disjunctive kriging revisited: part ii

Mathematical Geology, Vol, 18, No. 8, 1986

Disjunctive Kriging Revisited: Part II t

M. Armstrong 2 and G. Matheron 2

In order to find distributions other than infinitely divisible distributions which are suitable for disjunctive kriging, infinitesimal generators are used. In addition to distributions developed in Part I, this leads to development of suitable models for the beta (~), hypergeometric, and binomial distributions.

KEY WORDS: Nonlinear geostatistics, disjunctive kriging, beta distribution, hypergeometric distribution, binomial distribution, isofactorial.

INTRODUCTION

In the decade since the first paper on non-linear geostatistics appeared (Math- eron, 1976), geostatisticians have had time to test the method and find its strengths and weaknesses. One problem to date has been that, in its present form, disjunctive kriging always has been associated with a transformation to a normal distribution, which is unsuited for use with data like uranium, which has a large peak of zero values, with discrete variables, or with grouped data as is found in size or density distributions. So a real need exists for new types of disjunctive kriging, particularly for "discrete disjunctive kriging."

The first note on disjunctive kriging (Matherson, 1973) gives the theory behind the method and shows how it can be used for data having one of the following distributions: normal distribution, gamma (2/), Poisson, or negative binomial. More importantly, general conditions for finding distributions suitable for disjunctive kriging are presented. These are that the joint distribution f(x, y) can be expressed in an isofactorial form; that is

f(x, y) = ~ T,,x,,(x) X,,(Y) g(x) g(y) It = 0

~Manuscript received 10 June 1985; accepted 31 March i986. 2Centre de Geostatistique et de Morphologie Mathematique, Ecole Nationale Superieure des Mines de Paris, 35 Rue Saint-Honore, 77305 Fontainebleau, France.

729

0882-8121/86/1100-0729505.00/0 ,~9 [986 Plenum Publishing Corporation

730 Armstrong and Matheron

where g(-) is the marginal distribution, T,, are constants, and X,~ ( ' ) are factors which, for simplicity, must be polynomials. A recent translation of this work is presented in a previous paper (Armstrong and Matheron, 1986).

One unfortunate limitation of this method for finding distributions suitable for disjunctive kriging was that it could be applied only to infinitely divisible distributions. This limitation can be overcome by using a different approach (infinitesimal generators). This was developed first for continuous distributions (Matheron, 1975a) and afterward for discrete distributions (Matheron, 1975b). The objective of this paper is to present an updated translation of those parts of these two research notes which are relevant directly to alternative types of disjunctive kriging.

THE INFINITESIMAL GENERATOR METHOD APPLIED TO CONTINUOUS DISTRIBUTIONS

In Part I, four distributions (normal, 7, Poisson, and negative binomial) were shown to have required isofactorial properties together with polynomial factors. Unfortunately, the method used for finding suitable models could be used only with infinitely divisible distributions.

Here, infinitesimal generators and the theory of semigroups are used to find other suitable models. In this section, the method is applied to continuous distributions; the discrete case is treated in the following section. In both cases, we give only an outline of the proof. Readers who wish to fill out the proof may find it helpful to consult a text on functional analysis (such as, Brezis, 1983), for the Hille-Yosida theorem in particular.

Let g(x) be the marginal distribution. Working in the space L 2 (~, g), we want to find a function a(x) so as to write the differential operator Afassociated with the stochastic process in the form

f ' c I Af = af" + -- (ag) g d x

w h e r e f ' = Of/Ox. Given this function a(x), we can show that operator A is a negative Her-

mitian operator (in the sense that the scalar product (Af, f ) = - ( ~ a f ' , ~aaf') ~< 0. (See Appendix A.) The evolution equation

Of A f = l _ O I Of 1 Ot g Ox ag Ot ] (1)

is a type of heat equation. Now, provided that A is closed and dense in L 2 (~, g), a semigroup Pt =

e At with A as its infinitesimal operator exists, and because we are dealing with

Disjunctive Kriging Revisited: Part II 731

a heat equation, this is a diffusion semigroup. Because ~ gAf = 0, the density g is the ergodic limit o f Pt(x; dy) as t -~ ~ .

The next step is to see whether the eigenfunctions associated with operator A include a series Xn which forms an orthonormal basis for L 2 (I~t, g). I f this is the case

P, xn = e-X"tXn

where Xn is the eigenvalue corresponding to Xn- Consequently for any function f e L 2 (~, g)

Pt f = ~ e-X"t ( f , X~)Xn

The bivariate distribution F,(dx, dy) = g(x) P,(dx, dy) dx can be written, therefore, in an isofactorial form

<(dx , dy) = ~ e-X"' X~(X) X~(Y) g(x) g(y) dx dy.

In other words, the Markov process associated with this semigroup is an isofactorial model.

A few examples should help clarify this approach.

Gauss ian Process on the W h o l e Line

This corresponds to the case a(x) = 1 and g(x) = (1 /~a~) e - x2 /2 . Here

Af = f " - x f '

which clearly is equal to ( l /g ) (d /dx) (g f ' ) . The orthogonal polynomials relative to the normal distribution are Hermite polynomials H,, (x). Moreover, because Akin = -nHn, eigenvalues are X A = n and so the bivariate distribution F t (dx, dy) is ~ o~H~(x) H~(y) g(x) g(y) dx dy where the correlation coefficient o equals e - ' .

y Process on R +

This time a(x) = x and g(x) = [1/r(c~)l x " - ' e-*. Consequently,

Af = xf" + (oe - x ) f '

The Laguerre polynomials

L ( - n , c ~ , x ) = 1 + ~] ( - 1 ) e n ( n - 1) . . - (n - k + 1)x ~ ~=1 a((x + 1) " ' " (c~ + k - 1) k!

are eigenfunctions associated with eigenvalue - X , = - n . Letting In denote the normed polynomial, the representation for the bivariate distribution is

e-n'l,,(x) I n ( y ) x '~ - l y '~-' e - x - y F,=

F(cO r(oe)


- - N O T E - -

The definition given here for Laguerre polynomials is not the same as in earl ier notes. They differ by a mult ipl icat ive factor (-1)n(o~ + n - 1)(~ + n

- 2) . . . . ~ . The new definition is used because it simplifies considerably the form of the recurrence relation used to obtain Laguerre polynomials from the preceding two polynomials .

p Proces s on (0, 1)

In this case, a(x) = x(1 - x) and

r ( ~ + ~ ) x ~ - l ( 1 - x) ~-~ g(x) = 0 <_ x <_ 1

p(~) r(/~)

The expression for the operator A is

A f = x(1 - x ) f " + [or - (o~ + ~) x l f '

The orthogonal polynomials are Jacobi polynomials

F(n, ~ + 13 + n - 1, o~; x) r l

= 1 + ~ ( - 1 ) k n ! (c~ + /3 + n + k - 2)! (c~ - 1 ) !x k

k=l (n - k)! (c~ + /3 + n - 2)! (c~ + k - 1)!k!

which are associated with eigenvalues

- X , , = - n ( n + oL + ~ - 1)

If Xn denotes the normed polynomial

F,(dx, dy) = ~ e-n<n+~e--l~xn(X ) Xn(Y) g(x) g(y) dx dy

The form of normed factors is rather curious

( - 1 ) ~ P(cQ P(c~ + /3 + 2n - 1) 1 + ' ' ' + x n

F(c~ + n) F(c~ + 13 + n - 1)

The family of polynomials can be expressed in another way

d" [ x ~ + n _ l ( 1 _ x ) ;~+ ._ l ] x 1-=(1 - x) 1 -~

In this case, the constant needed to normalize polynomials is

r(o~ + ~) F(~ + n) I ' (~ + n)

F(c~) F(/3) F(c~ + /3 + 2n - 1)


One more important difference between this distribution and the preceding

two is that we no longer have

Tn : pn = e - n r

Instead

Tn = e-n(c~ +~3+n-I)t

INFINITESIMAL GENERATOR METHOD APPLIED TO DISCRETE DISTRIBUTIONS

In the preceding section, isofactorial models with polynomial factors were found for continuous distributions from Markov processes where the infinitesimal generator was local. In this section, s imilar models are obtained for discrete distributions. The local nature of infinitesimal generator A is replaced by a condition limiting possible transitions to adjoining states, that is i ---, i + 1 or

i ~ i - 1. Consequently

( A f ) i = - ( a i + bi ) f + a i r + 1 + b i f - r

where ai is the probabil i ty of transition i ~ i + 1 and bi is the probabil i ty of transition i --+ i - 1.

Three different types of processes can be dist inguished, depending on values of ai and b i.

1. I f a i and bi are strictly posit ive for i = 0, _+ 1, + 2 . . . . . i can vary from - ~ to + w. This case will not be treated here (reasons for this choice will become apparent in the next paragraph.)

2. I f b 0 = 0 and b i ~> 0 for i = l , 2 . . . . . and ai > 0 for all i, i varies from 0 to ~ (infinite case).

3. I f b o = 0 a n d a N = 0, a n d i f b i > 0 f o r i = 1 . . . . . N a n d a ~ > 0 f o r i = 0 . . . . . N - 1, i varies from 0 to N (finite case).

In addit ion to this, the process must be ergodic, so a probabil i ty W = {wi } must exist such that

On rewriting

and clearly

- ( a i + b i ) w i + a i _ l W i _ 1 + bi + l wi + i = O

bi+i wi+ I - a i w i = b i w i - a i _ l w i _ 1

bi+l wi+ ~ = a i w i (2)


If ai and b i a r e strictly posit ive, eq. 2 is true provided that aiwi and b i w i

tend to 0 as i tends to - ~ . However , in the two cases of interest to us (finite and infinite cases), we have bo = 0 and so eq. 2 is true. No additional hy- potheses are required. Therefore, the l imiting distribution (if it exists) is defined by

ao al an - - 1 wn = . . . . . . w0 (3)

bl b2 bn

and the condit ion for its existence is that r. wn < ~ . Condit ion (2) also means that the process is reversible, because wiPij (t) =

wjPj i ( t ) , where Pa( t ) is the probabil i ty of going from i t o j in t ime t.

Condition for Polynomial Factors

Because the process is reversible, polynomial factors exist if and only if

I. Polynomials belong to L2(~ , W); that is

E(i n) = ~ wi in < co

II. For each n, A~ is a polynomial of degree n in i (where n = 0, 1, 2 . . . . . c¢ or n = 0, 1 . . . . . N as the case may be).

Taking condit ion II first

A'] = ai[(i + 1) n - i n] + b i [ ( i - 1) n - i n]

So for n = 1

A i = a i - b i

and this must be a 1st degree polynomial in i. For n = 2

A~ = ai[2i + 11 + b i [ - 2 i + 1]

and this must be a 2nd degree polynomial in i. These two conditions are satisfied if

ai - bi is l inear and

a~ + bi is quadratic

that is, if a i and bi are of the form

ai = ao + a i + .y2 (4) b i = b o + 13i + " / 2

Moreover , b o = 0 because we are not concerned with cases where i goes from - ~ to oo. If ai and bi are of this form, condit ion (2) is satisfied for all n.

Condit ion I is satisfied also in the infinite case. (Obviously it is for the


finite one). In the case where a i and bi are both strictly positive, i.e., where i varies from - c ~ to + e , , we find that E [ i " ] = cc for sufficiently large n and consequently that no models with polynomial factors exist. This is the reason that this case has not been considered.

Here

L i n e a r M o d e l s : y = 0

a i = a 0 + ai

bi = fl i

Now ao must be strictly positive or else the process stops; similarly, so must/3. Three cases must be considered: c~ > 0, c~ = 0, a < 0 which lead, respectively, to the negative binomial, Poisson, and binomial distributions. (This progression is hardly surprising when limit relations between different distributions are remembered).

C a s e o~ > O: N e g a t i v e B i n o m i a l D i s t r i b u t i o n

From eq. 3, clearly

w0

Putting p = a / f l , v = ao/c~

1 Z w,,s" = (1 - p s ) -~ w o s < -

P

Therefore, the limiting distribution Wexists i f fp < 1; that is, if c~ < ft. In that case, the generating function is

q~ C,(s) -

(1 - p s ) °

where q = 1 - p. Because G(s) can be differentiated as many times as is desired, the polynomials belong to L2(i~, W). Provided c~ < /3, an isofactorial model therefore exists with polynomial factors and with the negative binomial

P(u + n ) p ~ W ~ = q ~

r (u ) n !

as its marginal distribution. Polynomials have to be determined explicitly. To do this, we need eigen-

values o f operator A that can be found by finding the coefficient o f i" in the


expression for A~

A~ = (% + c~i) [(i + 1) n - i ~] - ~i[i ~ - (i - 1) ~]

= n(~ - ~)i n + polynomial of degree < n.

Consequently

X~ = - n ( ~ - c~) = -nt3q

So if X~ (i) denotes the normed factor of degree n, the bivariate distribution Fij(t) is

Fo(t) = W~Wj ~ p ~ x . ( i ) x . ( j ) (5) n=O

where P = e ~qt. Moreover,

F,y(t) = W~P,j (t)

where Pij (t) is the translation matrix which is the resolution of the second Kol- mogorov equation

d dtP~i(t) = - ( a j + bj) Pij(t) + aj_ ,Pi j ,(t) + bj+~Pij+l(t) (6)

Put t ing

bi(s, t) = ~ Po(t)sJ J

Equation 5 then becomes

(OG/Ot) + (1 - s)(c~s - 13)(Ob/Os) = - a o ( 1 - s)

with Gi(s, a) = s i. Integrating this gives

[ l I_(1 _ - p s ~ p(1 - ~1 ~ q Gi(s, t) L1 - p s - p o ( 1 s)J 1 - p s - p p ( 1 - s)

Bivariate generating functions in terms of s and o (t is implicit) can be deduced from eq. 6

G(s, o) = ~ F~isi ~i ,,j

: ~] w i a i G i ( s , t)

= I( q~ )1 ~ 1 - p s ) ( 1 - p a ) p p ( 1 - s ) ( 1 - a

This can be expanded in terms of P as


l~i q2 7~ ~0 F(V + n ) p ' ~ O p ( 1 - s ) n ( 1 - a) r' G ( s , o ) = 1 - p s ) ( l - p o I'(v) n! (1 - ps) ' (1 - po)"

From eq. 5 we also have

G(s, a) = Z p ~ Z Wix~(i )s i ~ W j x ~ ( j ) a j n i j

Because p = e -~3qt varies from 0 to oo, we can equate terms in these two expansions. This shows that the generating function of x ~ ( i ) W i is

q , [ I ' ( v + n)p" (l - s)" Z X,,(i)Wis i P (~ )n ! (1 - p s ) ~+~

Polynomial X,,(i) can be deduced from this to be

IlP(v + n) p~]l/2 ~ ( - 1 ) ~"CkI ' (v + i + n - k) i(i - 1) " ' " ( i - k + 1)

[- F(-~ ~.~ J k=O F(V + i) pk

Case c~ = O: Poisson Distribution

I f c~ = 3' = bo = 0 in (4), then ai = ao and bi = /3i where 13 > 0. From (3), the limiting distribution still exists and is a Poisson distribution with parameter 0 = (ao//3). The corresponding stochastic process describes a queuing process with an infinite number of servers, when arrival times follow a Poisson distribution with parameter a 0 and the service time distribution is exponential with parameter/3. The infinitesimal generator is

(Af) i = a o ( f + : - f ) - b i ( f - f _ j )

Consequently, eigenvalue X~ associated with polynomial factor X, (of degree n) is X,, = -n/3.

So bivariate distribution Fgj (t) = WiPii (t) can be written as co

f/./ = W/Wj Z # ' x n ( i ) X , , ( j ) (7) n = 0

where p = e -~ ' Using Kolmogorov ' s second equation

Z Pij( t)s j = (1 - p + pp)J e -°(1-°)(1 -.,) J

and hence the generating function for the bivariate distribution is deduced

G(s, ~) = ~ FijsJa j q

: E w , . p j ' s i ij

e 0 ( i - s ) - O ( 1 - a ) + p ( l - s ) ( 1 - a )


We now compare this with the expression for the generating function obtained from eq. 7

G(s, a) ~ on ~ • i = WiXn(t)a ~ Wjxn ( j ) s j n i j

Identifying coefficients of terms in pn gives

• i= 0 ~ Wix~(t)s (1 - s) ~ e °O -s)

= ( - 1 ) " 0~ d" ~ e0(1-s)

Consequently

Therefore

Wixn(i ) = ( - 1 ) n d-~ e -°

O~nO~ " i! e ° d ~ ( 0 ~ ) x~(i) = ( - 1 ) ~ 07 dO n e -°

Case ~ < O: Binomial Distribution

If c~ is negative, a i = ao + cd would become negative for sufficiently large values of i. As {i } therefore must be finite, ai must be zero for some value of i = N, and the process is restrained to interval (0, N). We therefore have

ai = a(N - i) b i = bi

From eq. 3, the limiting distribution is a binomial distribution with parameters p and N where

p __

The infinitesimal generator is

a

a + b

(Af)i = a(N - 1) [f/+~ - f / ] - bi[fi - f - l ]

and so the eigenvalue associated with Xn is

hn = - n ( a + b)

Consequently, the bivariate distribution is

Fij(t) = WiWj ~] p~xn(i) xn(J)

where p = e -(a+b~ and W/ = (U) pi(1 -p)N-i. (8)


The expression for E~=o Pij(t) s j also can be obtained directly to be

[q(1 - p) + (p + pq)s]i[q + OP + p(1 - p)s] N - i

w h e r e q = 1 - p . Consequently

G(s, a) = ~ WiPij( t)sJ a i

= [(q + p s ) ( q + pa) + ppq(1 - s)(1 - a)] N

Expanding this in powers of O gives

G(s, a) = (q + ps )N(q + pa) N ~ p~p~q" [ -- -- |'~ N F (1 s)(1

q

o n L-(q + P ~ ( q + pa)J

Another expression for G(s, a) can be obtained directly from eq. 8. Iden- tifying terms in the two expressions gives

W/x~(t)s = p~q~ (1 - s)"(q + ps) N - n

n! ~ ) p. q. d i_ ~ N! dq" [q(1 - s) + s]N b

Consequently,

W~xo( i ) = ~..

Hence

x . ( i ) =

In this case

dn q N -- i 1 pnq~ I ( N ) ( 1 _ q)i

! (N - n)! (1 - q)i q N - i dq" [qN- i (1 -- q)i]

Case 3' =/: O: Hypergeometr ic Distribution

a i = a o + c~i + "yi 2

bi = 13i + 3`i 2

where 3, :~ 0. In the infinite case 3, > 0, a 0 > 0, and ai, b i > 0 for i = 1, 2, . . . , a l imiting distribution W may be obtained provided that certain conditions are satisfied, but even then polynomials of degree n need not belong to L 2 (R, W) when n becomes large. Consequently, no polynomial factors exist.

However , in the finite case, a solution exists. Suppose 0 < i < N. Then,

740

changing notation slightly, we have

a i = ( N - i) [a + v ( N - i)1

b i = i[b + 3,i]

where a + 3' > 0, a + N 3 " > 0 ; b + 3' > 0, b + N 3 ' > 0. From (3), the limiting distribution is of the form

w~=

Armstrong and Matheron

w0 n, )

Putting (a/3,) + N = c~ and (b/3,) = /3 gives

( - N ) ' ' ' ( - N + n - 1) ( - c~ ) ( - c~ + 1) . . . ( - a + n - 1) w.= w0

n!

Because E Wn = 1

/3(/3 + 1)(/3 + n - 1)

P(c~ + N) P(/3 + N) w0=

r(/3) r(~ +/3 + N)

Hence

N ) P(/3 + N) r ( a + N) Wn = P(/3 + n) P(c~ + fl + N) C~(~ - 1) " ' ' ( a - n + 1)

Polynomial factors still have to be found explicitly, but as this is much more complicated than in preceding cases, it will be left until it is needed for practical applications.

C O N C L U S I O N

Isofactorial models presented in this paper and its predecessor permit disjunctive kriging when data are not normally distributed. Continuous distributions such as 3, and 13 distributions may be useful for modeling long-tailed distributions, thus obviating need for an anamorphosis as used in traditional gaussian distinctive kriging. Discrete isofactorial models have a wide range of potential uses with discrete data or grouped data. Only experience will show how useful they really are in practice.

Before concluding, potential users should note two limitations in work presented in this paper. The first is of a numerical nature. Whereas isofactorial models with polynomial factors are good to have, these are of little practical interest unless polynomials can be calculated quickly and efficiently. Some sort


of recurrence relation is needed therefore. These exist and can be found in any standard reference on orthogonal polynomials.

The second limitation concerns "change of support." One great advantage of the normal distribution is that it facilitated models of change of support, which are needed for point-block models and block-block models. In this work, nothing has been said about how these can be developed for isofactorial models. Interested readers can consult some recent work (Matheron, 1982, 1984, and 1985).

APPENDIX A

We want to show that operator A defined by

f ' d l d A f = af" + --g dx (ag) = -g dx (agf ')

is a negative Hermitian operation; that is

( A f , f ) = - ( ~ a f ', ,ffaaf ' ) <~ 0

Suppose that the marginal density g(x) is sufficiently regular and is con- centrated on the (possibly infinite) interval [b, c]. Let a(x) be a strictly positive function on this interval and let a(b) = a(c) = 0. Moreover we suppose that product a(x) g(x) is differentiable in this interval.

I f f is twice differentiable on [b, c] and if function ~ is differentiable on the interval, then using integration by parts we can show that

a(x) f " (x) + i f ( x ) d b g(x~-) dxx [a(x) g(x)] ~,(x) g(x) dr

f c = - f ' ( x ) ,p' (x) a(x) g(x) dx b

The result then follows.

R E F E R E N C E S

Brezis, H., 1983, Analyze fonctionnelte: Thdorie et applications: Masson et Cie, Paris, 233 p. Matheron, G., 1973, Le krigeage disjonctif, Internal note N-360: Centre de G~ostatistique, Fon-

tainebleau, 40 p. Matheron, G., 1975a, ComplEments sur les modbles isofactoriels, Internal note N-432: Centre de

GEostatistique, Fontainebleau, 21 p. Matheron, G., 1975b, ModUles isofactoriels discrets et modbles de Walsh, Internal note N-449:

Centre de GEostatistique, Fontainebleau, 24 p. Matheron, G., 1976, A simple substitute for conditional expectation: The disjunctive kriging, in

M. Guarascio, Advanced geostatistics in the mining industry, Proceedings, NATO A.S.I.: Reidel & Co., Dordrecht, p. 221-236.


Matheron, G., 1982, La destructuration des hautes teneurs et la krigeage des indicatrices, Internal report N-761: Centre de G6ostatistique, Fontainebleau, 33 p.

Matheron, G., 1984, Isofactofial models and change of support, in G. Verly, (Ed.), Geostatistics for natural resources characterization, NATO A.S.I.: Reidel & Co., Dordrecht, Netherlands, p. 000.

Matheron, G., 1985a, Une m6thodologie gdn6rale poir les modules isofactoriels discrets: Sci. Terre- Inform., vol. 21, p. 1-78.

Matheron, G., 1985b, Change of support for diffusion-type random functions: Math. Geol., vol. 17, no. 2, p. 137-166.

Armstrong, M. and Matheron, G., 1986, Disjunctive kfiging revisited, Part I: Math. Geol., vol. 18, no. 8, p. 711-728.

disjunctive kriging revisited: part ii

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