;$Id: moment.pro,v 1.22 2001/01/15 22:28:08 scottm Exp $ ; ; Copyright (c) 1994-2001, Research Systems, Inc. All rights reserved. ; Unauthorized reproduction prohibited. ;+ ; NAME: ; MOMENT ; ; PURPOSE: ; This function computes the mean, variance, skewness and kurtosis ; of an N-element vector. IF the vector contains N identical elements, ; the mean and variance are computed; the skewness and kurtosis are ; not defined and are returned as IEEE NaN (Not a Number). Optionally, ; the mean absolute deviation and standard deviation may also be ; computed. The returned result is a 4-element vector containing the ; mean, variance, skewness and kurtosis of the input vector X. ; ; CATEGORY: ; Statistics. ; ; CALLING SEQUENCE: ; Result = Moment(X) ; ; INPUTS: ; X: An N-element vector of type integer, float or double. ; ; KEYWORD PARAMETERS: ; DOUBLE: IF set to a non-zero value, computations are done in ; double precision arithmetic. ; ; MDEV: Use this keyword to specify a named variable which returns ; the mean absolute deviation of X. ; ; SDEV: Use this keyword to specify a named variable which returns ; the standard deviation of X. ; ; MAXMOMENT: ; Use this keyword to limit the number of moments: ; Maxmoment = 1 Calculate only the mean. ; Maxmoment = 2 Calculate the mean and variance. ; Maxmoment = 3 Calculate the mean, variance, and skewness. ; Maxmoment = 4 Calculate the mean, variance, skewness, ; and kurtosis (the default). ; ; NAN: Treat NaN elements as missing data. ; (Due to problems with IEEE support on some platforms, ; infinite data will be treated as missing as well. ) ; ; EXAMPLE: ; Define the N-element vector of sample data. ; x = [65, 63, 67, 64, 68, 62, 70, 66, 68, 67, 69, 71, 66, 65, 70] ; Compute the mean, variance, skewness and kurtosis. ; result = moment(x) ; The result should be the 4-element vector: ; [66.7333, 7.06667, -0.0942851, -1.18258] ; ; ; PROCEDURE: ; MOMENT computes the first four "moments" about the mean of an N-element ; vector of sample data. The computational formulas are given in the IDL ; Reference Guide. ; ; REFERENCE: ; APPLIED STATISTICS (third edition) ; J. Neter, W. Wasserman, G.A. Whitmore ; ISBN 0-205-10328-6 ; ; MODIFICATION HISTORY: ; Written by: GGS, RSI, August 1994 ; Modified: GGS, RSI, September 1995 ; Added DOUBLE keyword. ; Added checking for N identical elements. ; Added support for IEEE NaN (Not a Number). ; Modified variance formula. ; Modified: GGS, RSI, April 1996 ; Modified keyword checking and use of double precision. ; GSL, RSI, August 1997 ; Added Maxmoment keyword. ; Modified: Wed Jan 28 13:28:07 1998, Scott Lett, RSI Added ; NAN keyword. ;- FUNCTION Moment, X, Double = Double, Mdev = Mdev, Sdev = Sdev, $ Maxmoment = Maxmoment, NaN = nan ON_ERROR, 2 if keyword_set( nan ) then begin ;If NaN set, remove NaNs and recurse. whereNotNaN = where( finite(X), nanCount) if nanCount gt 0 then begin return, moment( X[whereNotNan], Double = Double, Mdev = Mdev, $ Sdev = Sdev, Maxmoment = Maxmoment ) endif endif TypeX = SIZE(X) IF NOT keyword_set( Maxmoment ) THEN Maxmoment = 4 IF Maxmoment gt 1 and N_ELEMENTS(x) lt 2 THEN $ ;Check length. MESSAGE, "X array must contain 2 OR more elements." ;If the DOUBLE keyword is not set then the internal precision and ;result are identical to the type of input. IF N_ELEMENTS(Double) EQ 0 THEN $ Double = (TypeX[TypeX+1] EQ 5 OR TypeX[TypeX+1] EQ 9) nX = TypeX[TypeX+2] Mean = TOTAL(X, Double = Double) / nX Var = !VALUES.F_NAN Skew = !VALUES.F_NAN Kurt = !VALUES.F_NAN IF Maxmoment GT 1 THEN BEGIN ; Calculate higher moments. Resid = X - Mean ; Var = TOTAL(Resid^2, Double = Double) / (nX-1.0);Simple formula ; Numerically-stable "two-pass" formula, which offers less ; round-off error. Page 613, Numerical Recipes in C. Var = (TOTAL(Resid^2, Double = Double) - $ (TOTAL(Resid, Double = Double)^2)/nX)/(nX-1.0) ;Mean absolute deviation (returned through the Mdev keyword). if arg_present(Mdev) then $ Mdev = TOTAL(ABS(Resid), Double = Double) / nX ; Standard deviation (returned through the Sdev keyword). Sdev = SQRT(Var) IF Sdev NE 0 THEN BEGIN ;Skew & kurtosis defined IF Maxmoment GT 2 THEN $ Skew = TOTAL(Resid^3, Double = Double) / (nX * Sdev ^ 3) ; The "-3" term makes the kurtosis value zero for normal distributions. ; Positive values of the kurtosis (lepto-kurtic) indicate pointed or ; peaked distributions; Negative values (platy-kurtic) indicate flat- ; tened or non-peaked distributions. IF Maxmoment GT 3 THEN $ Kurt = TOTAL(Resid^4, Double = Double) / (nX * Sdev ^ 4) - 3.0 ENDIF ENDIF RETURN, [Mean, Var, Skew, Kurt] END
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