## Re: GEOSTATS: Summary of Linear Model of Coregionalization

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• 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
Message 1 of 5 , Jun 4, 1999
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

Hope now it is clear

Pierre
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________ ________
| \ / | 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/

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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
> E-mail: Steven.I.Citron-Pousty@...
> *^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^*^
>
>
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• 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
Message 2 of 5 , Jun 4, 1999
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