- Thanks for the overwhelming response to my question

concerning the Linear Model of Coregionalization!!!

Here is a short summary of the responses I got from several

people.

Special thanks go to Pierre Goovaerts, Phaedon C. Kyriakidis,

Carme Hervada i Sala, Gilles Bourgault, John Kern, Kirsten

de Beurs, John Kern, Syed Abdul Rahman, and Denis Allard.

Linear Model of Coregionalization:

"The key point in modeling coregionalization

is to ensure that the matrix of auto and cross covariance

models is positive semi-definite for any possible lag

distance and direction" (P. Goovaerts).

>Question 1:

Yes!

>if I use e.g. a spherical model for the cross-variogram of

>variables A-B, I have to use a spherical model for both, the

>variogram of variable A and the variogram of variable B?

>Question 2:

Yes!

>The ranges of both variogram models and the range of the cross-

>variogram model have to be identical?

>Question 3:

Cokriging matrices might not be invertible.

>What are the consequences of violating the Linear Model of

>Coregionalization?

Cokriging variances might be negative.

Alternatives to the Linear Model of Coregionalization

exist:

1) Fast Fourier Transformation (FFT)

was mentioned by Phaedon C. Kyriakidis, Carme Hervada i Sala,

John Kern, and Pierre Goovaerts. The references are:

- Yao, T. and A.G. Journel. 1998. Automatic modeling of (cross)

covariance tables using Fast Fourier Transform.

Mathematical Geology 30 (6): 589-616.

- Borgman and Kern, 1996. pg.155. Proceedings of Spatial Accuracy

Assessment in Natural Resources and Environmental Sciences,

Second International Symposium, May 21-23 1996. Mowrer,

Czaplewski and Hamre, USDA, Forest Service, Rocky Mountain

Forest and Range Experiment Station, Fort Collins, CO.

General Technical Report: RM-GTR-277.

- John Kern. PhD dissertation. University of Wyoming.

2) combinatorial optimization schemes

was mentioned by Syed Abdul Rahman

3) hope (!?)

was mentioned by Pierre Goovaerts:

"Model independently the 3 variograms and hope that in the

subsequent cokriging all matrices will be positive definite.

Many people who modeled independently their variograms

reported negative kriging variances and numerical instabilities.

I remember that some authors proposed to perturb (slightly!)

any troublesome cokriging matrix to make it positive definite,

but I wouldn't recommend this practice.

In summary, my advice is to stay with the well proven and yet

reasonably flexible linear model of coregionalization."

(P. Goovaerts)

Literature recommended by Pierre Goovaerts:

1. Morisette J. 1997. Examples using SAS to fit the model of linear

coregionalization. Computer and Geosciences, 23:(3) 317-323.

2. http://www.agro.ucl.ac.be/biom/recherche/projets/agromet/

for public-domain programs to perform coregionalization analysis

and cokriging.

3. Goovaerts, P., 1994. On a controversial method for modeling a

coregionalization. Math. Geol. 26: 197-204.

4. Goovaerts, P., 1998. Ordinary cokriging revisited.

Math. Geol. 30: 21-42.

Literature recommended by Denis Allard:

1. Wackernagel, H. (1999) Multivariate Geostatistics (2nd ed.),

Springer Verlag.

2. Goulard M. and Voltz M. (1992) Linear coregionalization model:

tools for estimation and choice of multivariate variograms,

Math. Geol. 24: 269-286.

With best regards!

Gerhard

--

******************************************************************

| Gerhard Hunner |

|----------------------------------------------------------------|

| Ph.D. candidate |

| Geostatistics, GIS, Remote Sensing |

|----------------------------------------------------------------|

| GIS and Remote Sensing Program |

| Department of Forest Sciences |

| Colorado State University |

| Fort Collins, CO 80523 |

| USA |

| Tel.: (970) 221-1826 |

| Fax: (970) 491-6754 |

| Email: gerhard@... |

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DO NOT SEND Subscribe/Unsubscribe requests to the list! - Dear Gerhard:

I have also started reading about coregionalization and cokriging and my

take on question 2 from the literature doesn't match...

--------->Question 2:

Yes!

>The ranges of both variogram models and the range of the cross-

>variogram model have to be identical?

----------

In Isaaks and Srivastava and in Goovaerts (1998) in Biol. Fert. of Soils

they have models with different ranges. Actually in the Goovaerts he has

a nested variogram model of which one model shares a common range with

the other models variograms but the short scale model has a different

range from the other semivariogram models. He gives the same type of

example in his book on pg 211. Does this mean that the

coregionalization is only occuring at the larger spatial scale.

In Issaks and Srivastava I may also be confusing their use of symbols.

Any feedback on this issue would be appreciated...

Steve

*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^

Steven I. Citron-Pousty

Dept. of Ecology and Evol. Biology

University of Connecticut

75 N. Eagleville Rd, U-43

Storrs, CT 06269

P: 860-486-4157 F: 860-486-6364

NOTE NEW E-MAIL ADDRESS!!

E-mail: Steven.I.Citron-Pousty@...

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DO NOT SEND Subscribe/Unsubscribe requests to the list! - Steven et al.
I think it is incontestable that in the context of a linear model of co-regionalisation that the answer to question 2 is 'YES' !!!

Denis' suggestion about looking at Hans Wackernagel book deserves upgrading. Hans worked quite a bit on these issues and presents a good discussion of these topics (including a generalisation of the linear model due to Michel Grzebyk)

Two side issues are (possibly) worth mentioning

(1) Pierre Goovaerts mentions that variables that are cross correlated will tend to show similar patterns of variability. In this he is quite correct when in comes to scalar variables. It is worth noting (for the very few who may care) that for vector random variables - the cross correlation between components of the vector will

*not*look anything like the variogram of the components but*can not*be ignored. There are a class of variogram/cross variogram models for this situation which does not follow the linear model of co-regionalisation.(2) The FFT algorithm of Yao and Journel is a fairly heurestic one. It is very interesting and may offer good models in many cases. However it is an iterative algorithm and there is no proof offered that the iterations (fixing the amplitudes but varying the phase) will converge. My experience (in the rather nasty case of the vector fields mentioned above) is that it did not give a good solution. This may be a problem with the algorithm or simply my bad programming! Too early to tell!

Have a good weekend

Colin Daly

Steven I . Citron-Pousty wrote:Dear Gerhard:

I have also started reading about coregionalization and cokriging and my

take on question 2 from the literature doesn't match...

---------

>Question 2:

>The ranges of both variogram models and the range of the cross-

>variogram model have to be identical?Yes!

----------

In Isaaks and Srivastava and in Goovaerts (1998) in Biol. Fert. of Soils

they have models with different ranges. Actually in the Goovaerts he has

a nested variogram model of which one model shares a common range with

the other models variograms but the short scale model has a different

range from the other semivariogram models. He gives the same type of

example in his book on pg 211. Does this mean that the

coregionalization is only occuring at the larger spatial scale.

In Issaks and Srivastava I may also be confusing their use of symbols.

Any feedback on this issue would be appreciated...

Steve*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^

Steven I. Citron-Pousty

Dept. of Ecology and Evol. Biology

University of Connecticut

75 N. Eagleville Rd, U-43

Storrs, CT 06269

P: 860-486-4157 F: 860-486-6364

NOTE NEW E-MAIL ADDRESS!!

E-mail: Steven.I.Citron-Pousty@...

*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^--

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DO NOT SEND Subscribe/Unsubscribe requests to the list! - Hello,

As I explained in the Biology and Fertility paper, the

linear model of coregionalization is built by considering

a common set of basic variogram models for all direct and

cross variograms. The condition for the model to be permissible

is that, for each basic structure g_l(h), the matrix of coefficients

b^l_ij (sill for bounded models such as spherical and exponential,

or slope for the linear model) must be semi-definite positive.

It means that some of the coefficients can actually be equal

to zero! In case of two variables, if one structure is missing from

a direct variogram (say b^l_ii=0) it should also be missing for

the cross variogram to keep the matrix semi-definite positive,

e.g. b^l_ij=0 to ensure that |b^l_ij| <= sqrt(b^l_ii * b^l_jj)

(Note that the absolute value is not mentioned in my book

but it's fixed in forthcoming printing).

As I mentioned before, if you consider several structures besides

the nugget effect (i.e. nested model), you have some flexibility to model

a set of variograms with different ranges. As Steven rightly interpreted,

a missing basic structure in a cross variogram means that the two

variables are not correlated at that scale, although we have to be

cautious because of the underlying assumptions of the Linear

model of coregionalization, such as independence of different

spatial processes (phenomena are rarely additive).

Since Gerhard considered the case of a single structure

(besides the nugget effect) and that I assumed that the

cross variogram was not a pure nugget effect, the answer

was yes: direct and cross variograms must include the

same structures with the same range. But, it is not

a general answer.

Hope now it is clear

Pierre

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

________ ________

| \ / | Pierre Goovaerts

|_ \ / _| Assistant professor

__|________\/________|__ Dept of Civil & Environmental Engineering

| | The University of Michigan

| M I C H I G A N | EWRE Building, Room 117

|________________________| Ann Arbor, Michigan, 48109-2125, U.S.A

_| |_\ /_| |_

| |\ /| | E-mail: goovaert@...

|________| \/ |________| Phone: (734) 936-0141

Fax: (734) 763-2275

http://www-personal.engin.umich.edu/~goovaert/

<><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><><>

On Fri, 4 Jun 1999, Steven I . Citron-Pousty wrote:

> Dear Gerhard:

> I have also started reading about coregionalization and cokriging and my

> take on question 2 from the literature doesn't match...

> ---------

> >Question 2:

> >The ranges of both variogram models and the range of the cross-

> >variogram model have to be identical?

>

> Yes!

> ----------

> In Isaaks and Srivastava and in Goovaerts (1998) in Biol. Fert. of Soils

> they have models with different ranges. Actually in the Goovaerts he has

> a nested variogram model of which one model shares a common range with

> the other models variograms but the short scale model has a different

> range from the other semivariogram models. He gives the same type of

> example in his book on pg 211. Does this mean that the

> coregionalization is only occuring at the larger spatial scale.

> In Issaks and Srivastava I may also be confusing their use of symbols.

> Any feedback on this issue would be appreciated...

> Steve

>

> *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^

> Steven I. Citron-Pousty

> Dept. of Ecology and Evol. Biology

> University of Connecticut

> 75 N. Eagleville Rd, U-43

> Storrs, CT 06269

> P: 860-486-4157 F: 860-486-6364

> NOTE NEW E-MAIL ADDRESS!!

> E-mail: Steven.I.Citron-Pousty@...

> *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^

>

>

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> of any useful responses to your questions.

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>

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DO NOT SEND Subscribe/Unsubscribe requests to the list! - Dear Geostats folks:

After your responses and talking to a more geostats conversant friend, I

now have a better understanding of whats going on. Thanks for being

tolerant of a neophyte to the world of geostats...

Steve

*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^

Steven I. Citron-Pousty

Dept. of Ecology and Evol. Biology

University of Connecticut

75 N. Eagleville Rd, U-43

Storrs, CT 06269

P: 860-486-4157 F: 860-486-6364

NOTE NEW E-MAIL ADDRESS!!

E-mail: Steven.I.Citron-Pousty@...

*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^

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