[R-sig-Geo] Problem with gstat variogram estimation

[Carson, John]

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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