Vegetatio 82: 59-67, 1989. © 1989 Kluwer Academic Publishers. Printed in Belgium.

59

Multiscale ordination: a method for detecting pattern at several scales Jay M. Ver Hoef & David C. Glenn-Lewin Botany Department, Iowa State University, Ames, fA 50011-1020 USA Accepted 21.2.1989

Keywords: Association, Eigenanalysis, Landscape unit, Multivariate, Spatial analysis Abstract

Multivariate analyses of vegetation data have been restricted to a single scale of san1pling, or multiscale sampling has been restricted to a single species. However, vegetation scientists need to be able to explore spatial relationships of many species over many scales. We present a n10dification of Noy-Meir & Anderson's (1971) method of multiscale ordination by summing two-term local covariance matrices and smoothing the component profiles. The advantages of our method are: 1) results are less subject to the starting position of the transect, 2) n1atrices may be added at any block size, and 3) plots of factor scores are smoothed by a moving weighted average to better reveal patterns at a prescribed scale. This procedure provides statistical associations of species over a range of scales. The scales which exhibit the association to the maximum extent are then determined from n1ultiscale ordination. The relationships of different associations and their scales can then be examined. The application of the method to fabricated data proved successful in recovering the structure built into the data. When used on real vegetation data, from a community and a landscape, the n1ethod revealed the details of species associations over a range of scales, and of the relationships among associations. Abbreviations: PCA = Principal Con1ponents Analysis; DCA TTLC = Two-Term Local Covariance.

=

Detrended Correspondence Analysis;

Nomenclature: Gleason, H.A. & Cronquist, A. 1963. Manual of vascular plants of northeastern United States and adjacent Canada. D. Van Nostrand, New York.

Introduction

Techniques for examining associations of species (or other attributes) as a function of scale, and for detecting multispecies patterns, have emerged in the last two decades (e.g., Noy-Meir & Anderson 1971; Whittaker & Naveh 1979; Bouxin & Gautier 1982; Bouxin 1983; Galiano 1983; Gibson & Greig-Smith 1986). For a period, technical innovation preceded theoretical, resulting in methods and data that usually were case specific.

The recent development of hierarchy theory (Allen & Starr 1982) and patch dynamics (White & Pickett 1985) provides a broader theoretical basis for investigations of pattern, and emphasizes the importance of different scales in community structure. Our purpose is to introduce a method for analyzing the multiple, overlapping scales of patch structure in vegetation. For the single species case, Greig-Smith (1952) introduced a technique that en1ployed an analysis of variance of a nested grid of quadrat data. Sub-

60 sequent modifications have both expanded its applicability and made it more flexible (Kershaw 1957; Kershaw & Looney 1985; Usher 1969; Hill 1973). On the other hand, techniques such as Principal Components Analysis (PCA), Detrended Correspondence Analysis (DCA, Hill & Gauch 1980), and Non-Metric Multidimensional Scaling (Prentice 1977; Kenkel & Orl6ci 1986) provide ordinations for sample units of fixed scales, but their results depend on the scale at which the data were collected. It is apparent, however, that several environmental and biological factors may operate in a community, causing associations at various scales. Castro et al. (1986) showed how the association of species (as indicated by similar PCA scores) changed as plot size increased. Because associations vary depending on the size of the sample unit, the detection of many associations and patterns are not possible by conventional ordination techniques. If we accept that there are many factors or processes operating at different scales on the structure of vegetation, then we need to detect the associations with their related scales and their spatial patterns. A method for examining such multispecies, multiscale structure should result from a union of ordination for a fixed-size sample unit and a blocking technique in order to change the scale of the data. Our objective in this report is to introduce such a technique. Our method is built upon Noy-Meir & Anderson's (1971) multiscale ordination technique, and we demonstrate its utility with examples of fabricated, withinconlnlunity, and landscape data. The fabricated data set (Fig. 1) consists of 5 'species' in a 25 quadrat strip transect. The within-conlmunity data set conles from a basalt glade prairie community along the St. Croix River, which forms the Minnesota/Wisconsin border in Species 1 Species 2 Species .3 Species 4 Species 5

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20

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Fig. 1. Raw transect data for fabricated data set.

25

north central USA (Glenn-Lewin & Ver Hoef 1988). The cover presence for each species was recorded in each of 300 10 x 10 cm contiguous quadrats. The landscape ecology data set comes from an 8 km line transect through a hilly wooded and pasture landscape in central France, and is given by Forman & Godron (1986; p. 209, the 'Nante' 7 line). They divided the transect into 128 equal segments and noted presence for 7 different landscape units.

Multiscale ordination What follows is a brief description of the technique. Mathematical details may be found in the appendix. Data are collected from contiguous quadrats. In our field studies, we used 10 cm by 10 cnl quadrats in a transect 30 m long. From these contiguous quadrat data, covariance nlatrices for all species pairs are calculated for each block size using Greig-Smith's (1983) two-term local covariance (TTLC) statistic. This statistic computationally blocks the quadrats into increasingly larger sizes (in our case, block size 1 = 10 cm by 10 cm, block size 2 = 10 cnl by 20 cm, block size 3 = 10 cm by 30 cm, etc.). We blocked up to a size equal to one-third of the total transect length. The rest of the analysis then follows the original method of Noy-Meir & Anderson (1971). The use of TTLC solves two problems: 1) like Hill's (1973) two-term local variance statistic, results are less subject to the starting position of the transect, and, 2) covariance matrices may be calculated at any block size, rather than being limited to those of a geometric progression. The covariance matrices for each block size are summed to get one total covariance matrix. Eigenvectors and eigenvalues are found for the total covariance matrix. The eigenvalues may then be partitioned into the amount that each block size contributed to that eigenvalue. These partitioned eigenvalues are plotted against block size, and, in a manner analogous to the original single species technique (Greig-Smith 1952), the peak of such a curve reveals the scale at which the pattern was

61 most pronounced. One does not expect species or environmental patterns to be of exactly repeating din1ension, nor perfectly cyclical, yet the peak of the curve provides an 'optimum perspective' for detecting that pattern. The eigenvector provides species scores, and species with similar scores tend to be associated to the greatest extent at the scale indicated by the partitioning of the eigenvalues. To help visualize the behavior of the vegetation along the transect, a moving weighted average can be calculated. Noy-Meir & Anderson (1971) calculated a 'component profile' by multiplying the amount of each species in each quadrat by that species' eigenvector score, and adding these for each quadrat to give one quadrat score. All quadrat scores were then plotted by position (Whittaker & Naveh 1979 called this procedure a 'trace'). We calculate a moving average of the quadrat scores by first blocking the quadrats to the optimum perspective as indicated from the partitioning of the eigenvalues, and then calculating the average blocked quadrat scores, moving sequentially through the data one frameshift at a time. This removes much of the smallerscale noise and smooths the curves for the scale indicated.

(A) 3

2

(])

~

1

o > c

/e

Fig. 1 shows a fabricated data set with 3 primary features. First, species 1 through 3 show a general trend; species 2 occurs in higher abundance to the left and species 1 and 3 are more abundant to the right. Second, species 4 and 5 occur at alternating patches of block size 3. Finally, although species 1 and 2 have opposite trends over the whole transect, they occur together with species 4 at the scale of block size 3. The first 2 partitioned eigenvalues were plotted for the fabricated data set (Fig. 2). For the first eigenvalue, there is a continual increase, i.e., each block size contributed more than the previous one (Fig. 2A). Correspondingly, species 1 and 3 have high positive eigenvector scores, while species 2 has a low negative eigenvector score (Table 1).

e

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/

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6

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10

Block Size

Fig. 2. For the fabricated transect data, the first eigenvalue (A) and second eigenvalue (B) are partitioned as described in the text.

Table 1. First two eigenvectors for fabricated data set. Eigenvector loading

Species

Results and discussion

/

e

e

e

Eigenvector 1 Species Species Species Species Species

3

1 5 4 2

0.711 0.604 0.019 - 0.019 - 0.358

Eigenvector 2 Species Species Species Species Species

4 2 1 3 5

0.637 0.346 0.262 - 0.015 - 0.637

The moving weighted average scores (Fig. 3A) are negative to the left, indicating a relative abundance of species 2, and become positive to the right, indicating an increasing abundance of species 1 and 3 and a decreasing amount of species 2. Multiscale ordination correctly identi-

62 (A)

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-1.100

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25

Plot Sequence

Fig. 3. Moving average of weighted plot scores along the fabricated data transect. (A) shows the first eigenvector with the moving average at block size 10. (B) shows the second eigenvector with the moving average at block size 3.

tied the overall trend (species 2 vs. species 1 and 3) as the first feature of the data. Block size 3 contributes the most to the second eigenvalue (Fig. 2B), with a secondary peak at block size 9. Species 4 has the highest positive score for the second eigenvector (Table 1), with species 1 and 2 having somewhat lower positive scores. Species 5 has the lowest negative score for the second eigenvector. The moving weighted average scores (Fig. 3B) show that species 4, 2 and 1 (positive scores) occur in cyclical alternation with species 5 (negative scores) at a scale of block size 3. Therefore, multiscale ordination correctly detected the second and third features of the fabricated data set; the alternation of species 4 and 5 at block size 3 and the association of species 1 and 2 with species 4 at this scale. The peak at block size 9 (Fig. 2B) is a resonance effect; that is, at this scale, 2 patches of one type occur with one of the other, creating a secondary pattern which is really dependent on the block size 3 pattern.

Thus, multiscale ordination successfully identified all 3 primary features on which the fabricated data set was constructed. Particularly noteworthy is the way in which it demonstrated that, while species 1 and 2 have opposite overall trends, they are associated at a smaller scale of block size 3. While this at first seems counter-intuitive, it is likely that sonle real plant populations may exhibit this kind of relationship. Each species has a unique evolutionary history and so has a unique response to the many environnlental and biotic factors operating at nlany scales in any given area. A positive correlation among some species for a particular scale does not mean that such a relationship will hold over all scales. As opposed to conventional ordination techniques, for which the sanlples are fixed at some scale, multiscale ordination has the advantage of exposing patterns at several scales. For the basalt glade data, block size 50, corresponding to a patch size of 5 m, contributed the most to the first eigenvalue (Fig. 4A). Crustose lichens have the lowest negative score for the first eigenvector (crustose lichen patch type) (Table 2), (A)

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10 20 30 40 50 60 70 80 90 100

Block Size

Fig. 4. For the basalt glade transect, the first eigenvalue (A) and second eigenvalue (B) are partitioned as described in the text.

63 (A)

Table 2. First two eigenvectors for basalt glade data set. Only the species with the most positive and negative eigenvector scores are given.

0.800

Species

0.400

Eigenvector loading (l)

Eigenvector 1 Polytrichum juniperinum Andropogon scoparius Polytrichum commune Panicum lanuginosum Fruticose lichens Crustose lichens

01 0

0.623 0.280 0.269 0.170 0.155 - 0.612

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Eigenvector 2 Andropogon scoparius Panicum lanuginosum Polytrichum commune Cladina rangiferina Fruticose lichens Polytrichum juniperinum

0.000

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2

0.428 0.391 0.305 - 0.321 - 0.363 - 0.415

0.000

-0.700

a

50

100

150

200

250

300

Plot Sequence

in contrast to virtually all other species. The crustose lichens occurred where the basalt outcrops were exposed and there was no soil development. The nloving weighted average scores are negative in 3 regions along the transect (Fig. 5A), corresponding to the positions of the basalt outcrops covered with crustose lichens. Notice that the patch size of crustose lichens is usually less than 5 m, while the interpatch regions, composed of all other species, are somewhat larger than 5 nl. Multiscale ordination provides an intermediate scale for the detection of both patch types. The partitioning of the second eigenvalue peaks at block sizes 45 and 90 (Fig. 4B). The corresponding eigenvector contrasts primarily Andropogon scoparius, Panicum lanuginosum, and Polytrichum commune (graminoid patch type), having positive scores, with Polytrichumjuniperinum, fruticose lichens, and Cladina rangiferina (cryptogam patch type) at the negative end (Table 2). The moving weighted average scores for this eigenvector (Fig. 5B) may be compared with that for the first eigenvector (Fig. 5A). It is apparent that the patches between the basalt outcrops

Fig. 5. Moving average of weighted plot scores along the basalt glade transect. (A) shows the first eigenvector with the moving average at block size 50. (B) shows the second eigenvector with the moving average at block size 45.

are dominated almost exclusively by either the graminoid or cryptogam patch type. This accounts for the partitioned eigenvalue peak at a scale of 4.5 m, which is very close to that of the first eigenvalue. Notice that the moving weighted average scores for the second eigenvector are close to zero over areas where the first eigenvector is negative. This is reasonable since none of the species with high negative or positive scores on the second eigenvector grew on the basalt outcrops. The peak at block size 90 for the second partitioned eigenvalue is a secondary pattern, corresponding to a patch of 9 m. This can be visually determined from the nloving weighted average scores for the second eigenvector (Fig. 5B). If the quadrats were 9 m rather than 4.5 m, plotting the scores would smooth the minor peaks; scores would begin around 0, rise slightly, then become negative through the center of the transect, and finish positive. This secondary pattern is a larger scale effect, with the graminoid species pre-

64 dominant at the ends of the transect, and the cryptogam species more abundant toward the center. The performance of multiscale ordination for the basalt glade data can best be evaluated by how readily the results have ecological interpretations. We were able to identify a mosaic of 3 patch types on the transect, the species which compose the patches, the appropriate scales over which the mosaic of patches occur, and the relationships of the patches. It should be noted from this example that, while merely graphical, plotting the moving weighted average scores is a useful and necessary part of the technique, since information can be gained about the relationships of the patches. Multiscale ordination has provided a powerful means to dissect the component patterns of this particular basalt glade prairie. For the landscape ecology data, partitioning the first eigenvalue indicates peaks at block size

(A)

3

0 (])

(8)

0.750

> c (])

(J'l

W (])

c

Pastures Homes Streams and rivers Shrubland Dirt roads Paved roads Woods

0.200

Shrubland Paved roads Streams and rivers Homes Dirt roads Pastures Woods

0.100

Eigenvector 3

L

0

Eigenvector loading

-

0.767 0.030 0.016 0.021 0.024 0.043 0.640

-

0.869 0.251 0.176 0.008 0.150 0.207 0.293

-

0.733 0.236 0.021 0.029 0.244 0.323 0.493

Eigenvector 2 0.250

+i +i 0...

Landscape Element

0.500

-0

0

Table 3. First 3 eigenvectors for "Nante 7' (Forman & Godron 1986) landscape data set.

Eigenvector 1

2

::J 0

12 (0.75 km) and 44 (2.75 km) (Fig. 6A). The first eigenvector contrasts pastures and woods (Table 3). The 0.75 km perspective indicates a small-scale process such as average pasture or pasture cluster size. These patterns can be seen in the moving weighted average (Fig. 7A). The 2.75 km perspective, the largest tested, shows an inverse trend resulting from the large-scale gradient from more urban on the left to more wild on the right. The second partitioned eigenvalue peaks at block size 24 (1.5 km) (Fig. 6B). Shrubland is contrasted to pastures and woods on the second eigenvector (Table 3). The regions of abundant shrubland are shown by high scores for the moving weighted average (Fig. 7B).

0.000 (C)

0.300

0.000 0

10

20

30

40

Block Size

Fig. 6. For Forman & Godron's (1986) "Nante 7' transect, the first (A), second (B), and third (C) eigenvalues are partitioned as described in the text.

Paved roads Homes Pastures Woods Shrubland Streams and rivers Dirt roads

65 (A)

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Q)

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32

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Plot Sequence

Fig. 7. Moving average of weighted plot scores along Forman & Godron's (1986) 'Nante 7' transect. (A) shows the first eigenvector with the moving average at block size 12. (B) shows the second eigenvector with the moving average at block size 12. (C) shows the third eigenvector with the moving average at block sizes 2 and 38,

There are two peaks for the third partitioned eigenvalue, at block sizes 2 (0.125 km) and 38 (2.375 km) (Fig. 6C). The third eigenvector shows association between paved roads and homes, and contrasts them to an association of dirt roads, streams, and rivers (Table 3). The 0.125 km perspective indicates the small scale association ofindividual honles with paved roads, and can be seen as the high positive scores in Fig. 7C. The 2.375 km perspective indicates that the large scale process is once again a gradient of the more urban area on the left to the more wild area on the right. Fig. 7C also demonstrated how the moving average at block size 38 smooths the smaller scale noise at block size 2. A comparison of the moving weighted average scores gives a more complete picture of the relationships of the patterns indicated by the first three eigenvectors. At the extreme left, there are

pastures in an agricultural, rural setting. Then, there are high anlounts of shrubland, houses, paved roads, and woods followed by more pastures and shrubland. Finally, toward the right, woods become predominant, with fewer areas of pasture. There are several processes, occurring over a range of scales, which produce the composite pattern, ranging fronl the small scale association of homes and paved roads to the large scale gradient from urban-agricultural on the left to nl0re wild on the right. In fact, with their firsthand knowledge of the area, Forman & Godron (1986) concluded that paved roads and homes are situated to the left, which is an area of human activity. Surrounding this is an area of land abandonment where there is invasion by shrubland. Forests predonlinate to the right, where a national forest is located. Multiscale ordination makes explicit the scale of the processes producing the observed landscape patterns, and demonstrates that there are several processes occurring over a range of scales. Multiscale ordination proved to be successful in all three exanlples. Other recent nlethods that have attempted to quantify nlultispecies patterns (Whittaker & N aveh 1979; Galiano 1983; Gibson & Greig-Smith 1986) are all similar in that they perform ordination (Reciprocal Averaging, DCA, etc.) before pattern analysis. That is, by ordinating on the smallest plot size, the association structure is fixed for only that scale, and anyone of several pattern analysis techniques is performed on weighted average quadrat scores based on the ordination. The patterns of these associations can then be examined in relationship to ecological or environmental phenomena. As Castro et ale (1986) and others have demonstrated, however, associations change with increasing plot size. Multiscale ordination, in contrast to other methods, examines associations by explicitly changing the scale ofthe data, then ordinating and partitioning the ordination in search of the scale which contributed the most to that ordination. In comparison then, multiscale ordination is scale explicit, while other techniques are pattern explicit, and hence our choice of nomenclature. The technique is flexible. For instance, three-

66 term local covariance (see three-term local variance, Greig-Smith 1983) can be used to construct the covariance matrices. These would be less sensitive to quadratic trends in the data. Other detrending methods could also be used. One might also wish to weight the covariance matrices in some way before adding them. This nlight be useful in one case: it appears that the matrices at larger block sizes inherently contribute more to the partitioning of the eigenvalue, due to the fact that the larger block sizes have large, squared values. Simulations based on random runs, where no pattern is built into the data, will identify the rate at which the larger block sizes contribute to the partitioned eigenvalue. This rate then could be removed by weighting the covariance nlatrices inversely proportional to the rate of increase for that block size. Multiscale ordination is a successful means of assessing species associations over a range of scales, and for examining relationships among associations. Multiscale ordination recovered the assQ.ciation structure built into a fabricated data set, and provided ecologically meaningful results from real within-community and landscape data sets. It associates species through ordination and finds the optimum perspective at which the association occurs. If the association is the result of a comnlon response to some process or factor, or combination offactors, then the association scale is likely to be the same as that at which that process or factor operates. A plot of the superimposed moving weighted average scores provides a useful means of exanlining the scale and pattern of the associations, as well as how patches at similar or various scales are related.

Acknowledgements We appreciate the comments of Noel Cressie. Part of this research was supported by the United States National Park Service and by the North Atlantic Treaty Organization (NATO).

References Allen, T.F.H. & Starr, T.B. 1982. Hierarchy. Perspectives for ecological complexity. Univ. Chicago Press, Chicago. Bouxin, G. 1983. Multi-scaled pattern analysis: An example with savanna vegetation and a proposal for sampling design. Vegetatio 52: 161-169. Bouxin, G. & Gautier, N. 1982. Pattern analysis in Belgium lin1estone grasslands. Vegetatio 49: 65-83. Castro, I., Sterling, A. & Galiano, E.F. 1986. Multi-species pattern analysis of Mediterranean pastures in three stages of ecological succession. Vegetatio 68: 37-42. Forman, R.T.T. & Godron, M. 1986. Landscape ecology. 10hn Wiley and Sons, New York. Galiano, E.F. 1983. Detection of multispecies patterns in plant populations. Vegetatio 53: 129-138. Gibson, D.J. & Greig-Smith, P. 1986. Community pattern analysis: A method for quantifying community mosaic structure. Vegetatio 66: 41-47. Glenn-Lewin, D.C. & Ver Hoef, I.M. 1988. Prairies and grasslands of the St. Croix National Scenic Riverway, Wisconsin and Minnesota. Prairie Naturalist 20: 65-80. Greig-Smith, P. 1952. The use of random and contiguous quadrats in the study of the structure of plant communities. Ann. Bot. 16: 293-316. Greig-Smith, P. 1983. Quantitative plant ecology. 3rd ed. Blackwell Sci. Publ., Oxford. Hill, M.O. 1973. The intensity of spatial pattern in plant communities. 1. Ecology. 61: 225-235. Hill, M.O. & Gauch, H.G. 1980. Detrended correspondence analysis: An improved ordination technique. Vegetatio 42: 47-58. Kenkel, N.C. & Orl6ci, L. 1986. Applying nonmetric multidimensional scaling to ecological studies: Some new results. Ecology 67: 919-928. Kershaw, K.A. 1957. The use of cover and frequency in the detection of pattern in plant con1munities. Ecology 38: 291-299. Kershaw, K.A. & Looney, I.H.H. 1985. Quantitative and dynamic plant ecology. Edward Arnold, London. Noy-Meir, I. & Anderson, D.l. 1971. Multiple pattern analysis, or multiscale ordination: Towards a vegetation hologram? In: Patil, G.P., Pielou, E.C. & Waters, W.E., (eds), Many species populations, ecosysten1s and systen1s analysis. Stat. Ecol. Ser. 3: 207-231. Pennsylvania State Univ. Press, University Park. Prentice, I.C. 1977. Non-metric ordination methods in ecology. 1. Ecology 65: 85-94. Usher, M.B. 1969. The relation between mean square and block size in the analysis of similar patterns. 1. Ecology 57: 505-514. White, P.S. & Pickett, S.T.A. 1985. Natural disturbance and patch dynamics: An introduction. In: Pickett, S.T.A. & White, P.S. (eds), The ecology of natural disturbance and patch dynamics: 3-13. Academic Press, Orlando, Fl. Whittaker, R.H. & Naveh, Z. 1979. Analysis of a two-phase

67 pattern. In: Patil, G.P. & Rosenzweig, M.L., (eds), Contemporary quantitative ecology and related econometrics. Stat. Ecol. Ser. 12: 157-165. Internat. Coop. Publ. House, Burtonsville, Md.

Now, (4) can be written v/Cv = I.

(6)

Substituting (3) into (6) yields

I

Appendix Covariance matrices are calculated for any block size using Greig-Smith's (1983) two-term local covariance statistic, a modification of Rill's (1973) two-term local variance statistic; e.g., for all species pairs, for block size 1 calculate:

for block size 2: Ave. of {*(x 1 + X2 - X3 - X4 )(Yl + Y2 - Y3 - Y4)' *(x2 + X3 - X4 - XS )(Y2 + Y3 - Y4 - Ys), etc.}

(2)

and so on for larger block sizes. Let C i be the two-term local covariance n1atrix for all species pairs at block size i, i = 1, 2, ... n. Let (3) Eigenvectors and eigenvalues are then computed for C. Any eigenvalue I and its associated eigenvector v for C satisfy: Cv = vi

(4)

and v is made to unit length such that v/v = 1 .

(5)

+ C 2 + ... + Cn)v = = V'C1V + V/ C 2 V + ... + V/CnV

= v I (C 1

(7)

which shows how each eigenvalue can be partitioned into contributions from covariance matrices at each block size. Each v' Civ is then plotted against i, and local peaks indicate which block sizes contributed the most to that eigenvalue. The eigenvector is interpreted as in any ordination - species with similar distributions will load similarly on that eigenvector. Local peaks indicate the scales at which the pattern of species loading on the eigenvector are most evident; that is, an optimal signal-to-noise ratio. In order to visualize this pattern, a weighted average for each plot (weighted average plot score) can be calculated using species weights from the eigenvector (called 'component profiles' by Noy-Meir & Anderson, 1971, and 'trace' by Whittaker & Naveh 1979). Patterns may be made more evident by smoothing these scores by taking a moving weighted average, which averages the weighted average plot scores for any nun1ber of weighted average plot scores on either side of the plot in question. We have chosen to take weighted average plot scores on each side of a given plot such that the total number of plots equals the block size indicated from partitioning the eigenvalue. A moving weighted average simply 'frameshifts' along the transect, calculating an average from the plots surrounding it, and smooths the curves considerably.

Multiscale ordination: a method for detecting pattern at ...

starting position ofthe transect, 2) n1atrices may be added at any block size, and ... method to fabricated data proved successful in recovering the structure built ...

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Computer Science, University College London, UK; [email protected] ... algorithm are that (i) it is simple, (ii) it is fast, (iii) it lacks data-dependent parameters ...

A Multiscale Mean Shift Algorithm for Mode Estimation 1. Introduction
Computer Science, University College London, UK; [email protected]. 2 ...... 36(4): p. 742-756. 13. Everitt, B.S. and D.J. Hand, Finite Mixture Distributions.

A multiscale approach for biofilm modelling
Jun 28, 2011 - situations, where neither a completely microscopic IBM or a PDE system are possible ...... Communications, 150(21-22):1009–1022, 2010. doi: ...

Eliminating Dependent Pattern Matching - Research at Google
so, we justify pattern matching as a language construct, in the style of ALF [13], without compromising ..... we first give our notion of data (and hence splitting) a firm basis. Definition 8 ...... Fred McBride. Computer Aided Manipulation of Symbol

Pattern A for Loyalty Card.PDF
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Traffic Pattern at Maercker.pdf
Page 1 of 1. Maercker School Traffic Pattern. To enhance student safety and traffic flow around the campus, please note the drop-off and pick-up of. students at ...

Pattern forming method including the formation of an acidic coating ...
Apr 29, 1996 - Foreign Application Priority Data. Dec. 9, 1991 [JP] Japan . .... form ?ner patterns by using electron beam or X-ray with a shorter wavelength.

A Topological Approach for Detecting Twitter ... - Semantic Scholar
marketing to online social networking sites. Existing methods ... common interest [10–12], these are interaction-based methods which use tweet- ..... categories in Twitter and we selected the five most popular categories among them.3 For each ...

A Layered Architecture for Detecting Malicious Behaviors
phishing web sites or command-and-control servers, spamming, click fraud, and license key theft ... seen in the wild [9,10]. Therefore, it is .... Each behavior graph has a start point, drawn as a single point at the top of the graph ..... C&C server

Deacon Jerome Green Ordination Letter.pdf
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