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1858RE: [ai-geostats] Regression vs. Kriging vs. Simulation vs. IDW

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  • jyarus
    Jan 3, 2005
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      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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