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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