O3O1 + 6O2O12 - 3O14 C4 = 143,80702 - 4.37.3 + 6.10,122807.32 - 3.34 C4 = 3,44 N4 = 3,44/1,1228072
N4 = 2,73
N4 ~ 3 the optimal Gaussian curve
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v) Normal distribution – Gaussian curve 30 25
20 15 10 5 0 0
1
2
3
4
5
6
vi) The non-parametric test Interval division of frequency xi
ni ni/n
∑ ni/n
xi.ni xi2ni xi3ni xi4ni
(-∞;1,5> 6 0,105263 0,105263 6
6
6
6
(1,5;2,5> 9 0,157895 0,263158 18
36
72
144
(2,5;3,5> 26 0,45614 0,719298 78
234 702 2106
(3,5;4,5> 11 0,192982 0,912281 44
410 704 2816
(4,5;∞)
125 625 3125
∑
5 0,087719 1,00
25
57 1
171 811 2109 8197
Expression of p1 to p5 surfaces using the distribution function F(x) (when choosing the five elements range): p1 = F (1,5) p2 = F (2,5) – F (1,5) p3 = F (3,5) – F (2,5) p4 = F (4,5) – F (3,5) p5 = F (∞) – F (4,5)
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Transformation of normal distribution to standardized normal distribution u=(x – O1)/Sx, UL – upper limit of interval u1=(x1,UL – O1)/Sx u2=(x2,UL – O1)/Sx u3=(x3,UL – O1)/Sx u4=(x4,UL – O1)/Sx u5=(x5,UL – O1)/Sx
χ2- test of good conformity: 2 χ exp
ni npi 2 npi
xi
interval ni
ui
F(ui)
pi
npi
1
(-∞;1,5> 6
-1,4
0,92
0,08
4,56
2
(1,5;2,5> 9
-0,47
0,32
0,24
13,68
3
(2,5;3,5> 26
0,47
0,68
0,36
20,52
4
(3,5;4,5> 11
1,4
0,92
0,24
13,68
5
(4,5;∞)
∞
1
0,08
4,56
5
n i npi 2 npi 1
0,45
2
1,6
3
1,46
4
0,52
5
0,04
χ square
4,07
Critical theoretical value and degrees of freedom:
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α = 0,05
ν=2 2 χ 2teor χ k-r-1 α 0, 05 χ 22 α 0, 05 5,99
Right-hand critical field W:
W χ 22 0, 05 ; 5,99; 2 χ exp W
We can accept zero hypothesis H0. The empirical distribution can be substituted by theoretical normal distribution
vii) Parametric testing – two-sided t-test: comparison the full-time study and the combined study: The full-time study (FTS) xi
ni
xini
xi2ni
1
1
1
1
2
3
6
12
3
12
36
108
4
7
28
112
5
2
10
50
∑
25
81
283
O1,FTS = 81/25 = 3,24 O2,FTS = 283/25 = 11,32 C2,FTS = 11,32– (3,24)2=11,32–10,50 = 0,82 Sx,FTS = 0,91
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The combined study (CS) xi
ni
xini
xi2ni
1
5
5
5
2
6
12
24
3
14
42
126
4
4
16
64
5
3
15
75
∑
32
90
294
O1,CS = 90/32 = 2,81 O2,CS = 294/32 = 9,19 C2,CS = 9,19 - (2,81)2= 11,32 - 7,896 = 1,29 Sx,CS = 1,14
Two-sided t-test will be used in the form (index FTS – full-time study, index CS – combined study) t exp
O1,FTS O1,CS
n FTS n CS n FTS n CS 2
n FTS 1 Sx,FTS2 n CS 1 Sx,CS2
n FTS n CS
2 t
W (; t n1 +n 2 -2 α
n1 +n 2 -2
,
α 2 ; )
After application of the respective values we can get the value texp: texp =
25.32 57 −2
3,24 −2,81
57
25 −1 0,912 + 32 −1 1,14 2
texp =
0,43 24.0,82+31.1,29
800.55 57
texp = 1,33 W = −∞; −1,96 ∪ 1,96; ∞
The value texp is not ielement of critical field W, we can accept zero hypothesis H0 → between full-time and combined study is not the difference for the significance level α = 0,05
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C1.6. Conclusion The algorithm of curricular process is given by the arranged succession of curriculum variant forms in the shape (see Chapter C1.1.) i) Conceptual curriculum ii) Intended curriculum iii) Projected curriculum iv) Implemented curriculum-1 v) Implemented curriculum-2 vi) Attained curriculum The application of curriculum process algorithm was investigated on the basis of the concrete problem solving (see Chapter C1.2.): The investigation of physics principles of classical circular accelerators for radiology students. The solution of this problem was connected with the verification of two hypotheses: H1 An adequate educational text in the sphere of physical bases of circular accelerators can be created by application of curriculum process H2 Students´ knowledge in the sphere of physical bases of circular accelerators acquired on the base of worked out educational text will have normal distribution. The verification process worked on gradual application of the individual curriculum variant forms (the sequence of these individual curriculum variant forms creates the curricular process). The application of conceptual curriculum was connected with description of scientific system of radiological physics in the area of classical circular accelerators (see Chapter C1.3). The application of intended and projected curriculum was connected with creation of education text „The physics Base of Cycloctron for Radiology Students“ (see Chapter C1.4). The application of implemented curriculum-2 was connected with creation of education test (see Chapter C1.4.1.). The application of implemented curriculum-2 was connected also with statistical assessment of education test (see Chapter C1.4.2.). On the basis of application of the curriculum process algorithm was verified not only hypothesis H1 (the applicability of curriculum process algorithm was substantiated) but also hypothesis H2 (the concrete research was successfully finalized). In addition, two-sided t-test pointed at approximately the same educational level of full-time and combined study. This discovery indicates the good parameters of accreditation materials which were approved by the Accreditation Commission for students of radiology in the Czech Republic.
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C1.7. Literature Zaskodny,P. (2006) Survey of Principles of Theoretical Physics (with Application to Radiology) Ostrava, Czech Republic, Luzern, Switzerland: Algoritmus, Avenira ISBN 80-902491-9-1 Zaskodny,P. (2009) Curricular Process of Physics Ostrava, Czech Republic, Luzern, Switzerland: Algoritmus, Avenira ISBN 978-80-902491-0-3. Vesela,B. (2013a) Základy fyziky klasických kruhových urychlovačů pro radiologické asistenty (Bachelor thesis) České Budějovice: University of South Bohemia Vesela,B. (2012) How to Explain Physics Base of Cyclotron for Radiology Students In Proceedings of OEDM-SERM´11 (The 1st International e-Conference on Optimization, education and Data Mining in Science, Engineering and Risk Management, Bratislava, Slovakia) Praha, Czech Republic: Curriculum 2012 ISBN 978-80-904948-1-7 Vesela,B. (2013b) How to Explain Physics Base of Cyclotron for Radiology Students Part II In Proceedings of OEDM-SERM´12 (The 2nd International e-Conference on Optimization, Education and Data Mining in Science, Engineering and Risk Management, Bratislava, Slovakia) Praha, Czech Republic: Curriculum 2013 ISBN 978-80-904948-4-8 Vesela,B. (2013c) How to Explain Physics Base of Cyclotron for Radiology Students Part III In Proceedings of OEDM-SERM´12 (The 2nd International e-Conference on Optimization, Education and Data Mining in Science, Engineering and Risk Management, Bratislava, Slovakia) Praha, Czech Republic: Curriculum 2013 ISBN 978-80-904948-4-8
Tarabek,P., Zaskodny,P. (2011) Educational and Didactic Communication 2010 Bratislava, Slovak Republic: Didaktis. ISBN 978-80-89160-78-5
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C2. Curricular Process of Radiological Physics within Higher Education Level
Author Miroslava Bartonova University of South Bohemia [email protected]
Abstract The work “Curricular Process of Radiological Physics within Higher Education Level” presents the attempt to apply the universal theory of curricular process of physics (sequence of individual variant forms of curriculum) specifically for radiological physics and also specifically for higher education (University) level. Defined curricular process expresses the sequence of individual curriculum variant forms – the specific algorithm of curricular process. The conceptual curriculum is connected with analysis of Universities education in area of radiology and radiological physics The intended curriculum is connected with comparison of Universities education in area of radiology and radiological physics The projected curriculum is connected with selection of University education in area of radiology and radiological physics and with illustration through concrete textbook and expert book. The implemented curriculum-1 is connected with the description of preparation of concrete University research in the area of radiological physics. It is necessary to remind – the implemented curriculum-1 can be also connected with the preparation for University education. The implemented curriculum-2 is connected with the description of results of presented University research in the area of radiological physics. It is necessary to remind – the implemented curriculum-2 can be also connected with the description of results within University education. Attained curriculum is connected with applicability of gained results of University research in the area of radiological physics. It is necessary to remind – the attained curriculum can be also connected with an applicability of acquired results of University education. In the framework of presented work the research “Comparison of knowledge of laic and expert public in the area of radiological physics” was used for the demonstration of individual variant forms of curricular process of radiological physics within higher education level.
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Key Words Universal curricular process of physics, Variant form of curriculum, Curricular process of radiological physics, Higher education level, Comparison of knowledge, Laic public, Expert public, Conceptual curriculum, Intended curriculum, Projected curriculum, Implemented curriculum-1, Implemented curriculum-2, Attained curriculum
C2.1. Universal Curricular Process of Physics – Universal Algorithm The result data preprocessing within educational data mining approach in the area of physics education can be described by following way (see P.Tarábek, P.Záškodný, 2008): • The transformation T1 is transformation of scientific system of physics to communicable scientific system of physics (association with the first variant form of curriculum), • The transformation T2 is transformation of communicable scientific system of physics to educational system of physics (association with the second variant form of curriculum), • The transformation T3 is transformation of educational system of physics to both instructional project of physics and preparedness of educator to education (association the third and fourth variant form of curriculum), • The transformation T4 is transformation of both instructional project of physics and preparedness of educator to results of education (association with the fifth variant form of curriculum), • The transformation T5 is transformation of results of physics education to applicable results of physics education (association with the sixth variant form of curriculum)
The result data processing (as curricular process of physics) within educational data mining approach in the area of physics education can be described by following way (see Tarábek,P, Záškodný,P. (2008)): conceptual curriculum as the first variant form of curriculum (output of T1, the first form of education content existence) - the communicable scientific systém intended curriculum as the second variant form of curriculum (output of T2, the second form of education content existence) - the educational system of natural science
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projected curriculum as the third variant form of curriculum (output of T3, the third form of education content existence) - the instructional project of natural science implemented curriculum-1 as the fourth variant form of curriculum (output of T3, the fourth form of education content existence) - the preparedness of educator to education implemented curriculum-2 as the fifth variant form of curriculum (output of T4, the fifth form of education content existence) – the results of education attained curriculum as the sixth variant form of curriculum (output of T5, the sixth form of education content existence) - applicable results of education
C2.2. Application of Radiological Physics Curricular Process to Higher Education Level – Specific Algorithm of Curricular Process Within higher education level the curriculum variant forms can be briefly described by following way (the description of specific algorithm of curricular process): Conceptual curriculum – Analysis of Universities education in area of radiology and radiological physics Intended curriculum – Comparison of Universities education in area of radiology and radiological physics Projected curriculum – Selection of University education in area of radiology and radiological physics Implemented curriculum-1 – Preparation for University education or University research in the area of radiological physics Implemented curriculum-2 – Results of University education or University research in the area of radiological physics Attained curriculum – Applicability of determined results of University education or University research in the area of radiological physics The more detailed description of radiological physics curricular process (for University education) can be realized by means of following additional information: Conceptual curriculum – conception of what is to be a radiological physics content of University education, it is also a synonym of transmissibility of scientific system. This form can be connected with T1 transformation. Conceptual curriculum is described by an analysis of Universities education. Intended curriculum – the planned purposes and education contents with the clear definition for university students through the adjustment of conceptual curriculum. This form can be connected with T2 transformation. Intended curriculum can be identified by comparison of Universities education in the area of curriculum documents (syllabi, textbooks, expert books)
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Projected curriculum and implemented curriculum-1 – selection and preparation of University education can be given by means of choosen textbook or expert book in the area of radiological physics or by means of preparation way for successful radiological physics education. It is project and realization form which can be connected with results of T3 transformation. The selection of University education can be given by means of choosen textbook or expert book in the area of radiological physics. The concrete textbook or expert book is creating the basic element of projected curriculum. The basic result of T3 transformation - „projected curriculum“ (this one finds itself in well written textbook or expert book) can be extended by a new variant form of curriculum “implemented curriculum-1” associated with pedagogical work of University teacher in the area of radiological physics. Ergo „implemented curriculum-1“is distinguishable from a form of implemented curriculum-2 which is connected with the reached results of University education. Preparation of higher education teacher for communication with students of radiological physics can be associated with preparation of researcher for concrete problem solving in the area of radiological physics (e.g. comparison of knowledge of laical and expert community in the area of radiological physics, such comparison may be expressed by means of the formulated hypotheses) Implemented curriculum-2 – the results of University education or University research in the area of radiological physics find in student´s mind or in the conclusions of realized research. This form can be connected with T4 transformation. The results of higher education process in the area of radiological physics ocan be again associated with the results of problem solved in the area of radiological physics (e.g. the results of knowledge comparison of laical and expert community in the area of radiological physics, such results should come into being by means of the verification of formulated hypotheses) Attained curriculum – the applicability of determined results of University education or University research (in the area of radiological physics) represents form of developed conceptual knowledge systems acquired by addressees of University education on the basis of their personal and out-of-school experiences and interests. It guides not only absolvents of radiological physics education during their profession career but it also „lives“ in all society. This form can be connected with T5 transformation.
C2.3. Conceptual Curriculum of Radiological Physics for Higher Education Level The conceptual curriculum will be described by the analysis of Universities education in the area of radiology and radiological physics.
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Education in Czech republic – Radiological assistant Law no. 96/2004 Sb., about non-medical professions says: Professional qualification to exercise a profession of radiological assistant is gained through graduation in accredited medical bachelor study programme for radiological assistants´ preparation three-year study in field of radiologic assistant with a diploma on college (not an university!), if first semester of studies began at the least at school year 2004/2005, or secondary medical school in field of radiologic laboratory technician, if first semester of studies began at the least at school year 1996/1997. Reference: Zákon č. 96/2004 Sb. o podmínkách získávání a uznávání způsobilosti k výkonu nelékařských zdravotnických povolání a k výkonu činností souvisejících s poskytováním zdravotní péče a o změně některých souvisejících zákonů (zákon o nelékařských zdravotnických povoláních) - §8 Odborná způsobilost k výkonu povolání radiologického asistenta
In the framework of education in Czech republic is possible to reach position of radiological assistant or radiologic laboratory technician only through graduation of course Radiological assistant with bachelor degree in present time. Mentioned branch can be studied at following czech universities: Charles University in Prague Czech Technical University in Prague Masaryk University in Brno University of West Bohemia in Pilsen University of South Bohemia in České Budějovice Palacký University Olomouc University of Ostrava At all mentioned universities is radiological physics education for branch of radiological assistant set for two semesters. Education in abroad – Radiologic Technologist Education is slightly different all over the world. To entrance to the radiologic technologies study programme it´s necessary to pass high school, which is finished with graduation. Also the fulfilment of entering admission requirement and rightdown criminal record is need. Official education programs in radiologic sciences lead to obtaining assistancy certificate or bachelor grade. International tendency of safe care for patients prefers bachelor titles, ergo higher education, that´s why the assistentship programmes are retreated. In many countries master degrees are offered too. Educational curriculum is basically worldwide similar. Usually, during official education, students have to gain knowledge from human anatomy and physiology, general, nuclear and radiation physics sciences, general chemistry, mathematics, radiopharmacology, pathology, biology, medical imaging and diagnostics, radiologic instrumentation, medical first aid treatment, nursing, medical ethics, technologies, computer programming and management.
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USA Radiographic examination are usually conducted by radiologic technologists, who undergo two-year education for assistant title or four-year education for BSc. title in USA. Europe Radiologic technologists´ education is avalilable in various forms in 30 countries from total number 46 European countries. It´s possible to study two, three or four-year programme. Students follow basic educational courses of diagnostic imaging, radiotherapy, nuclear medicine and ultrasound – in any combination of these branches - during the study time. E.g. in Czech republic students learn all of mentioned sights, but e.g. in Iceland educational sylabus includes diagnostic imaging and nuclear medicine only. Great Britain Radiologic technologists are in United Kingdom known as „Radiographers“. Secondary terms „Diagnostic Radiographer“ and „Therapy Radiographer“ are in Great Britain protected titles so they cannot be used by people who didn´t undergo official education or people who aren´t registered by Health Professionals Council. Absolvents of education, who got university title BSc. in diagnostic imaging, must be registered by Health Professions Council before the begining of operating medical exposure or imaging in medical institute. Diplomas are offered in the framework of universities and their obtaining lasts 3 years in England and Wales, 4 years it lasts in Scotland. Student, future Radiologic technologist, has to spend a lot of time working in college hospital during his or her studies (form of clinical training). As soon as students obtain qualification, they are able to practise radiologic imaging and they can work as a part of diagnostic team. Radiologic technologists can get postgraduate specializations in CT, MRI, ultrasound or nuclear medicine with possibility to obtain master degree (MSc.) in especial field during home study or university courses – both are finished with exams. Radiologic technologists in Great Britain participate on working in tasks, which where performed only by Radiologists in past as well (Radiologist – medicine doctor, who specializes in interpretation of images, which were scanned by any radiologic method). Absolvent may write reports and diagnose pathologies, which are clear at images and he or she can consults them with physicians after graduation and registration by Health Professions Council and Society of Radiology. Australia Australian universities students decide about their specialization when they entry the study programme. Like in other countries is this specialization concerned to three basic radiology fields – nuclear medicine, radiotherapy, or diagnostic imaging. Additional is possibility to enroll for course to study diagnostic ultrasound. All these specializations come under the category Medical Radiation Sciences. Courses which come under these science are accredited by Australian Institute of Radiography (AIR). In some cases absolvents have to
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undergo one year of paid structured clinical practice training after finishing academic courses – National Professional Development Programme (NPDP). When is this practice training finished, graduates are qualified for getting an accreditation. That must be made in Australia in authorized radiologic institute. Education of medical imaging, radiotherapy and nuclear medicine can subsist: - graduation in three-year bachelor study programme with NPDP - graduation in four-year bachelor study programme - graduation in two-year master study programme with NPDP - graduation in two-year master study programme Yearlong practice and inscription in official NPDP includes official registration of radiologic technologist and access to informations of AIR also. Study is possible for expamle at following Australian universities: The University of Newcastle Charles Sturt University RMIT University University of South Australia University of Tasmania
It is also possible to study ultrasound, as the imaging method, but in Australia only as a postgraduate qualification. To entry this programme student has to have qualification in field of radiology imaging, radiotherapy, biomedicine science or nursing. Plenty of universities offer ultrasound education as a part of diagnostic radiography studies. Absolvent of course ultrasound imaging gains the Diploma of Medical Ultrasound from Australian Society for Ultrasound. Specialists, who work in this specialization after graduation in any study programme, are called Diagnostic Radiographers or Medical Imaging Technologists, Radiation Therapists and Ultrasonographer or Sonographer. Lenght of radiological physics education is different. It depends on the total lenght of study programme. In case of four-year programme physics is taught for four semesters, two semesters in case of three-year studies.
C2.4. Intended Curriculum of Radiological Physics for Higher Education Level The intended curriculum will be described by the comparison of Universities education in the area of radiology and radiological physics. It can be expressed by means of
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investigation of position of radiological physics in preparation of radiological assistants or technologists at Czech and abroad universities. In Czech Republic is a graduate of Application ionizing radiation studies called Radiological assistant. Abroad are graduates of this programme called Radiologic Technologists, Diagnostic Radiographers Medical Imaging Technologists, Therapy Radiographers or Radiation Therapists, Ultrasonographers or Sonographers. General physical base in curriculum essentially matches foreign study structures. Dissimilarities can be found in structure of physical base components in profile of radiological assistant. This diference is obviously connected with possibilities of narrowed study specialization (e.g. Medical Imaging Technologist/Therapy Radiographer). On the basis of the present state investigation of knowledge from radiological physics was shown, that contents of radiological physics courses at Czech and abroad universities are similar. Only specializations variety (various graduates´ titles in preparation of radiological assistants in abroad) reflects on richer spectrum of orientation in abroad curriculum. It´s also possible to speculate about considerable society weigh of e.g. „Radiologic Technologist“ in abroad. Analysis was done for three universities in USA, five Australian universities and four universities in Great Britain. In all of these cases were found study programs which are related to Application ionizing radiation in medicine. Evidentally abroad study programs related with Czech preparation of radiological assistants are rather called Application ionizing radiation.
C2.5. Projected Curriculum of Radiological Physics for Higher Education Level Projected curriculum is coming into being through selection of University education in area of radiology and radiological physics. The selection of University education can be given by means of choosen textbook or expert book in the area of radiological physics. The concrete textbook or expert book is creating the basic element of projected curriculum. The example of selected textbook and expert book can be demonstrated by books P.Zaskodny, 2006, 2009. The radiological physics is in P.Zaskodny (2006) structured through “Ionizing Radiation and its Applications in Radiology” – this part of mentioned book is segmented into following Chapters 13-18 (previous Chapters 1-17 are connected with explanation of Statistical and Non-Statistical Physics on the basis of their classical, relativistic and quantum dimension):
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13. Ionization of Medium, Ionization Tools and Methods, Ionizing Radiation 13.1. Process of Ionization of Medium 13.2. Outline of Ionizing and Non-Ionizing Types of Radiation and Wave Motion Employed in Radiology
14. Ionizing Radiation Sources and Interactions of Ionizing Radiation with Medium 14.1. Description of Sources 14.2. Description of Interactions
15. Measurement of Ionizing Radiation 15.1. Quantities and Units in Field of Ionizing Radiation 15.2. Ionizing Radiation Detection and Dosimetry
16. Physical Description of Radiodiagnostics and Radiotherapy 16.1. Description of Radiodiagnostics 16.2. Description of Radiotherapy
17. Physical Description of Imaging Techniques 17.1. Nuclear Medicine 17.2. Roentgenology 17.3. Thermography 17.4. Nuclear Magnetic Resonance 17.5. Sonography
18. Physical Principles of Magnetic Resonance 18.1. Physical Foundations 18.2. Physical Basis of Techniques Used
The presented book is also refilled by the supplements required for understanding radiological physics. The survey of supplements is possible briefly to present: Supplement 1 Supplement 2 Supplement 3 Supplement 4 Supplement 5 Supplement 6
Knowledge of Mathematics Required for Application to Radiology Practising Statistical Physics Examples - Statistical Physics Practising Non-Statistical Physics Examples - Non-Statistical Physics Practising Ionizing Radiation and its Applications in Radiology
An illustration of presented book will be given table which is presented within Chapter 13.1. Process of Ionization of Medium. This table was created for a schematic outline of types of radiation and wave motion used in radiology – see table below:
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a) Gamma radiation (indirectly ionizing radiation, ionization by photon absorption) Wave length: 10-13 – 10-12 m Frequency: 1020 1021 Hz Natural source: Transitions in atomic nuclei Artificial source: Accelerators, radio-nuclides Detection: Gas, spark, scintillation detectors Field of radiology: Nuclear medicine b) X-rays (indirectly ionizing radiation, ionization by photon absorption) Wave length: 10-10 m Frequency: 1018 Hz Natural source: Transitions in electron envelope of atom Artificial source: X-ray tube Detection: Gas, crystal, scintillation, photochemical detectors Field of radiology: X-ray diagnostics (X-ray apparatus, computerized tomography) X-ray therapy c) Infrared radiation (non-ionizing radiation) Wave length: 10-5 m Frequency: 1013 Hz Natural source: Vibration and rotation of molecules Artificial source: Bodies with temperatures higher than 0 K Detection: Radiothermometers, thermographic cameras Field of radiology: Thermography d) Radio waves (non-ionizing radiation) Wave length: 100 – 104 m Frequency: 104 – 108 Hz Natural source: Motion of almost free electrons Artificial source: Transmitter of high-frequency electromagnetic signals Detection: Receiver of high-frequency electromagnetic signal Field of radiology: Nuclear magnetic resonance e) Ultrasound waves (mechanical wave motion) Wave length: 10-3 m Frequency: 106 107 Hz Natural source: Vibrations of bodies Artificial source: Magnetostriction and piezoelectric oscillator Detection: Magnetostriction and piezoelectric oscillator Field of radiology: Sonography f) Corpuscular radiation (directly and indirectly ionizing radiation, impact ionization) Wave length: de Broglie wave length = h / mv Frequency: = mc2 / h Natural source: Natural radioactive nuclides Artificial source: Artificial radioactive nuclides, accelerators Detection: Gas, spark, crystal and scintillation detectors Field of radiology: Nuclear medicine, radiotherapy
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C2.6. Implemented Curriculum-1 of Radiological Physics for Higher Education Level Implemented curriculum-1 is connected with preparation for University education or University research in the area of radiological physics. Preparation of higher education teacher for communication with students of radiological physics is often associated with preparation of researcher for concrete problem solving in the area of radiological physics. Demonstration of implemented curriculum-1 will be realized by means of preparation of concrete problem research which is connected with comparison of knowledge of laic and expert public in the area of radiological physics, such comparison will be expressed by means of the formulated hypotheses. Hypotheses To accomplish the fundamental goal of the research (see M.Bartonova, 2011, 2013a, 2013b) was needed to create a structure of radiological physics model in education to radiological assistants. Creation of this model questionnaire was then used on respondents of the laic and expert public. It was verified as a normal existence of knowledge at the laic public while the existence of the knowledge at expert public was Poisson’s separation. There was measured the difference between the laics’ and the experts’ knowledge. Essential condition for realization and description of the procedure was the analysis of radiological assistants’ preparation in the Czech Republic and abroad. To reach the fundamental goal were set up three hypotheses: H1. Theoretical division of knowledge at the laic public will be closer to normal distribution. H2. Theoretical division of knowledge at the expert public will not have normal distribution. H3. Compare of knowledge at both public with the help of parametrical tests will lead to acceptation of alternative hypothesis. The entire three hypotheses were checked and positively accepted. For their verification it was mainly used, testing non-parametrical and parametrical hypotheses. The finding that the radiological physics structure for radiological assistants complies with the radiological physics structure used abroad for preparation of experts like radiological technologists, diagnostic radiographers, the medical imaging technologists, the therapy radiographers or the radiation therapists, was a good solution. This discovery was primarily found out from the analysis of curriculum at universities in the USA, Great Britain and Australian universities. From this point the validity of this constructive questionnaire rose. It needs to be mentioned that this questionnaire was made in terms to prepare radiological assistants in which the radiological physics has an only supportive role.
Methodology Data to process thesis were gained by using quantitative research methods. Questionnaire construction was effected to reflection of physics structure. Questionnaire was created with form of alternative test consisting from 20 questions. Expert community was chosen in
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cooperation with Society of radiological assistants in Czech republic. This society organized radiological physics courses for its radiological assistants. Sent away checklists were filled by 54 respondents. Laical community was not chosen by random selection. Questionnaire was given to unrepresentative sample of 50 respondents from thesis author´s social surroundings. Qualified hypotheses H1, H2 and H3 will be prove through descriptive and mathematical statistics methods. From descriptive stastistics methods the following steps were effected: scaling, measuring and elementary statistical processing. From mathematical statistics methods they were used non-parametrical testing ( χ2 test of fit) and parametrical testing (twosample parametric testing). Questionnaire construction and its processing Questionnaire created by author reflects structure of chosen questions from radiological physics, which is needed for university preparation of radiological assistants. Constructed questionnaire then is not structural reflection special radiological physics. It is reflection of conceptual and intended curriculum of radiological physics, which is needed for radiological assistants´ university education. Same questionnaire was given to laical and expert community respondents. Comparison of knowledge from radiological physics of these two groups will be effected through parametric two-sample t-test.
C2.7. Implemented Curriculum-2 of Radiological Physics for Higher Education Level Implemented curriculum-2 is given by the results of University education or University research in the area of radiological physics. Demonstration of implemented curriculum-2 will be realized by means of the presentation of results within research “Comparison of knowledge of laic and expert public in the area of radiological physics”. These results will be connected with the verification and discussion of formulated hypotheses. Verification and discussion of hypotheses H1 and H2 Hypothesis H1 (knowledge from radiological physics at laical community have normal distribution) was proved and accepted. Receiving of this hypothesis means, that there is central number of mistakes from 20 possible mistakes which has the biggest probability at the laical respondents. Counts of mistakes which are lesser and bigger than is this central number of mistakes fall to both sides from middle number with Gauss reductive probability. At transmission of 1st order general moment as an arithmetic mean from elements scale to statistical sign value, this central number of mistakes is 11 from 20 possible mistakes at moderate laical respondent. Appearance to that the questionnaire for knowledge from radiological physics research was at laical community constructed from aspects of radiological assistants´ preparation (not from aspects of elementary radiological physics education) and so it was possible to expect
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using of some general educational knowledge at laical respondents. This result is relatively good station of laical community. Regarding to analysis of separate questions it is evident, that the knowledge about fundamental particles and radiation does exist at laical community. Hypothesis H2 (knowledge from radiological physics at expert community haven´t normal distribution) was proved and accepted, even in two ways. Normality testing led to negative conclusion (too great number of respondents had only small mistake counts) so it had to be replaced by non-parametrical test for Poisson distribution. This transition was useful – Poisson distribution as “rare cases distribution“ was positively proved. Bigger mistakes counts at professional radiological assistants, who pass courses of Society of radiological assistants, were really only rare cases. Receiving hypothesis H2 means, that there is a central number of mistakes from 20 possible mistakes at the expert respondents, which has the biggest probability. The count of mistakes which were bigger than central number of mistakes falls from middle number with Poisson reductive probability. At transmission of 1st order general moment as an arithmetic mean from elements scale to statistical sign value, this central number of mistakes is 3 from 20 possible mistakes at moderate expert respondent. Professional radiological assistants proved very good knowledge especially at questions dealing with physical principle of separate components radiological assistant´s profile – courses ordered by Society of radiological assistants brought good results. This result shows that the professional radiological assistants erred almost 4 times less than laical respondents at average. Relative stability of professional respondents´ knowledge agree with small standard deviation, which reached less than 40% of weighted arithmetic mean (1st order general moment). Analogous standard deviation appeared at average laical respondent also, but here it´s indeed need to count from average mistakes count 11. Discussion bound to hypotheses H1 and H2 can also be confirmed by analysis of obliqueness parameters´ empirical values (3rd order normed moments). At laical community is obliqueness parameter of knowledge from radiological physics nearly equal with zero value. At radiological assistants´ knowledge it´s value is 0,5 – it means left bevelling. Lower scale elements (i.e. scale elements with small mistake counts) have essentially higher frequency (i.e. respondents´ counts) than higher scale elements. Verification and discussion of hypothesis H3 Hypothesis H3 researched comparison of knowledge from radiological physics at laical and expert community. It showed that the experimental value of used test standard (twosample t-test) lies deep inside critical line. Value of experimental test standard (7,9) was approximately 4 times greater than critical testing standard (1,96). Thereby was hypothesis H3 proved and accepted – at statistical relevancy 0,5 are professional radiological assistants´ knowledge essentially higher than knowledge of laical community.
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This conclusion accord with findings which were reached at verification of hypotheses H1 and H2 – professional radiological assistants erred almost 4 times less at average than laical respondents. Because the questionnaire extracted from general physics base part and part which concerned to physical components of radiological assistant´s profile, it would be good to perform separate parametric testing of these two parts in future.
C2.8. Attained Curriculum of Radiological Physics for Higher Education Level Attained curriculum is linked with applicability of delimited results of University education or University research in the area of radiological physics. In presented case with delimited result of University research “Comparison of knowledge of laic and expert public in the area of radiological physics”. Totally it is possible to state that hypotheses H1, H2 and H3 were proved and accepted. Knowledge of laical community from radiological physics have normal distribution, knowledge of professional radiological assistants from radiological physics have Poisson distribution. Knowledge of professional radiological assistants and laical respondents can be characterized by number 4 approximately – mistakes counts in questionnaire investigation were 4 times higher at average laical respondent than at average professional radiological assistant. Maximal count of mistakes was 20. Number 4 resulted from non-parametric testing verification of hypotheses H1 and H2, and from parametric testing of hypothesis H3 also. Goals of research can be considered as fulfilled. In this research were used data mining models of components radiological assistants´ studies as well as data mining models of physical structure as a whole (recensed by many Czech and abroad reviewers within book P.Zaskodny, 2006, 2009). These models enabled to find a place of radiological physics as a whole and in framework of radiological assistants´ preparation also. Through these models it was possible to define elements of radiological physics in framework of radiological assistants´ preparation and to construct the questionnaire. When the questionnaire was given to respondents and when it was statistically analysed, then the knowledge quantification of laical and expert community was possible as well as their comparison.
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C2.9. Conclusion Within the delimitation of individual curriculum variant forms in the course of preceding chapters C2.1. – C2.8. it is needful to remind the basic applicable result of work “Curricular Process of Radiological Physics within Higher Education Level”. This applicable result was created through the transformation of universal curricular process to the specific curricular process of radiological physics. Such specific form of curricular process represents the algorithm of curricular process within University level. The algorithm of curricular process within University level will be now reminded by means of individual stages (without association with radiology and radiological physics): i) Conceptual curriculum – Analysis of Universities education in the area of relevant scientific branch ii) Intended curriculum – Comparison of Universities education in the area of relevant scientific branch iii) Projected curriculum – Selection of University education in the area of relevant scientific branch iv) Implemented curriculum-1 – Preparation for University education or University research in the area of relevant scientific branch v) Implemented curriculum-2 – Results of University education or University research in the area of relevant scientific branch vi) Attained curriculum – Applicability of determined results of University education or University research in the area of relevant scientific branch The order of presented stages of the algorithm shoud be kept in the framework of higher education level.
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C2.10. Literature Tarábek,P. Záškodný,P. (2008) Educational and Didactic Communication 2007, Vol.3 – Applications Bratislava, Slovak Republic: Didaktis ISBN 987-80-89160-56-3 Záškodný,P. (2006) Survey of Principles of Theoretical Physics (with Application to Radiology) (in English) Lucerne, Switzerland, Ostrava, Czech Republic: Avenira, Algoritmus ISBN 80-902491-9-1 Záškodný,P. (2009) Curicular Process of Physics (with Survey of Principles of Theoretical Physics) (in Czech) Lucerne, Switzerland, Ostrava, Czech Republic: Avenira, Algoritmus ISBN 978-80-902491-0-3 Bartoňová,M. (2012) Comparison of Knowledge from Radiological Physics at Laical and Expert Community (in Czech) Bachelor diploma thesis University of South Bohemia, Faculty of Health and Social Studies, Czech Republic Bartoňová,M. (2013a) Comparison of Knowledge from Radiological Physics at Laical and Expert Community (in English) In Proceedings OEDM-SERM 2012 ISBN 978-80-904948-1-8 Bartoňová,M. (2013b) Analysis of Imaging Methods Studies for Universities in Czech Republic, USA, Great Britain and Australia (in English) In Proceedings OEDM-SERM 2012 ISBN 978-80-904948-1-8
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PART D The Algorithm of Problem Solving in Option Pricing and Hedging
Author of Abstract and Orientations Premysl Zaskodny University of South Bohemia, University of Finance and Administration [email protected]
Authors of Chapters (according to order of the single chapters) Vladislav Pavlat University of Finance and Administration, Curriculum Studies Research Group [email protected] Premysl Zaskodny, Ivan Havlicek University of South Bohemia, University of Finance and Administration [email protected] [email protected] Martin Pasta University of Finance and Administration [email protected] Vladimir Risky University of Finance and Administration [email protected] Martin Soucek University of Finance and Administration [email protected] Miroslav Sebest University of Finance and Administration [email protected]
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Abstract The problem of option pricing and option hedging is solved primarily using theoretical studies on the basis of financial markets regularities. The reached results of theoretical studies are given from the point of view of financial mathematics by continuous and discrete models (Black-Scholes model and its innovation, Figlewski model, multinomial models). Secondarily, the reached results of problem solving should be described through suitable algorithms (see also Part A “The Algorithm of Problem Solving in Data Mining Approach”, Chapter A9). And the finalization of option pricing solution and option hedging solution should be tertiarily connected with software creation. This succession of primary, secondary and tertiary phase enables to formulate the main goals of Part D “The Algorithm of Problem Solving in Option Pricing and Hedging”. The formalization of goals is associated with three goals. The first goal of Part D is to validate the hypotheses that it is possible to obtain an algorithm for option pricing by means of the application of multinomial models properties. For example, the algorithm of option pricing on the basis of trinomial model can be determined by extension of the algorithm for the binomial model – likewise the algorithm on the basis of quadrinomial model, pentanomial model, etc. The second goal of Part D is to validate hypotheses that it is possible to obtain an algorithm of discrete option hedging by means of application of continuous option hedging properties. For example, the Greeks in Black-Scholes model are representing the way how to describe the continuous option hedging and how to realize the transfer from continuous option hedging to discrete option hedging on the basis of a multinomial model (see also Part A “The Algorithm of Problem Solving in Data Mining Approach”, Chapter A9). The third goal of Part D is to show the way how to refill the delimited algorithms by software for option pricing and option hedging.
Key Words Option pricing, Discrete option hedging, Continuous option hedging, Option portfolio hedging, Algorithm, Software, Multinomial model, Binomial model, Trinomial model, Quadrinomial model, Pentanomial model, Black-Scholes model, Greeks
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D. Initial Orientation The goals of Part D are as follows: - The first goal of Part D is to validate the hypotheses that it is possible to obtain an algorithm for option pricing by means of the application of multinomial models properties. - The second goal of Part D is to validate hypotheses that it is possible to obtain an algorithm of discrete option hedging by means of application of continuous option hedging properties. - The third goal of Part D is to show the way how to refill the delimited algorithms by software for option pricing and option hedging. The goals of Part D were completed using by the follow-up chapters: D1. The Research, Historical and Economical Substantiations (according to V.Pavlat) D2. The Algorithms Identified in Option Portfolio Hedging (according to P.Zaskodny, I.Havlicek) D3. From Algorithm to Programming and Way of Verification (according to M.Pasta) D4. From Algorithm to Software for Trinomial Option Pricing (according to V.Risky) D5. From Algorithm to Software for Quadrinomial Option Pricing (according to M.Soucek) D6. From Algorithm to Software in Binomial Delta Hedging Based on Call Option (according to M.Sebest) The titles of single chapters and sub-chapters were adjusted by author of abstract and orientations, taking into consideration the needs of logical structure of Part D. The first chapter justified why Part D came into being by means of research, historical and economical reasons. The second chapter showed the survey of existing algorithms within option portfolio hedging and simultaneously the relation to option pricing algorithms. The third chapter tried to describe the general coherence between algorithm and software for option hedging and the way of this coherence verification. The fourth chapter found the way how to create software (on the basis of relevant algorithm) of option pricing by means of trinomial model and how to remind the relevance of algorithms in the area of statistics. The fifth chapter substantiated the possibility to create software (on the basis of relevant algorithm) of option pricing by means of quadrinomial model and at the same time differentiated between the various multinomial models. The sixth chapter processed the essential problem – how to create software (on the basis of algorithms described through the second chapter) for option hedging, although only in the light of the simplest multinomial model.
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D1. Research, Historical and Economical Substantiations D1.1. Summary of Chapter At the University of Finance and Administration (Prague, Czech Republic), recently a new Research Group on developing a new option portfolio hedging has been started. The research team consists of several staff members of the Department of Informatics and Mathematics and of the Department of Finance. Selected students are expected to be able to participate in the Group as well. In the first part of this chapter (see V.Pavlat (2013)), economical and historical knowledge in the field of option portfolio hedging is summarized. In the second part of paper the main characteristic of the on-going Research Group are described. The second part of the chapter (see V.Pavlat (2013)) mainly draws on works written by P. Zaskodny.
D1.2. Main Characteristics of Economical and Historical Dimension The beginnings of present-day U.S. financial derivative markets date back to 1972 – 1982 and the U.S. has been at the forefront in the field ever since that time. The period is nicknamed the “decade of innovation.” During this time, Chicago’s exchanges – the Chicago Board of Trade (CBOT) and the Chicago Mercantile Exchange (CME) – introduced new financial derivative products: stock options, US Treasury bond futures, stock-index-based derivatives, Eurodollar futures and options on futures. The introduction of those products into the financial markets revolutionized the U.S. financial markets of the time. Historians consider this to be the most far-reaching event that presaged the long-term development of financial derivative markets in the U.S. and across the globe. As the history of financial derivative markets unfolded, there was groundbreaking progress in the contemporaneous theory of financial derivatives, particularly financial options. The “decade of innovation” saw the greatest option theory discoveries of the day: the works of Black and Scholes (1973) and Merton (1973), which came to be known as the Black-Scholes model. The Black-Scholes options´ world can be described by means of three quotations (necessity of discrete form of option hedging delta, gamma, vega). (1) The first quotation: Hedging an option is easy in the basic Black-Scholes world. The only stochastic variable is the stock price, and by holding a short position in the stock equal to minus the partial derivative of the call price with respect to the stock, a momentarily riskless hedge of a call option is achieved. With no transactions costs or impediments to trading, the hedge can be rebalanced continuously through the entire life of the option. But delta hedging
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like this isn’t possible in the real world, where it only makes sense to rebalance a hedge periodically, and the change in the delta as the stock price moves - the option’s gamma becomes a source of unhedged risk. Delta-gamma hedges can manage both delta risk and the change in the delta, but volatility also changes over time, necessitating another Greek letter (or quasi-Greek in this case) - and vega was invented. (2) The second quotation: The financial markets have successfully established hedging instruments to protect against market risk, interest rate risk, currency risk, commodity risk and credit risk. The classic work in option-pricing by Black & Scholes [1973], and Merton [1973] helped mitigate equity risk through the prudent use of equity options. In the past three decades, derivatives in interest rates, foreign exchange, weather, and real estate have all grown in popularity. Each of these instruments allows for the isolation of an element of risk, and the transfer of this risk to a willing market participant. (3) The third quotation: Most option pricing models are set in continuous time in order for it to be (theoretically) possible to follow an option replication strategy that continuously rebalances a delta-neutral hedge. One big problem in applying such a model to the real world is that perfect replication theoretically entails trading an infinite amount of the underlying asset. With transactions costs, no matter how small, the cost of this strategy is also infinite. A delta hedge can not be rebalanced continuously, so how should one rebalance periodically to achieve the best replication at minimum cost. Originally designed as a European-options-only model, the Black-Scholes model later was adapted by other authors to fit American options, and it can also be applied to other derivative products. The Black-Scholes model has a number of practical applications and has remained in use to the present day. The model has had critics as well as advocates: the debate of the model’s applicability has been there ever since. The current argument was sparked by the recent global financial crisis: while the critics (Bouchaud (2008); Derman&Taleb (2008); Triana (2009); and, most recently, Haug&Taleb (2011)) warn against the risks of applying the model, which they believe are huge during crises, other authors (Wilmott (2008)) advocate the model’s potential. The fact is that large-scale failures on the financial derivative market during the last decades were mostly due to market participants’ negligent attitude to risk management. Theoretical works have expanded on the binomial model implied by the Black-Scholes model (see, e.g., Pavlát, Záškodný & Budík (2007)) and there are software products that rely on the “new implied generation”, i.e., the trinomial model (for an algorithm to apply the trinomial model see, e.g., Pavlát&Záškodný (2012)).
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Another “new implied generation”, the quadrinomial model, is one of the topics of the proposed project and may provide a further step towards risk reduction in option hedging applications. Algorithms and software are scarce in this area. Algorithms, and attempts at software products, for the various types of hedging, especially option hedging, are based on software products for the implied forms of the simpler versions of the multinomial model (the binomial, trinomial and quadrinomial models). In particular, there are algorithms for binomial delta hedging, binomial gamma hedging and some signs of vega hedging algorithms (see, e.g., Pavlát&Záškodný (2012); Havlíček&Záškodný (2012); and Havlíček&Budínský (2011–2012)). There has been a large increase in the quantity of the various types and forms of financial derivatives traded over the long term. Financial derivatives can serve the purpose of hedging, speculation and arbitrage and they are a tool that calls for investors with some level of theoretical knowledge and a good knowledge of how the tool can be applied in practice; in addition, this tool expects the investor to be aware of the fairly large risk involved in financial derivative trading. This risk went up over the last decade, hand in hand with the development of structured products that contain financial derivatives, one of the causes being that, in fact, some financial engineering products have pretty much escaped regulation and efficient oversight, including in developed financial markets (the U.S., in particular). CSO (calendar spread option) and CDS (credit default swap) are two examples. There is an analysis of the development trends of financial derivative markets (see, e.g., estimates of the future development of financial derivative trading generated by the World Federation of Exchanges and the International Swaps and Derivatives Association that is attached to the Federation) which indicates that further increase in financial derivative trading is envisaged. In the longer run, the Czech Republic can be expected to experience a gradual development of financial derivatives, on OTC markets, in particular. Given the steady improvement of the EU’s common market and the relatively close ties between the Czech Republic and the European Economic Area, research focusing on the highly relevant topic of financial derivatives may be considered of major importance. Positive results of theoretical research that can have practical applications in the form of new software solutions and user manuals – which is the case of the proposed financial option pricing and hedging research – may help the Czech Republic to get closer to countries with developed financial markets in this province.
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D1.3. Main Characteristics of Research Dimension The leading idea of the research group is to develop financial option pricing algorithms of a form that can be connected to the relevant discrete option hedging algorithms. Applying the algorithms, as much as possible, will enable to develop the software with a suitable programming language, considering the reality of the financial markets. The Objectives of the described idea can be summarized in the following way: (i) Theoretical (i.e., mathematical and statistical as well as financial and economic) explanations of the possibility of applying the simplest versions of a multinomial model to price options. (ii) Reflecting the theoretical explanations in suitable option pricing algorithms. (iii) Connecting the suitable option pricing algorithms to suitable option hedging algorithms. (iv) Developing software using the algorithms. (v) Exemplifying the software in relation to the reality of the financial markets and attempting to establish that the software is financially and economically exploitable. The way how to solve the above problems is proposed as follows: (i) The solution is to identify option pricing algorithms and related option hedging algorithms. (ii) As regards both option pricing algorithms and option hedging algorithms, four separate algorithms can be identified: a conceptual algorithm, a theoretical algorithm, a practical algorithm and a model algorithm. (iii) The four option hedging algorithms can be described as follows: (1) Conceptual algorithm based on fixed and proportional hedging of an option portfolio. (2) Theoretical algorithm based on option hedging derivation structure. (3) Practical algorithm based on financial risk management using a specific option hedging approach. (4) Model algorithm based on links between related models of the successive application of various option hedging types. (iv) The algorithms should incorporate a detailed series of steps to allow us to choose, and apply, a suitable programming language (e.g., the high-level object oriented programming language C#, with the software’s graphics developed in Windows Forms and data stored with ADO.NET technology).
D1.4. Conclusion Developing a software applying the trinomial and quadrinomial models of financial option pricing, and developing a software for binomial (and, if appropriate, trinomial and quadrinomial) delta and gamma hedging in relation to call and put options, will enable to attempt to develop a vega hedging software.
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The added value of the vega hedging software consist in getting more precise results for options´ valuation. However, the practical use of the new software tool mainly depends on two factors: first, the degree of precision achieved, and second, on the operability of the vega hedging software.
D1.5. Selected Literature Havlíček, I., Záškodný,P. (2010) Application of Mathematical Data Mining Tools – Greeks of First Order In: Tarábek, P. and P. Záškodný: Educational and Didactic Communication, 60–85 Bratislava: Didaktis ISBN 978-80-89160-78-5 Havlíček, I., Záškodný,P. (2010) Application of Mathematical Data Mining Tools – Greeks of Second Order In: Tarábek, P. and P. Záškodný: Educational and Didactic Communication, 86–104 Bratislava: Didaktis ISBN 978-80-89160-78-5 Havlíček, I., Záškodný,P. (2010) Application of Mathematical Data Mining Tools – Greeks of Third Order In: Tarábek, P. and P. Záškodný: Educational and Didactic Communication, 105–114 Bratislava: Didaktis ISBN 978-80-89160-78-5 Pavlát,V., Záškodný,P. (2012) Od finančních derivátů k opčnímu hedging (From Financial Derivatives to Option Hedging), 343 ps. Prague: Curriculum ISBN 978-80-904948-3-1 Havlíček, I.,Budínský, P. (2012) Continuous Delta Option Hedging and Its Affinity to Discrete Form In Proceedings (OEDM SERM 2011), 77-83, 6 ps Bratislava: Curriculum ISBN 978-80-904948-1-7. Záškodný,P., Havlíček,I., Budínský,P. (2010) Partial Data MiningTools in Applied Statistics – in Greeks and Option Hedging In Educational&DidacticCommunication, 41-50, 10 ps Bratislava: Didaktis ISBN 978-80-89160-78-5 Pavlát,V. (2013) On Developing a New Option Hedging Software In Proceedings (OEDM SERM 2012), 6 ps Bratislava: Curriculum ISBN 978-80-904948-4-8
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D2. Algorithms Identified in Option Portfolio Hedging D2.1. Summary of Chapter The chapter (see P.Zaskodny, I.Havlicek, (2013)) is seeking the way how to realize the transfer from continuous option hedging to discrete option hedging. The basic assumption for execution of the demarcated principle is given by the specification of algorithms which can be identified in option portfolio hedging. The main principle of chapter: Discrete form of option hedging The main goal of chapter: Algorithms applicable for option hedging discrete form The procedure of chapter: Conceptual algorithm, Theoretical algorithm Practical algorithm Model algorithm The results of chapter: 1. Structure of conceptual algorithm 2. Structure of theoretical algorithm 3. Structure of practical algorithm 4. Structure of model algorithm
D2.2. Introduction In mathematical finance, the Greeks are the quantities representing the sensitivities of derivatives such as options to a change in underlying parameters on which the value fiction of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are often denoted by Greek letters. The Greeks in the Black-Scholes model are relatively easy to calculate, a desirable property of financial models, and are very useful for derivatives traders, especially those who seek to hedge their portfolios from adverse changes in market conditions. For this reason, those Greeks which are particularly for Hedging Gamma are well-defined for measuring changes in Hedge Ratio. The Greeks are representing the way how to describe the continuous option hedging. The affinity of continuous option hedging to an option hedging discrete form should work on the algorithms identified in option portfolio hedging. The algorithms should incorporate a detailed series of steps to allow us to choose, and apply, a suitable programming language (e.g., the high-level object oriented programming language C#, with the software’s graphics developed in Windows Forms and data stored with ADO.NET technology). The structures of algorithms identified in option portfolio hedging labour under the work Pavlát,V., Záškodný,P. (2012), From financial derivatives to option hedging, Prague, Czech Republic: Curriculum, ISBN 978-80-904948-3-1.
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D2.3. Structure of Conceptual Algorithm The conceptual algorithm, which is based on fixed and proportional hedging of option portfolios, is defined by the following series of concepts (a) to (i), which are algorithmically connected to each other: (a) Hedged option portfolio: A fixed-value portfolio resulting from option hedging. (b) Fixed hedge relating to a single hedged option portfolio, and a proportional hedge that relates to a set of hedged portfolios. (c) Four input positions to apply option hedging to: Call option buyer, put option buyer, call option seller and put option seller. (d) Constructing “call option seller” and “put option seller” positions to finance a hedged option portfolio during option hedging implementation. (e) Transition from continuous option hedging to discrete option hedging, i.e., a link between continuous and discrete option pricing models. (f) The role of discrete option pricing models as a means to go from a fixed to a proportional hedge. (g) Implementing a discrete option hedge in two steps: Developing dynamic hedging strategies and rebalancing. (h) Developing a dynamic hedging strategy: Forming a lattice that represents a chosen type of random walk, putting relevant option hedge factor values at the nodes. (i) Rebalancing the hedge: Transforming the original hedged option portfolio into new hedged portfolios within a predetermined random walk along the time series of consecutive nodes.
D2.4. Structure of Theoretical Algorithm The theoretical algorithm, which is based on option hedging derivation structure, is defined by the following series of steps (a) to (e), which are algorithmically connected to each other: (a) Selecting a random walk type: The selection is based on the choice of an s-fold multinomial model.
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(b) Developing a lattice with nodes and time layers, i.e., taking the selected type of random walk: E.g., developing a binomial lattice (binomial tree), a trinomial lattice (trinomial tree) or a quadrinomial lattice (quadrinomial tree). (c) Developing a dynamic hedging strategy: E.g., developing a dynamic delta hedge and developing a dynamic gamma hedge. (d) Selecting a specific random walk, i.e., a time series of consecutive nodes on each time layer: E.g., selecting a specific binomial random walk on a binomial tree, selecting a specific trinomial random walk on a trinomial tree and selecting a specific quadrinomial random walk on a quadrinomial tree. (e) Rebalancing, and developing a set of hedged option portfolios at specific nodes on consecutive time layers of the lattice: E.g., developing a set of binomial hedged option portfolios, developing a set of trinomial hedged blocks of trinomial option portfolios and developing a set of quadrinomial hedged blocks of quadrinomial option portfolios.
D2.5. Structure of Practical Algorithm The practical algorithm, which is based on a specific option hedging approach to manage financial risk, is defined by the following series of steps (a) to (i), which are algorithmically connected to each other: (a) Selecting a suitable type of a discrete option pricing model, defining the parameters of the model selected, determining the values of the input parameters (growth and decline indices, in particular) and describing the selected type of random walk. (b) Defining the origins of the growth and decline indices in terms of their links to the implied model or the autonomously implied model or the autonomous model. (c) Calculating underlying stock prices and theoretically correct option prices at the nodes of a lattice resulting from the implementation of a selected type of random walk. (d) Selecting the appropriate equations and formulae to prepare the implementation of a delta hedge. (e) Developing a dynamic delta hedge strategy. (f) Delta rebalancing. (g) Selecting the appropriate equations and formulae to prepare the implementation of a gamma hedge. (h) Developing a dynamic gamma hedge strategy. (i) Gamma rebalancing.
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(j) Preparing and implementing other option hedging types that have been selected (e.g., speed hedging and vega hedging).
D2.6. Structure of Model Algorithm The model algorithm can be defined by three branches of models (a) to (c) employed: (a) The Black-Scholes model and its innovations: The implied binomial model; the implied trinomial model; and the implied quadrinomial model. (b) The autonomous binomial model; the autonomously implied trinomial model; and the autonomously implied quadrinomial model. (c) The autonomous binomial model; the autonomous trinomial model; and the autonomous quadrinomial model. Note: A pentanomial model may be applied.
D2.7. Description of Potential Way of Algorithms Application (i) Option pricing algorithm using the trinomial model. (ii) Software based on the trinomial model (e.g., the high-level object oriented programming language C#, with the software’s graphics developed in Windows Forms and data stored with ADO.NET technology). (iii) Option pricing algorithm using the quadrinomial model. (iv) Software based on the quadrinomial model (e.g., the high-level object oriented programming language C#, with the software’s graphics developed in Windows Forms and data stored with ADO.NET technology). (v) Greeks derivation and discrete option hedging transformation algorithm. (vi) Call-option-related hedging algorithm. (vii) Call-option-related hedging software (e.g., the high-level object oriented programming language C#, with the software’s graphics developed in Windows Forms and data stored with ADO.NET technology). (viii) Put-option-related hedging algorithm. (ix) Put-option-related hedging software (e.g., the high-level object oriented programming language C#, with the software’s graphics developed in Windows Forms and data stored with ADO.NET technology). (x) Using comparative analysis to contrast the software with the reality of the current financial markets and the financial and economic applicability of the algorithms.
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D2.8. Selected Literature Albrecher,H., Dhaene,J., Goovaerts,M., Schoutens,W, (2005) Static Hedging of Asian Options under Lévy Models The Journal of Derivatives, Spring 2005 Vol. 12, No. 3, 63-72. Anson,M.J.P. (2001) Hedge Fund Incentive Fees and the “Free Option” The Journal of Alternative Investments, Fall 2001 Vol. 4, No. 2, 43-48. Barnea,A., Hogan,R. (2012) Quantifying the Variance Risk Premium in VIX Options The Journal of Portfolio Management Vol. 38, No. 3, 143-148. Bhaduri,R., Meissner,G., Youn,J. (2008) Hedging Liquidity Risk Potential Solutions for Hedge Funds The Journal of Trading, A Guide to Multi-Asset Trading Strategies, Fall 2008 Vol. 2008, No. 1, 72-82. Black,F., Scholes,M. (1973) The Pricing of Options and Corporate Liabilities Journal of Political Economy 81 (3): 637–654. Bodurtha,J.N., Thornton,D.B. and Jr. (2002) FAS 133 Option Fair Value Hedges. Financial Engineering and Financial Accounting Perspectives The Journal of Derivatives, Fall 2002 Vol. 10, No. 1, 62-79. Bouchaud,J.P. (2008) Economics Needs a Scientific Revolution Nature 455: 1181 http://arxiv.org/abs/0810.5306v1 Broadie,M. Jain,A. (2008) Pricing and Hedging Volatility Derivatives The Journal of Derivatives Vol. 15, No. 3, 7-24. Chance,D.M. (2008) Essays in Derivatives: Risk-Transfer Tools and Topics Made Easy 2nd Edition, Hoboken, NJ, USA: Wiley ISBN 978-04-700862-5-4 Chang,C.-C., Liao,T.H., Tsao,C.Y. (2011) Pricing and Hedging Quanto Forward-Starting FloatingStrike Asian Options The Journal of Derivatives, Summer 2011 Vol. 18, No. 4, 37-53. Dai,T.S., Lyuu,Y.D. (2010) The Bino-Trinomial Tree: A Simple Model for Efficient and Accurate Option Pricing The Journal of Derivatives, Summer 2010 Vol. 17, No. 4, 7-24.
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Derman,E., Taleb,N.N. (2005) The Illusions of Dynamic Replication Quantitative Finance Vol. 5, No. 4, August 2005, 323–326. Dixon,M., Oreto,P., Starr,D., Zheng,Ch. (2008) Implied Volatility Term Structure and Vega Hedging in the Q-Alpha-Sigma Model www.stanford.edu/msande444/2008/ivts1report.pdf. Eriksson,A., Ghysels,E., Wang,F. (2009) The Normal Inverse Gaussian Distribution and the Pricing of Derivatives The Journal of Derivatives, Spring 2009 Vol. 16, No. 3, 23-37. Etheridge,A. (2000) Stochastic Calculus for Finance Oxford, Oxford University. Golts,M., Kritzman,M. (2010) Liquidity Options The Journal of Derivatives, Fall 2010 Vol. 18, No. 1, 80-89. Hamza,O., L'Her,J.F., Roberge,M. (2007) Active Currency Hedging Strategies for Global Equity Portfolios The Journal of Investing, Winter 2007 Vol. 16, No. 4, 146-166. Haug,E.G., Taleb,N.N. (2011) Option Traders Use (very) Sophisticated Heuristics, Never the Black-Scholes-Merton Formula Journal of Economic Behavior and Organization Vol. 77, No. 2, 2011. Havlíček,I., Budinský,P. (2011a) Continuous Delta Option Hedging and Its Affinity to Discrete Form. In Proceedings (OEDM SERM 2011). Bratislava: Curriculum, 2011, 77-83 ISBN 978-80-904948-1-7. Havlíček,I., Budinský,P. (2011b) Continuous Gamma Option Hedging and Its Affinity to Discrete Form In Proceedings (OEDM SERM 2011). Bratislava: Curriculum, 2011, 70-76 ISBN 978-80-904948-1-7. Hull,J.C. (2006) Options, Futures, and Other Derivatives 6th Edition, 769 pp. New York: Prentice Hall ISBN 0-13-149908-4. Keith,H.B. (2006) Improving Hedge Fund Risk Exposures by Hedging Equity Market Volatility, or
How the VIX Ate My Kurtosis The Journal of Trading, Spring 2006 Vol. 1, No. 2, 6-15 Krause,A. (2003) Crashes in Bond Markets and the Hedging of Mortgage-Backed Securities The Journal of Fixed Income, December 2003 Vol. 13, No. 3, 19-32.
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Mallaby,S. (2011) More Money Than God: Hedge Funds and the Making of a New Elite Kindle Edition, USA, Penguin Press ISBN 0-14-311941-9 Martinsek,M. (2012) Charm-Adjusted Delta nad Delta Gamma Hedging The Journal of Derivatives Vol. 19(3), 69-76. Martellini,L, Priaulet,P. (2002) Competing Methods for Option Hedging in the Presence of Transaction Costs full access Institutional Investor Journals (The Journal of Derivatives) Vol. 9, No. 3, 25-38. Merton, R.C. (1973), Theory of Rational Option Pricing. Bell Journal of Economics and Management Science (The RAND Corporation), Vol. 4 (1), 141–183. Merton,R. (1974) On the Pricing of Corporate Debt: The Risk Structure of Interest Rates The Journal of Finance Vol. 29, 449–470 Nalholm,M., Poulsen,R. (2006) Static Hedging of Barrier Options under General Asset Dynamics Unification and Application The Journal of Derivatives, Summer 2006 Vol. 13, No. 4, 46-60. Orosi,G. (2011) A Multi-Parameter Extension of Figlewski’s Option-Pricing Formula The Journal of Derivatives, Fall 2011 Vol. 19, No. 1, 72-82. Partnoy,F. (2009) Infectious Greed: How Deceit and Risk Corrupted the Financial Market Kindle Edition. USA: Public Affairs ISBN 1-58-648784-1. Pavlát,V., Záškodný,P., Budík,J. (2007), Finanční deriváty a jejich oceňování Praha: VSFS ISBN 978-80-86754-73-4 Pavlát,V., Záškodný,P. (2012) Od finančních derivátů k opčnímu hedging (From Financial Derivatives to Option Hedging), 343 pp Prague, Curriculum ISBN 978-80-904948-3-1. Rebonato,R. (1999) Volatility and Correlation in the Pricing of Equity, FX and Interest-Rate Options Wiley ISBN 0-471-89998-4 Rebonato,R., Pogudin,A., White,R. (2008) Delta and Vega Hedging in the SABR and LMM-SABR Model Risk Magazine, 1, December 2008
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Reinhart,C.M., Rogoff,K. (2009) This Time is Different: Eight Centurie sof Financial Folly Kindle Edition, USA, Princenton University Press ISBN 978-06-91142-16-6. Rendleman,R.J. and Jr. (2004) Delivery Options in the Pricing and Hedging of Treasury Bond and Note Futures The Journal of Fixed Income, September 2004 Vol. 14, No. 2, 20-31 Triana,P. (2009) Lecturing Birds on Flying: Can Mathematical Theories Destroy the Financial Markets? Wiley ISBN 978-0-470-40675-5 Wallace,J. (2001) Option Hedging Strategies Under FAS 133 Special Issues, FAS 133 and the New Derivatives Accounting Landscape, Fall 2001 Vol. 2001, No. 1, 72-76. Wilmott,P. (2008) Financial Models Must Be Clean and Simple Business Week, December 31, 2008 Witzany,J. (2010) Credit Risk Management and Modeling, 214 pp. Prague: Oeconomica ISBN 978-80-245-1682-0 Záškodný,P., Havlíček,I., Budinský,P. (2010a) Partial Data Mining Tools in Applied Statistics – in Greeks and Option Hedging In Educational and Didactic Communication, Bratislava, Didaktis, 2010, 41 - 50 ISBN 978-80-89160-78-5 Záškodný,P., I. Havlíček,I., Budinský,P., Hrdlička,L. (2010b) Where Will Be Used the Partial Data Mining Tools in Statistics Education? In Greeks In Educational and Didactic Communication, 51 - 59 Bratislava: Didaktis ISBN 978-80-89160-78-5 Záškodný,P., Havlíček,I. (2010a) Application of Mathematical Data Mining Tools – Greeks of First Order (Delta, Dual Delta) In: Tarábek,P., Záškodný,P.: Educational and Didactic Communication 2010, 60 – 69 Bratislava: Didaktis ISBN 978-80-89160-78-5. Záškodný,P., Havlíček,I. (2010b) Application of Mathematical Data Mining Tools – Greeks of First Order (Theta, Rho, Vega) In: Tarábek, Záškodný,P.: Educational and Didactic Communication 2010, 70 – 85 Bratislava: Didaktis. ISBN 978-80-89160-78-5
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Záškodný,P., Havlíček,I. (2010c) Application of Mathematical Data Mining Tools – Greeks of Second Order (Gamma, Dual Gamma, Vomma) In: Tarábek,P., Záškodný,P.: Educational and Didactic Communication 2010, 86 – 95 Bratislava: Didaktis ISBN 978-80-89160-78-5 Záškodný,P., Havlíček,I. (2010d) Application of Mathematical Data Mining Tools – Greeks of Second Order (Vanna, Charm, DvegaDtime) In: Tarábek,P., Záškodný,P.: Educational and Didactic Communication 2010, 96 – 104 Bratislava: Didaktis ISBN 978-80-89160-78-5. Záškodný,P., Havlíček,I. (2010e) Application of Mathematical Data Mining Tools – Greeks of Third Order (Speed, Zomma, Color, Ultima) In: Tarábek,P., Záškodný,P.: Educational and Didactic Communication 2010, 105 – 114 Bratislava: Didaktis ISBN 978-80-89160-78-5 Záškodný,P., Havlíček,I. (2013) The Algorithms Identified in Option Portfolio Hedging In Proceedings (OEDM SERM 2012), 6 ps Bratislava: Curriculum ISBN 978-80-904948-4-8 Zuckerman,G. (2009) The Greatest Trade Ever: The Behind-the-Scenes Story of How John Paulson Defied Wall Street and Made Financial History Kindle Edition, USA, Crown Business ISBN 0-38-552994-5
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D3. From Algorithm to Programming and Way of Verification D3.1. Summary of Chapter The presented chapter (see M.Pasta (2013a) describes option hedging algorithms which are ready to implement in software applications. Option hedging is one of the risk management strategies. The importance of risk management is growing in last years thanks to turbulences on financial markets. The objective of the chapter is to prove that it is possible to get structure of algorithms suitable for programming by applying set of financial option models. First part of chapter contains theoretical basis for option hedging. It includes option pricing models and option hedging strategies. There are two basic model categories depending whether computation is done for continues or discrete time. Continuous time models described in the chapter are Black-Scholes pricing model and its prerequisites as well as Greeks describing option price sensitivity to various underlying parameters and delta hedging strategy. Discrete time model described in the chapter is the Trinomial model and its application to discrete delta hedging that is deduced from continues delta hedging. Second part of chapter contains algorithms for applying delta hedging with trinomial model. The autonomous version (Pavlat and Zaskodny, 2012b) of the model is used. This means that input parameters for the model are obtained by descriptive statistics methods instead of deriving from continuous models. Final part includes a demonstration how to use these algorithms in practical scenario with real market data and conclusion.
D3.2. Basic Concepts Investing on capital markets involves various forms of risk e.g. credit risk, market risk, volatility risk etc. Hedging is a strategy how investor can minimize the exposure to potential risk. In simple explanation hedging can be seen as a form of insurance. The basic principle of various hedging strategies is that one investment is hedged by another one. In real world it is achieved by creating of investment portfolio that is composed of several investment assets in negative correlation (Pavlat and Zaskodny, 2012a: 36). This means that if the value of one asset increases the value of other asset decreases and vice versa. The important is to realize that profit is in correlation with the risk taken. Decreasing risk and potential losses means decreasing potential profit as well. It follows that hedging is not intended to make profit but to minimize potential losses. In case that investment is profitable the profit will be lower for the amount equals to cost of the hedge and if the investment will be in loss this loss will be minimized in case that chosen hedging strategy is effective (Hull, 2009: 49). Typical instruments used for hedging strategies are financial derivatives (Pavlat and Zaskodny, 2012a: 36). Best known derivatives are futures and options. Future is an agreement about buying or selling underlying asset (e.g. stock) in pre-agreed time and price in the future. The risk of price change is same for both counter-parties. The profit of one party equals to loss of the other party. Option is right to buy or sell underlying asset in pre-agreed time and price in
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the future. Fundamental difference between futures and options is that the owner of call or short option owns right to buy or sell underlying asset unlike the writer of call or short option which has obligation to sell or buy the underlying asset (Hull, 2009: 181). Hence the owner of an option can use this advantage for creating several hedging strategies. The example can be protective put option strategy. In this strategy the investor wants to buy some asset (e.g. stock) and want to hedge the investment against fall bellow certain price. The investor goes long with the stock together with long put option for the same stock. In case of stock price falling under the specific value set by strike the investor can exercise option and get strike price (Hull, 2009: 220). From this example we can see again that the owner of an option has an advantage against the writer of an option therefore the owner pays option premium to the writer. It is reward for the risk taken. There are three types of investors on derivatives markets. One types of investor are speculators. They are willing to take higher risk and get higher profit as a reward. They are important part of the market because they create liquidity and it couldn’t be possible to create hedging strategies without them. Next are hedgers. Their target is to minimize risk and they are willing to pay for this by decreasing their potential profit. Third are arbitrageurs. They benefit from inefficiency of the market. There are differences between markets lasting for very short period of time that leads to risk-less profit. This is usually available only for financial institutions because it is profitable only with very fast transaction systems with low transactional cost. Arbitrage ensures effectiveness of the market (Pavlat and Zaskodny, 2012a: 34-37).
D3.3. Algorithms, Programming, Future Verification The objective of the chapter is to create option hedging algorithms in form which is suitable for programming. The thesis of the chapter is “It is possible to get structure of algorithms suitable for programming by applying set of financial option models”. The future verification of thesis will be composed of two parts. First part will contain necessary theoretical knowledge that will be used in the application part. This part will start with description of Black-Scholes formula and its prerequisites, which are Wiener process and Ito’s lemma. The Black Scholes formula is fundamental model for computing fair option price for continuous time (Black and Scholes, 1973). The next step will be description of Greeks, which are math variables derived from Black-Scholes formula describing the option price sensitivity to change in various underlying parameters. The example can be delta Greek which describes option price sensitivity to price of the underlying asset. With this knowledge we are able to create a portfolio, which is risk-less to underlying price movements by applying delta Greek calculus. This is called delta hedging (Hull, 2009: 360). As the continuous time models have limited usage in real world scenarios because we are not able to change input parameters continuously we need to move towards discrete time models. This is done by deriving delta hedging formula for discrete time (Pavlat and Zaskodny, 2012b: 77) and applying multinomial trees. The continuous time is replaced by finite periods of time and the option price is computed for possible outcomes in every period. The easiest version of model is binomial tree, which has two possible outcomes in one period. If the market is much more volatile we need to use trinomial model with three possible outcomes instead. The decision, which model to use, is based on descriptive statistics measurements of underlying
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price volatility (Pavlat and Zaskodny, 2012b: 54). As volatility increases we need to apply higher orders of multinomial trees e.g. quadrinomial etc. The verification will describe usage of trinomial model for discrete delta hedging. Second part of the future verification will contain algorithms for implementing discrete delta hedging strategy using trinomial model. Creation of algorithms is important part of future verification because it should be possible to create software implementation of model based on it. Usage of algorithms will be demonstrated on real data from US financial markets. The hedged option portfolio will be created for writer of the call option. First step will be performing discrete statistics measurement for volatility of the price of underlying asset. Correct form of multinomial model will be chosen based on this measurement. Next step will be to compute evolving of option prices, stock prices and hedge ratios for possible scenarios in every period of time. The computation will be demonstrated only for few periods of time to reduce the complexity. Last step is computing dynamic hedging schemas based on previous computations and demonstrate rebalancing the hedged option portfolio on the end of every period. The concrete verification of delimited thesis was realized by means of identified algorithms and indicated programming in M.Pasta (2013b).
D3.4. Selected Literature Hull, J.C. (2009) Options, futures and other derivatives J. C. Hull, ed., Pearson Prentice Hall Pavlát,V., Záškodný,P. (2012a) Od finančních derivátů k opčnímu hedging (From financial derivatives to option hedging), První díl: Finanční opce a infrastruktura finančních obchodů Praha: Curriculum ISBN 978-80-904948-3-1 Pavlát,V., Záškodný,P. (2012b) Od finančních derivátů k opčnímu hedging (From financial derivatives to option hedging), Druhý díl: Oceňování finančních opcí a opční hedging Praha: Curriculum ISBN 978-80-904948-3-1 . Black,F., Scholes,M. (1973) The Pricing of Options and Corporate Liabilities The Journal of Political Economy Vol. 81, No. 3 (May - Jun., 1973), 637-654 Pašta,M. (2013a) Preparation of Option Hedging Algorithms In Proceedings (OEDM SERM 2012) Bratislava: Curriculum, 4 ps ISBN 978-80-904948-4-8. Pašta,M. (2013b) Preparation of Option Hedging Algorithms Bachelor Diploma Thesis University of Finance and Administration, Prague, Czech Republic
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D4. From Algorithm to Software for Trinomial Option Pricing D4.1. Summary of Chapter The goal of this chapter (see [7]) is to validate the hypotheses that “It is possible to obtain an algorithm for the trinomial model by extention of the algorithm for the binomial model” (hypothesis H1), and that “Trinomial model software can be obtained by a projection of the binomial model software using trinomial parameters” (hypothesis H2). Furthermore this algorithm can be used in calculations of empirical statistics, the Black-Scholes model, the binomial model, the trinomial model compatible with Black-Scholes model, the autonomous trinomial model, a valuation of options and premature assertions, and premature sales of options.
D4.2. Introduction This chapter aims at clarifying multinomial models applied to financial option pricing. More exactly, it aims at validation of hypothesis that it is possible to obtain an algorithm of the trinomial model by extension of algorithm of the binomial model, and application of the BlackScholes model, the binomial model, the option pricing, the premature assertions, the premature sales of options, and the autonomous trinomial model as well. Initially, we have to learn and understand specific aspects of the Czech Republic – market size, limited amount of subjects being authorized to use the financial options. These are mostly large enterprises that ought to learn how to use various financial instruments and how to handle risks. I think there is insufficient knowledge of these products in the Czech Republic. The vast majority of enterprises do not have any complete information on these products. Employees of economic and foreign departments of large enterprises do not usually use these instruments because they do not know them very well, and they are afraid of these processes. They have not understood importance and significance of financial derivatives. Before 1989, these instruments used to be applied to a limited extent. Therefore, the financial derivatives were not largely taken into account, or included into ordinary working processes. Banks mediating financial operations were not prepared enough to provide these services in the 1990’s. In many cases, they did not find their way to potential users of these services. Nowadays, such banks are prepared enough to provide the financial derivatives to their clients. To introduce and explain the instruments to other subjects – it is their most important duty. Price is the most limiting factor of such financial instruments and processes. Concluded contracts must be helpful and financially acceptable for enterprises. The question of price is to play a significant role in the future. This topic highly deserves our attention, as it belongs to up-to-date elements of the market economy. Unfortunately, subjects concerned are not informed enough on them. Higher option pricing models (it means higher than the binomial and the Black- Scholes ones) are used to a very limited extent in the Czech Republic. Monitoring the market, we notice
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traders use the Black- Scholes model to different purposes, than it has been designed and projected to. The Black- Scholes model assumes an option price is lognormally distributed. If it is not, the Black- Scholes model produces incorrect option prices. The lognormal distribution is based on a hypothesis a share price runs a constant volatile process. Taking market option prices into account, we realize that this volatility (implied by the pricing formula) is more determined by strike and time period, until the option expires, than constant quantity. Nowadays, Bloomberg terminal is the most famous product applying the trinomial model to counting. Bloomberg terminal is a computer system allowing us to monitor and analyse data produced by the financial market in real time. It also provides up-to-date information and news concerning the financial market fluctuations, and various investment instruments trading too. It provides detailed analyses. This is a commercial product, which is sold for the price of $ 1,500 per month (see [3]). Book called Measuring and Controlling Interest Rate and Credit Risk (see [1]), and its page 188, demonstrate Bloomberg terminal using the trinomial model in its 2002 version. MBRM - MB Risk Management is another interesting product. This is a commercial product too, which was sold for the individual user’s price of £ 49,999 per computer in 2011 (see [4]). Option Matrix: The advanced Derivates Calculator is another interesting application. It supports more than 136 theoretical models projected to the financial derivatives (see [5]). The application has been launched under GNU GPLv3 license; it is available in OS Windows (DOS) version, as well as Linux/Unix version. Mac version is also to be developed in a short time.
D4.3. Validation of Hypothesis H1 In “From financial derivatives to option hedging” (see [2]), chapter 1.5. “Multinomial model and its options”, the multinomial model, its definition, methods and steps are described in a pregnant and readable way. From the very beginning of the chapter, the general multinomial financial option pricing model is based on an algorithm of discrete s-fold multinomial theoretical distribution. Authors of this book also state, this model can be approximated by multidimensional Poisson distribution. In certain case, the model can be approximated by multidimensional Hypergeometric distribution. D4.3.1. Comparison of Binomial and Trinomial Model S-fold multinomial distribution is a discrete theoretical distribution s-Multi(n,p1,….,ps-1) having s-theoretical parameters of n, p1,…, ps-1 of a random vector X=[X1,...,Xs-1] (values of the random X1,…,Xs-1 quantities the random vector comprises of are expressed as i1,…,is-1 = 0,1,…,n) (see [2]).
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Comparison of the models:
binomial model:
o it is based on s-fold multinomial distribution for s=2, o the binomial distribution, expressed as Bi(n,p), is related to this model during n period (∆t); there are two profit options available for each of these periods (with p probability and 1-p probability) which are characterized by index of decrease (d) and increase (u) of base share (S) in t=0 time. o probability function of the binomial distribution (for n ≥ j ≥ 0):
n n n! n j Π j = p j 1 p , where j j j ! n j ! Probability function for Bi(n,p) trinomial model: o it is based on s-fold multinomial distribution for s=3, o the trinomial distribution, expressed as Tr(n,p1,p2), is related to this model for n period (∆t); there are three profit options available for each of these periods (with p1 probability, p2 probability and p3=1–p1–p2 probability) which are characterized by index of decrease and increase of base share (S) in t=0 time; these indexes options are expressed as I1, I2 and I3. o probability function of the trinomial distribution: 2
i1 ,i2
2 p1i1 p2i2 1 p j 2 j 1 i1 ! i2 ! n i j ! j 1
n!
n i j j 1
.
Probability function for Tr(n,p1,p2) Resemblance of these two relations (the probability function of the binomial distribution and the probability function of the trinomial distribution) is obvious and not random. Probability function of the quadrinomial distribution follows in order to make the probability functions complete: 3
i1,i2 ,i3
3 n! p1i1 p2i2 p3i3 1 p j 3 j 1 i1 ! i2 !i3 ! n i j ! j 1
Probability function for Qu(n,p1,p2,p3)
n i j j 1
.
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Comparing the other relations, we develop a higher degree model by extension of parameters: Base share prices for Bi and Tr, after n period expires:
Si1 ,i2 I1i1 I 2i2 I 3
S j u j d n j S
n
2
i j j 1
S
Theoretically correct price of purchase option for Bi and Tr:
C
1 qn
n
jC j
C
j 0
C j max 0, S j X
1 qn
n
n
i i Ci i
i1 0 i2 0
12
12
Ci1i2 max 0, Si1i2 X
Theoretically correct price of sales option for Bi and Tr:
P
1 qn
n
j Pj
P
j 0
Pj max 0, X S j
1 qn
n
n
i i
i1 0 i2 0
12
Pi1i2
Pi1i2 max 0, X Si1i2
Internal value of purchase option for Bi and Tr:
IVC kj max 0, S kj X
IVCik1i2 max 0, Sik1i2 X
Internal value of sales option for Bi and Tr:
IVPjk max 0, X S kj
IVPi1ki2 max 0, X Sik1i2
Relation between call and put American options for Bi and Tr in k time period:
Ci1k,i2
Pi1k,i2
1 p1Ci1k1,1 i2 p2Ci1k,i211 1 p1 p2 Ci1k,i21 q 1 p1 Pi1k1,1i2 p2 Pi1k,i211 1 p1 p2 Pi1k,i21 q
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Pi1k,i2 ,i3
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1 p1Ci1k1,1 i2 ,i3 p2Ci1k,i211,i3 p3Ci1k,i21,i3 1 1 p1 p2 p3 Ci1k,i21,i3 q 1 p1 Pi1k1,1i2 ,i3 p2 Pi1k,i211,i3 p3 Pi1k,i21,i3 1 1 p1 p2 p3 Pi1k,i21,i3 q
Adding another parameters into the model (decrease and increase i – index, {whereas j in the binomial model} and p - probability), we develop a higher degree multinomial model. Number of parameters we have to add is determined by s-parameters (or s–1 parameters) of s-fold multinomial distribution. Calculation of parameters is based on the secondary empiric-statistical analysis of empirical standard deviations and empirical parameters of obliqueness of decrease or increase index of the autonomous binomial model. Previous text validates the hypothesis that it is possible to obtain an algorithm of the trinomial model by extension of algorithm of the binomial model. I came to conclusion that, extending (or distributing) parameters of share price increase or decrease, converting increase or decrease probabilities, calculating increase or decrease index and substituting the binomial relations by the trinomial relations, the trinomial model is calculated in similar way to the binomial model (except for certain little differences). Indexes of share price increase or decrease I1, I2, I3 of the autonomous trinomial model are derived by the secondary empiric-statistical analysis of the empirical standard deviations and the empirical parameters of obliqueness of increase or decrease index of the autonomous binomial model (see [2]). There are two options of distribution of share price increase or decrease index resulting from the analysis: - ub increase index is divided into two trinomial increase indexes (I1=u1t and I2=u2t); db index is kept as the trinomial decrease index and expressed as I3=dt; p-probability is divided into p1 and p2; p3=1–p probability stays unchanged. - db decrease index is divided into two trinomial decrease indexes (I2=d1t and I3=d2t); ub index is kept as the trinomial increase index and expressed as I1=ut; 1–p probability is divided into p2 and p3; p1=p probability stays unchanged (see [2]). D4.3.2. Validation of Hypothesis in Practice and Brief Description of Application In practice, the hypothesis H1 has been validated by developing the application calculating elementary statistical measures and the financial derivatives. The option pricing according to the Black-Scholes model, the binomial model and the trinomial model. Statistical application has been developed there; it is logically and intuitively controlled. We can turn back in every single data processing step, revise and recount the results. Data may be converted into a graph, if meaningful.
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Financial derivatives programming has been developed to count the financial derivatives (as its name indicates). The program is implemented in order to count the Black-Scholes model of European and American no-dividend share option, the Black-Scholes model of European continuous-dividend share option, the Black-Scholes model of American discrete-dividend purchase share option, the binomial model of pricing of European and American no-dividend share option, and the trinomial autonomous model of pricing of no-dividend share option. i) Black-Scholes model Hedge ratio calculation, probability of application in time period, and call option and put option values. ii) Binomial model Individual sections are calculated during the expiry period. In table u-column, there are increase indexes; in d-column, there are decrease indexes; in p-column, there are probabilities of increase; in 1–p-collumn, there are probabilities of decrease; in S-column, there are base share acceptable prices; in C-column, there are purchase share acceptable prices; in P-column, there are sales share acceptable prices; in the last Π-column, there are probabilities of purchase option acceptable price. Call option, put option and Put-call parity values are also calculated there; Put-call parity means a relation between option bonuses of the corresponding call options and put options (the call options and the put options having the same basic instrument, the same price of implementation and the same expiry date). Using Put-call parity method, we calculate e.g. option bonus, or we deduce put option features from the already calculated option bonus, or from derived call option features. iii) Trinomial model Individual sections are calculated during the expiry period. In table Πij-column, there are probabilities of purchase option acceptable prices; in Sij-column, there are base share acceptable prices; in Cij-column, there are purchase option acceptable prices; in Pij-column, there are sales option acceptable prices. Call option, Put option and Put-call parity values are also calculated there; Put-call parity means a relation between option bonuses of the corresponding call options and put options (the call options and the put options having the same basic instrument, the same price of implementation and the same expiry date). Using Put-call parity method, we calculate e.g. option bonus, or we deduce put option features from the already calculated option bonus, or from derived call option features. Increase indexes or decrease indexes are put down there (I1, I2 and I3). The application also includes Tree of development of share prompt price, Tree of development of put/call option internal price, and Tree of development of purchase/sales option price. In a text field, the applied model puts down value, date and time, when the calculation has been done, particular calculation parameters and the calculation result.
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D4.4. Validation of Hypothesis H2 The hypothesis H2 “Trinomial model software can be obtained by a projection of the binomial model software using trinomial parameters” can be considered as the confirmed hypothesis by means of Master Diploma Thesis (see [6] – programming language C++ was used). The trinomial model software was created by the projection of binomial model software and testified by the installment of various values of trinomial parameters. Validation of the hypothesis H2 was also substantiated by means of brief description of the applications connected with the confirmed hypothesis H1 (see chapter D4.3.2.).
D4.5. Conclusion In conclusion, we have to be aware of a little attention paid to the option pricing by the trinomial and the quattronomial models in the Czech Republic. Nowadays, there is a lack of specialized literary sources dealing with the option pricing by the trinomial and the quattronomial models. Therefore, I highly appreciate the title From financial derivatives to option hedging by Pavlát and Záškodný; it deals with the above-mentioned issue in a theoretical way in its second part. There are no obstacles hindering us from launching the multinomial models of a higher degree in practice. No matter, these option pricing models are more precious and more exact. Czech institutions operating the financial derivatives ought to focus on the application of the multinomial models in practice. The application has significantly contributed to calculations of the financial derivatives, but also to implementation of analytic and synthetic models of the empiric statistics; such models have a significant cognitive dimension. It shows how the analysis, the synthesis and the abstraction are used and applied. The analysis, the synthesis and the abstraction are crucial for us to reach a correct result.
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D4.6. Selected Literature [1] Fabozzi,F.J., Mann,S.V., Choudhry,M. (2003) Measuring and Controlling Interest Rate and Credit Risk England: John Wiley & Sons ISBN 0-471-26806-2. [2] Pavlát,V., Záškodný,P. (2012) Od finančních derivátů k opčnímu hedging (From financial derivatives to option hedging) Praha: Curriculum ISBN 978-80-904948-3-1. [3] http://www.bloomberg.com/professional/software_support/ [4] www.mbrm.com/index.shtml [5] opensourcefinancialmodels.com [6] Říský,V. (2011) Application for Processing Statistical Data and Option Pricing by Trinomial Model
Master Diploma Thesis University of Finance and Administration, Prague, Czech Republic www.vsfs.cz [7] Říský,V. (2013) Application for Processing and Option Pricing by Trinomial Model In Proceedings (OEDM SERM 2012) Bratislava: Curriculum, 6 ps. ISBN 978-80-904948-4-8
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D5. From Algorithm to Software for Quadrinomial Option Pricing D5.1. Summary of Chapter This chapter (see M.Soucek, 2013) is dedicated to describe the software for using quadrinomial model for evaluating options. When I was creating this software I kept the focus on friendly user interface and on synoptic graphic representation of model. The software is created using high-level object-oriented programming language C#. Graphic design is provided by Windows Forms. Data storage technology ensures ADO.NET.
D5.2. Introduction As mentioned this chapter concerns on the software which I created as a part of my diploma thesis (see M.Soucek (2012)). Main purpose of this software is to quickly calculate and clearly display quadrinomial option pricing model of call and put options. On the internet can be found a couple of programs dealing with the simpler binomial model (from which is quadrinomial model based on). However, freely available software for use of trinomial or quadrinomial option pricing model does not exist. These types of software can be found in some institutions (mostly banks), but no one without connections to them can’t have access to the software. The main goal is to create software that will apply quadrinomial model for the evaluation of call and put options. Emphasis is placed on a clear graphical representation of the model. To achieve this, is needed to create an algorithm for the application of quadrinomial option pricing model, which is constructed using quadrinomial trees. The program also includes a warm intuitive and user environment in which users can be able to choose quadrinomial tree according to their wishes. Finally, these models will be saved, to be able quickly reload them when the software is restarted later on. Of course there is also user guide. The main goal of this chapter is to introduce the description of the algorithm of quadrinomial model used in this software. Based on defined objectives and a description of the current state of software development for use quadrinomial model for the valuation of call and put options is no longer possible to formulate hypotheses which will validate by the thesis: H1: Extending the algorithm application binomial pricing model of call and put options can create applications of quadrinomial algorithm model. H2: By projection of software for the use of binomial option pricing model, using discrete structures multinomial division can create software for use of quadrinomial option pricing model, including graphic module. Software is created using high-level object-oriented programming language C#. As the development environment is used Microsoft Visual C# Express Edition 2010th Program uses
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the forms that are created by using the Windows form applications (Windows Forms). Saving models will ensure ADO.NET database technology.
D5.3. Algorithm of Quadrinomial Model The main purpose of quadrinomial model is to compile these three quadrinomial trees: Quadrinomial tree of share prices Quadrinomial tree of call/put option prices Quadrinomial tree of call/put option inner values The software creates all these trees simultaneously including both call options and put options. All three trees are merged in one complex tree. Every node contains these values: Share value
S in, kj ,k ,
Probability
in,kj ,k
Call option
Cin, kj ,k ,
Put option
Pi ,njk,k
Internal value of call option
IVCin, kj ,k ,
Internal value of put option
IVPi ,njk,k
Quadrinomial model calculation is performed in three steps: STEP 1.: Calculation of the probability parameters and growth/loss parameters STEP 2.: Creation of quadrinomial tree STEP 3.: Calculate the missing option prices values Quadrinomial model is based on the assumption that all parameters in function f = (S, X, q, u, d, n) are known.
STEP 1: Calculation of the probability parameters and growth/loss parameters In the first step is necessary to calculate the probability parameters and growth/loss parameters. This step varies in dependency of the type of quadrinomial tree. There is therefore one option step for implied autonomously and implied models. Then second option contains an autonomous model. Option 1: Implied and autonomously implied model Parameters calculation according the following formulas:
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qd ud
1 p
I1 = u3
210
uq ud
p1 = p3
I2 = u2d
p2 = 3p2(1-p)
I3 = ud2
p3 = 3p(1-p)2
I4 = d3
p4 = (1-p)3
Option 2: Autonomous model First, we calculate the autonomous binomial model. Then perform its secondary empirical statistical analysis and empirical standard deviations and skewness parameters of indices of growth and loss. Based on the results of this analysis, we obtain the parameters of probability and growth / loss for autonomous quadrinomial model.
STEP 2: Creation of quadrinomial tree In the second step is the tree created from the root. Thus creation of the first node [0,0,0,0] where parameter values are i = 0, j = 0, k = 0, l = 0 and nk = 0 follows these formulas: nk = i + j + k + l
Sin, kj ,k I1i I 2j I 3k I 4l S ni k,i
1 2 ,i3
n k p1i1 p2i2 p3i3 p4i4 i1i2i3
IVCin, kj ,k max( 0, S in, kj ,k X ) IVPi ,njk,k max( 0, X Sin, kj ,k )
Make sure that the node is not already in the tree. If the node is in, there is no need to continue. Otherwise next steps depend on the height of the tree (= value of parameter n): a) If n > nk, then parameter nk is incremented by one. It continues recursively, i.e. similarly to node [0,0,0,0], with the creation of four other related nodes, but with different parameters. This way:
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1st follow-up node has the same values of parameters j, k, l as the original node and the parameter i is incremented by one. (e.g. if the original tree is [0,0,0,0] then this follow-up tree is [1,0,0,0]). 2nd follow-up node has the same values of parameters i, k, l as the original node and the parameter j is incremented by one. (e.g. if the original tree is [0,0,0,0] then this followup tree is [0,1,0,0]). 3rd follow-up node has the same values of parameters i, j, l as the original node and the parameter k is incremented by one. (e.g. if the original tree is [0,0,0,0] then this followup tree is [0,0,1,0]). 4th follow-up node has the same values of parameters i, j, k as the original node and the parameter l is incremented by one. (e.g. if the original tree is [0,0,0,0] then this followup tree is [0,0,0,1]). b) If n = nk, then the death end node was reached. It allows filling up the missing values of call and put option prices, which are given by the following relations:
Cin, kj ,k IVCin, kj ,k
for n = nk,
Pi ,njk,k IVPi ,njk,k
for n = nk.
The tree is almost finished in this step.
STEP 3: Calculate the missing option prices values In this last step are missing values of call option prices and put options computed. So those values that are not in the final nodes (all values in the death end nodes are already calculated in previous step). This time the tree is browsed by levels. Compared to the second step this time is tree browsed from death end nodes to the root, respectively from the nodes whose are direct descendants of the death end nodes. It follows that the starting position is nk = n-1. For each floor are step by step computed all values of each node:
Cin, kj ,k ,l Pi ,njk,k ,l
1 ( p1Cink1, 1j ,k ,l p2Cin, kj 11,k ,l p3Cin, kj ,k11,l p4Cin, kj ,k1,l 1 ) q 1 ( p1 Pin1k ,j1,k ,l p2 Pi ,njk11,k ,l p3 Pi ,njk,k11,l p4 Pi ,njk,k1,l 1 ) q
Then descend to a lower level. We reduced nk by one and values of call and put options are computed same way as in previous level. Repeat this until the root is reached (nk=0). The quadrinomial tree is finally completed right now.
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D5.4. Graphics Interface To describe data in the most complex way, the software displays them as many as possible. Each pictured node thus contains the index, the value list (S: for the tree of stock prices, C: for the tree of call options, IVC: for the tree of internal values of call options, P: for the tree of put options, IVP: for the tree of internal values of put options), the probability, index position (i, j, k, l), the call option status and the put option status. The layout of all these data in the worksheet is best illustrated by the following figures (see Figure 1 and Figure 2 below). Figure 1: Node description
Figure 2: Quadrinomial tree
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D5.5. Conclusion The main objective of this chapter was to create software that will apply quadrinomial model for the valuation of call and put options. This objective has been met in full Furthermore, the algorithm was created to enable applications of quadrinomial model expansion by algorithm of application binomial pricing model of call and put options. On the basis of this result can be considered as hypothesis H1 is confirmed. With this software the algorithm was developed for the use of quadrinomial option pricing model, including graphic module, based on projection from software for use of binomial option pricing model, using discrete structures of multinomial distribution. On the basis of this result can be also considered and confirmed hypothesis H2. Theoretical and practical benefits of the work can be summarized as follows: When creating software I have used modern technology that allows a relatively quick and easy to create the desired software. During the processing I learned a variety of methods and technologies. To create a program of this magnitude has brought me a lot of experience and insights to the field of software development, as well as the pricing. The program allows appreciating faster and better option pricing. Also offers the possibility of graphical representation of quadrinomial and binomial trees. Finally, the application can compare quadrinomial model with binomial. Based on the achieved results I can suggest other possible follow-up work (e.g. grants and collaboration with other experts): Software can be upgraded. In particular, it would be good to optimize secondary empirical statistical analysis of empirical standard deviations and skewness parameters of indices of growth and decline of the autonomous binomial model for calculating autonomic quadrinomial model. Another possibility is the evolution of software development on software for use multinomial option pricing model with the help of multinomial algorithm. A comprehensive comparison of more than two models (in this case, the binomial, quadrinomial) could also be beneficial. Potential can cause conversion of current program to web application or an application designed for mobile devices (mobile phones, PDAs, etc.).
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D5.6. Selected Literature Black,F., Scholes,M.S. (1973) The Pricing of Options and Corporate Liabilities Journal of Political Economy Vol. 81, No. 3, ps. 637-654. Cox,J.; Ross,S., Rubinstein,M. (1979) Option pricing: A simplified Approach Journal of Financial Economics, 7, 229-263. Cyhelský,L.; Kahounová,J.; Hindls, R.(1996) Elementární statistická analýza Praha: Management Press ISBN 80-85943-18-2 CSRG 2011. Curriculum Studies Research Group [online]. c2011 [cit. 2011-08-08] Available on WWW: Dvořák,P. (2008) Deriváty, Druhé přepracované vydání. Praha : Nakladatelství Oeconomica ISBN 978-80-245-1435-2 Gregoriou,G.N., Pascalau,R. (2011) Financial Econometrics Modeling: Derivatives Pricing, Hedge Funds and Term Structure Models Palgrave Macmillan ISBN 978-0-230-28363-3 Hull,J. (2011) Options, Futures, and Other Derivatives 6th Edition, 769 pp, New York: Prentice Hall ISBN 0-13-149908-4 Liberty,J.; Xie,D. (2008) Programming C# 3.0. O'Reilly Media ISBN 0-596-52743-8 Madan,D.B., Milne,F., Shefrin,H. (1989) The Multinomial Option Pricing Model and Its Brownian and Poisson Limits Review of Financial Studies, Oxford University Press for Society for Financial Studies Vol. 2, No. 2, 251-265 ISSN 0893-9454 Pavlat,V., Zaskodny,P. (2012) Od finančních derivátů k opčnímu hedging (From financial derivatives to option hedging) Praha: Curriculum ISBN 978-80-904948-3-1 Petzold,Ch. (2006) Programování MS Windows Forms v jazyce C Brno: Computer Press ISBN 80-251-1058-3 Rebonato,R. (1999) Volatility and Correlation in the Pricing of Equity, FX and Interest-Rate Options Wiley ISBN 0-471-89998-4
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Soucek,M (2012) Software for using quattronomial model for evaluating options Master Diploma Thesis University of Finance and Administration, Prague, Czech Republic Soucek,M (2013) Software for using quattronomial model for evaluating options
In Proceedings (OEDM SERM 2012), 6 ps Bratislava: Curriculum ISBN 978-80-904948-4-8 Virius,M. (2006) C# -- Hotová řešení Brno: Computer Press ISBN 80-251-1084-2 Yamada,Y., Primbs,J. (2004) Properties of multinomial lattice with cumulants for option pricing and hedging Asia-Pacific Financial Markets 11, 335-365 Zaskodny,P., Pavlat,V., Budik,J. (2007) Finanční deriváty a jejich oceňování Praha: VŠFS 2007 ISBN 978-80-86754-73-4 Zaskodny,P., Tarabek,P. (2011) Educational & Didactic Communication 2010 Bratislava: Didaktis ISBN: 978-80-89160-78-5
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D6. From Algorithm to Software in Binomial Delta Hedging Based on Call Option D6.1. Summary of Chapter Since creation of financial tool, such as shares, the investors have paid attention to effective distribution of their investment portfolio between bonds and shares. Shares can be profitable for the investor and furthermore, they can bring more profit in form of dividends. On the other hand, they represent a risk type of financial instrument (see M.Sebest, (2013b)). The basic tool protecting the investor from risk is so called financial derivative. The first simple financial derivatives date back to the 19th century; they were used in agricultural contracts on purchase of crops. These types of contracting of purchasing prices during winter season gave the farmers the possibility to invest and estimate the land area to sowing. This type of business can be compared to one of the basic financial derivatives such as so called forward (see M.Sebest, (2013b)). Since then, much more complicated financial derivatives have been developed. They are used to protect investment portfolios from risks resulting from fluctuations of prices of shares. One of the basic types of financial derivatives is option, and also interest rate derivative. The development of prices in time is often unstable and is liable to more or less significant fluctuations. These changes arise from both stock and off-the-board markets influencing price of asset. Supply and demand after a given asset form its development in time. In analysis of time data, we can often see the possibility to define a certain trend on one side and an attempt to determine the fluctuation component of development of price on the other. While the first component is a result of a long-term trend influenced mainly by the position and strategy of the company, the fluctuation component arises from the market mechanism of balancing supply and demand, future expectations and the like. Investors want to minimize possible loses resulting from a sudden drop in prices of shares. One of effective tools to attain this goal is hedging instrument such as different asset derivative (see M.Sebest, (2013b)). Options represent a type of contract in which their owner has the right (but no obligation) to buy or sell an asset for a price determined beforehand in a given time limit. Thus, option gives no obligation to its owner who is free to use it at their will. If the price of underlying share rises during the life of option, ownership of this option brings us profit. We say that the option is in the money because its exercise brings us profit. On the other hand, if he price of the underlying share falls, this option becomes useless to its owner and there is no reason to exercise it. We say that such an option is out of the money because it cannot bring us any profit. The right but no obligation to exercise an option gives its owner certain advantage. The right to buy itself becomes a value. The caller asks for so called option
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premium for the owner’s right to exercise it in future. The main problem of the theory of financial derivatives is to assess the value of this right so that neither contract party was damaged (see M.Sebest, (2013b)). The protection from risks by an investment to a suitable instrument is called hedging. The investor not only invests into the chosen asset but he also takes a position in a hedging asset that minimizes the changes in price of the investment. A badly chosen hedging instrument can increase the risk of the investment. This protection can be seen as a sort of investment whose profit is lowered risk (see M.Sebest, (2013b)). Effectiveness of the protection depends on correlation between the possessed and the hedging assets. Investors try to find an instrument that would replicate the revenue of their portfolio as accurately as possible and thus lower the risk. Options are one of these instruments (see M.Sebest, (2013b)). Different factors influence changes in price of options, mainly current value of underlying asset, risk-free interest rate, strike price, volatility of underlying asset or remaining time before expiration. In the real market environment, these factors have effect in the same time. In risk management, the factors are usually analyzed separately so change in only one parameter is considered and the rest stays constant (see M.Sebest, (2013b)). Price sensitivity of the option to these factors is measured in hedge parameters. Delta (Δ), gamma (Γ), theta (Θ) a rho (ρ) are belonging among the main parameters. Parameter delta measures sensitivity of value of an option using the price of underlying asset. The investor’s goal is to lower the sensitivity and thus the parameter Δ as much as possible, preferably to zero. A portfolio with zero parameter Δ is called delta neutral and this kind of hedging is called delta hedging. A good instrument to lower Δ to zero is the underlying asset itself because its Δ is one. The value of Δ changes in time so the investor’s position is assured for only a short time period. That is the reason why it is necessary to adjust the hedging (rebalance it) and why it has to be monitored all the time (see M.Sebest, (2013b)).
D6.2. Basic Terms Option hedging is realized with the help of discrete models. We carry out this hedging by creating dynamic hedging schemes and process of rebalancing. Dynamic hedging scheme represents such an overview of composition of option portfolios in different time periods that these option portfolios become secured option portfolios. Rebalancing mans adjusting the hedging. It transforms original secured option portfolios into future secured option portfolios. Dynamic hedging scheme delta and delta rebalancing are closely related to the hedging ratio H. We can speak about dynamic hedging scheme H and rebalancing H. Within the chosen type of random walk, dynamic hedging scheme delta requires a scheme of values of H between different nodes in network that would be created by effectuation of the chosen type of random walk. Delta rebalancing requires realization of changes in hedging on the basis of the given type of random walk and the chosen sequence of nodes. It is not possible to effectuate
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rebalancing continuously, the discrete s-tuple discrete models are very important in this case (see M.Sebest, (2013b)).
D6.3. Creation of Dynamic Hedging Schemes Delta The creation of dynamic hedging schemes delta is based on the sensitivity of the value function V to spot price S within a suitable continuous system of evaluating options. Considering the relatively simple calculation of Delta Greek, it is better to choose the Black-Scholes model. Delta Greek is defined within a suitable continuous model as: (1)
Δ=
∂V ∂S
This partial derivative can be compared to a fixed hedge – that is to say with the hedging within the first time period Δt from n time periods. The fixed hedge is represented by hedging ratio H which is defined for a call option as: (2)
H CO=
Cu− Cd ( u− d )S
This can be expressed using value function V as: (3)
H=
ΔV ΔS
This relation represents a discrete form of the partial derivation given in relation (1). Within the chosen type of random walk, the dynamic hedging scheme delta requires creation of a scheme of values H between neighboring nodes in network which was created by realization of the chosen type of random walk. Random walk is composed of a selection of subsequent values of discrete or continuous quantity that changes in dependence on parameter t. In the field of option hedging, time t is used for parameter t, the random quantities are also used in their discrete form. The nodes in network consist of values of discrete random quantity that change in time (price of the underlying asset, probability of this price), that change in dependence on the parameter t. These values are usually gathered at the end of a specific time period. The set of nodes in network at the end of a time period is called a time layer. Suitable types of random walks can be chosen with help of discrete models of evaluating financial options. The basic discrete models are s-multiple multinomial models. Random walk within a binomial model (s = 2) was used for the software processing. Network and nodes created by binomial random walk constitute a binomial tree with n time layers. A time layers is composed of a set of nodes at the end of the time period Δt.
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Binomial hedging scheme delta also has the form of binomial tree that consists of values
H ik, j ,
where k is the serial number of the time layer and subscript i and j represent two
neighboring nodes in network with the number of growth indexes i, j ≤ k. Hedging ratio for two neighboring nodes in n-th time layer can be calculated by the triangular method using hedging ratios between neighboring nodes from the following time layer. This can be expressed by the relation: k+1 (4) H kj , j+1= p' H k+1 j +1, j +2 +(1− p' ) H j , j+1 1 Another generalization is the calculation of the hedging ratio H 0.1 using the first time layer and hedging ratios from the last n-th time layer: n− 1
(5)
H 10,1 = ∑ n− 1 p ' j (1− p ')n − 1− j H nj , j +1 j j= 0
D6.4. Delta Rebalancing Delta rebalancing requires necessary adjustments in hedging within the chosen type of discrete random walk and the given time sequence of nodes in network. Suitable types of discrete random walks are related to s-multiple multinomial models. In the same way as during the creation of the dynamic hedging scheme, binomial random walk (s = 2) will be examined. Firstly, it is necessary to determine the input data for binomial rebalancing. Rebalancing consists of changes in hedging that are attained by changing the structure of option portfolio. The examined position will be the issuer of call option. The changes in hedging can be realized by changing the number of option contracts (we will work with the reversed value of hedging ratio 1/H) or by changing of the number of underlying shares (we will work with the hedging ratio H). The time τ remaining to the expiration of the option will be divided into n equally long time periods Δt. Index k represents the serial number of the time layer from the total n. Index (j = a) represents the number of time layers from the number of layers k in which the price S of underlying share increased with the growth index u. Value (k – j = b) represents the number of decreases with the decrease index d. The structure of secure option portfolio for node [j, k] = [a, b for k] in binomial network is determined by hedging ratio
H kj.j11 between prices S kj 1 and S kj11 . The secured option portfolio has the same value when the price S kj of underlying share changes into the price S kj 1 (when the price is decreased by the decrease index d) and when it changes into the price S kj11 (when the price rises by the growth index u).
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The binomial delta rebalancing itself for n time periods is done by several steps. The binomial random walk was chosen for the random walk model. Next step is to determine the specific time sequence of nodes in network created by realization of all possible random walks. In general, the development of the price of the underlying share will be determined by the time sequence of nodes jk , k ak , bk for k within a binomial tree with k = 0,1,2,…, n, where
ak bk k . Hedging of the node
jk , k ak , bk
for k is related to the hedging ratio H kj.j11 .
Rebalancing during the transition to the node different ways.
jk , k ak , bk
for k can be done in two
Firstly, by an order to buy or sell the underlying shares for the price S kj , in which case we work with the hedging ration H, or secondly, by order to buy or sell the option contracts for the price
C kj , in which case we work with the reversed value of the hedging ratio 1/H.
D6.5. Selection of Algorithm for Programming Relevant Option Hedging According to Chapter D2 there are four algorithms to distinguish when hedging of option portfolios is concerned: Conceptual algorithm – based on fixed and ratio hedge of option portfolios. Theoretical algorithm – based on structure of deriving of the type of option hedging Practical algorithm – based on financial risk management by a specific option hedging. Model algorithm – based on relations between subsequent models of sequential application of different types of option hedging. In the case of Binomial Option Hedging for Call Option two algorithms will be essential for creation of software – theoretical and practical algorithms. The structure of practical algorithm will be suggested with utilization of general structure from Chapter D2: Practical algorithm – all the described steps should be connected with call option a) Choice of a suitable type of discrete model of evaluation of options, determination of parameters of the specific model, determination of values of the input data, description of the chosen type of random walk. It will be given by binomial model, binomial tree, and by random walk within the binomial tree. b) Determination of sources of growth and decrease indexes from the node of view of the used model, autonomously implied model or autonomous model. It will be given by determination of increase index u and decrease index d in the association with implied and autonomous type of binomial model (see Model Algorithm in Chapter D2).
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c) Calculation of prices of the underlying share and of theoretically correct option prices in nodes of network that was created by realization of the chosen random walk. It will be given by construction of binomial trees for S kj and C kj and by specification of these trees for concrete random walk. d) Preparation of the execution of delta option hedging by choosing the right formulas and relations. It will be given by formulas (1), (2), (3), (4), (5) and by specification of these formulas for selected random walk. e) Creation of a delta dynamic hedging scheme. It will be given by application of formulas (1), (2), (3), (4), (5) on the basis of description of binomial dynamic hedging scheme by means of Chapter D6.3. f) Execution of delta rebalancing. It will be given by application of formulas (1), (2), (3), (4), (5) on the basis of description of binomial delta rebalancing by means of Chapter D6.4.
D6.6. Software Processing of Practical Algorithm of Relevant Option Hedging The algorithm was processed in the Java programming language (see M.Sebest 2013a). All the steps of the option hedging practical algorithm were executed in it. Below are given results of processing of all the steps of the algorithm:
a) Choice of a suitable type of discrete model of evaluation of options, determination of parameters of the specific model, determination of values of the input data, description of the chosen type of random walk.
a1) Determination of binomial model parameters The binomial model of evaluation of financial options was chosen for the software processing. We suppose that the issuer of the option called the call option on an underlying share. The aim of using the binomial model is to secure the position of the issue of the call option by option delta hedging. The chosen type of random walk is the binomial random walk. The result of all specific binomial random walks is a binomial tree, binomial network of nodes. a2) Determination of binomial model parameters Binomial model to evaluate financial option is related to the functional dependency of the theoretically correct price on parameters:
6
C = f S, X,u,d,q,n
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a3) Assigning and determining values of input parameters Following values were entered: S = 100 cu (currency unit), X = 100 cu, τ = 6 months, r = 5% p.a., n = 3, there is either a rise of underlying share price by 15,32% or decrease by 13,29% for every time period. Values of input parameters have following meanings: S = 100 cu – spot price S for which the shares are traded in the moment, X = 100 cu – strike price X as a forward price of the share, τ = 6 months – time to expiration of the option r = 5% – annual risk-free interest rate, n = 3 – the number of equally long time periods Δt into which the time period Δt, is divided τ Δt = 2 months u = 15,32%, d = 13,29% - respective indexes of growth and decrease of the price of the underlying share in one time period Δt. After that, it is necessary to convert the given values into suitable units.
S = 100 cu, X = 100 cu, τ = 0,5 year, Δt = 1/6 year = 0,1667 year, r = 5% = 0,05, q = 1 + rΔt = 1,0083, n = 3, u = 1,1532, d = 0,8671, p = 0,4937, 1 – p = 0,5063
b) Determination of the origin of growth and decrease indexes considering the implied model or the model autonomously implied model or the autonomous model. The product ud = 1, the binomial model is the implied model.
c) Calculation of prices of the underlying share and the theoretically correct prices of option in the nodes of the network that was created by realization of the chosen type of random walk. A binomial tree was created by realization of random walk. The tree has three time layers. By subsequently assigning into relations:
7
S j = u j d n j S,S kj = u j d k j S
8 9
C j = max 0,S j X
C kj =
1 1 k+1 pC k+ j+1 + 1 p C j p
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We obtained following values: Fig. 1: Spot prices of underlying share
Fig. 2: Prices of call option
The last value C00 = C = 11,7905 is also theoretically correct price of the examined call option.
d) Preparation of execution of delta option hedging by choosing suitable formulas and relations. In order to build the delta binomial hedging scheme, we need to consider following relations:
10
1 H k+ j, j+1
=
1 k+1 C k+ j+1 C j 1 k+1 C k+ j+1 C j
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1 k+1 H kj , j 1 = p'H k+ j+1,j+2 + 1 p' H j, j+1
The values inserted into the relations are based on binomial trees of prices of the underlying share and of the prices of call option in different nodes of binomial network. e) Creation of delta dynamic hedging scheme The results of calculation are as follows: Fig. 3: Hedging ratios
f) Execution of delta rebalancing Let the issuer of call option sell 105 option contracts. Let a specific binomial random walk be chosen; it consists of a time sequence of nodes [j, k]. The first component j represents the number of rises of price of the underlying share with growth index, the second component k represents a specific time layer of the binomial network. Let the time sequence of nodes which form the random walk, be as follows:
0,0 0,1 1,2 2,3 Rebalancing consist in creating secured option portfolios to hedge different nodes of the chosen specific binomial random walk. 1 = 0,5819 . The f1) The initial node [0,0] is in time t = 0 secured by the hedging ratio H 01 structure of secured option portfolio:
[105 call options –, 53 000 shares +] The structure of the portfolio corresponds with sale of 105 option contracts and purchase of 58190 underlying shares. The loan at the beginning of the first time period given by the relation
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HdS q
Sale of 105 option contracts with price C00 = 11,7905 leads to obtain of 1 117 905 cu. The loan is given by the relation
13
1 1 1 H 01 S0 C01 H 01 dS C01 = q q
It leads to the sum for one option contract 46,4012 cu. Thus, for 105 option contracts, the final sum will be 4 640 012 cu. The total sum of 5 819 917 cu is enough to buy 58 190 underlying shares whose price is S = 100 cu for one underlying share. 2 f2) The initial node [1,1] in time t = Δt is secured by the hedging ratio H12 = 0,7976 . The structure of secured option portfolio:
[105 call options – , 79 760 shares + ] The structure of the portfolio corresponds with purchase of additional 21 570 underlying shares. The additional purchase of 21 570 shares will be funded by first giving back the loan of 4 640 012 cu multiplied by interest rate q = 1 + rΔt = 1,0167 for one time period. It means 4 717 500 cu would be returned. The amount needed to pay back the loan can be obtained by selling 40 907 underlying shares. Price of one underlying share is S11 = 115,32 cu. Than another loan would be taken using relation (13) again in the version for node [1,1] and purchase of additional 21 570 shares. 3 f3) The initial node [1,2] in time t = 2Δt is secured by the hedging ratio H12 = 0,5353 . The structure of secured option portfolio:
[105 call options – , 53 530 shares + ] The structure of the portfolio corresponds with sale of 26 230 underlying shares. Price of one share when sold would be S12 = 99,9940 cu. Funding is again based on paying back the loan from the second time period and it is also necessary to sell 26 230 shares to create a new secured option portfolio. f4) Node [2,3] in time t = 3Δt represents the end of the specific binomial random walk through binomial network for three time periods.
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D6.7. Conclusion The procedure “From Practical Algorithm to Software in Binomial Delta Hedging Based on Call Option” was realized by means of sub-chapters D6.1. to D6.6. This procedure can be according to sources structured into three levels i), ii), iii): i) From Practical Algorithm ii) to Software
(see P.Zaskodny, V.Pavlat (2012) and see Chapter D2) (see Java Programming Language used in M.Sebest (2013a))
iii) in Binomial Delta Hedging Based on Call Option
(see J.C.Hull (2009)).
In sub-chapter D6.6., above mentioned procedure was substantiated by means of concrete illustrating example. The dynamic hedging scheme was represented in concrete shape, the delta rebalancing, connected with chosen random walk by binomial tree, was realized. The transformation of “Practical Algorithm” to “Software by Java Programming Language” was successfully finalized in M.Sebest (2013a).
D6.8. Selected Literature Zaskodny,P., Pavlat,V. (2012) Od finančních derivátů k opčnímu hedging (From Financial Derivatives to Option Hedging) Praha: Curriculum ISBN 978-80-904948-3-1 Ševčovič,D., Stehlíková,B., Mihula,K. (2009) Analytické a numerické metódy oceňovania finančných derivátov Bratislava: Nakladateľstvo STU, First Edition ISBN 978-0071389976 Hull,J.C. (2009) Options, Futures and Other Derivatives Upper Saddle River: Pearson Prentice Hall, Seventh Edition ISBN 978-01-360-1586-4 Zaskodny,P., Pavlat,V., Budik,J. (2007) Finanční deriváty a jejich oceňování Praha: Eupress ISBN 978-80-86754-73-4 Sebest,M. (2013a) Software Processing of Binomial Delta Hedging Based on Call Option Master Diploma Thesis University of Finance and Administration, Prague, Czech Republic Sebest,M. (2013b) Software Processing of Binomial Delta Hedging Based on Call Option, 6 ps In Proceedings (OEDM SERM 2012) Bratislava: Curriculum ISBN 978-80-904948-4-8
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D. Closing Orientation Three goals of Part D were fulfilled by following way: - The first goal and the second goal were connected with the delimitation the algorithms for option pricing and option hedging. The most useful algorithm, linking both algorithm of option pricing and algorithm of option hedging, was given by Practical Algorithm. The Practical Algorithm was described in Chapter D2 by P.Zaskodny and I.Havlicek and justified in Chapter D1 by V.Pavlat. - The third goal was connected with the way how to refill the most useful algorithm (Practical Algorithm) by software for option pricing and option hedging. The relation between algorithm and software was generally described in Chapter D3 by M.Pasta. The concrete programming languages were applied in Chapters D4, D5 for option pricing by V.Risky and M.Soucek. The concrete programming language was applied in Chapter D6 for both option pricing and option hedging by M.Sebest.
The completion of the first goal and the second goal can be reminded through the structure of Practical Algorithm, defined by the following series of steps (a) to (i): (a) Selecting a suitable type of a discrete option pricing model, defining the parameters of the model selected, determining the values of the input parameters (growth and decline indices, in particular) and describing the selected type of random walk. (b) Defining the origins of the growth and decline indices in terms of their links to the implied model or the autonomously implied model or the autonomous model. (c) Calculating underlying stock prices and theoretically correct option prices at the nodes of a lattice resulting from the implementation of a selected type of random walk. (d) Selecting the appropriate equations and formulae to prepare the implementation of a delta hedge. (e) Developing a dynamic delta hedge strategy. (f) Delta rebalancing. (g) Selecting the appropriate equations and formulae to prepare the implementation of a gamma hedge. (h) Developing a dynamic gamma hedge strategy. (i) Gamma rebalancing.
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The follow-up works within the first and second goal could be linked with developing a dynamic vega hedge strategy and vega rebalancing in the field of underlying financial instrument volatility. To enrich Practical Algorithm by vega hedging would constitute a major contribution to the theory of financial and commodity markets.
The completion of the third goal can be reminded through used programming languages: i) Object-oriented programming language C++ with graphic design KDE and with OS Linux (used in Chapter D4 for software creation within trinomial option pricing) ii) High-level object-oriented programming language C# with graphic design by Windows Forms and with data storage technology by ADO.NET (used in Chapter D5 for software creation within quadrinomial option pricing) iii) Java programming language (used in Chapter D6 for software creation within binomial option hedging based on call option) The follow-up works within the third could be linked with application of mentioned programming languages partly to the pentanomial option pricing, partly to the trinomial, quadrinomial and pentanomial option hedging.
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Epilogue The monograph “Educational & Didactic Communication 2013 (Vol.1: Algorithms as Significant Result of Data Mining Approach)” has operated the applications of educational data mining approach (see Part C) and the applications of general data mining approach (see Part B and Part D). The conception of this monograph has represented the concrete attempt to postulate the joint base of educational and general data mining approach (see Part A). Such joint base has been given by the detection of algorithms enabling problem solving. Part C was connected with problem solving from the area of educational sciences. On the basis of content pedagogy theory the science education was selected as a representative of the area of educational sciences. The problems solving has had the character of mediated solution of problems. The role of mediator should belong to suitable educator, the role of education addressee can be connected with pupil, student, etc. Cumulatively, the addressee of education is any person interested in mediated cognition. Part B and Part D were connected with problem solving from the area of two concrete sciences – statistics and financial derivatives theory. The problems solving has had the character of real solution of problems. The researcher of identified problem can be defined as any person interested in the cognition of concrete scientific branch – in presented monograph in the cognition of statistics and financial derivatives theory. Mediated solution of problems and real solution of problems – the joint base of seemingly different procedures has been determined in Part A of presented monograph. This joint base, how it is mentioned above, has been given by the detection of algorithms enabling problem solving. The creation of needful algorithms was connected with the realization of three stages and it was not important if the relevant problem was solved by mediated or original way. The first stage was given by the construction of analytical synthetic model of solved problem (see Chapter A5 of Part A). The second stage was given by the construction of matrix model of solved problem (see Chapter A6 of Part A). The third stage was given by the description of groups if definition line elements of matrix model (see Chapters A7, A8, A9 of Part A). This definition line of respective matrix has expressed the detected algorithm. Analytical synthetic modeling was taken as the significant partial tool of data mining approach, the matrix modeling was taken again as the significant partial tool of data mining approach. The joint base of mediated solution of problems and real solution of problems was clarified. The consequences of mediated and real solution should be differentiated. In the field of real solution of problems the detected algorithm is very often representing the substantiation for programming. The appropriate programming language must be applied. In the field of mediated solution of problems the detected algorithm is predominantly representing the substantiation for textbook creation. The basic result of submitted monograph “Educational & Didactic Communication 2013 (Vol.1: Algorithms as Significant Result of Data Mining Approach)” can be expressed very briefly. This result is given by the joint base of educational data mining approach and general data mining approach (the definition line of matrix as the sought algorithm). This result is also connected with the different way of its application.
