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

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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 1 of 16 , Jan 3, 2005
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      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 2 of 16 , 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
      • 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 3 of 16 , Jan 3, 2005
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          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 4 of 16 , Jan 4, 2005
          • 0 Attachment
            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 5 of 16 , Jan 4, 2005
            • 0 Attachment
              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 6 of 16 , Jan 4, 2005
              • 0 Attachment
                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 7 of 16 , Jan 4, 2005
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                  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 8 of 16 , Jan 4, 2005
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                    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 9 of 16 , Jan 4, 2005
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                      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 10 of 16 , Jan 5, 2005
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                        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 11 of 16 , Jan 5, 2005
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                          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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