[R-sig-Geo] Positive Definite Covariance Matrix for Grid Sampled Data
Edzer J. Pebesma
e.pebesma at geo.uu.nl
Wed Sep 12 09:01:10 CEST 2007
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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