gaussian approximations to conditional distributions for multi-gaussian processes

Mathematical Geology, Vol. 19, No. 5, 1987

Gaussian Approximations to Conditional Distributions for Multi-Gaussian Processes 1

M i c h a e l S t e i n 2

Suppose a multi-Gaussian process is observed at some set o f sites, and we wish to obtain the conditional block grade distribution given some observations. We show that this conditional distribution is approximately Gaussian under certain conditions. In particular, given a single observation from a continuous multi-Gaussian process, the conditional distribution under a small change of support is approximately Gaussian unless, roughly speaking, the observed process is twice differentiable and the observation site is at the center o f mass of the support region. A Gaussian approximation for the conditional prediction error of the total ore in a fixed region is considered also, although an example demonstrates that a naive analysis can give incorrect limiting conditional means.

KEY WORDS: Change of support, conditional prediction error, random field.

I N T R O D U C T I O N

Estimation of block distributions for non-Gaussian processes is a pressing and largely unresolved problem in geostatistics. Matheron (1985) has developed a powerful technique for approximating the unconditional distribution of the av- erage of a process over a small block when the process is non-Gaussian. In practice, we would often be interested in the conditional block grade distribution given some observations. The purpose of this paper is to show that, under certain circumstances, this conditional distribution is approximately Gaussian even if the underlying process is non-Gaussian.

Specifically, assume that the observed process z ( ' ) is multi-Gaussian (Verly 1983, 1984); that is

z(x) : e ) [y(x)] (1) where y ( • ) is a Gaussian process and 0 ( " ) is a "smooth" strictly monotonic function. Cressie (1985) suggests oh(s) = e S or O ( s ) = s ~ as a possible set of

~Manuscript received 3 June 1986; accepted 11 November 1986. 2Department of Statistics, The University of Chicago, Chicago, Illinois 60637.

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0882 8121/87/0700-0387505.00/I © 1987 International Association for Mathematical Geology

388 Stein

transformations. Whereas taking z ( • ) to be a pointwise transformation of a Gaussian process is a strong assumption, more complicated models for z(" ) usually would be difficult to use. Cressie (1985) suggests a simple method for estimating 4~ ( • ) based on local variogram estimates. Other approaches to estimating 4~( ') include Hermite expansions and graphical means (Journel and Huijbregts, 1978, p. 472-479).

In two different settings, we consider the conditional distribution of

Iv z(x) dx

where V is a specified region, given a set of observations. We approximate the conditional distribution of ~v z(x) dx given z(0) as the region V shrinks, i.e., the conditional distribution under a small change of support. If V does not have a center of mass at the origin, the conditional distribution is approximately Gaussian for a large class of variograms. I f V does have center of mass at the origin, the conditional distribution is approximately Gaussian for a large class of variograms corresponding to the process y ( ' ) not being twice differentiable. Explicit formulas for the limiting (as V shrinks) conditional mean and variance are derived for V, a ball centered at the origin in one, two, and three dimensions.

We consider approximating conditional distributions of a multi-Gaussian process given z(x~ ) . . . . . z(xn), where xl, x2 . . . . is a sequence of points in a bounded region. For large n, the conditional distribution of z (x) given z (xj), . . . . z (xn) is approximately Gaussian, and the limiting conditional mean and variance are obtained easily. We consider extending this result to approximation of the conditional distribution of ~ v z (x) dx, where V is a fixed region of positive but finite volume. However, in a simple example, whereas conditional distribution of ~v z(x) dx is asymptotically Gaussian as n increases, the limiting conditional mean is different than what a naive analysis suggests.

An important advantage of being able to approximate the conditional distribution, as opposed to just the conditional mean and variance, is that we can now approximate other functionals of the conditional distribution, such as tonnage and benefit functions (Matheron, 1985, p. 141). Thus, we avoid Mather- on's criticism of distribution-free geostatistics (Matheron, 1985, p. 138), which he considers unrealistic for modeling distributions under a change of support.

The multi-Gaussian model in Eq. (1) has been considered also by Journel and Huijbregts (1978, p. 566) and Verly (1983, 1984). Journel and Huijbregts (1978, p. 566) note that the conditional distribution of z (x) given z (x~) . . . . . z(x~,) can be obtained exactly under this model. Verly considers the problem of estimating ~v z(x) dx; he suggests either using Monte Carlo simulations or approximating the joint distribution of ~v z(x) dx and z(xj ) . . . . . z(xn) using an assumption that Verly notes cannot be true for nonlinear ~ ( • ) and V having block support (Verly, 1984, p. 502).


THE SMALL CHANGE OF SUPPORT PROBLEM

General Results

We suppose z(x) = 0 [ y ( x ) ] , where y ( ' ) is a continuous, stationary Gaussian process in 61d with

Ey(x) =- 1~ and

Coy [ y ( x ) , y ( x ' ) ] : K(x - x ' )

and 4 ( • ) is a strictly monotonic continuously differentiable function. Let V be some bounded region in 6td with positive volume which we denote by IV I. Define for a positive number e

~v = { ~ x : x e v }

that is, the set of points obtained by multiplying each point in V by e. In this section, we consider the limiting conditional distribution of I~v z(x) dx given z(0) as e decreases toward zero.

For any x, using a Taylor series approximation, we have

z(x) - z(0) : { 0 ' [ y ( 0 ) ] + R ( x ) } [ y ( x ) y(0) ]

where R(x) ---, 0 as y(x) ~ y(O), (Rao, 1973, p. 386). Integrating this expres- sion over V, we obtain

f ~ v z ( x ) d ~ - z(0)~dl vt

S : 0 ' [ y ( 0 ) ] ,v[Y(X) - y ( 0 ) ] d x + ~vR(X)[y (x ) - y ( 0 ) ] d x (2)

Using standard results from multivariate analysis (Anderson, 1984, Sec. 2.5), conditional on y(0) , y( • ) is a Gaussian process with conditional mean

_ ~ ( x ) E[y(x) Iy(O)] K(0) [/x - y (0) ] + y(0) (3)

and conditional covariance function

v(x) v(~') Cov [ y ( x ) , y(x ' ) l y(O ) ] = 7(x) + "y(x' ) - 7(x - x' ) K(0)

(4) where 7(x) = K(0) - K(x) is the unconditional semivariogram of y( - ). Thus, the first term on the right-hand side of Eq. (2) is conditionally Gaussian with conditional mean

M(~V) = ~ ' [y (O) ] ~ - y(O) I ~ ( ~ 5 o~ ~(x) dx (5)

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and conditional variance

- "y(x - x ' ) - 7 ( x ) " y ( x ' ) ] dx ,:Ix' (6) K(o) j

Let us suppose that 3, ( . ) is isotropic and its behavior at the origin is of the form

~ ( x ) = C l x l ~ + o(Ixl °) as Ixl ~ 0 (7)

where C > 0. We can show easily (Appendix A), as e + 0

M(eV) = Cq~'[y(O)] /~ - y(O) cd+~ I K(O) v [xl dx + o(e d+~) and

S(eV) = C~b ' [y(0)] 2 eZd+~J(V, c~) + o(e 2a+~) where

~(~' ~ ) : f~ f~ [Ixl° + Ix'l° - I x - x'l°l ~ d x ' (8)

For 0 < ee < 2, if w(x) is a stochastic process with semivariogram y ( x ) =

Ix[ ~

For 0 < ee _< 1, the integrand in Eq. (8) is positive and, therefore, J ( V, a ) is strictly positive. For 1 < c~ < 2, I am unaware of a proof that J ( V, oe) is strictly positive. [This result would follow from Journel and Huijbregts (1978, Remark 1, p. 307), but they give no proof o f their claim.] For c~ = 2

~(v, 2 / = Iv Iv IIxl2 + Ix'12- Ix - x'I2] ~ ~ '

= 2 ~,, xix[ dx dx' V V i = 1 )2

= 2 xi dx >_ 0 i = l V

where x = (xl . . . . , Xd ), X' = (Xl, . . . , X~ ), with equality if and only if the center of mass of V is at the origin. We show (Appendix B) that the remainder term in Eq. (2) is of lower order in probability if J ( V, c~) > 0. Further, if J ( V,

) is positive and ~b' [ y (0) [ ~e 0, we also have as e $ 0

M(eV) = O(e d+'~) = o[S((~V) 1/2]


That is, to a first approximation, the limiting conditional distribution of f~v z (x) dx has mean z(o)cdl V I" We can conclude:

T h e o r e m 1. I f for 0 < (x _<_ 2, 7 ( x ) s a t i s f i e s Eq. (7), ( h ' [ y ( 0 ) ] :~ 0, and J(V, c~) > 0, then conditional on y ( 0 ) , as e$ 0

fovz(x) ~ - z(o)~L vl 53 O,[y(O)]!ed+(~/2)[CJ(V ' c~)],/2 --* (I) (9)

53 where -~ means "converges in distribution" and • is a standard Gaussian random variable.

Now consider the case where c~ = 2 and Vhas center of mass at the origin, in which case J ( V, 2) = 0 and relation (9) does not apply. When c~ = 2, y ( . ) [ and hence z ( • )] is differentiable and, not surprisingly, a "cancellat ion of errors" occurs when integrating over a region with center of mass at the origin. Suppose

v(x) = c lx l : - D i x i e + o(]xl ~) as I x l - - 0

where C > 0, D > 0, a n d 2 < c~ < 4. I f D < 0, t h e n y ( . ) i s n o t a v a l i d semivariogram. In this situation, y ( . ) has one, but not two, derivatives. We have

S(eV) = - D ( h ' [ y ( 0 ) ] 2 eZd+~J(V, (x) + O(e 2d+~) (10)

For 2 < a < 4, - Ix I ~ is a valid generalized covariance of order one [follows from Matheron (1965, p. 173) and Matheron (1973, Theorem 2.1)], and Iv [w(x ) - w ( 0 ) ] dx is an authorized increment of order one when Vhas center of mass at the origin (Delfiner, 1976). Hence, if w( • ) has generalized covariance - Ix i ~

V a r I l v [ W ( X ) - w ( O ) ] d x l = -J(V, o~) >__ 0

I f -J (V , a ) > 0, we have, conditional on y ( 0 )

f z(x) dx - z(O)edv eV £

~)1,/2 -~ m (11) ~ ' [ y ( 0 ) ] ed+( '~/2)[-DJ( V,

if ~ " ( ' ) exists and is sufficiently well-behaved (Appendix B). Suppose

~(x) = c l x l 2 - D t x q 4 + o ( I x l 4) as Ix1 -~ 0

which implies that y ( • ) is twice differentiable. In this case, the remainder term in Eq. (2) (Appendix B) is of the same order as the first term, which suggests that the limiting conditional distribution of Joy z(x) dx given z ( 0 ) is no longer Gaussian.

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Finally, we note that all results in this section can be extended to the case of 3, ( • ) having a geometric anisotropy. We simply take a linear transformation of the coordinates that leaves 3' ( " ) isotropic in the new coordinates and trans- form V using the same transformation. The region V will have center of mass at the origin in the new coordinate system if and only if it has center of mass at the origin in the original coordinate system.

Formulas W h e n V Is A Ball Centered at the Orig in

We now evaluate J(V, e~) when V is a ball of radius 1 centered at the origin in one, two, and three dimensions. From Eq. (8)

o'= 2' 1 S S f 4 V V V

If V = {x: Ix] ~ 1 } in 6l d, by changing to polar coordinates

S S' v I x [ ° dx = A~ o r ° + ~ - ' d r • A~/ (~ + d )

where Ad is the surface area of a unit sphere x ' , t = x

in d dimensions. Letting, s = x -

2~pv n Iv + s)llsl d,

F J(V, ~) = &~2IvI / (c~ + d)

F o r d = 1, so V = [ - 1 , 1]

where V + s = { t: t - s ~ V }. Changing to polar coordinates

j f Ix - x'l~ a~d°c' = A~ IV ('l (V +ru)lr~+d-l dr V V 0

where u = ( 1 0 . . . 0) is a vector o f length d. Thus

o l v o ( v + r u ) l r ~ + ~ - ' d r

For d = 2

I 2~ 21 J(V, o~) - 8 1 o ~ + 1 o~+

47r 2 ( J(V, cQ - ~1

~ + 2 t

r(~ + 3) -~ (c~ + 4) F[(ce + 4 ) / 2 ] 2

(12)

(13)

(14)


where I ' ( . ) is the gamma function. For d = 3

J(V, c~) = 32rr2(c~2 + 10c~ + 24 - 12 • 2 ~) ( i 5 ) 3(~ + 3)(~ + 4)(~ + 6)

In each case, J(V, oe) > 0 for 0 -< c~ < 2, J(V, 2) = 0, and J(V, oe) < 0 for c~ > 2 (Appendix C), as expected [relation (9) and Eq. (11)].

Application to Intrinsic Random Functions of Order Zero

Suppose y ( ' ) has Gaussian increments and a stationary semivariogram y ( ' ) , but T ( x ) ~ oo as Ixl ~ oo, so that y ( ' ) is not a stationary process. Results of the previous sections seemingly should hold if we set K ( 0 ) = oo [ or l / K ( 0 ) = 0] in Eqs. (5) and (6). However , specifying the joint distribution of all increments is not sufficient to obtain conditional distributions given y (0) . I f we make the stronger assumption that conditional on y (0) , y ( • ) is a nonstationary Gaussian process with

E [ y ( x ) l y ( O ) ] = y ( 0 ) and

Cov [ y ( x ) , y ( x ' ) l y ( O ) ] = 7 ( x ) + y ( x ' ) - 7 ( x - x ' )

then Eqs. (5) and (6) are correct if we take 1 / K ( O ) = 0. For example, Brown- Jan motion often is modeled as a Gaussian process with a fixed origin, a constant mean, and nonstationary covariance function

x(x,x,) = + I 'l- Ix - x ' t

which implies that y(x , x' ) = Ix - x ' I for all values of y ( 0 ) ,

A Higher-Order Approximation for the Conditional Expectation

The conditional expectation of ~ v z (x) dx given y (0) is often of particular interest; hence, a higher-order approximation to it than the value z (0 )e J] V] given by Theorem 1 is worthwhile to obtain. To derive the next-order term for the conditional expectation, the second term on the right-hand side of Eq. (2) must be considered. More specifically, if ~b ( . ) has three derivatives

fovZ( ) - z(0)odl vI = 'Iy(0)j t vIy(x)- y(0)l 1

4 / ' [ y ( 0 ) ] I [ y ( x ) - y ( 0 ) ] 2dx (16) + 2 eV

plus a remainder that can be shown to have an asymptotically negligible impact on the conditional expectation if ~h " ( " ) is sufficiently well-behaved. And, using Eqs. (3) and (4)

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E{[y(x) - y ( 0 ) ] 2 y ( 0 ) } = 2"y(x) - K(0---~ - -

I f 3' ( " ) is of the form gwen in Eq. (7), as e $ 0

EI. I z(x)dxlz(O)l = z(0)# lvI + c~ d+~ ~ b ' [ y ( 0 ) ] / x - y ( 0 )

+ 0"[y(0)] f Ixt dx + v

So, both the first and second terms on the right-hand side of Eq. (16) contribute terms of the same order of magnitude to the conditional expectation.

PREDICTION OF TOTAL ORE WITHIN A FIXED REGION

Some Heuristics

Here, we consider using Taylor series expansions to approximate the conditional distribution of prediction errors for total ore within a fixed region given a large number of point-support observations. Specifically, let x~, x 2 . . . . be a sequence of points in 6 t" that is dense in a fixed, bounded region V; we are interested in the conditional distribution of I v z (x) dx given zn = [ z (xl ) . . . . . z (x . ) ] ' for large n, where z ( • ) is a stochastic process of the form given in Eq. (1).

Let us define

o(x) ; e[y(x)lz ] and

K~(x, x ' ) = C o v [ y ( x ) , y ( x ' ) l z ~ ]

Because xl, x2 . . . . is dense in V and y ( ' ) is continuous, ~gn (x) is a consistent predictor for y(x) as n ~ oo (Yakowitz and Szidarovszky, 1985). We have, similar to Eq. (2)

z ( x ) - ~b[~,(x)] = {qS'[3~n(x)] + R~(x)}[y(x) - yn(x)] (17)

where R,~(x) ~ 0 as p , ( x ) ~ y(x). Now, conditional on z,, ~b' [ p , ( x ) ] [ y ( x ) - pn(x)] is Gaussian with mean zero and variance ~b' [p~(x)] 2 Kn(x, x). For the observed realization of the process, suppose 29n(x) ~ c, which will occur with probability one (c will depend on the realization and will equal y(x) with probability one). I f qS(. ) is continuously differentiable, ~b' (c ) :~ 0, and K~(x, x) > 0 for all n; similar to the proof of Theorem 1, we can show, as n ~

q ~ ' [ ~ ( x ) ] K.(x, x) '/2 z. --+ • (18)


That is, conditional on z~, for n large, z (x) is approximately Gaussian. Because relation (18) holds conditionally with probability one, it also holds uncondi- tionally. Note that the unconditional distribution of z (x) is certainly not Gauss- ian for nonlinear 4 ( - ) .

We can integrate Eq. (17) over a region V to obtain

+ i R n ( X ) [ y ( x ) -- 3~n(X)] dx

(19)

Conditional on zn the distribution of the first term on the right-hand side of Eq. (19) is exactly Gaussian with conditional mean zero and conditional variance

v v 0 ' ~ dx'

If, conditional on z., the remainder term in Eq. (19) is of lower order in probability as n --+ oo we would have with probability one

s (v) ® (20) V

Unfortunately, relation (20) does not hold in a simple case.

An Example

Suppose z ( x ) = ~ [ y ( x ) ] , where y ( x ) is Brownian motion; that is, y ( . ) is Gaussian, y ( 0 ) = 0, E y ( x ) =- 0, and Coy [ y ( x ) , y ( x ' )] = rain (x, x ' ) for x, x ' > 0. Consider approximating the conditional distribution of I~ z ( x ) (Ix g i v e n z , = { z ( 0 ) z [ ( n - 1) l ] z [ 2 ( n _ 1) 1] . . . . , z ( 1 ) } , w h e r e 6 ( ' ) i s strictly monotonic with a continuous second derivative. We show (Appendix D), that with probability one

0 {Z(X) -- ~ [ y n ( x ) ] } dX -- 2H ,~l=

111} 111} 2 ,j2

t3 z~ + db (21)

We also have with probability one

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S' o 0" [y (x ) ] ~z

(22)

Although this formulation of the limiting conditional distribution is more ele- gant, relation (21) is stated only in terms of/~ z(x) dx and z n and would be the form of the result used in practice.

Limit (20) does not hold in this example; we need to take into account the second term in the Taylor series expansion [thus the appearance of ~" ( • ) in relations (21) and (22)] to get the correct limiting conditional mean, although the limiting conditional distribution is Gaussian. Whether a similar correction to the conditional mean is needed in a more general setting is unknown.

CONCLUSIONS

A method for approximating the conditional distribution under a small change of support for multi-Gaussian processes has been introduced. The method is based on using a Taylor series expansion to show that the conditional distribution is approximately Gaussian for a small change of support. Roughly speaking, the conditional distribution is approximately Gaussian whenever the point observation is not the center of mass of the block support or if the observed process is not twice differentiable.

Use of Taylor series for approximating the conditional distribution of the error of a predictor of the total ore within a fixed region is also examined. In an example, this conditional distribution is found to be asymptotically Gaussian as the number of observations in the region of interest increases. Unfortunately, the conditional mean is not obtained by using just the first term in a Taylor series approximation. Whether this result is typical is unknown.

Approximating conditional error distributions for non-Gaussian processes by Gaussian distributions is appealing because of the simplicity of the approximation. Also, by approximating the conditional error distribution as opposed to just the first two moments, estimates of nonlinear functions of block grades, such as the tonnage and benefit functions, can be obtained. The fact that this approach approximates conditional rather than unconditional error distributions also is appealing, because conditional error distributions are often more relevant for mining problems.

Besides the obvious unanswered questions, several other problems relating to this method also need further investigation. In particular, whether or not these results can be extended to non-Gaussian processes other than multi-Gaussian


processes is unclear. Also, the effect on the results of using an estimated transformation 4) ( " ) or estimated mean and covariance functions is unknown. Prac- tical aspects of applying these results, such as computing integrals in the various formulas for the conditional variance, also need attention, although explicit formulas for the limiting conditional variance for a small change of support are given when the block is a ball. A strong restriction on results presented here is that the observed process has no nugget effect. Although the small change of support results are dependent on the continuity of the underlying process, approximately Gaussian conditional distributions for the prediction error of the ore in a fixed region may be possible even when a nugget effect is present; however, no such results are known to the author.

A C K N O W L E D G M E N T S

This manuscript was prepared using computer facilities supported in part by National Science Foundation Grant No. DMS-8404941 to the Department of Statistics at the University of Chicago.

APPENDIX A

Approximations for M ( e V ) and S (e V ) are derived. Substituting C I x [~ for "y(x) in Eq. (5)

f Ix1 dx (o5 ov c °

= C4~'[y(0)] • ~-(~- y(0) ed+~ f v I W i ~ dw

by letting w = ex. Given a > 0, we can find rl > 0 such that, for all e <

- cl l l < 6I I ° for all x e eV

using Eq. (7) and the fact that V is bounded. Thus, for all e <

Because 6 is arbitrary we obtain, as e $ 0

M(eV) = C O ' [ y ( 0 ) ] " - y(0) d + ~ f ix[~dx + °(e~+~)

Similarly

398

feV feV

Stein

b'(-~) +.y(x,)-~,(x-~,)]~,~,=co~d+o f,. Iv [ Ixls + Ix'IS

- Ix - x, I s] d~ ,~' + o(~ ~'+o) Also

fcv f~v'g(x) "g(x') dx dx' = I f~vT(X) dxl 2

Substituting these results into Eq. (6)

S(eV) = CO'[y(O)]2e 2d+'~ Iv Iv [ Ixls + Ix ' Is

- I ~ - ~' I'~] dx d~' ÷ O('2d+c~)

= o(,2d+~)

Because X is arbitrary

E.LO I . x ~ e V

in probability as e ~ O. Thus

sup IR(x)l--* o x E e V

Hence

APPENDIX B

First we show that the remainder term in Eq. (2) is, conditional on y (0), of lower order in probability if 7(" ) is of the form given in Eq. (7), J( V, c~) > 0, and 0' [ Y (0) ] :~ 0. The remainder term is

fEvR(X) [y(x) -- y(0)] dr

where R(x) --* 0 as y(x) ~ y(O). That is, given X > 0, 6 > 0 exists such that

ly(x) - y(O)l <6 implies IR(x) I < x Now, y( • ) is continuous with probability one, so

l i m P I s u p ] y ( x ) - y ( 0 ) ] < 61 = 0 E ~O L x ~ e V


[ I~vR(x)[y(x) - y(0)] dx

We can show easily that

<_ sup IR(x)l I ly(x) - y(O)tdx x ~ e V eV

fov lY(X) - y(0)[ d~ = op(0 d+~/2)

and Theorem 1 follows under the stated conditions. Now consider the case where V has center of mass at the origin and

7(x) = C xl 2 - D t x l ~ + o([x] ~) as {x I ~ 0

for 2 < c~ < 4. In this case, J ( V, 2) = 0, so to show that the remainder term in Eq. (2) is asymptotically negligible, a more careful analysis is required. If 0 (" ) is twice differentiable, we have

1 0" [Y(0)] 2 f [y(x) - y(0)] 2 dx (23) + 2 eV

plus a remainder that can be shown to be asymptotically negligible if 0" ( • ) is sufficiently well-behaved. Conditional on y (0), we have

E ( I 0 " [ y ( 0 ) ] 2 X¢v[Y(X) - y(O)]2dxl 2 y(O))

= o,,ly~o~l 4 o(~ S~ ~{ly~- y~o~; ly~'~ - y~o~; y~o~} ~ ~,

-< ~"ly~°~l 4 I~ S~v~{Ly~'o - y(o~1' y~o~} ''~

• E{[y(x' I -Y(Ol] 4 y ( 0 ) } ' / 2 d x d x '

: 4,"[y(O)]4(f, vE{ [y ( x ) - y(0)] 4 y (0 )} ' /2 dx) 2

400 Stein

where the first inequality is by Cauchy-Schwarz, and the second is true for some finite M [possibly dependent on y(0)] , using the fact that y(x) - y(O) is conditionally Gaussian given y (0) with conditional mean and variance given in Eqs. (3) and (4). Thus, conditional on y(0), the second term on the right- hand side of Eq. (23) i s Op(6d+2). And, i fJ (V, ~) < 0, the first term on the right-hand side of Eq. (23) is conditionally Gaussian with conditional variance of order C 2d+ee, which implies that this term is n o t Op(ed+2). Equation (11) follows. I f J (V, 4) < 0, and

~(x) = c l x l ~ - O l x [ 4 + o(Ixl 4) as Ix[ ~ 0

the first term on the right-hand side of relation (25) is Op(ea+2), so that the second term may be of the same order.

APPENDIX C

Equations (13)-(15) are derived. In one dimension, from Eq. (12)

J(V, c~) = 214/(c~ S 2 l + 1) - o (2 - r)r ~dr

In two dimensions

I 2°j 8 1 o e + l o e + 2

J(V, c~) = 27rI27r/(c~ , 2 , siI 2cos' (2) rEl (2)il'~21r~+l~rl

r ~+1 dr = (~ + 2 ) -

Integrating by parts

Sicos-,(~) Letting u = ( r / 2 ) 2

flr~+2E1 (2)21-1j2

f~r~+2E1 (2)2] -~j2

dr

dr

I i (2u'/2)~+2(1 - u) - ' /Zu - ' / 2 du

I 1 = 2 c~+2 U(C~+l)/2(l - - /t) -1/2 du 0


using the definition of the beta function (Carrier, Krook, and Pearson, 1966, p. 191)

using F ( 1 / 2 ) = ~r L/2 and the duplication formula for the gamma function (Carrier, Krook, and Pearson, 1966, p. 187). Similarly

f2 i ( 2 ) 2 ] I/2 reP(e~ + 3) r ~ + 2 1 - dr =

o (oe + 4) I'[(o~ + 4 ) /2 ] 2

Thus

J (V , o~) = 2rrI2rr/(oe 1 7rP(c~ + 3)

+ 2 ) - + 2 p [ ( ~ + 4 ) /2 ] 2

7rV(c~ + 3) ] + (c~ + 4) V[(c¢ + 4 ) /2 ] 2 ]

471"2 [1 -- r(c~ + 3)

c~ + 2 (c~ + 4) P[(c¢ + 4 ) / 2 ] 2

In three dimensions

S 2 1 J (V , c~) = 4~r 7r/(c~ + 3) - ~r [4 _ r + ( r 3 / 1 2 ) ] r ~+2 dr 0

32~r2(c¢ 2 + 10c~ + 24 - 12 • 2 ~)

(c¢ + 3)(e~ + 4)(c~ + 6)

We now show that J(V, c~) > 0 forO _< c~ < 2, J ( V , 2) = O, and J(V, c~) > 0 for c~ > 2 when V is a ball of radius 1 centered at the origin in one, two, and three dimensions. In one dimension, J ( V , O) > O, J ( V , 2) = O, and 1 - [2~/(c~ + 2)] has a negative second derivative, so the result follows. Simi- larly, in three dimensions, J ( V , O) > O, J ( V , 2) = O, and c~ 2 + lOc~ + 24 - 12 • 2 ~ has a negative second derivative. In two dimensions, the result follows by noting that J(V, O) > O, J ( V , 2) = O, and showing that

log I F (~ + 3) 1 (c¢ + 4) F [ ( ~ + 4 ) /2 ] 2

has a positive second derivative. We have

402 Stein

d2 E dot2 log r (o~ + 3 ) - log (~ + 4 ) - 2 log I ' ( C ~ @ ) ]

where = xIt'(OZ q- 3 ) q- (Or q- 4 ) -2 --

d • (w) = ~ log r ( w )

Now o o

,'(w) = k~o (w + k) -2

(Carder, Krook, and Pearson, 1966, p. 190), so

~'(c~ + 3) + (c¢ + 4) -2 - - -

= k=OZ (0~ q'- 3 -t- k ) -2 - ~ [ ( ~ / 2 ) + 2 + k] + (c~ + 4) -2

1 1 1 + (c~ + 4) -2 = (c~ + 4 ) -2 > 0 c~ + 4 2 (c~/2) + 2

where the first inequality follows from

1 ~ ~ 1

3 w + l t o o

(W "Jr .)--2 du <-- ~ ( w + k) -2 _< (w + u) '-2 du = - k=0 0 W

A P P E N D I X D

Derivation of relations (21) anti (22) are provided. Define y ( x ) and zn as previously (in an example). Also, for 0 <_ x < 1, define

In(x ) = i if (i - 1 ) / (n - 1) _< x < i / ( n - 1)

Then straightforward calculations yield

fen(x) = ( n - 1 ) { y [ i / ( n - 1 ) ] [ x - ( i - 1 ) / ( n - 1)]

+ y [ ( i - 1 ) / ( n - 1)][i/(n- l ) - x]} forln(x) = i, and

Imin [x - (i - 1 ) / (n - 1), x ' - (i - l ) / ( n - 1)1

K~(x,x,) = l i ( n - l / I x - ( i - 1)fin-1)l[x'- ( i - 1)/(n-,1)] for In(x) = I,~(x ) = i

for 1,,(x) 4= I , , ( x ' )


F o r V = [0, 1]

i=1 I) ( - l ) , Y ( i - - 1 ) / ( n - l ) +,[~.(x)] ~,[y.(~,)lK.,(x, x'1 dr dr'

n L, (,~,_,, (i~,o_~, ' K.(x, x') dx dx' - , = , ~ { y [ ; / ( , - 1 ) ] } 2 ~,_,~/~°_; , ~ , - , , / ~ o - , ,

n 1 = ~ ' }2 - 12(n 113 2 { y [ i / ( n - 1)1 (24)

where a(n) ~ b(n) means a ( n ) / b ( n ) --+ 1 as n --* co. Equation (24) holds for all realizations of y ( . ) that are continuous on [0, 1 ], which is true with probability one. Now consider a second-order Taylor series expansion for I~ z(x) dx. We have

~ ( x / - 4 ) [ y . ( x ) ] -- + , [ y , ( ~ ) ] [ y ( x ) - y.(~)]

+ -'~"[y.(xtl[y(x)=- - y,,(~)]=

where the remainder will be asymptotically negligible for 4)"( ' ) sufficiently well-behaved. Thus

f l

o {z(xt - 4)[~n(xtl} dr ~ - f l 4)'[Y"(xt][y(x) - ~.(x)] dr

S' 1 41" [ ~ . ( x ) ] [ y ( x ) - )~.(x)] 2 dx +2 o

(25)

Now

fl - n x,l 2 = I' ~,,[~,,(x)] K.(x, ~1 dr 0

n - I

1 c)"{y[ i / (n 1t1 } n- ~ i=~

(26)

again, with probability one. And

404 Stein

Var I Ii 4;'[P~(x)]ly(x) - p~(x)]=dxlzn 1

= 1 ( P " [ ' g n ( X ) l d P " [ y n ( x ' ) I C o v { [ y ( x ) - ; n ( x ) ] 2 ' 0 0

t 2 [y (x ' ) - 3 ~ ( x 11 [z~}dxdx '

i t s ' = 2 , ; b " [ f ~ ( x ) ] ~b " r ^ " x ' )2 dx ' o o LY,,L ) ] K,,(x, x' dx

n k l I i / ( n - 1 ) f i / ( n - 1 ) = (~)] <(~, x a ~ 2 4/ , [ 3~n ( x ) ] 4,,,[3~ n 2 ,)2 doe'

i = 1 o(i - 1 ) /(n - 1 ) J(i - 1 )/(n -- 1 )

. ,-I 2 fi/(n 1) - 2 ~ ch"{y[ i / (n - 1)]} /

i= 1 O(i-- l ) / (n-- l)

n--I 2 ( / ' / - 1) - 4 ~ ~ " { y [ i / ( n - 1)]} 2

i=1

~ i /(n -- 1)

Kn (x, x ' )2 d,x d x ' J(i-- 1)/(n-- 1)

Thus, the conditional variance of the second-order term in relation (25) is of lesser order than the conditional variance of the first-order term. However, the conditional mean of the second-order term is of the same order of magnitude as the standard deviation of the first-order term, so it will have a nonnegligible impact on the limiting conditional mean. More specifically, with probability one

n -- I

o {z(x) - ~b[33.(x)l} dx - 2-n i=,

) / ( 1 ~k' '{ 2) ' /21 ~ - 1 ) ] } ~ n i=i ~ y [ i / ( n - 1 ) ] } z n ~ ~b (21)

Because

- I

l n ~ 1 0 " { y [ i / ( n - I)]}/I F/ i=1 0

1 2 o ' { y [ i / ( n - 1 F/ i=1 0

O"[y(x)] dx --+ 1

qS'[y(x)] 2 dx ~ 1

and

with probability one, relation (22) also holds with probability one.


R E F E R E N C E S

Anderson, T. W., 1984, An Introduction to Multivariate Statistical Analysis: John Wiley & Sons, New York, 675 p.

Carrier, G. F.; Krook, M.; and Pearson, C. E., 1966, Functions of a Complex Variable, Theory and Technique: McGraw-Hill, New York, 438 p.

Cressie, N., 1985, When Are Relative Variograms Useful in Geostatistics?: Math. Geol., v. !7, p. 693-702.

Delfiner, P., 1976, Linear Estimation of Non-Stationary Phenomenon, in M. Guarascio et al. (Eds.), Advanced Geostatistics in the Mining Industry: Reidel, Dordrecht, p. 49-68.

Joumel, A. G. and Huijbregts, C. T., 1978, Mining Geostatistics: Academic Press, London, 600 p.

Matheron, G,, 1965, Les Variables Rdgionalisdes et Leur Estimation: Masson, Paris, 305 p. Matheron, G., 1973, The Intrinsic Random Functions and Their Applications: Adv. Appl. Prob.,

v. 5, p. 439-468. Matheron, G., 1985, Change of Support for Diffusion-type Random Functions: Math. Geol., v.

17, p. 137-166. Rao, C. R., 1973, Linear Statistical Inference and Its Applications (2nd ed.): John Wiley & Sons,

New York, 625 p. Verly, G., 1983, The Multigaussian Approach and its Applications to the Estimation of Local

Reserves: Math. Geol., v. 15, p. 259-286. Verly, G., 1984, The Block Distribution Given a Point Multivariate Normal Distribution, in G.

Verly et al. (Eds.), Geostatistics for Natural Resources Characterization: Reidel, Dordrecht, Netherlands, p. 495-515.

Yakowitz, S. and Szidarovszky, F., 1985, A Comparison of Kriging with Nonparametric Regres- sion Methods: J. Multivar. Anal., v. 16, p. 21-53.

gaussian approximations to conditional distributions for multi-gaussian processes

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