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

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  • Syed Abdul Rahman Shibli
    Perhaps there is some confusion here. Simple kriging, for instance, can be decomposed to the familiar multilinear regression equation since if one assumes all
    Message 1 of 16 , Jan 4, 2005
      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

      LiC(Xi,Xi)=C(Xi,Xo)

      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.

      Cheers

      Syed

      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
      >
      >
      >
      > * By using the ai-geostats mailing list you agree to follow its rules
      > ( see http://www.ai-geostats.org/help_ai-geostats.htm )
      >
      > * To unsubscribe to ai-geostats, send the following in the subject or in the
      > body (plain text format) of an email message to sympa@...
      >
      > Signoff ai-geostats
    • Darla Munroe
      Just to get the group s opinion on this - When do you use IDW? When is it an advantageous technique, or what purposes does it well serve? Darla Munroe ...
      Message 2 of 16 , Jan 4, 2005
        Just to get the group's opinion on this -

        When do you use IDW? When is it an advantageous technique, or what purposes
        does it well serve?

        Darla Munroe

        -----Original Message-----
        From: Syed Abdul Rahman Shibli [mailto:sshibli@...]
        Sent: Tuesday, January 04, 2005 2:19 PM
        To: jyarus; 'Seumas P. Rogan'; ai-geostats@...
        Subject: Re: [ai-geostats] Regression vs. Kriging vs. Simulation vs. IDW


        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

        LiC(Xi,Xi)=C(Xi,Xo)

        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.

        Cheers

        Syed

        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
        >
        >
        >
        > * By using the ai-geostats mailing list you agree to follow its rules
        > ( see http://www.ai-geostats.org/help_ai-geostats.htm )
        >
        > * To unsubscribe to ai-geostats, send the following in the subject or in
        the
        > body (plain text format) of an email message to sympa@...
        >
        > Signoff ai-geostats
      • Isobel Clark
        Agrred, IDW is a good rough way to visualise your data before embarking on more objective (?) approaches. If your data is pretty regularly spread out, small
        Message 3 of 16 , Jan 4, 2005
          Agrred, IDW is a good rough way to visualise your data
          before embarking on more 'objective'(?) approaches.

          If your data is pretty regularly spread out, small
          nugget effect and you use the semi-variogram to choose
          the search radii, there is little difference between
          an IDW-squared map and kriging.

          Isobel
        • Digby Millikan
          Seumas, I was probably a bit misleading to say regression is not an estimation technique. The word regression meaning to revert back to the original, or find
          Message 4 of 16 , Jan 5, 2005
            Seumas,

            I was probably a bit misleading to say regression
            is not an estimation technique. The word regression
            meaning to revert back to the original, or find the
            underlying real equation for a set of data. "Kriging"
            is a form of what is called "generalised linear regression"
            which is one of the most advanced forms of regression.
            The simpler forms of regression can be used to fit
            parametrics equations to data, such as linear regression
            to fit an equation of a line to a set of data points,
            or non-linear regression to fit a polynomial surface
            to a scattered set of say topography data points.
            Not really estimation, but equation fitting. I use non-linear
            regression to fit equations to drillhole survey points
            to plot their curves. In it's more advanced form when
            you wish to fit equations to say a set of two dimensional
            data points, or three dimensional orebody samples,
            this is called trend surface fitting. Unfortunately normally
            the equations developed from trend surface fitting
            become massively too complex to handle to be practical,
            and hence estimation is opted for.

            Digby
          • Digby Millikan
            For ore resource modelling I ve used IDW on a highly skewed lognormally distributed deposit, where no variograms could be produced. With lognormally
            Message 5 of 16 , Jan 5, 2005
              For ore resource modelling I've used IDW on a highly skewed lognormally
              distributed deposit, where no variograms could be produced. With lognormally
              distributed data often found in ore resources, having a good variogram is
              important, to avoid large errors in kriging hence it may be preferential to
              use
              IDW and a topcut. However if your data is not so highly skewed even
              approximating
              a variogram can provide superior results. I used to model topography
              surfaces
              and Kriging with a 'guessed' variogram produced good results compared to
              IDW which produced highly spiked and erroneous results.

              Digby
              www.users.on.net/~digbym
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