From the desk of BILL LANE
From the desk of BILL LANE
From the desk of BILL LANE
Method of Moments Approach To Historic Data REVISED 1/10/95 Giving Equal Weight to Each Observation Simple Case Which Could be Easily Expanded Given: Systematic Values, s1, i = 1 to S Historic Values, h1, i = 1 to H Number of Historic Period Years, M, where flows were below truncation level, T A. Basic Approach to the Problem (Philosophy and Goal). All observed values are to be given the same weight and the unrecorded historic period years are properly taken into account as being below the threshold, T. The desire is to arrive at a set of LP-III parameters that are consistent with the moments of the data. The following moment sum equations are the basis for this approach. Note that the values of s, h, T, and x (below) are logarithms of flows. S
M F(T)
E
h
+
j=1
fyf(y)dy
T
H
S =
T
H
+
M
F(T)
j
y2 f(y) dy .
s 1=1
T
H
S =
+
+ 1=1
M
F'(T)
f
y3 f(y) dy
The sample size is now equal to S+H+M, and f(y) is the probability density function for the fitted distribution. The moment sum equations, when used in the standard method of moments parameter estimation equations of Bulletin 17B, must result in the same parameters as used to evaluate the integrals.
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From the desk of BILL LANE
From the desk of BILL LANE
From the desk of BILL LANE
B. Computational Options. Normally, one would like to directly solve for the parameters, but this does not appear possible. A second approach would be to evaluate the full range of parameter sets and select the best set, but this would be very cumbersome. The last and recommended approach is an iterative approach. .
. .
.
C. Computational Approach. 1. Obtain an initial set of parameter estimates, probably equal to the systematic record values. 2. Evaluate the above moment equations and recalculate the parameter estimates. 3. Repeat step 2 until the parameter estimates converge. D. Required Computer Routines. The expected values of the moments from a truncated LP-III are needed. These are the integral terms in the equations. The computations may require some programming work but are not technically difficult. I would not be surprised to find that this work has already been done. Values for the moments of the truncated distribution could be calculated using tabled LP-III cumulative distribution function values, calculating values at equal probability spacing over the truncated range, and averaging the results. The finite integration approach would be difficult by hand but easy with a computer. E. Advantages. The approach has advantages over current Bulletin l7B in that all observations are given equal weight and also that the unmeasured values that are below the observation level are also given their due. For confidence limits, it is not clear what should be used for the sample size. It is difficult to evaluate the value of the N years of values for which the only knowledge is that they are below the truncation level. Certainly the number of years for the confidence limits should be larger than S+H, but probably not as great as S+H+M. The approach is compatible with other Bulletin 17B procedures such as skew weighting with regional values. F. Examples. I would have liked to have several examples, especially for actual sets of data which have already had parameters estimated by maximum likelihood methods. I would like to see this for the Pecos River, Bradbury Damsite, the Salt River, among others. These must be deferred until time permits.
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