ELSEVIER

Journal of Statistical Planning andInference 69 (1998) 275-294

journal ofstatistical planningand inference

Constructing and fitting models for cokriging andmultivariable spatial prediction

Jay M. Ver HoeP' b, *, Ronald Paul Barry b

a Alaska Department of Fish and Game, 1300 College Road, Fairbanks, AK 99701, USAb Department of Mathematical Sciences, University of Alaska, Fairbanks, AK 99775, USA

Received I J September J996; revised 15 August 1997

Abstract

We consider best linear unbiased prediction for rnultivariable data, Minimizing mean-squared­prediction errors leads to prediction equations involving either covariances or variograms. Wediscuss problems with multivariate extensions that include the construction of valid models andthe estimation of their parameters. In this paper, we develop new methods to construct validcrossvariograms, fit them to data, and then use them for multi variable spatial prediction, includ­ing cokriging. Crossvariograms are constructed by explicitly modeling spatial data as movingaverages over white noise random processes. Parameters of the moving average functions maybe inferred from the variogram, and with few additional parameters, crossvariogram models areconstructed, Weighted least squares is then used to fit the crossvariogram model to the empir­ical crossvariogram for the data. We demonstrate the method for simulated data, and showa considerable advantage of cokriging over ordinary kriging. © 1998 Elsevier Science B.V.All rights reserved.

A MS Classification: 62M30

Keywords: Spatial statistics; Geostatistics; Cokriging; Kriging; Variogram, Crossvariogram

1. Introduction

When sampling the environment, measurements are often collected on more thanone variable type. For example, in geostatistics, measurements may be taken on manydifferent minerals, or, in ecology, data may be collected on many different species. Theidea of autocorrelation, or correlation within a variable type, is central to time seriesand spatial statistical models. Spatial prediction from autocorrelated spatial models hasbeen called kriging in the geostatistical literature (for a review, see Cressie 1991). Thenotion that there may also be spatial correlation among variable types, crosscorrelation,

* Corresponding author. Tel.: (907) 459-7278; fax: (907) 452-6410; e-mail: ffjmv@aurora.alaska.edu.

0378-3758/98/$19.00 © 1998 Elsevier Science BY All rights reserved.PII S0378-3758(97)00 162-6

276 1. M. Ver Hoe! R.P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294

has led to generalizations of kriging which make use of both autocorrelation and cross­correlation (Ver Hoef and Cressie, 1993). In this paper, we will use the term cokriging

to mean prediction of a single random variable at a single location from multivariate

spatial data, and the tem1 multivariable spatial prediction when predicting a vector of

random variables (of different types) at a single location.

1.1. Quick revie~v of kriging

Kriging is a spatial prediction method that is formulated by minin1izing the mean­

squared-prediction errors (MSPE). Define stationarity as follows: assume that the kthspatial random variable Zk(S) is defined at each location S in some region !?2 C [Rd, that

E[Zk(S)] == Jlk for all S E ~, and that the variogram,

(1)

exists. Data are collected at nk locations, and assume the data are a realization of the

random vector Zk == [Zk(Sl k), Zk(S2k)"'" Zk(snk )]. Let h'Zk be a linear predictor for thekth random variable type Zk(SO) at location So, subject to the unbiasedness constraint

E(h'Zk) == E[Zk(SO)]. Then kriging uses Eq. (1) to minimize the MSPE; that is, one

wishes to minimize E[h'Zk - Zk (SO)]2 for h subject to unbiasedness, or find an a such that

E[h' Zk - Zk(So )]2 - E[a' Zk - Zk(So )]2 ~ 0 (2)

for all h such that h'Zk is unbiased (see Cressie, 1991, pp. 119-123). As seen here,

kriging is simply best linear unbiased prediction (BLUP). In general, using variogramsEq. (1) for kriging is better than using covariances because there is less bias in esti­

mating the variogram than in estimating the covariogram [(Cressie, 1991, p. 71) seeSection 2.5 below] and there is a larger choice of models (Cressie, 1991, p. 68). Krig­

ing has proven to be very popular. However, extensions to cokriging and multivariable

spatial prediction, while extremely appealing, have proven difficult to apply in practice.

1.2. Cokriging and multivariable spatial prediction

When there are L spatial variable types, assume that they all follow the stationarityassumption. Cokriging is formulated as BLUP by minimizing the MSPE for the linear

predictor h'z (with Z == [z~, z~, ... , z~]'), and finding an a such that a'Z is unbiased(E(a'z) == E[Zk(SO)]) and

E[h'z - Zk(SO)]2 - E[a'z - Zk(SO)]2 ~O (3)

for all h such that h'z is unbiased. For multivariable spatial prediction, BLUP is gen­eralized by Ver Hoef and Cressie (1993) as follows. The linear predictor B'z is an

unbiased predictor of z(so) (where z(so) == [Zl(SO),Z2(SO), ... ,ZL(SO)]') if E(B'z) ==E[z(so)], and A'z is the BLUP if A'z is unbiased and

E{[B'z - z(so)][B'z - z(so)]'} - E{[A'z - z(so)][A'z - z(so)]'} (4)

is nonnegative-definite for every B such that B'z is unbiased.

J.M. Ver Hoef, R.P. Barry/Journal of Statistical Planning and Inference 69 (1998) 275-294 277

There are three main problems with cokriging as traditionally applied. First, whilethe variogram can be used to minimize Eq. (2), its multivariate generalization has

caused problems. The traditional crossvariogram has been defined as,

(5)

Next, define the covariance function, or crosscovariogram as,

(6)

and if k == m then we call it the covariogram. Then, as discussed by Journel and

Huijbregts (1978, p. 326), Myers (1982), and Ver Hoef and Cressie (1993), MSPE forcokriging Eq. (3) is minimized using Eq. (5) only when Ckm(h) is an even function;

C/an(h) == Ckm (-h) (also called reflection symmetry (Lu and Zinlnlerman, 1994)). The

reflection symmetry condition, C/an(h) == Ckm( -h), is very restrictive and nlakes the

use of Eq. (5) questionable in nlany cases. Most applications that we reviewed either

failed to mention this condition, or used crosscovariograms, Eq. (6), for cokriging.The second problem with the traditional formulation of cokriging is that to estimate

the crossvariogram, Eq. (5), data of both variable types need to be measured at the

same location. It was largely this deficiency that led Clark et aI. (1989) to propose

a new quantity based on E[Zk(Si) - Zm(Sj )]2. By adapting it to account for unknown

means, this new crossvariogram becomes,

(7)

(Cressie, 1991, p. 140) and has also been called a 'pseudo-crossvariogram' (Myers,

1991). There appears to be some confusion about the validity of Eq. (7). For example,Wackernagel (1995, p. 136) says 'it usually does not nlake sense to take the difference

between two variables measured in different physical units'. However, simple expansion

of Eq. (7) shows that 2Y/an(Si, Sj) == var[Zk(si)] + var[Zm(sj)] - 2COV[Zk(Si), Zm(Sj )], so

Eq. (7) is no less objectionable than COV[Zk(Si), Zm(Sj )].The third problem with the traditional formulation of cokriging is that it is difficult to

produce valid crossvariogram models that are consistent with the variogram models. By

valid, we mean that the prediction variances remain positive: a condition on variogram

models is that they must be conditionally nonnegative definite (see Cressie, 1991, p.

86). Journel and Huijbregts (1978, p. 326) recommend a coregionalization model whereall crossvariograms are a weighted sum from a common set of basic variogram models.

A literature review showed, however, applications which apparently mix variogramand crossvariogranl nlodels of different types. When mixing models of different types,

Myers (1982) states that the crossvariogram and variograms must obey the Cauchy­

Schwartz inequality. Although necessary, this may not be sufficient to guarantee validmodels yielding nonnegative prediction variances. Again, a review of the literature

showed applications where the Cauchy-Schwarz inequality was violated (e.g., Nash et

aI., 1992).

For cokriging, and multivariable spatial prediction in general, Ver Hoef and Cressie(1993) show that the generalization of the variogram to the crossvariogram, Eq. (7),

278 J.M. Ver Hoej, R.P. Barry/Journal of Statistical Planning and Inference 69 (1998) 275-294

allows minimization of the MSPE in Eqs. (3) and (4) without the need to assumeCkm(h) == Ckm( -h), elin1inating problem one listed above. As Clark et al. (1989), VerHoef and Cressie (1993) and Papritz et al. (1993) all noted, use of Eq. (7) eliminatesthe second problem of requiring both data types at the same location. However, theproblem remaining for the crossvariogran1, Eq. (7), as for the traditional crossvari­ogram, Eg. (5), is the construction and estimation of valid models for 2Ykm(.). Papritzet al. (1993) consider coregionalization n10dels for Eg. (7). However, Papritz et al.( 1993) and Helterbrand and Cressie (1994) show that coregionalization is often far toorestrictive to model data, and it is often difficult to apply (Isaaks and Srivastava, 1989,p. 391). Also, although Wackemagel (1995, p. 172) does not consider 2Ykm(Si,Sj),Eq. (7), he introduces the bilinear model for covariances, which is a slight variationon coregionalization to allow for uneven functions. However, Wackemagel's model(1995, p. 174) is again restrictive by forcing each component variogran1 to be identi­cal, and he gives no models for the crosscovariogram function Ckm(h); presumably heuses covariance models derived from variogram models that have sills. In this paper,we introduce a new way to construct valid crossvariogram models.

In Section 2, we concentrate on developing valid crossvariogram models that areconsistent with variogram models. We give several examples, show how to estimatethe crossvariogram, and then, as for variograms, fit a valid crossvariogram to the em­pirical crossvariogram. In Section 3, we review how the crossvariograms are used forcokriging and multivariable spatial prediction. Section 4 gives two simulated examples,and discussion and conclusions are given in Section 5.

2. Crossvariogranl nlodels and estinlation

2.1. A moving average representation

Barry and Ver Hoef (1996) show that many variograms can be developed by in­tegration of a moving average function over a white noise random process. In fact,their methodology allows easy construction of many variograms beyond those com­monly used. This same methodology can be used to construct valid crossvariograms.Start with the following spatial processes: Wk ( x) is a zero mean white noise randomprocess with x E ~d; k == 0,1,2, ... ,L; i.e., E[Wk(X)] == 0, var[JA Wk(x)dx] == IAI,COV[JA Wk(x)dx,JB Wk(x)dx] == °when AnB == 0 and Wk(X) is independent of Wm(x)when k i= m. Now, define

Then

1. M. Ver Hoef, R.P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294 279

for -1 ~ Pk ~ 1 for k == 1,2, ... ,L. Note that Ak == (.1kp ... , L1 kd )' allows a spatial 'shift'in the correlation between the two Y -processes. That is, if Ak == Am == 0 then we have

bivariate correlation at each location x with independence among locations x and twhen x -I- t. Now, let Uk( e) be another white noise process with E[Uk(X)] == 0 and

var[Uk(x)] == I and let it be independent of Uk(t) for all x -I- t, and let Uk(x) be

independent of Um(t) for all k -I- m and all x and t.

Then for some s E !» C [Rd let

Zk(sllh, Vk, Ilk, pk, Ak ) == I:···I: fk(X - sllh )Yk(xlpk, Ak)dx + Vk Uk(S) + Ilk,

(8)

where !k(xIOk) is Riemann integrable and is called the moving average function. For

simplicity, suppose that J~oo[!k(xIOk)]2dx < 00, which induces second-order station­

arity for Zk(e). Then, using the result of Yaglom (1987, p. 418),

Cov ( r fk(X)W(X) dx, r fm(x - h)W(X)dX) = rh(x)fm(x - h)dx, (9)~ ~ ~

for some white-noise process W(x), and from Eqs. (1) and (8), the stochastic process

Zk( e) has variogram,

2Ykk(S, S + h )IOk, Vk)

{

0 for h == 0

= I: ...I: [fk(xlfh) - fk(X - hl()dfdx + 2vZ for h 7'c 0 (10)

where v~ is the nugget effect. We have found that representation of the variogram,

Eq. ( 10), from the moving average construction in Eq. (8) includes nlany of the

commonly used variograms (e.g., linear, linear-with-sill, spherical, exponential, ratio­

nal quadratic, wave, power, and cosine), at least for one dimension. In practice, we

will need to develop the crossvariogram for the quantities Rk(s) == Zk (s) - Jik. The

crossvariogram for the mean-centered processes Rk ( e) is, from Eqs. (7) and (8), again

using Eq. (9),

2Ykm(S,S + hlO, v,Pk,Pm,Ak,A m)

= I:···I: fl(xl()k)dx +I:··· I: f;(xl()m)dx

-2PkPmI: ...I: h(xl()k )fm(x - h + Am - Ak I()m )dx + vz + V~, (11)

where 0 == (O~,O~)' and v == (Vk,Vm)'. Notice that there are d(L-I) free parametersfor Ak; k == 1,2, ... ,L. In practice, we can let Al == 0 and all subsequent Ak; k -I- 1 are

relative to A 1. Also note that for L == 2 variables, Pk and Pm will not be identifiable,

but their product is identifiable, and for L > 2, Pk and Pm will be identifiable. In fact,

the number of crossvariograms grows as (L - 1)L/2, while the number of parameters

Pk grows as L.

280 J.M. Ver Hoej, R.P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294

There are several important features of the variogram, Eq. (10), and crossvariogram,Eq. (11) constructions. First, note that we do not need to prove their 'validity' (e.g. thevariogram must be conditionally negative definite, see Cressie (1991, p. 86) becausethey must be valid by construction. That is, we are taking a real physical process,white noise, and smoothing it by a real function, so the resulting variograms andcrossvariograms must be valid. Second, we do not claim that Eqs. (10) and (11)contain all possible valid n10dels; they only provide a convenient means to constructcrossvariogram models that are compatible with the variogram models. Third, notethat in Eq. (11) it is easy to combine different variogram models simply by choosing

!k(X), which in1plies a certain variogram model for Zk( e), to be different from !m(x),which implies a certain variogram model for Zm( e). In fact, this same methodologycould be used to obtain the traditional crossvariogram 2Tkm( e), given by Eq. (5), sothat it is consistent with its component variograms. However, when the shift parameterAk :I Am, then Ckm(h) :I Ckm (-h), so use of the traditional crossvariogram, Eq. (5),for cokriging is not optin1al (Ver Hoef and Cressie, 1993). Finally, notice that Eq. (11)shows that 2Ykm(Si, s)), Eq. (7), can be used for models with negative crosscorrelation(Pkm <0) despite Wackemagel's (1995, p. 136) claim to the contrary.

2.2. Example in [Rl

Let the ZI-process be defined by,

(12)

and Lh == 0 in Eq. (8), where 81,)?0 and I(expression) is the indicator function, equalto 1 if expression is true, otherwise it is O. Let the Z2-process be defined by,

(13)

where {82,)? 0 and let L1 2 == L1 (because L1 1 == 0 we suppress use of subscripts).The n10ving average functions, Eqs. (12), and (13), imply variogran1s for variables

ZI(e) and Z2(e). From the integration in Eq. (10), Eq. (12) yields an exponentialsemivariogram,

(h) _ { 0 for h == 0Yll -v~ + PI (1 - exp( - 812 1hi) for h :I 0

where PI == 8Tl/(2812 ), while Eq. (13) yields the linear-with-sill semivariogram,

(14 )

(15)

J.M. Ver Hoej, R.P. Barry/Journal of Statistical Planning and Inference 69 (1998) 275-294 281

where 132 == e~l. Then, from Eq. (11), the semicrossvariogran1 n10del is

YI2(h) == (vi + v~ )/2

{

/31+fh()22 _ P()ll o21 [e()12(h-J) _ e()l2(h-J-()n)] for h < Ll2 el2

+ /31+/32()n _ poll()21 [1 - e()12(h-J-8n )] for L1 ~h < e22 + L12 ()12

/31 +g2()n for e22 + L1 ~ h( 16)

where p == P12. To obtain Y2I(h), note that Y21(h) == YI2( -h). In Eq. (16) the 'nugget'

of the sen1icrossvariogram is the average of each of the semivariogram nuggets, and

the 'sill' of the semicrossvariogram is the average of each of the semivariogram sills.

By the moving average construction in Section 2.1, it is generally true that semicross­

variogran1 nuggets will be the average of the component semivariogram nuggets, and

likewise for sills. Specific examples of this model will be used in simulations later in

this paper.

2.3. Example in ~2

Let the Zk-process be defined by

fk(xI8k ) == (Jk3/( -(JkI < Xl ~ (JkI )/( -(Jk2 < X2 ~ (Jk2) (17)

in Eq. (8), where ek,j ~O. The moving average function, Eg. (17), is simply a box

of constant height. From the integration in Eq. (10), Eq. (17) yields the following

anisotropic sen1ivariogram,

o for hI == 0 and h2 == 0

Ykk(h)= V~+f3k U~~~ + ~~~; - d:;'~:kl,] for [hll < 28kl and Ih21<28k2 (18)

v~ + 13k otherwise

where 13k == 4e~3 ekI ek2 . Fron1 Eqs. (11) and (17), the semicrossvariogram is

Ykm(h) == (v~ + v~ )/2

{

for minI> maxI

+ Ih~flm - Pkm8kJ8m3(minl - maxI )(min2 - max2) and m~n2 > max2/3k+/3m otherwIse

2 (19)

whereminI == min(8kl , 8mI -+ hI - L1 ml + L1 kl ),

n1axI == max( -8kI , -8mI + hI - L1 ml + L1kI),

min2 == mineek2 , (Jm2 + h2 - Ll m2 + Llk2),

maX2 == max( -8k2 , -8m2 + h2 - L1 m2 + L1 k2 ).

Let ZI have the following parameters, VI == 0, 01 == (2, 2, 2)', and AI == (0, 0)', let Z2 have

the following parameters, V2 ==0, O2==(3,7,1)', and A2==A ==(5,7)' and PI2 ==0.7. Then

282 1. M. Ver Hoej, R.P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294

D

BA.-.(9ZI

•r-0::0Z'-" -

C'\I.c-1

C

6'zIr-0::0Z'-"

C'\I.c

-1

-10 -5 0 5 10 -10 -5 0 5 10

h1 (EASTING) tl 1 (EASTING)

Fig. 1. Semivariograms and semicrossvariograms. The hatched lines indicate contours are decreasing invalue with respect to the next inner contour, and the solid lines indicate contours are increasing in value.(A) semivariogram for ZI with VI = 0, (Jl = (2,2,2)', (B) sen1ivariogram for Z2 with V2 = 0, (J2 = (3,7,1 )',(C) semicrossvariogram with and Al = (0,0)', A2 = (5,7)' and PI2 = 0.7, and (D) semicrossvariogramwith and Al = (0,0)', A2 = (5,7)' and PI2 = -0.7

the semivariogram for the Zl-process, Eq. (18) is given in Fig. lA, the semivariogramfor Z2-process is given in Fig. IB, and their semicrossvariogram, Eq. (19), is givenin Fig. 1C. In Fig. 1A, the sill is 64, in Fig. 1B, the sill is 84, and in Fig. 1C, thesill is 74. The same semicrossvariogram is given in Fig. ID, but now Pl2 == -0.7.Again it has a sill of 74, but because the two processes are negatively correlated, thesemicrossvariogram is greater than 74 in the vicinity of h == A.

2.4. Simulation from moving averages

Data can be simulated from moving averages based on the following results. Noticethat

Zk(S) =I:· ..I: h(x - S)Yk(x)dx

= f ... f 1. . fk(X - s)Yk(x)dx,il =-CX) id=-CX) A(ll ,...,ld)

J.M. Ver Hoef, R.P. Barry/Journal of Statistical Planning and Inference 69 (1998) 275-294 283

where A(il , ... , id) is the small area (iI/aI, (i l + 1)/al) x ... x (id/ad, (id + 1)/ad) and

IA(il,,,.,id)1 == (l/al)(lja2)".(1/ad) for all {il,,,.,id}, so call it IAI. Now, if aj;j ==1, ... ,d, are quite large, then A(i l , ... , id) will be a small area and fk(X) will be

relatively constant over A(il, " ., id), so

with the approximation improving as all aj -----+ CX) for j == 1, 2, " . ,d because fk (x) is

Reimann integrable. But,

from the central limit theorem and the definition of white nOIse. Basically, we can

approximate white noise with a set of densely packed independent random numberswith small variance. Hence, we have

where Yk(i 1,,,., id) ~N(O, IA I).In practice, we use L~:=bj for LZ:-CXJ for some bj and Cj whenever Ifk(xj - sj)1

becomes relatively small or °below bj and above Cj' In order to get crosscorrelation

and shift, we simulate an (L+ 1)-variate vector [wo(x), WI (x), ... ,WL(X)J' of independentrandom variables Wk(X) rv N(O, IAI) at each location on the densely packed grid, and

create shifted correlation among the L-variates at each location by taking Yk(X) ==

VI - piWk(X) + Pkwo(x - Lf k). Finally, each variable can be given an additive whitenoise component and nonzero mean,

where Uk(S) is a simulated standard normal random variable. Notice that in order to

accommodate shifts and the moving average function, the original grid of densely

packed points may need to be substantially larger than f!fi.

2.5. Estimation of 2Ykm(h)

The method of moments estimator for the variogram is

(20)

(21 )

(22)

284 1. M. Ver Hoe}: R.P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294

where h == Sj - Si, N(h) == {(Si, Sj) : Sj - Si == h} and IN(h)1 is the number of distinctelements in N(h). The en1pirical variogram, Eq. (20), is unbiased for Eq. (1). Recall

Rk(s) == Zk(S) - f.1k, so let rk(s) =- Zk(S) - Zk, where Zk is the mean of the kth variable.Then the n1ethod of moments estimator of the covariogram is,

" 1 '""Ckk(h) = IN(h)1 L/k(sdrk(Sj),N(h)

which is biased for Eq. (6). The estimator given by Eq. (21) is biased because it is

necessary to estimate the mean f.1k of the spatial process with the mean of the realization

Zk (Cressie 1991, p. 70). It is the lack of bias that makes variogram estimation 'better'than covariogram estimation.

It is also necessary to estimate the crossvariogram for the zero-mean residual process

Rk(-). Then a method of moments estimator of 2}'km(h) for the Rk(-) is,

"h 1 '"""" 22Y/an( ) = IN(h )1 L..)rk(Si) - rm(Sj)JN(h)

where again h == Sj - Si. Papritz et al. (1993) also suggest this estimator, and note thatit is biased. Another problem that we encountered with Eq. (22) is that estimation is

poor when the variance of Rk (.) is orders of magnitude different from that of Rm( - ),

and (Cressie 1991, p. 141) suggests standardizing the data by (Zk(S)-f.1k)/Vvar(zk(s)).

We propose a two-step method for estimating crossvariogram parameters. As dis­cussed in the previous section, estimation for the variogram parameters () and v is

generally good (unbiased), and is likely better than trying to re-estimate them in the

crossvariogram (biased). Thus, we might as well use the parameters () and v as esti­

mated for the variogram and plug them into the formula for the crossvariogram. That

leaves only the parameters Pk, Pm, Ak , and Am to estimate in the crossvariogram. Onepossibility is to minimize a weighted least squares,

WLSkm == L IN(h)l[empirical- model]2.all h

(23 )

For the variogram, empirical is 2Ykk(h) from Eq. (20) and model is 2Ykk(hl()k, Vk) fromEq. (10), and then Eq. (23) is minimized for ()k and Vk. For the crossvariogram, en1pir­

ical is 2Ykm(h) from Eq. (22) and model is 2Ykm(hl(), v,Pk,Pm,Ak,A m) from Eq. (11),

and then Eq. (23) is minimized for Pk, Pm, Ak , and Am, taking () and v as estimated fromthe variogram. Recall that there is some nonidentifiability in Pk (for L == 2) and Ak,. As

suggested earlier, one sin1ply identifies P as PI P2 and one could choose Al == 0, and

shifts in Z2 are relative to ZI. For L? 3, the remaining parameters Ak , k == 2, ... ,L, are

identifiable, as are Pk, k == 1, ... ,L . For L? 3, the two-step procedure is: (1) estimate

() and v for each variogram separately using WLSII, ... ,WLSLL , and then, (2) use theredundancy in the crossvariograms to minimize WLS 12 +...+WLSL ,L-I simultaneously

1. M. Ver Hoef, R.P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294 285

for the p'sand j's. Another possibility is to minimize all parameters simultaneously

for all WLS ll + ... + aLLWLSLL + al2WLS l2 + ... + aL, L-I WLSL,L-I, where weights

akm can adjust for different sample sizes, etc. This may be computationally difficult if

L is large.

In the absence of more research, use of Eq. (22) in Eq. (23) for crossvariogram

fitting is both intuitive and sin1ple. There are other possibilities, however. For example,

Cressie (1985) shows that, for kriging, minimizing,

L IN(h)1 ( 2Ykk(h) _ 1)2,2Ykk(hIOk,Vk)

all h

is better than Eq. (23), as it puts more weight on a closer fit near the origin, which is the

most critical area for kriging (Stein, 1988). However, accurate estimation of the sill for

each 2Ykk(h) is also needed in order to estimate the Pk'S properly in the crossvariogram,

which is important because Pkm controls how much 'information' cokriging uses from

each covariate. Hence, it is not clear that we should use increased weighting near the

origin for variogram and crossvariogram estimation.

3. Cokriging and multivariable spatial prediction

After estimating all parameters, the full crossvariogram is defined and can be used for

cokriging and multivariable spatial prediction as follows. Suppose we want to predict

the full vector Z (so), and call the predictor p(So Iz), where the data Z =: (z~ , z~, ... , ZL)'

and Zk =: [Zk(Slk ),Zk(S2k), ... ,zk(snk)J' with nk not necessarily equal to nm . Then the

prediction weights

lOll £012 £OIL

£021 £022 W2LQ=:

lOLl WL2 lOLL

are a solution to

(~:) (~) (~) , (24 )

where

TIl T I2 TIL

T21 T 22 T 2LT=:

T LI T L2 T LL

286 J.M. Ver Hoef, R.P. Barry/Journal of Statistical Planning and Inference 69 (1998) 275-294

and r km has as its (i,j)th element 'Ykm(Si, Sj \0, V, Pk' Pm' 1k,1m) and r kk has as its (i,j)th

element 'Ykk(Si, Sj 10k, \\); and where

G=

1'L1 1'L2 1'LL

and 1'km has as its ith element 'Ykm(Si, So 10, V, Pk' Pm' 1k,1m), and 1'kk has as its ith element

'Ykk(Si, So 10k, Vk); and where

l(nl Xl) 0 0 1 0 0

0 1(n2 X 1) 0 0 1 0x= 1=

0 0 l(nL x 1) 0 0 (LxL)

and A is a diagonal matrix of Lagrange multipliers. Eq. (24) yields the multivariable

predictor

(25)

with MSPE matrix,

where

D=

"Ill (So, so) 'Y12(SO, so)

'Y21(SO,SO) 'Y22(SO,SO)

'Y1L(SO, so)

'Y2L(SO, so)

Details of results given in Eqs. (24 )-(26) are given in Ver Hoef and Cressie (1993).

Each element of peso Iz) is the cokriging predictor. The cokriging mean-squared-predic­

tion errors are the diagonal elements of Eq. (26), and crossvariable prediction covari­ation is given by the off-diagonal elements of Eg. (26).

4. Examples

We will illustrate these methods with simulated data because all of the properties

(i.e., the true parameter values) of the data are known. The simulations are in one

dimension, using the moving average functions, Egs. (12) and (13), given earlier. These

examples illustrate simulating data using moving averages, estimating the variogram andcrossvariogram, fitting models to the empirical variograms and crossvariograms, and

J.M. Ver Hoef, R.P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294 287

then using the fitted variogram and crossvariograms for cokriging and multivariablespatial prediction.

Independent pairs of correlated data, Yl (Xi) and Y2(Xi) were sin1ulated from a bi­variate standard normal distribution with some P (because we have only 2 variables,

here we suppress the subscript on Plan), Xi == 0.25,0.50,0.75, ... ,400, so i == 1, ... , 1600such pairs were independently generated. Two other sets of data, U1 (s) and U2(S),

s == 101, 102, ... ,400 were also simulated independently, each from a standard normaldistribution. Then the Zl -process was constructed from,

1600

Zl(S) == L VO.25Yl(Xi)!1(Xi - sI8 11 ,812 ) + V1 U1(S) + /11,

i=l

for S == 101, 102, ... ,400; also notice that J 1 == O. The Z2 -process was constructed from

1600

Z2(S) == L VO.25Y2(Xi - J)!2(Xi - s1821 , ( 22 ) + V2 U2(S) + /12,

i=l

for S == 101, 102, ... ,400, where 82, j ~ O. Notice that because J 1 == 0 we suppress useof subscripts for J 2. The {S == 1, ... , 100} were not used to allow for the shift incorrelation structure and to minimize edge effects. The data sets were then re-indexedon the integers fron1 1,2, ... ,300.

Two examples are given. The second example has a nugget effect, while the firstdoes not, and the first example illustrates cokriging (prediction of a single component)while the second example illustrates multivariable spatial prediction (bivariable, in thiscase ).

4.1. Example 1: empirical estimation and fitting models

Data were simulated with P == 0.9 and J == 50, with 811 == 5, 812 == 0.1, VI == 0 and/11 == 10 for Zl (.), and with 821 == 5, 822 == 10, V2 == 0, and /12 == 50 for Z2(.) (Fig. 2).Notice that no nugget effect was added to these simulations. The empirical variogramswere calculated for Zl (.) and Z2(.) using Eq. (20), and are given in Fig. 3A andFig. 3B, respectively. The empirical crossvariogran1 was calculated using Eq. (22), andis given in Fig. 3C. The model given by Eq. (14) was fit to the empirical variogramof Zl ( .); and the model given by Eq. (15) was fit to the empirical variogram of Z2 ( • ),

using Eq. (23) for both minimizations. The crossvariogran1 n10del, Eq. (16), was fitto the empirical crossvariogram using the variogram estimates 0 of f) and minimizingEq. (23) for P and J. All minimization used the downhill simplex method of NeIderand Mead (1965). Table 1 gives the true values and the estimated values for allparameters. The fitted variograms and crossvariograms are also shown in Fig. 3. Thissimulated example demonstrates that estimation of the variogram and crossvariogramparameters also estimates the moving average parameters. The shift parameter J causesthe semicrossvariogram to be asymmetric in h and not centered on 0 (Fig. 3C). Webelieve this is the first time such a model has been presented.

288 J. M. Ver Hoef, R.P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294

LEGEND

Z2 simulated data

z1 simulated data100

90

80

70

60W::> 50....J~

> 40«I- 30~0

20

10

0

-10

-20

0 50 100 150 200TRANSECT LOCATION

Fig. 2. Simulated data for Example 1.

250 300

4.2. Example 1: kriging and cokriging

For kriging and cokriging, the matrices and vectors defined for Eq. (24) were de­

veloped with the estimated parameter values in Table 1. To compare predictions andMSPE, of the original data pairs simulated at 300 locations, 100 locations were ran­

don1ly selected from each variable to be used as data for kriging and cokriging. LetPk be the set of 200 locations removed for each simulated variable, k = 1,2. We

used kriging and cokriging to predict all So E PI (Fig. 4). For all predictions, the cok­riging equations were well-behaved, yielding valid MSPE estimates that were alwayspositive.

Along with the visual comparison in Fig. 4, MSPE may be evaluated in two ways.The first is the average squared error between the predicted value and the actualsimulated value, that is calculated as,

(27)

where Pk(solz) is the kth element of Eq. (25). Also define,

(28)

where for cokriging a~(solz) is the ith diagonal element of Eq. (26) and for kriginga~(solz) is the single-variable version of Eq. (26). If we have estimated parameters

J.M. Ver Hoef; R.P. Barry/ Journal of Statistical Planning and Inference 69 (1998) 275-294 289

Legend

• Empirical Semivariogram

Fitted Semivariogram

200 A:E~0::C>0~ 100~

~:Ewen 0

0 50 100 150LAG

400 B~~~ 300(90(i: 200

~ 100~WCJ)

0

0 50 100 150LAG

~«300~

(900:: 200 •~

••CJ)CJ) 1000~

0~ aw(f) -200 -100 0 100 200

LAG

Fig. 3. Empirical semi(cross )variograms fitted for a random subset of 100 data from 300 simulated values inExample 1; (A) exponential semivariogram fit for variable 1, (B) linear-with-sill variogram fit for variable2, (C) crossvariogram and fitted model.

well, then ESEk , the average 'estimated' MSPE, Eq. (28), should be close to ASEk ,

the average 'actual' squared errors, Eq. (27) (a realization of MSPE).Results for Eqs. (27) and (28) are given in Table 2, where we class ASE and ESE

by location. Overall, cokriging had a lower ASE 1 and ESE 1 than for kriging. Also,for cokriging, both ASE l and ESE} essentially agree. MSPE was overestimated forkriging because ASE l was considerably lower than ESE l . Examining ASE l and ESE]

290 1. M. Ver Hoe;; R.P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294

Table 1Parameters and their estimates for Example 1, 300 simulated data

Param eter

50

True value

5.0

0.1

5.010.00.9

50.0

Estimate

5.44

0.101

5.3910.10.806

50.0

w:::>....J

~

40•

30

20

10

o

-10

•

+

• ••

e+o

Legend

Actual Simulated Values z1

Kriging Predictions

Cokriging Predictions

•

o 50 100 150 200 250TRANSECT LOCATION FOR PREDICTION SET P1

300

Fig. 4. Two hundred kriging and cokriging predictions compared to actual simulated values for the 2} -process

of Example 1, based on a random subset of 100 data.

Table 2Mean-squared-prediction errors for So E PI for simulated data in Example 1.

ASE} ESE}

Location Kriging Cokriging Kriging Cokriging

so~ 50 32.6 34.6 39.5 26.5

50 < so:( 100 30.7 25.3 44.8 30.2

100 < so:( 150 35.1 26.8 38.7 25.6

150 < so:( 200 27.6 22.7 23.4 17.2

200 < so:( 250 17.3 19.9 21.5 17.5250 < so:( 300 30.7 30.7 36.1 36.1

Overall 29.5 27.0 34.8 26.1

1. M. Ver Hoe/: R. P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294 291

Table 3Mean-squared-prediction errors for So E P2 for simulated data in Example 1.

ASE2 ESE}

Location Kriging Cokriging Kriging Cokriging

so~ 50 40.0 40.1 55.1 55.0

50 < So ~ 100 55.4 56.5 62.6 48.7

100 < So ~ 150 32.3 28.8 48.7 42.6

150 < So ~200 40.5 39.9 51.8 41.6

200 < So ~250 37.6 33.3 40.8 30.5

250 < So ~300 91.0 32.4 60.0 36.7

Overall 49.2 38.4 53.2 42.7

by location class for both kriging and cokriging, notice that both methods did more

poorly at the ends, as opposed to the middle, as expected (Table 2). An interesting

feature of Table 2 is that, for So > 250, cokriging became essentially the same as

kriging, and this may be seen in Fig. 4 as well. This makes sense, because all of

the 'information' in the Z2-process, for predicting the ZI-process, was shifted forward

Ll = 50. Hence, there is no reason to use data Z2(Si) when predicting so> 250 for the

ZI -process. In fact, we might expect that because of the shift Ll, cokriging would do

quite a bit better than kriging for smaller So, but that this advantage would taper off

as So increases; in general, Table 2 does show this pattern.

The same analysis was carried out for prediction of the set P2 for the Z2 -process. We

used kriging and cokriging to predict all So E P2 . The MSPE results, in Table 3, show

that cokriging was a dramatic improvement over kriging because ASE2 for cokriging

is much lower than ASE2 for kriging. Both kriging and cokriging slightly overesti­

mated MSPE because ASE2< ESE2 for both. Also notice that in Table 3, kriging and

cokriging are essentially the sanle for So ~ 50, as expected.

4.3. Exanzple 2: en1pirical estbnation and ,tittin?] lnodels

For the second example, data were simulated with p = 0.8 and L1 = 50, and with

011 =2,0 12 =0.1, VI =4, and III = 10 for ZI(-), and with (]21 = 1, (}22 = 15, V2 =3, and

Il2 = 20 for Z2 (-). Of these, 100 locations were randomly selected and both data values

were retained. In this example, we added a nugget effect and estimated parameters for

all 300 data and the random subset of 100. As in the first exanlple, we use Eqs. (20)and (22) for empirical estimation of semivariograms and the semicrossvariogram, and

fit models of Eqs. (14) and (16) to the empirical estimates by minimizing Eq. (23).The parameter estimates are given in Table 4.

4.4. Exalnple 2: lnultivariahle spatial prediction

Using only bivariate data at the 100 data points and the paranleter estinlates based

on these 100 data, the 200 bivariate data that were set aside were predicted. For each

292 J.M. Ver Hoef, R.P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294

Table 4Parameters and their estimates for simulated data in Exan1ple 2.

Parameter True value est n = 300

811 2.0 2.16812 0.1 0.129

VI 4.0 4.03821 1.0 0.962

822 15.0 12.5

V2 3.0 2.91p 0.8 0.817L1 50.0 48.7

est n = 100

2.03 phantomO0.1224.091.06

13.32.580.834

48.7

Note: Estimation was for all 300 simulated data (colun1n 3) and a random subset (column 4).

prediction, we obtained the MSPE matrix, Eq. (26). For each location, assuming pre­diction errors are normally distributed, bivariate prediction regions may be developed,as shown by Ver Hoef and Cressie (1993). The average of each element in the n1atrix,Eq. (26), over the 200 prediction sites, was

(23.5

- 0.0298- 0.0298)

10.6 '(29)

showing a very slight negative correlation between the two predicted values. Note thatESE 1 and ESE2 are the first and second diagonal elements of Eq. (29). For the firstvariable ASE 1 was 23.9 and for the second variable ASE2 was 12.3, indicating reason­able closeness between estimated and actual squared-prediction errors, even though allestimation and prediction was based only on the random subset of 100. By comparison,kriging only variable 1 gave ESE 1 == 24.8 and ASE 1 == 26.2, and kriging only variable2 gave ESE2 == 11.0 and ASE2 == 12.0. Because of the rather large nugget effect thetwo variables are essentially independent, so the advantage of cokriging over krigingis not as apparent as in Example 1.

5. Discussion and conclusions

The use of integration of a moving average function over white noise random pro­cesses, Eq. (8) leads to construction of valid variograms and crossvariograms. Theresulting crossvariogram, Eq. (11), can allow for modeling crossvariograms for a widevariety of component variograms. The examples illustrate the development and fittingof models, and then using them for spatial prediction. The first simulation exampleillustrates fitting models to data, and demonstrates that cokriging can have a consid­erable advantage over kriging. The second simulation example illustrates multivariablespatial prediction, also showing that estimation on a subset of 100 locations is still quitegood, and that the nugget effect diminishes the advantages of cokriging over kriging.Both of these examples contain a nonzero shift parameter L1 that causes asymmetry in

J.M. Ver Hoef, R.P. Barry/ Journal of Statistical Planning and Inference 69 (1998) 275-294 293

the covariogram, i.e., Ckm(h) # Ckm ( -h), so the use of the traditional crossvariogram,Eq. (5), would not be optinlal. Real-world processes that might be expected to containsuch a shift would be things like weather temperatures, where Z1 (.) are temperatureson one day and Z2(.) are temperatures on the following day. Air mass movementwould cause crosscorrelation to be shifted.

With the methods outlined in this paper, use of the crossvariogram, Eq. (7), in placeof the traditional crossvariogram, Eq. (5), eliminates all three of the main problemslisted in the introduction: (1) the symnletrical covariance assumption, Ckm(h )== Ckm (-h),(2) data on each variable do not need to come from the same location in order to es­timate the crossvariogram, and, (3) we can consider models beyond those of coregion­alization. Coregionalization assumes that the crossvariogram has a functional form likea variogram; i.e., it is linear, exponential, spherical, etc. Figs. 1 and 3 show why core­gionalization is inflexible and has performed poorly in the past. Crossvariogram modelshave functional forms that may be considerably different than those of variograms!

So far, we have only considered variogram models with sills. Now consider themoving average function,

where J~cxJ.fk(xIOk)]2 d.x == 00. Still, we can use Eq. (10), which implies a linearvariogram. However, now the integral in Eq. (11) will be finite only if !k(X) and.f~l(X) have the same height (fh == 8m ). In general, we see that asymptotically as x ----+ 00

there can not be too nluch area in [!k(X)- !m(X )]2. This leads us to the sanle conclusionas Papritz et al. (1993), that a linear variogram for Z1 (.) implies that the variogramfor Z2 (.) must have the same slope for the crossvariogranl to exist. It should notbe too surprising that variogranl models without sills cause potential difficulties whencokriging, considering that they have unbounded variation; i.e., the linear variogramdescribes Brownian nlotion.

More research is needed on estimation and fitting methods for the crossvariogranlused in cokriging and multivariable spatial prediction. For exanlple, unlike variogramversus covariogram estimation, for crossvariograms it is necessary to subtract the meanfrom each variable, and possibly rescale them, so there may be no advantage to es­

timating crossvariograms as opposed to estimating crosscovariances. In addition, wepropose a weighted least squares criterion to fit variograms first, and fit any additionalcrossvariogram paranleters as a second step, but simultaneous fitting of all parametersis also attractive.

Acknowledgenlents

Support for this work was provided by Federal Aid in Wildlife Restoration to theAlaska Department of Fish and Game and the Eli-Lander Foundation. We thank NoelCressie, Andy Royle, and one anonymous reviewer for their comments.

294 1. M. Ver Ho~t: R. P. Barry / Journal of Statistical Planning and Inference 69 (1998) 275-294

References

Barry, R.P., Ver Boef, J.M., 1996. Blackbox kriging: spatial prediction without specifying variogram models.

J. Agricul. Biological Environment. Statist. 1, 297-322.Clark, L Basinger, K.L .. Harper, W.V., 1989. MUCK -- A novel approach to cokriging. In: Buxton,

B.E. (Ed.), Proc. Conf. Geostatistical, Sensitivity, and Uncertainty Methods for Ground-Water Flow andRadionuc1ide Transport Modeling, Batelle Press, Columbus, pp. 473-479.

Cressie, N., 1985. Fitting variogram models by weighted least squares. J. Internat. Assoc. Math. Geology17,693--702.

Cressie, N., 1991. Statistics for Spatial Data. Wiley, New York.

Helterbrand, J.D., Cressie, N., 1994. Universal cokriging under intrinsic coregionalization. Math. Geol. 26,205-226.

Isaaks, E.H., Srivastava, R M., 1989. An Introduction to Applied Geostatistics. Oxford University Press, NewYork.

JourneL A.G., Huijbregts, C.1., 1978. Mining Geostatistics. Academic Press, London.Lu, H., Zimmennan, D.L., 1994. A nonparametric approach to testing for directional sYlnmetry of spatial

correlation. Technical Report 234, Depa11ment of Statistics and Actuarial Science, University of Iowa,Iowa City, Iowa.

Ivlyers, D.E., 1982. Matrix formulation of cokriging. Math. Geol. 14, 249-257.Myers, D. E., 1991. Pseudo crossvariograms, positive-definiteness, and cokriging. Math. Geol. 23, 805--816.Nash, M.S., Toonnan, A., Wierenga, PJ., Gutjahr, A., Cunninghaln, G.L., 1992. Estinlation of vegetative

cover in an arid rangeland based on soil-moisture llsing cokriging. Soil Sci. 154, 25-~36.

NeIder, .J.A., Mead, R., 1965. A simplex method for function minimization. Comput. J. 7,308-313.Papritz, A., Kunsch, H.R., Webster, R., 1993. On the pseudo cross-variogram. Math. Geo!. 25, 1015-1026.Stein, M.L., 1988. Asymptotically efficient prediction of a random field with a misspecified covariance

function. Ann. Statist. 16, 55--63.Ver Hoef, J.M .. Cressie, N., 1993. Multivariable spatial prediction. Math. Geol. 25, 219-240. [Erratum, 1994.

Math. Geol. 26, 273-275.]Wackemagel, H., 1995. Multivariate Geostatistics. Springer, Berlin.Yaglom, A.M., 1987. Correlation Theory of Stationary and Related Random Functions I: Basic Results.

Springer, New York.