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

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  • Ben Fang
    Hi: We had participated in AI-GEOSTATS SIC 2004 exercise in the past September. I think that several techniques (Kriging, regression, etc.) may have been used
    Message 1 of 16 , Dec 31, 2004
      Hi:

      We had participated in AI-GEOSTATS SIC 2004 exercise in the past September.
      I think that several techniques (Kriging, regression, etc.) may have been
      used by different participants. It will be illuminating to read the final
      report of SIC 2004 when available.


      K.K. (Benjamin) Fang
      (We had used an interpolant/estimator similar to "IDW" and "nonparametric
      regression".)


      ----- Original Message -----
      From: "Seumas P. Rogan" <sprogan@...>
      To: <ai-geostats@...>
      Sent: Friday, December 31, 2004 11:14 AM
      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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    • Gali Sirkis
      When comparing kriging versus regression, I meant using linear regression between sparse and exhaustive datasets to interpolate the sparse one, since as Digbi
      Message 2 of 16 , Jan 3, 2005
        When comparing kriging versus regression, I meant
        using linear regression between sparse and exhaustive
        datasets to interpolate the sparse one, since as Digbi
        Milligan pointed out in general case regression is not
        an estimation method.

        --- Gali Sirkis <donq20vek@...> wrote:

        > Seumas,
        >
        > see few practical points that you may find useful:
        >
        > 1. kriging vs regression:
        >
        > a) kriging honors original data points, while
        > regression does not
        > b) kriging allows to account for anizotropy
        > c) kriging allows to control the influence of the
        > data
        > points
        >
        > 2. Kriging versus other interpolation technics
        >
        > a) Kriging allows to decluster data
        > b) kriging allows to estimate uncertainty of
        > estimation
        > c) kriging allows to use for estimation secondary
        > information from another exhaustive dataset
        >
        > 3. Kriging vs simulations
        >
        > a) Kriging produces smoother version than real
        > distribution, while simulation gives more details
        > b) simulations allow to estimate joint uncertainty,
        > for example probability that values in several
        > adjacent points are above certain level.
        > c) simulation allows to estimate risk of various
        > scenarios - while kriging only shows the most
        > probable
        > one.
        >
        > All the best,
        >
        > Gali Sirkis.
        >
        >
        > >
        > > 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
        >
        >
        >
        >
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      • jyarus
        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
        Message 3 of 16 , Jan 3, 2005
          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
        • Volker Bahn
          Hi all, picking up on Jeff s point about collocated cokriging: what is the difference between this technique (which I m not familiar with) and an
          Message 4 of 16 , Jan 3, 2005
            Hi all,

            picking up on Jeff's point about collocated cokriging: what is the
            difference between this technique (which I'm not familiar with) and an
            autoregressive regression models such as CAR, SAR etc?

            Thanks

            Volker


            _______________________________

            Volker Bahn

            Dept. of Wildlife Ecology - Rm. 210
            University of Maine
            5755 Nutting Hall
            Orono, Maine
            04469-5755, USA
            Tel. (207) 581 2799
            Fax: (207) 581 2858
            volker.bahn@...
            http://www.wle.umaine.edu/used_text%20files/Volker%20Bahn/home.htm-----
            Original Message -----
            From: "jyarus" <jyarus@...>
            To: "'Seumas P. Rogan'" <sprogan@...>; <ai-geostats@...>
            Sent: Monday, January 03, 2005 12:34
            Subject: RE: [ai-geostats] Regression vs. Kriging vs. Simulation vs. IDW


            | 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
          • 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 5 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 6 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
              • Pierre Goovaerts
                Well... I would say that IDW is still being used by a few consultants that think that kriging is too complicated to apply and that the client will pay them as
                Message 7 of 16 , Jan 4, 2005
                  Well... I would say that IDW is still being used by a few consultants that
                  think that kriging is too complicated to apply and that the client will pay
                  them as long as the map looks pretty...
                  and less cynically IDW could give OK results if your data are gridded
                  and the pattern of variability is ostropic.

                  Pierre


                  Pierre Goovaerts

                  Chief Scientist at Biomedware

                  516 North State Street

                  Ann Arbor, MI 48104

                  Voice: (734) 913-1098
                  Fax: (734) 913-2201

                  http://home.comcast.net/~goovaerts/

                  -----Original Message-----
                  From: Darla Munroe [mailto:munroe.9@...]
                  Sent: Tue 1/4/2005 3:06 PM
                  To: ai-geostats@...
                  Cc:
                  Subject: RE: [ai-geostats] Regression vs. Kriging vs. Simulation vs. IDW




                  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
                  >
                  >
                  >
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                • Edzer J. Pebesma
                  ... I use IDW to plot a smooth surface, fitted through the data points. This may serve as another spatial visualisation of the data; I see it as an exploratory
                  Message 8 of 16 , Jan 4, 2005
                    Darla Munroe wrote:
                    > 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?

                    I use IDW to plot a smooth surface, fitted through the data points.
                    This may serve as another spatial visualisation of the data; I see
                    it as an exploratory step towards building a statistical model for
                    spatial variation.
                    --
                    Edzer
                  • Isobel Clark
                    Syed The term independent variables is confusing in the context of regression. It does not mean that the variables are independent of one another. It means
                    Message 9 of 16 , Jan 4, 2005
                      Syed

                      The term "independent variables" is confusing in the
                      context of regression. It does not mean that the
                      variables are independent of one another. It means
                      that they are independent of the error incurred in the
                      estimation. The variance-covariance matrix is
                      classically produced directly from the data and does
                      not need to be diagonal.

                      The difference between simple kriging and regression
                      is solely that the covariances are derived from a
                      model rather than directly from the data.

                      Isobel
                      http://geoecosse.bizland.com/books.htm
                    • 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 10 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 11 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 12 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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