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

Keith Dunnigan keith at statkingconsulting.com
Wed Sep 12 15:00:39 CEST 2007


Edzer,

  Thanks again, sorry if I misread you.  Thanks also to James Holland
Jones and Christopher Paciorek who responded with helpful comments.  I
will read over all of these and give it another try.

  Keith

-----Original Message-----
From: Edzer J. Pebesma [mailto:e.pebesma at geo.uu.nl] 
Sent: Wednesday, September 12, 2007 3:01 AM
To: Keith Dunnigan
Cc: r-sig-geo at stat.math.ethz.ch
Subject: Re: [R-sig-Geo] Positive Definite Covariance Matrix for Grid
Sampled Data

Keith, read my suggestion good; I didn't suggest replacing all zeroes in

the distance matrix! You either rearrange points and recompute 
distances, or modify off-diagonal zero-distance entries in the 
covariance matrix.

Sounds like you modified the zero entries on the diagonal as well.
--
Edzer

Keith Dunnigan wrote:
> Edzer,
>
>   Thanks for your help!  I tried the first suggestion.., I replaced
all
> the zero's in the distance matrix with a different value.., but I
still
> have the same problem.  I get negative eigenvalues.  I tried various
> constants, replacing the zero with positive numbers up to 5, with no
> luck.
>
>   Anyone have any other ideas?
>
>   Keith
>
> -----Original Message-----
> From: Edzer J. Pebesma [mailto:e.pebesma at geo.uu.nl] 
> Sent: Tuesday, September 11, 2007 2:30 PM
> To: Keith Dunnigan
> Cc: r-sig-geo at stat.math.ethz.ch
> Subject: Re: [R-sig-Geo] Positive Definite Covariance Matrix for Grid
> Sampled Data
>
> 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
>>
>>  
>>
>>
>> 	[[alternative HTML version deleted]]
>>
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>>   
>>     
>
>
>




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