# Problems estimating anisotropy parameters for a spatial model

I am looking for someone with experience in how to handle the anisotropy-parameters in the likfit() function which is part of the GeoR package in R. I am using likfit() to generate the obj.m-parameter for the function krige.conv(). The data I use consists of scattered points (5 to 30) on a ~50*50 grid.

In likfit(), I want the parameters psiA and psiR to be estimated from the available data. This works fine by stating:

fix.psiA = FALSE, fix.psiR = FALSE


However, the estimated values show quite a huge range for psiR, given that the datasets I used are not fundamentaly different from each other. (It is a set of soil moisture measurements with values from 0 to ~45.)

psiR ranges from 1 to about 8000... can that be right? Most of the values are in a range from 1 to 10, but I can not tell why some datasets produce these very large values for psiR.

I was unable to find further information on what this parameter exactly does. I do understand that this parameter regulates the dependency of an estimated value according to it's location relative to the ambient measured values. But I do not know in which way this is accomplished.

I am sorry for the possibly bad english, not a native speaker. I am also sorry for not posting my code, but it is quite long and I felt that it is not necessary in behalf of my question.

Thank you very much for your interest. I will post additional details, if required.

• This question is squarely on topic here, even though it is couched in the context of particular software. It asks about estimating a spatial generalized linear model (and this R package happens to be about the only one that does it). – whuber Aug 29 '12 at 12:34
• @whuber, as a close voter, I'll just say my thought was that this seems to be asking what a particular argument, psiR, does, not about the statistical model (btw: it's not quite clear to me what package the OP is using but I don't think it's the only one that fits spatial GLMs - spBayes and geoRglm both can). – Macro Aug 29 '12 at 12:38
• @Macro:I am using GeoR, as stated in the title. I am sorry for not unsing more appropriate tags, but there were no. There is no tag for "GeoR", left alone "likfit". – rumpel Aug 29 '12 at 12:53
• @Macro I appreciate your reasoning and agree with it. However, I hope that my answer demonstrates the sense in which this question goes beyond any particular package. (I lump geoRglm in with geoR because they are typically used together and documented together. spBayes uses a different approach.) – whuber Aug 29 '12 at 12:55

In short, identifying anisotropy is hopeless with these sparse data.

The two parameters in question, psiA and psiR, describe anisotropy (the angle and ratio, respectively, of a "geometric anisotropy": consult GSLIB or Journel & Huijbregts for details, because the geoR documentation in Diggle & Ribeiro Jr is indeed inadequate concerning anisotropy). With relatively few datapoints it is quite possible--indeed, with soils data (which can be notoriously variable) it is quite likely--that in some directions almost no spatial correlation is detected while in other directions there appears to be some correlation. This can result in near-infinite ratios. Also, if there is a trend in just one direction and it is not removed, this trend will create a strong anisotropy.

Your problem is that five points are way too few for any kind of parameter estimation and $30$ are still too few to identify anisotropy reliably. Rules of thumb in the literature suggest you need at a minimum between $30$ and $100$ points just to get started with estimating the parameters and computing the predictions (that is, kriging). (All rules of thumb have exceptions, but it sounds like these data are not nice enough to qualify.) If you do not assume an isotropic model, you need to explore directional variograms in at least four cardinal directions, whence each such variogram would be based on approximately $5$ to $10$ points each, which again is too small. To identify anisotropy, figure on needing about $100$ points.

The cure is to impose isotropic variograms (or determine anisotropy from considerations independent of the data) and hope for the best. Expect the prediction errors to be large.

• @ whuber: Thank you very much for your detailed post. I spend the last half hour looking on contour plots of the interpolated data unsing a variety of values for psiR. It looks as if any value above 5 is due to an unfortunate spatial relation between measurement points and a general lack of points. All datasets with ~30 points give an estimation for psiR of about 1-2. So it looks like as if I just have to use an average value obtained from the "better" datasets. I was hoping I could avoid that. Again, thank you very much for your post and the literature, I will dig in it. – rumpel Aug 29 '12 at 13:03
• If you have multiple related datasets (such as data sampled over time), you can "borrow strength" by combining them for estimating a common variogram. At a minimum, related data might suggest the direction (psiA) and amount (psiR) of anisotropy, which you can then impose on likfit. If you're getting psiR only in the 1-2 range, you might be best off assuming isotropy: the additional parsimony (the model uses two fewer parameters) can be an advantage. – whuber Aug 29 '12 at 13:09
• The datasets are related as follows: I have five datasets for 10 plots each. the datasets for each plot represent 2d soil moisture for a vertical profile, distance between profiles is 20 cm. I will try do combine them, even though I don't think that this will solve the problem, since "blank spots" in the measurements are mostly present in the same spots of the five profiles, due to soil compaction etc. Regarding the assumtion of isotropy: I don't think I should do this here, since the project is about the infiltration-front, which is - by definition - not isotrop. Thanks for the suggestions. – rumpel Aug 29 '12 at 14:17
• Addendum: Ok, I think I understand what you mean: Put the datasets in the same grid (in their original spatial allignment with the original grid resolution), but with lots of space between the points for each profile, right? That is a very very good idea, I will put that into execution immediately. – rumpel Aug 29 '12 at 15:39
• Update: I was able to produce reasonable values by unsing the suggestions from whuber, even though some plots (with sparse data) still don't show any signs of anistropy. But I guess I have to live with that. – rumpel Aug 29 '12 at 17:41