Dear list -
This is something that has concerned me as well, since I am frequently
interested in metrics that are non-Euclidean on a plane, such as travel
time/cost. This question is very relevant for geographers/economists. It is
my understanding that the p.d. property of covariance models needs to be
checked every time we change the metric. The textbook theoretical results are
for Euclidean metrics. I mention below a few observations, that I have had
trouble reconciling, and maybe someone could tell me where my misunderstanding
lies.
An interesting example is given by Christakos ("Modern" Spatiotemporal
Geostatistics, pg. 65). The Gaussian covariogram, p.d. in infinite dimensional
Euclidean space, is not p.d. on a planar surface with a "Manhattan" metric.
Compare this with an interesting mathematical result due to John Nash. I am
not a mathematician, but if I understand it correctly, it claims that any space
with a Riemannian metric can be embedded in a finite (but possibly very large)
dimensional Euclidean space. To me, this suggests that if I pick a suitable
travel time/cost metric, I can proceed if I pick (just to be safe), any
covariogram valid in infinite dimensional Euclidean space.
My difficulty come from "What does a Riemannian metric "look like," and how can
I avoid metrics like the Manhattan. Does it have to do with the fact that
there is no unique shortest path between two points using a Manhattan metric?
(There are an infinite number of shortest paths).
What practical advise can be given for odd distance metrics. To me, it seems
sufficient (but definitely not necessary and likely too stringent) to (1) make
sure that your metric creates a unique shortest path between any two points,
and (2) pick a covariance model valid in an infinite dimensional Euclidean
space.
Is this advise legitimate, or am I misunderstanding something?
Nicholas N. Nagle, Assistant Professor
University of Colorado
Department of Geography
UCB 260, Guggenheim 110
Boulder, CO 80309-0260
phone: 303-492-4794
---- Original message ----
>Date: Mon, 9 Apr 2007 08:59:34 -0400
>From: "Marc Serre" <[EMAIL PROTECTED]>
>Subject: AI-GEOSTATS: positive-definiteness needed for kriging
>To: <[email protected]>
>
>
>Dear list :
>
>Following the intuition of Roderik Lindenbergh, I would actually like to ask
>as question to the whole list whether :
>
>"The positive definiteness between a set of points, as needed for kriging,
>is not spoiled by changing the distance metric between these points, as long
>as one uses a valid distance metric as defined for example in
>http://mathworld.wolfram.com/Metric.html"; ?
>
>Note that the above statement refers to the positive-definiteness needed for
>kriging, not the merely positive-definiteness of a covariance model. I am
>not sure which of these was Roderik refering to.
>
>I don't think that everybody would agree with the above statement, but what
>work exists either proving it, or providing counter examples?
>
>Marc Serre
>Assistant Professor,
>Environmental Sciences and Engineering
>The University of North Carolina at Chapel Hill
>Chapel Hill, NC 27599-7431, USA
>
>----- Original Message -----
>From: "Roderik Lindenbergh" <[EMAIL PROTECTED]>
>To: <[email protected]>
>Sent: Friday, April 06, 2007 3:16 PM
>Subject: Re: AI-GEOSTATS: Kriging a non-planar surface lacking natural
>parametization
>
>
>Dear list,
>
>my intuition is that as long as you define a valid notion of distance on the
>object of interest, in this case a bone, you can do Kriging.
>The idea is that the variogram/covariance function does not `know` how
>your point pair distance is obtained anyway, so the positive definiteness is
>not spoiled by changing your distance function.
>
>A valid notion of distance is obtained by making sure that your distance
>function
>fulfills the requirement of a metric:
> http://mathworld.wolfram.com/Metric.html
>that is, triangle inequality should hold, etc.
>
>good luck, Roderik Lindenbergh
>
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