[R-sig-Geo] Positive Definite Covariance Matrix for Grid Sampled Data

[Edzer J. Pebesma]

Keith,

indeed kriging usually fails when one or more point pairs have zero 
distance. One solution in terms of distances would be to shift these 
points a bit, such that no zero distances occur anymore. In terms of the 
covariances, the solution would be to lower the corresponding 
off-diagonal entries with a small amount.

If you have measurements with a known measurement error variance, it may 
make sense to use this variance as the amount to subtract from all  
off-diagonal elements of the covariance matrix.

Hope this helps,
--
Edzer

Keith Dunnigan wrote:
> Hello all,
>
>  
>
>   First I would like to apologize if this question is inappropriate for
> this list.  I am new here, I found this list doing a web search and it
> seemed like the members here would have knowledge in this area.  If
> there are more appropriate lists of forums for this question, I would
> appreciate that information.
>
>  
>
>   I do the majority of my work as a biostatistician in the
> pharmaceutical industry, so I am new to this area.  I am working on a
> couple of small projects in this area though.  I have consulted a couple
> of basic texts ("Introduction to Geostatistics" by Kitanidis, and "An
> Introduction to Applied Geostatistics" by Isaaks & Srivastava).
>
>  
>
>   The gist of what I have gathered from my reading is that standard
> practice is not to use the actual covariance matrix calculated from the
> data.  This is because this matrix may in general not be positive
> definite.  Instead standard practice seems to be to pick from one of
> several standard covariance models, which are guaranteed to be positive
> definite.  After fitting the most appropriate model then, one generates
> the covariance matrix from this model and the distance matrix.  The
> resulting matrix should be positive definite.
>
>  
>
>   The only problem is, I am not finding that to be true.  For instance,
> when I apply the exponential model to my distance matrix and calculate
> the eigenvalues, I find that some of them are negative.  Very, very
> small, but negative (For example -1.2 x 10exp-13).  I applied a couple
> of models and found this to be true. Could someone help me with this?
>
>  
>
>   This is a small data set.  I have a distance matrix that is 20 by 20.
> The exponential model I have used has range parameter R = 14 and sigma
> squared parameter 86.618.  Letting the distance be x, the exponential
> model then is c(x) = sigmasq * exp( ((-3)*x)/R .  
>
>  
>
>   My distance matrix is such that most of the covariances have very
> small values (effectively zero), except for the first couple of
> distances.  That may be the trouble, what do geo folks usually do in
> situations such as this?  I have copied the distance matrix below in the
> case any of you wants to take a look at this.
>
>  
>
>                  0 162 232 246 474   0 162 232 246 474   0 162 232 246
> 474   0 162 232 246 474
>
>          162   0  70  84 312 162   0  70  84 312 162   0  70  84 312 162
> 0  70  84 312
>
>          232  70   0  14 242 232  70   0  14 242 232  70   0  14 242 232
> 70   0  14 242
>
>          246  84  14   0 228 246  84  14   0 228 246  84  14   0 228 246
> 84  14   0 228
>
>          474 312 242 228   0 474 312 242 228   0 474 312 242 228   0 474
> 312 242 228   0
>
>            0 162 232 246 474   0 162 232 246 474   0 162 232 246 474   0
> 162 232 246 474
>
>          162   0  70  84 312 162   0  70  84 312 162   0  70  84 312 162
> 0  70  84 312
>
>          232  70   0  14 242 232  70   0  14 242 232  70   0  14 242 232
> 70   0  14 242
>
>          246  84  14   0 228 246  84  14   0 228 246  84  14   0 228 246
> 84  14   0 228
>
>          474 312 242 228   0 474 312 242 228   0 474 312 242 228   0 474
> 312 242 228   0
>
>           0 162 232 246 474   0 162 232 246 474   0 162 232 246 474   0
> 162 232 246 474
>
>          162   0  70  84 312 162   0  70  84 312 162   0  70  84 312 162
> 0  70  84 312
>
>          232  70   0  14 242 232  70   0  14 242 232  70   0  14 242 232
> 70   0  14 242
>
>          246  84  14   0 228 246  84  14   0 228 246  84  14   0 228 246
> 84  14   0 228
>
>          474 312 242 228   0 474 312 242 228   0 474 312 242 228   0 474
> 312 242 228   0
>
>           0 162 232 246 474   0 162 232 246 474   0 162 232 246 474   0
> 162 232 246 474
>
>          162   0  70  84 312 162   0  70  84 312 162   0  70  84 312 162
> 0  70  84 312
>
>          232  70   0  14 242 232  70   0  14 242 232  70   0  14 242 232
> 70   0  14 242
>
>          246  84  14   0 228 246  84  14   0 228 246  84  14   0 228 246
> 84  14   0 228
>
>          474 312 242 228   0 474 312 242 228   0 474 312 242 228   0 474
> 312 242 228   0
>
>  
>
>   Thanks in advance for any help you can provide!  Warmest Regards,
>
>  
>
>     Keith Dunnigan
>
>     Statking Consulting
>
>     Cincinnati Ohio
>
>  
>
>
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>
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