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1869Re: [ai-geostats] Regression vs. Kriging vs. Simulation vs. IDW

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  • Syed Abdul Rahman Shibli
    Jan 4, 2005
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      Perhaps there is some confusion here. Simple kriging, for instance, can be
      decomposed to the familiar multilinear regression equation since if one
      assumes all the Z(Xi)s are independent variables, then in the covariance
      matrix C all of C(Xi,Xj) would be zero except for C(Xi,Xi). So


      The lambdas here being the parameters of the regression equation. The
      intercept term is the sam, i.e. Lo=E(y)-LiE(xi).

      Not sure if the previous poster meant this or simply using the location as
      the "independent" variable.



      On 3/1/05 5:34 PM, "jyarus" <jyarus@...> wrote:

      > Hi Seumas:
      > I thought I would throw my 2 cents in regarding a comparison between kriging
      > and linear regression.
      > While some of the responses have hit a few important differences, like
      > Kriging is a spatial estimator and regression is not, or kriging will honor
      > the original data and regression will not (unless residuals are added back
      > in - not often done). For me, the critical point to be made is between the
      > collocated cokriging application and regression. In collocated cokriging,
      > like simple regression, two variables are being used, one independent and
      > one dependent (of course, this could be expanded to more than one
      > independent variable). The object is to predict a value of the dependent
      > variable from a relationship established between both the independent and
      > dependent observed values. In the ensuing regression equation, there is a
      > slope term. For example, in the equation, Y= c-bX, c is the intercept and b
      > is the slope. As pointed out by one of the contributors, regression by
      > itself is not a spatial estimator, it is a point estimator. As such, the
      > equation contains no information about the surrounding data or about the
      > relationship between the observed data and the unsampled location where a
      > desired estimate of the dependent variable is required. In kriging (or
      > cokriging), the slope term "b" is replaced by a covariance matrix that
      > informs the system not only about the behavior of the surrounding data
      > points and the unsampled location (similar to distance weighting if
      > omnidirectional), but also about the spatial behavior within the
      > neighborhood - that is, how neighbors are spatially related to other
      > neighbors. Thus, the slope term "b" is replaced with a sophisticated
      > covariance matrix containing the spatial information.
      > The ramifications of using simple regression instead of true spatial
      > estimator are significant if the results are presented in map form. While
      > this is often difficult to grasp for some, using simple regression as a
      > mapping tool will cause geographic portions of a map to consistently be
      > overestimated and others underestimated! For example, you may find that all
      > the values estimated in the upper left quadrant of the map to be
      > overestimated, and those in the lower right to be underestimated. We would
      > like to believe that a good spatial estimator will be unbiased, and the
      > distribution of the error variances over the area of a map will be uniform -
      > no one part of the map will preferentially over- or underestimated. The
      > bias brought about by the slope term in simple regression can be easily
      > tested and proved.
      > I have attached a short paper my partner Richard Chambers and I published in
      > the Canadian Recorder a few years back which addressed this issue. The
      > article talks about seismic attributes related to petroleum reservoir
      > characterization. However, beginning around page 10 or 11, we give an
      > example that demonstrates the above points.
      > I hope this is informative and useful.
      > King Regards,
      > Jeffrey M. Yarus
      > ------------------------------------
      > QGSI
      > Jeffrey M. Yarus
      > Partner
      > jyarus@...
      > 2900 Wilcrest, Suite 370
      > Houston, Texas 77042
      > tel: (713) 789-9331
      > fax: (713) 789-9318
      > mobile: (832) 630-7128
      > ------------------------------------
      > -----Original Message-----
      > From: Seumas P. Rogan [mailto:sprogan@...]
      > Sent: Friday, December 31, 2004 1:14 PM
      > To: ai-geostats@...
      > Subject: [ai-geostats] Regression vs. Kriging vs. Simulation vs. IDW
      > Hello everyone,
      > I apologize if this question is too elementary for this list;
      > I want to understand the key differences between linear regression,
      > kriging, conditional simulation and other interpolation techniques such as
      > IDW or splines in the analyses of spatial data. I would like to know the
      > assumptions, strengths and weaknesses of each method, and when one method
      > should be preferred to another. I browsed the archives and looked at some
      > of the on-line papers, but they are written at a level beyond my own
      > current understanding. It seems to me that this would be a great topic for
      > the first chapter of an introductory spatial analysis textbook. Can anyone
      > recommend any basic textbooks or references on this topic?
      > Any assistance you can offer would be appreciated.
      > Sincerely,
      > Seumas Rogan
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