Problem with gstat variogram estimation
John, thanks for sharing this with r-sig-geo. As Thierry mentioned, the default model fitting procedure (fit.variogram in package gstat) uses weighted least squares, with weights proportional to N_h/(h^2). This explains why the first lag gets so much weight. For pure nugget models, this of course makes little sense; for other models it often does. Argument fit.method gives you somewhat more control. Give it value 1 to have N_h weights; give it value 6 to do unweighted averaging (I agree that this information should be in the fit.variogram documentation). The SSErr values will be uncomparable accross different weighting schemes, as you might expect. -- Edzer
Carson, John wrote:
I have found anomalous behavior in gstat's variogram estimation. I have listed 3 example variograms below for small data sets. In order to better estimate the nugget effect, I slightly perturbed the locations (by 1 foot increments) of duplicate results. The empirical variograms are given below.
Before I did this (I averaged duplicate values initially), a Gaussian model with 0 nugget was selected for the second variogram and pure nugget models for the first and third. I am using the candidate model list ('Nug', 'Exp', 'Sph', 'Gau', 'Mat', 'Cir', 'Lin', 'Bes') and selecting the model based on SSErr for preliminary testing purposes. Afterward, the pure nugget models had the lowest SSErr and were selected. Note that the variogram fits are completely controlled by the short range variance, because even the original pure nugget models are substantially different in the estimate of the nugget. The fitted models are listed below. Just by inspection, based on the numbers of pairs in these examples, a pure nugget model should be about halfway between the empirical semivariance of the last lag and the average of the other lags. However, the fitted nuggets are almost identical to the semivariance of the first last (dist = 1.4).
It seems to me that this must be due to a bug in the GSTAT code. I pointed this out to Edzer Pebesma, and he asked me to post it here.
The variograms are
tmp.vgm
[[1]]
np dist gamma dir.hor dir.ver id
1 4 1.414214 0.14174537 0 0 PC1
2 2 44.742603 6.70989788 0 0 PC1
3 2 57.707880 1.76351594 0 0 PC1
4 4 59.987678 1.52197310 0 0 PC1
5 3 71.512518 1.21348268 0 0 PC1
6 1 84.852877 0.05381849 0 0 PC1
7 1 97.266495 1.21827622 0 0 PC1
8 3 112.237133 5.07947925 0 0 PC1
9 18 121.478856 1.93707676 0 0 PC1
[[2]]
np dist gamma dir.hor dir.ver id
1 4 1.414214 0.09725079 0 0 PC2
2 2 44.742603 0.33598072 0 0 PC2
3 2 57.707880 0.39088727 0 0 PC2
4 4 59.987678 0.87315735 0 0 PC2
5 3 71.512518 0.14944845 0 0 PC2
6 1 84.852877 0.19809863 0 0 PC2
7 1 97.266495 0.63557814 0 0 PC2
8 3 112.237133 1.92063948 0 0 PC2
9 18 121.478856 0.65468693 0 0 PC2
[[3]]
np dist gamma dir.hor dir.ver id
1 4 1.414214 0.035250817 0 0 PC3
2 2 44.742603 0.105299796 0 0 PC3
3 2 57.707880 0.020245674 0 0 PC3
4 4 59.987678 0.124159836 0 0 PC3
5 3 71.512518 0.008112554 0 0 PC3
6 1 84.852877 0.034337591 0 0 PC3
7 1 97.266495 0.053879459 0 0 PC3
8 3 112.237133 0.021922987 0 0 PC3
9 18 121.478856 0.085270969 0 0 PC3
But the fitted models are:
tmp.vgm.fit
[[1]]
model psill range
1 Nug 0.1483120 0
[[2]]
model psill range
1 Nug 0.09849419 0
[[3]]
model psill range
1 Nug 0.03535234 0
John H. Carson Jr., PhD
Senior Statistician
Applied Sciences & Engineering
Shaw Environmental & Infrastructure
16406 US Rte 224 East
Findlay, OH 45840
Phone 419-425-6156
Fax 419-425-6085
john.carson at shawgrp.com
http://www.shawgrp.com/
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Edzer Pebesma Institute for Geoinformatics (ifgi), University of M?nster Weseler Stra?e 253, 48151 M?nster, Germany. Phone: +49 251 8333081, Fax: +49 251 8339763 http://ifgi.uni-muenster.de/ http://www.springer.com/978-0-387-78170-9 e.pebesma at wwu.de