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 159 members
[aigeostats] Regression vs. Kriging vs. Simulation vs. IDW
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 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 online 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  Hi:
We had participated in AIGEOSTATS 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: <aigeostats@...>
Sent: Friday, December 31, 2004 11:14 AM
Subject: [aigeostats] 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 online 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 aigeostats mailing list you agree to follow its rules
> ( see http://www.aigeostats.org/help_aigeostats.htm )
>
> * To unsubscribe to aigeostats, send the following in the subject or in
the body (plain text format) of an email message to sympa@...
>
> Signoff aigeostats  Seumas
Linear regression : not really an estimation technique for spatial data,
though regression forms some of the basic
theory
for the derivation of kriging equations.
Splines : not really an estimation technique, just something
used to make curves smooth on contour plots.
IDW : after ploygonal technigues the simplest of spatial
modelling techniques, does not take into
account
clustering of data.
Kriging : a spatial modelling technique superior to IDW
when you have reasonable variograms, and
takes into account clustering of data and
instead
of using distance to weight samples uses your
variogram (sometimes referred to as
statistical distance).
Conditional simulation: a spatial modelling technique of a special sort.
Where kriging gives you the "best" estimate.
Simulation
is used to analyse other possible outcomes
for your
sample set, so you can see the effects if
reality varies
from the "best" model. Example use includes
planning
for chemical purchases at a metallurgical
plant, so you
can plan for variations from your kriged
model should
they occur.
Digby Millikan
www.users.on.net/~digbym  Seumas,
Linear regression : not really an estimation technique for spatial data,
though regression forms some of the basic
theory for the derivation of kriging equations.
Splines : not really an estimation technique, just something
used to make curves smooth on contour plots.
IDW : after polygonal techniques the simplest of spatial
modelling techniques, does not take into
account clustering of data.
Kriging : a spatial modelling technique superior to IDW
when you have reasonable variograms, and
takes into account clustering of data and
instead of using distance to weight samples
uses your variogram (sometimes referred to
as statistical distance).
Conditional simulation: a spatial modelling technique of a special sort.
Where kriging gives you the "best" estimate.
Simulation is used to analyse other possible
outcomes for your sample set, so you can
see the effects if reality varies from the
"best" model. Example use includes planning
for chemical purchases at a metallurgical
plant, so you can plan for variations from
your kriged model should they occur.
Digby Millikan
www.users.on.net/~digbym  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,
to
>
> 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 online 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 aigeostats mailing list you
> agree
> to
> > follow its rules
> > ( see
> > http://www.aigeostats.org/help_aigeostats.htm )
> >
> > * To unsubscribe to aigeostats, send the
> following
> > in the subject or in the body (plain text format)
> of
> > an email message to sympa@...
> >
> > Signoff aigeostats
>
>
>
>
> __________________________________
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> follow its rules
__________________________________
> ( see
> http://www.aigeostats.org/help_aigeostats.htm )
>
> * To unsubscribe to aigeostats, send the following
> in the subject or in the body (plain text format) of
> an email message to sympa@...
>
> Signoff aigeostats
Do you Yahoo!?
Send a seasonal email greeting and help others. Do good.
http://celebrity.mail.yahoo.com  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= cbX, 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) 7899331
fax: (713) 7899318
mobile: (832) 6307128

Original Message
From: Seumas P. Rogan [mailto:sprogan@...]
Sent: Friday, December 31, 2004 1:14 PM
To: aigeostats@...
Subject: [aigeostats] 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 online 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  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
044695755, 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@...>; <aigeostats@...>
Sent: Monday, January 03, 2005 12:34
Subject: RE: [aigeostats] 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= cbX, 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) 7899331
 fax: (713) 7899318
 mobile: (832) 6307128
 

 Original Message
 From: Seumas P. Rogan [mailto:sprogan@...]
 Sent: Friday, December 31, 2004 1:14 PM
 To: aigeostats@...
 Subject: [aigeostats] 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 online 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 aigeostats mailing list you agree to follow its rules
 ( see http://www.aigeostats.org/help_aigeostats.htm )

 * To unsubscribe to aigeostats, send the following in the subject or in
the body (plain text format) of an email message to sympa@...

 Signoff aigeostats  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= cbX, 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) 7899331
> fax: (713) 7899318
> mobile: (832) 6307128
> 
>
> Original Message
> From: Seumas P. Rogan [mailto:sprogan@...]
> Sent: Friday, December 31, 2004 1:14 PM
> To: aigeostats@...
> Subject: [aigeostats] 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 online 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 aigeostats mailing list you agree to follow its rules
> ( see http://www.aigeostats.org/help_aigeostats.htm )
>
> * To unsubscribe to aigeostats, send the following in the subject or in the
> body (plain text format) of an email message to sympa@...
>
> Signoff aigeostats  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'; aigeostats@...
Subject: Re: [aigeostats] 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= cbX, 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) 7899331
> fax: (713) 7899318
> mobile: (832) 6307128
> 
>
> Original Message
> From: Seumas P. Rogan [mailto:sprogan@...]
> Sent: Friday, December 31, 2004 1:14 PM
> To: aigeostats@...
> Subject: [aigeostats] 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 online 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 aigeostats mailing list you agree to follow its rules
> ( see http://www.aigeostats.org/help_aigeostats.htm )
>
> * To unsubscribe to aigeostats, send the following in the subject or in
the
> body (plain text format) of an email message to sympa@...
>
> Signoff aigeostats  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) 9131098
Fax: (734) 9132201
http://home.comcast.net/~goovaerts/
Original Message
From: Darla Munroe [mailto:munroe.9@...]
Sent: Tue 1/4/2005 3:06 PM
To: aigeostats@...
Cc:
Subject: RE: [aigeostats] 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'; aigeostats@...
Subject: Re: [aigeostats] 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= cbX, 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) 7899331
> fax: (713) 7899318
> mobile: (832) 6307128
> 
>
> Original Message
> From: Seumas P. Rogan [mailto:sprogan@...]
> Sent: Friday, December 31, 2004 1:14 PM
> To: aigeostats@...
> Subject: [aigeostats] 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 online 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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> Signoff aigeostats  Darla Munroe wrote:
> Just to get the group's opinion on this 
I use IDW to plot a smooth surface, fitted through the data points.
>
> When do you use IDW? When is it an advantageous technique, or what purposes
> does it well serve?
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  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 variancecovariance 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  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 semivariogram to choose
the search radii, there is little difference between
an IDWsquared map and kriging.
Isobel  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 nonlinear regression to fit a polynomial surface
to a scattered set of say topography data points.
Not really estimation, but equation fitting. I use nonlinear
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  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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