Dear All,

Is it true that estimation variance of standardize Ordinary Co-Kriging (SOCK) is always equal or smaller than Traditional Ordinary Co-Kriging (TOCK)?

What is the advantage of TOCK to SOCK (I think it is about negative weights) and are there any criteria to choice TOCK or SOCK?

Thanks

Behrang

- In the definition of the cross variogram you can see that it is not

adimentional (depend of units >> Km, %, ppm, etc.), you can avoid this

effect using standardize Ordinary Co-Kriging.

Adrian

-----Original Message-----

From: "Behrang Kushavand" <Kushavand@...>

To: <ai-geostats@...>

Date: Wed, 4 Jan 2006 19:55:01 +0330

Subject: [ai-geostats] Traditional OCK or Standardize OCK?

> Dear All,

____________________________________________________________________________________________

>

>

>

> Is it true that estimation variance of standardize Ordinary Co-Kriging

> (SOCK) is always equal or smaller than Traditional Ordinary Co-Kriging

> (TOCK)?

>

> What is the advantage of TOCK to SOCK (I think it is about negative

> weights) and are there any criteria to choice TOCK or SOCK?

>

>

>

> Thanks

>

> Behrang

>

>

Participe en el V Congreso Internacional de Educaci�n Superior

"Universidad 2006". La Habana, Cuba, del 13 al 17 de Febrero del 2006

http://www.universidad2006.cu

_____________________________________

Instituto Superior Minero Metal�rgico de Moa

Dr. Antonio N��ez Jim�nez

http://www.ismm.edu.cu - Hi,

The main difference between SOCK and TOCK is that, in the standardized

form, only one unbiasedness constraint is imposed, i.e. the sum of all

primary and secondary data weights is one, while in the traditional

version a separate constraint is applied for each variable, i.e.

sum of primary data weights is one and the sum of secondary data

weights is zero for each secondary variable. The traditional

constraints lead to larger and more frequent negative weights

for the secondary variables. The difference between SOCK and

TOCK estimates is expected to increase as differences between

the variance of primary and secondary variables increases.

The different types of cokriging are described and compared in the

following paper:

Goovaerts, P. 1998. Ordinary cokriging revisited.

Mathematical Geology, 30(1): 21-42.

Cheers,

Pierre

Pierre Goovaerts

Chief Scientist at BioMedware

516 North State Street

Ann Arbor, MI 48104

Voice: (734) 913-1098 (ext. 8)

Fax: (734) 913-2201

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

-----Original Message-----

From: Adrián Martínez Vargas [mailto:amvargas@...]

Sent: Wed 1/4/2006 12:53 PM

To: Behrang Kushavand; ai-geostats@...

Cc:

Subject: Re: [ai-geostats] Traditional OCK or Standardize OCK?

In the definition of the cross variogram you can see that it is not

adimentional (depend of units >> Km, %, ppm, etc.), you can avoid this

effect using standardize Ordinary Co-Kriging.

Adrian

-----Original Message-----

From: "Behrang Kushavand" <Kushavand@...>

To: <ai-geostats@...>

Date: Wed, 4 Jan 2006 19:55:01 +0330

Subject: [ai-geostats] Traditional OCK or Standardize OCK?

> Dear All,

____________________________________________________________________________________________

>

>

>

> Is it true that estimation variance of standardize Ordinary Co-Kriging

> (SOCK) is always equal or smaller than Traditional Ordinary Co-Kriging

> (TOCK)?

>

> What is the advantage of TOCK to SOCK (I think it is about negative

> weights) and are there any criteria to choice TOCK or SOCK?

>

>

>

> Thanks

>

> Behrang

>

>

Participe en el V Congreso Internacional de Educación Superior

"Universidad 2006". La Habana, Cuba, del 13 al 17 de Febrero del 2006

http://www.universidad2006.cu

_____________________________________

Instituto Superior Minero Metalúrgico de Moa

Dr. Antonio Núñez Jiménez

http://www.ismm.edu.cu - The downside of SOCK (often not mentioned) is that as a minimum requirement one must know the difference(s) between the population means (i.e., the means of the random functions) of the primary and secondary variables. In practice, one rarely knows these and uses the differences between the sample means instead, which is incorrect, unless one takes the associated estimation errors into account. However, when the BLUE of the differences between population means is used and the associated estimation errors are taken into account, then I suspect that SOCK boils down to something very close or identical to TOCK. Along similar lines, recall that substituting the BLUE of the population mean in the simple kriging equations yields a predictor that is identical to the ordinary kriging predictor (I think it is in Cressie's book, but in fact it is not that difficult to establish this result).

The main (only?) purpose of using ordinary kriging instead of simple kriging is that one often does not know the population mean and cannot simply assume that it is equal to the sample mean or some other combination of the sample data. That is why ordinary kriging is used much more often than simple kriging. It puzzles me why so many geostatisticians so easily replace TOCK by SOCK and ignore the problem above. It is not the right method to avoid large and many negative weights, there are much better ways for that (see discussion of one month ago).

Gerard

Gerard B.M. Heuvelink

Soil Science Centre

Wageningen University and Research Centre

P.O. Box 47

6700 AA Wageningen

The Netherlands

tel +31 317 474628 / 482420

email gerard.heuvelink@...

http://www.sil.wur.nl/UK/

-----Original Message-----

From: Pierre Goovaerts [mailto:Goovaerts@...]

Sent: donderdag 5 januari 2006 0:20

To: Adrián Martínez Vargas; Behrang Kushavand; ai-geostats@...

Subject: RE: [ai-geostats] Traditional OCK or Standardize OCK?

Hi,

The main difference between SOCK and TOCK is that, in the standardized

form, only one unbiasedness constraint is imposed, i.e. the sum of all

primary and secondary data weights is one, while in the traditional

version a separate constraint is applied for each variable, i.e.

sum of primary data weights is one and the sum of secondary data

weights is zero for each secondary variable. The traditional

constraints lead to larger and more frequent negative weights

for the secondary variables. The difference between SOCK and

TOCK estimates is expected to increase as differences between

the variance of primary and secondary variables increases.

The different types of cokriging are described and compared in the

following paper:

Goovaerts, P. 1998. Ordinary cokriging revisited.

Mathematical Geology, 30(1): 21-42.

Cheers,

Pierre

Pierre Goovaerts

Chief Scientist at BioMedware

516 North State Street

Ann Arbor, MI 48104

Voice: (734) 913-1098 (ext. 8)

Fax: (734) 913-2201

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

-----Original Message-----

From: Adrián Martínez Vargas [mailto:amvargas@...]

Sent: Wed 1/4/2006 12:53 PM

To: Behrang Kushavand; ai-geostats@...

Cc:

Subject: Re: [ai-geostats] Traditional OCK or Standardize OCK?

In the definition of the cross variogram you can see that it is not

adimentional (depend of units >> Km, %, ppm, etc.), you can avoid this

effect using standardize Ordinary Co-Kriging.

Adrian

-----Original Message-----

From: "Behrang Kushavand" <Kushavand@...>

To: <ai-geostats@...>

Date: Wed, 4 Jan 2006 19:55:01 +0330

Subject: [ai-geostats] Traditional OCK or Standardize OCK?

> Dear All,

____________________________________________________________________________________________

>

>

>

> Is it true that estimation variance of standardize Ordinary Co-Kriging

> (SOCK) is always equal or smaller than Traditional Ordinary Co-Kriging

> (TOCK)?

>

> What is the advantage of TOCK to SOCK (I think it is about negative

> weights) and are there any criteria to choice TOCK or SOCK?

>

>

>

> Thanks

>

> Behrang

>

>

Participe en el V Congreso Internacional de Educación Superior

"Universidad 2006". La Habana, Cuba, del 13 al 17 de Febrero del 2006

http://www.universidad2006.cu

_____________________________________

Instituto Superior Minero Metalúrgico de Moa

Dr. Antonio Núñez Jiménez

http://www.ismm.edu.cu - Hello,

It is indeed correct that as for simple cokriging, the standardized OCK

requires knowledge of the population means for both primary and

secondary variables, and as I mentioned in my book p. 232 "Provided the

data are representative of the study area, these means can be estimated

from the sample means". Of course, we could also account for the uncertainty

attached to those samples means.. but the same can be said regarding the

uncertainty attached to the parameters of the semivariogram model...

The main reason ordinary kriging is used instead of simple kriging is

its ability to accommodate changes in the mean across the study area

(what I called global trend in my book) through the use of local

search windows. The interesting fact for standardized OCK is that,

even if a global mean is used in the standardization, local means

are still re-estimated within each search window thanks to the

unbiasedness constraint. The main assumption however is that after

rescaling by their global means both primary and secondary variables

have the same local mean, see Goovaerts (1997, 1998). For me, this

might be the main weakness/limitation of the approach. As always, cross-validation is a good way to compare the prediction performances

of the different estimators.

Pierre

Pierre Goovaerts

Chief Scientist at BioMedware

516 North State Street

Ann Arbor, MI 48104

Voice: (734) 913-1098 (ext. 8)

Fax: (734) 913-2201

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

-----Original Message-----

From: Heuvelink, Gerard [mailto:Gerard.Heuvelink@...]

Sent: Thu 1/5/2006 4:31 AM

To: Pierre Goovaerts; Adrián Martínez Vargas; Behrang Kushavand; ai-geostats@...

Cc:

Subject: RE: [ai-geostats] Traditional OCK or Standardize OCK?

The downside of SOCK (often not mentioned) is that as a minimum requirement one must know the difference(s) between the population means (i.e., the means of the random functions) of the primary and secondary variables. In practice, one rarely knows these and uses the differences between the sample means instead, which is incorrect, unless one takes the associated estimation errors into account. However, when the BLUE of the differences between population means is used and the associated estimation errors are taken into account, then I suspect that SOCK boils down to something very close or identical to TOCK. Along similar lines, recall that substituting the BLUE of the population mean in the simple kriging equations yields a predictor that is identical to the ordinary kriging predictor (I think it is in Cressie's book, but in fact it is not that difficult to establish this result).

The main (only?) purpose of using ordinary kriging instead of simple kriging is that one often does not know the population mean and cannot simply assume that it is equal to the sample mean or some other combination of the sample data. That is why ordinary kriging is used much more often than simple kriging. It puzzles me why so many geostatisticians so easily replace TOCK by SOCK and ignore the problem above. It is not the right method to avoid large and many negative weights, there are much better ways for that (see discussion of one month ago).

Gerard

Gerard B.M. Heuvelink

Soil Science Centre

Wageningen University and Research Centre

P.O. Box 47

6700 AA Wageningen

The Netherlands

tel +31 317 474628 / 482420

email gerard.heuvelink@...

http://www.sil.wur.nl/UK/

-----Original Message-----

From: Pierre Goovaerts [mailto:Goovaerts@...]

Sent: donderdag 5 januari 2006 0:20

To: Adrián Martínez Vargas; Behrang Kushavand; ai-geostats@...

Subject: RE: [ai-geostats] Traditional OCK or Standardize OCK?

Hi,

The main difference between SOCK and TOCK is that, in the standardized

form, only one unbiasedness constraint is imposed, i.e. the sum of all

primary and secondary data weights is one, while in the traditional

version a separate constraint is applied for each variable, i.e.

sum of primary data weights is one and the sum of secondary data

weights is zero for each secondary variable. The traditional

constraints lead to larger and more frequent negative weights

for the secondary variables. The difference between SOCK and

TOCK estimates is expected to increase as differences between

the variance of primary and secondary variables increases.

The different types of cokriging are described and compared in the

following paper:

Goovaerts, P. 1998. Ordinary cokriging revisited.

Mathematical Geology, 30(1): 21-42.

Cheers,

Pierre

Pierre Goovaerts

Chief Scientist at BioMedware

516 North State Street

Ann Arbor, MI 48104

Voice: (734) 913-1098 (ext. 8)

Fax: (734) 913-2201

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

-----Original Message-----

From: Adrián Martínez Vargas [mailto:amvargas@...]

Sent: Wed 1/4/2006 12:53 PM

To: Behrang Kushavand; ai-geostats@...

Cc:

Subject: Re: [ai-geostats] Traditional OCK or Standardize OCK?

In the definition of the cross variogram you can see that it is not

adimentional (depend of units >> Km, %, ppm, etc.), you can avoid this

effect using standardize Ordinary Co-Kriging.

Adrian

-----Original Message-----

From: "Behrang Kushavand" <Kushavand@...>

To: <ai-geostats@...>

Date: Wed, 4 Jan 2006 19:55:01 +0330

Subject: [ai-geostats] Traditional OCK or Standardize OCK?

>

>

>

> Is it true that estimation variance of standardize Ordinary Co-Kriging

> (SOCK) is always equal or smaller than Traditional Ordinary Co-Kriging

> (TOCK)?

>

> What is the advantage of TOCK to SOCK (I think it is about negative

> weights) and are there any criteria to choice TOCK or SOCK?

>

>

>

> Thanks

>

> Behrang

>

>

Participe en el V Congreso Internacional de Educación Superior

"Universidad 2006". La Habana, Cuba, del 13 al 17 de Febrero del 2006

http://www.universidad2006.cu

_____________________________________

Instituto Superior Minero Metalúrgico de Moa

Dr. Antonio Núñez Jiménez

http://www.ismm.edu.cu