R. Dimitrakopoulos (ed.), Geostatistics for the Next Century, 18~29. © 1994 Kluwer Academic Publishers. Printed in the Netherlands.

In many areas, one has to deal not. only with quantitative variables but very often with categorical variables. Then a cornmon problem consists of estimating the probability for each category to prevail at any particular location. Using an indicator approach these probabilities can be established by three classes of algo­ rithms: coindicator kriging (coII\.), multiple indicator kriging (IK), and median in­ dicator kriging (mIK). The "theoretically" better algorithm is coIK which accounts for the transition probabilities between different categories and yields the smallest estimation variance. Its implementation, however, requires inference and modeling all category indicator (cross) variograms, and solving large and possibly unstable cokriging systems. This explains why IE which involves only modeling the direct indicator va.riograms has been the preferred algorithm so far. Further simplification consists of considering the same model for all sample indicator variograms, this is referred to as median indicator kriging. Many variants to these three basic algo­ rithms (Suro-Perez and Journel, 1990; Soares, 1992; Dimitrakopoulos and Dagbert, 1993) have been developed but will not be considered in this study.

The recent. development of iterative procedures (Goula.rcl and Voltz, 1992) or graphical procedures (Chu, 1993) for fitting a linear model of coregionalization, coupled with the increase in computing power have rendered coindicator kriging practical. There remains to assess the actual practical advantage brought by coIK.

18

INTRODUCTION

A performance comparison of three algorithms ( coindicator kriging, multiple in­ dicator kriging, median indicator kriging) for estimating conditional probabilities of categorical variables is presented. The reference soil data. set includes 2649 locations at which the soil type was determined. Three subsets of 50, 100 and 500 sample locations were randomly selected and used to estimate the probability for the dif­ ferent soil types at the remaining locations. In all cases the variograms required were modeled using the complete data set. The comparison of estimated vs true probabilities shows that, whatever the number of conditioning data, the theoreti­ cally better colK does not provide more accurate re-estimates than the two other algorithms. The latter two algorithms involve less variogram modeling effort and smaller computational cost. Furthermore, the number of order relation deviations is showed to be significantly higher for the coIK results.

P. GOOVAERTS Geology and Environmental Sciences Department Stanford University Stanford CA 94305 U.S.A.

COMPARISON OF COIK, IK AND MIK PERFORMANCES FOR MODELING CONDITIONAL PROBABILITIES OF CATEGORICAL VARIABLES

k = 1 to I< \:/a= 1 to n, k'=l f)=l

I\ n

L L >-dur3; Sk0) 11(u,, - us; Sk, s~) - µk

The weights Ak(ua; Sko) are obtained by solving the following ordinary cokriging system of K(n+l) equations

(3) [\' n

p;,,IJ<(u;_sk0) = LL Ak(u"'; Sk0).i(u"'; Sk) k=l or=l

Coindicator kriging

The coindicator kriging (coll<) estimate utilizes indicator data related to any category:

where the notation (n) represents the conditioning information utilized. The estimated probability for a given category sk0, p"'(u; Sk0), is computed as a

linear combination of neighboring indicator data. i(u"'; sk)·

(2) p(u; sk) =Prob {u E sk[(n)} =Prob {I(u;sk) = lj(n)}

The different algorithms aim at estimating the conditional probabilities of the K categories at any location u E D, i.e.

(1) . ( ) { 1 if u., E s k z uC(; 3k = 0 otherwise

Let { s,_, ; k = 1, ... , I<} be a set of K mutually exclusive categories observed over a field D. At each sample location u.,, a. vector of K indicator values i(u"'; sk) can be formed where

INDICATOR ALGORITHMS FOR ESTIMATING CONDITIONAL PROBABILITIES OF CATEGORICAL VARIABLES

The fact that the cokriging estimation variance is smaller than the kriging estima­ tion variance does not necessarily entail that the cokriging estimate is in practice the most accurate. Practice has recurrently nown (J•1urnel and Huijbregts, 1978, p. 326) that cokriging improves little over kriging when . 11 the covariables involved are equally sampled; such is the case in most indicator kriging applications. The recon­ ciliation between the theoretical promise of cokriging and its lesser performance in practice remains an open question in geostatistical theory, yet it has major practical implications.

This paper adds another case study to the file "cokriging vs kriging". It presents a performance comparison utilizing a. reference soil data set comprising 2649 loca­ tions at which the soil type was determined. Three subsets of 50, 100 and 500 sample locations were randomly selected and used to re-estimate by coIK, IK and mIK the probability for the different soil types at the remaining locations. The com­ parison criteria account for the following characteristics : number and magnitude of order relation deviations, and differences between true and estimated vectors of probabilities.

19 COMPARISON OF Co!K, IK, rnIK PERFORMANCES

Goulard and Voltz (1992) and Goovaerts (1992) describe in some detail the prac­ tice of choosing the number and type of models. Once selected the basic models g1(h)

• choosing the number (L) and characteristics (type, range) of the basic vari­ ogram functions g1(h), and

• fitting the selected model to the experimental values, i.e. estimating the coef­ ficients bL,, under the constraint of positive semi-definiteness of the coregion­ alization matrices B1.

(6) L

r1(h) = :L B1g1(h) l=l

where each B1 = [ bik'] is a K x K symmetric matrix called a coregionalization matrix. This matrix formulation allows the conditionally negative semi-definite condition

for r 1(h) to be expressed in practical terms: the condition holds true if the 91 (h) are licit variogram models and each matrix B1 is positive semi-definite (Journel and Huijbregts, 1978, p. 172). Therefore, modeling the coregionalization involves, in practice, the following steps:

(5) Vk,k' L

11(h; sk, sk') = L b~k'g1(h) l=l

where bik' is the weight applied to the function g1(h) in the model for -n(h; sk, sk' ). Using matrix notation, Eq. (5) is rewritten :

The only model that is of wide usage so far is the linear model of coregionalization which involves modeling all the variograms as linear combinations of the same set of L basic variogram functions g1(h) :

11(h; s«, sr;)

/1(h; S·1, .SJ)

where µk is the kth Lagrange multiplier and Okko is the Kronecker delta. This cokriging system calls for the joint modeling of K(K+l)/2 indicator (cross)

variograms 11(h; sk, sk' ). This large number of variograms to model is the main reason why coIK is rarely used in practice. In fact, the difficulty does not lie in the number of models to infer but in the fact that the matrix of variogram models, r7(h), must be conditionally negative semi-definite (Journel and Huijbregts, 1978, p. 171),

( 4) \;/ k = 1 to K

Tl

I: ,\k(uf3; Sk0)

{3:.o)

P. GOOV AERTS 20

(10) ( l ) 1 ~ 11(h; sk, sk)

/ml 1 =-; L , ]i k=l Pk(l - Pk)

where Pk = E[I(u; sk)] is the global proportion of fa.cies k, and Pk(l - Pk) is the corresponding indicator variance.

Since the indicator data configuration is the same for all categories, one single IK system needs to be solved with the resulting weights r(ua) being used in Eq. (9) for a.11 categories,

In addition, the mIK algorithm assumes that all K direct indicator variograms are proportional to each other (Deutsch and Journel, 1992, p. 74). In this paper, the common standardized indicator variogram /m1(h) was computed as the mean of the I< rescaled indicator variograrns :

(9) n

P~i!I<(u; Sk0) = L T(uo).i(uo; Sk0)

o=l

Median indicator kriging

Similarly to IK, the median indicator kriging (mIK) estimate accounts only for the indicator information about the category Sko being considered :

Unlike the linear model of coregiona.lization, there is no requirement for every indicator variogram to be modeled with the same set of basic variogram functions. Moreover, there is no constraint on the coefficients of the different models. This greater flexibility of modeling leads to models that generally fit the experimental values better.

(8) 1 n

L l!ko(u13; Sk0)

f)o=l

V a= 1 ton

n L vko(u13; sko) ~r1(Ua- - u13; Sko, Sko) - µk

13=1

The system to be solved is less cumbersome and requires only the K direct indicator variogram models 11(h; ski sk).

(7) n

p[g( u ; Sko) = L l/ko (Un; Sko ).i( Ua; Sko) o sc I

are fitted iteratively by weighted least squares.

Multiple indicator kriging

Unlike the coIK estimate, the multiple indicator kriging (IK) estimate utilizes only ind icator data. on the category Ska being considered :

21 COMPARJSON OF Co!K, IK, m!K PERFORMANCES

Variogram modeling

Whatever the number of conditioning data, the variograms required by the dif­ ferent algorithms were modeled using the complete reference data set. We decided to work in each case with the best structural information available because our pri­ mary interest is in comparing the performances of the algorithms, independently

• the number and magnitude of order relation deviations were computed, see later definitions (12) and (13),

• the departures between true and estimated vectors of probabilities were assessed using two criteria, defined hereafter in relations (14) and (15).

3. For each algorithm and set of N1 data locations,

General procedure

To assess the performances of col K, IK and mlK and investigate their dependence on the number Ne of conditioning data, we proceeded a.s follows:

l. Three subsets of Ne= 50, 100 a.nd 500 sample locations were randomly selected.

2. Ea.ch subset was used to estimate by the three algorithms the probability for the different soil types at the N1 remaining locations (Ni = 2649 - Ne)· In each case, the search radius was 25 km and the nearest data up to a maximum of 32 were retained.

This study relates to a survey undertaken over a 3500 km2 area located in south­ east Scotland (McBratney et al., 1982). The data set comprises 2649 locations at which the soil type was determined according to morphological soil properties. Six different soil types, denoted sk (k=l, ... ,6), were identified. Most locations (61.6%, exactly) pertain to the soil types s3 or s4. The four other soil types represent fairly similar proportions of the data.

An important feature of ea.ch soil type is its spatial distribution. Figure 1 shows that locations of type s1 are scattered in space whereas the other soil types form more spatially compact groups oriented NE-SW. Therefore, estimating the probabilities for soil s1 is expected to be more difficult than for other soils.

The data set

PERFORMANCE COMPARISON : A CASE STUDY

(11) 1

V o = 1 ton

n

L T(ufJ) {m1(u"' - UfJ) - µ f'mr(uo: - uo) /3=1

22 P. GOOV AERTS

tc L p'(u"; sk) = 1 (13) k=l

The first condition may not be satisfied because kriging weights can be negative, therefore the kriging estimate is a non-convex linear combination of the conditioning data. For each algorithm, the percentage of probabilities valued outside the interval ~O, l] a.s well as the magnitude of these order relation deviations were computed, see Table 1. Whatever the number of conditioning data, coIK has a.bout two to four times more order relation problems than the two other algorithms. This can be explained by the com: constraint that the sum of the weights applied to secondary variables is zero, which implies a larger number of negative weights and so a greater risk of getting probabilities outside the interval [O, l ].

Except for anisotropic indicator kriging (Ne= 50), the magnitude of order rela­ tion deviations is small.

The first constraint (12) was met by resetting the faulty probabilities to the nearest bound, 0 or 1. Once this correction has been performed, each value p'( 110; sk) was restandardized by the sum L;[(=J p*(u0.; s~,) to meet the second condition (13).

p*(u0; sk) E [0, l] (12)

At each location 110, the set of K estimated probabilities {p*(u0; sk) k = 1, ... , K} must satisfy the two following order rela.tions :

of questions of inference. Note that this decision privileges coIK because it is the algorithm that involves the most difficult inference.

For col K, a. linear model of coregionalization was fitted to the 21 indicator (cross) variograms, see Figures 2 and 3 (solid line), using the iterative procedure developed by Goulard and Voltz (1992). The fit appears quite satisfactory apart from the cross-variograms /s3s5 and /8386• These two cross variograms are zero up to 10 or 15 km, which indicates that the two soil types have no common border.

For II\, the 6 indicator variogra.ms were modeled independently, see Figure 2 (dashed line). Since there is no constraint on the parameters of the models, the fits are seen to be better. Note the small range of the indicator variograms lsi and /52,

which reflects the greater scattering of the two soil types in the region. The median indicator variogra.m required by mIK was computed from the mean

of the 6 direct standardized indicator variograms, see Figure 4. So fa.r only isotropic models have been considered. According to Figure 1, an

anisotropic behaviour is expected for most categories. Indeed, except from soil types s1 and s2, the indicator va.riogra.ms a.re anisotropic, especially for la.gs greater than 5 km. To investigate the possible loss of accuracy due to the assumption of isotropy, a.n anisotropic model was fitted to the four direct indicator variograms (s3, s4, s5, s6)

and to the median indicator variogram, see Figure 5. As the categorical variables do not share the same directions of anisotropy, it was not possible to fit an anisotropic linear model of coregionalization. Therefore the comparison between isotropic and anisotropic modeling is performed only for the IK and mIK algorithms.

Result 1 : Order relation deviations

23 COMPARISON OF Co!K, IK, m!K PERFORMANCES

The index C2 equals one minus the average estimated probability for the correct soil type. C2 measures the averct.gF pro&ability to draw a wrong category when using

[{

C2(Uo.) = LJJ*(ua; Sk).i(ua; Sk) k=l

where

(15)

The second criterion denoted C2 is defined as :

The index C1 measures the average absolute deviation between the true and esti­ mated vectors of probabilities. Its value is close to zero if the algorithm is accurate. The weighting system adopted gives rnore weight to categories with small indicator variance, i.e. to soils of small areal extent. Indeed, categories that are rare a.re often of higher interest.

According to criterion Ci, IK and mIK perform similarly and slightly better than col K, whatever the number of conditioning data, see Table 2. The accuracy of the estimation increases with the number of conditioning data but is not improved by taking :into account anisotropy.

For Ne = 50, Table 3 gives the values of C1 for each soil type. There is a strong relation between accuracy of the estimation and the extent (proportion) of the corresponding soil type. The lowest (best) C1 values are observed for the two most frequent categories, s3 and s4. The worst results are obtained for the least frequent categories, s1 and s2. For these last two categories, IK and especially mIK do a better job than col K. The anisotropic modeling improves the accuracy of the estimation only for soil type s5.

(L{~1 1/pk(1 -1h))-1

Jh(l-pk)

The weights Wk were computed as

t: (\(uo:) = L:;wk lp*(uoi sk) - i(ua;sk)I

k=l

where

(14)

For the N, test locations, both true indicator values i(ua; sk), with i(u,,; sk) = 0 or 1, and estimated probabilities p*(u"; sk), with p*(ua; sk) E [O, l], are available. These were compared using the two following criteria.

The first criterion denoted Ci is defined as :

Result 2 : Differences between true and estimated vectors of probabilities

P. GOOV AERTS 24

• Chu, .J. (1993) "XGAM user's guide", Stanford Center for Reservoir Forecast­ ing, Stanford University, Unpublished Annual Report n° 6.

• Deutsch, C.V. and Journel, A.G. (1992) GSLIB : Geostatistical Software Li­ brary and User's Guide, Oxford University Press, New-York.

• Dimitrakopoulos, R. and Dagbert , M. (1993) "Sequential modelling of relative indicator variables: dealing with multiple lithology types", in A. Soares (ed.), Geost atistics Troia '92, Kluwer Academic Publishers, Dordrecht, pp. 413-424.

• Goovaerts, P. (1992) Multivariate geostatistical tools for studying scale-dependent correlation structures and describing space-time variations, Ph. D. thesis, U C.L.. Belgium.

REFERENCES

The author thanks the East Scotland College of Agriculture, especially Mr H. M. Gray, Dr R. G. Mcl.aren, and Dr R. B. Speirs, for the data from their survey.

ACKNOWLEDGEMENTS

Even if better software makes coindicator kriging less demanding, this study draws our attention to the possibility that putting more partially redundant infor­ mation into a. (co )kriging system does not necessarily improve the accuracy of the estima.tion. It is conjectured that the linear model of coregionalization may artifi­ cially limit the potential of cokriging.

• The number of order relation deviations is found to be higher for coIK.

• The comparison of true vs estimated probabilities shows that coIK does not provide more accurate estimations than the other algorithms. It underperforms IK and mIK for the categories of smaller extent which are potentially of greater interest.

In theory, coindicator kriging should outperform the other algorithms because it accounts for the cross relations, i.e. the transition probabilities between the different categories. In practice, however, com: does not appear to do a better job than IK or mIK.

CONCLUSIONS

the estimated probabilities in the simulation process. Its value is close to zero if the algorithm is accurate.

When only 50 conditioning data are available, the average probability for mis­ drawing is 0.45, see Table 2. This probability decreases with increasing number of conditioning data. According to C2, no algorithm performs better than the others.

For Ne= 50, Table 3 gives the values of C2 for each soil type. As for C1, there is a strong correlation between the probability of misdrawing and the smaller extent of the corresponding soil type. Again, the highest values (0.7-0.8) are observed for the least frequent categories s1 and s2, with IK and especially mIK performing better than coll\. For soil type s5, the anisotropic modeling allows reducing the probability of misdrawing from 0.7 to 0.6.

25 COMPARISON OF Co!K, IK, m!K PERFORMANCES

co m S1 0.224 0.21 0. 10 .734 .71 .71 S2 0.349 0.330 0.323 0.757 0.724 0.736 S3 0.062 0.062 0.067 0.264 0.265 0.278 S4 0.09.5 0.091 0.108 0.375 0.408 0.435 S,5 0.214 0.206 0.188 0.716 0.732 0.618 56 0.152 0.161 0.156 0.509 0.534 0.509

Table 3. Values of criteria C1 and C2 for each category (Ne = 50) (best if values small},

j c = .50 Ne= 100 Ne= 500

Table 2. Values of criteria C1 and C2 for different number of conditioning data (best if values small).

0.029 O.Q15 0.030 0.007

1 c = 50 Ne = 100 Ne= 500

Table I. Percentage and magnitude of order relation deviations for different number of conditioning data. The deviation magnitude is defined as : (p" ~ 1) if p* > 1, and IP* I if p* < 0.

• Suro-Perez, V. and Journel, A.G. (1990) "Stochastic simulation of lithofacies: an improved sequential indicator approach", in Guerillot and Guillon (eds), Proc. of the 2nd European Conf. on the Math. of Oil Recovery, Publ. Technip, pp. 3-10.

• McBratney, A. B., Webster, R., McLaren, R. G. and Spiers, R. B. (1982) "Regional variation of extractable copper and cobalt in the topsoil of south­ east Scotland", Agronomie 2, 969-982.

• Soares, A. (1992) "Geostatistical estimation of multi-phase structures", Math. Geol. 24, 149-160.

• Coulard, M. and Voltz, M. (1992) "Linear coregiorralization model; tools for estimation and choice of cross-variogram matrix", Math. Geo!. 24, 269-286.

• Journel, A.G. and Huijbregts, C.J. (1978) Mining Geostatistics, Academic Press, London.

P. GOOV AERTS 26

Distance, km

0.6 E ~0.6 0

~ 0.4

J,...U-"--

Figure 2: Indicator variograms and models, as used in CoIK (solid line) or IK (clashed line).

E ~0.8

·~ > 0.4

1.0_ Soi/5

~ 0.6

f > 0.4

1.2

1.2

.0 .0

Soi/6

.0

0.2

~ 06

,g 0.4 ~ >

0.8

Soi13 .···. Soi/2

Distance. km

~ 0.8 g>

·~ > 0.4

1.2

Soi/4

Distance, km

0.2

Figure l: Location of the samples for the different soil types the proportion of which is indicated into brackets (units= km).

20.

40.

60.

80.~--------~ Soil 6 (12.2 %)

80.~--------~

Soil 3 (37.7 %)

27

Soil 5 (10.4 %) Soil 4 (23.9 %) 80. 80.

60. ·~-··.,: 60.

40. «: 40. ,,.,. ~, . 20. {1:! 20

1#- 0. 0.

80. 80.

60. 60. =-~· :.?;>'

''"· .;.. 40. ~ 40.

20. 20. ;'_·~ .• i·

0. 0.

Soil 2 (5.4 %)

COMPARISON OF ColK, IK, m!K PERFORMANCES

Soil 1 (10.4 %)

Fi. ca Figure 3: Cross indicator variogra.ms and the linear model of coregionalization fitted.

Distance. km

·0.4

~ Q ·0.2 .Q • > ·0.3

·0.1

-o.s_,_,_~~----~~~~~ 0.0 5.0 1 .0

Distance, km

·0-4

·· ... ·--;-'-~-= · .. ·

0.0

Soi/5-Soi/6

Distance, km

·0.4

E e ,g ·0.2

~ ·0.3

... ··· ·.·· -, -, ..

t-, ·0.1.

~ ·0.2 .. ~ > ·0.3

.................... -, ·· . ·0.1

·O.S+..-~--~~~~~~~~ 0.0 5.0 1 .0 1 .0 .0 . 2 .0 Distance.km

·0.4

'•• . . ·0.1

~ t ·0.2

> ·0.3

.. . .. 0.0

Soil 4 - Soil 5

·0.5,J.,,.-"'=T,.-...-.-,-TT-.,.,-,.....:..:tn"".......;"' 0.0 5.0 1 .0 .0

Distance, km

·0.4

···· -, -. ........

0.0

Soi/3-Soi/6

Distance, km

····· ... 0.0 ...........

·0.1

~-0.2 -~ > -0.3

. .. ···· ·······-:--...,-.'-- · .

0.0

Soi/4-Soi/6

-o.s odco~"""s"'o,.....~1 .. _o,....."""''7'__,,.,..,_-= Distance, km

·0.4

-0,1 E . i -0.2 ; > -0.3

·0.1

~ -0.2 . g ~ ·0.3

00

Soi/ 3 - Soi/ 4

Distance. km

·0.4

.. ·· ....... · ... --:.:- .. ~ .. ,... ~-­ ··.·· ·0.4

Soi/2-Soi/6

Distance, km

... ···. .. ·· -, 0.0

Soi/3-Soi/5

Distance, km

-0.4

E

-~ -0.2

~ -0.3

·0.1 .. ········· E ~-0.2 .2 ~ > -0.3

·0.1 ····· ... ···· 0.0

Soil 2-Soil 5

-o.so,,._0~~5~.0--1 ".o~~, ~.o~~~-~

Distance, km

·0.4

-0.1

~ .g-0.2

> ·0.3

. ... 0.0

·0.1 E 5, -0.2 ~ > ·0.3

·0.4

·0.5 .o 2 .0 0.0 5.0

Soi/2-Soi/3

Distance, km

·0.4

E ~ ·0.2 -~ > ·0.3

·0.1

... .... ··

0.0

Soi/ 2-Soil 4

-0.s.o"'.-0--s~.o~-,~.o--""-~~~.o Distance, km

-0.4

-0.1

l-0.2 ~ > -0.3

........

0.0

0.0

Soil 1-Soil 6

Distance, km

E ~ ·0.2:!"'~ ........ ,_-~-;'""'..-'--"0-··_ ... _ •• _ •• _o ......

~ -0.3 • •••••••.••••••• •

·0.1

00

Soi/ 1-Soil 5

-o.50"".o~~s"'.o,.....~1·.o~~~~=-~ Distance, km

·0.4

-0.1 }"-"""-'-"''--,,-- -, -··"'···~-.~. ~--~ •• ..=·~··. •••·••·••• ~ f -0.2

> -0.3

Soil 1-Soil 4

P. GOOV AERTS

0.0

Soil 1-Soil3

.. · .. ···

00

Soil 1 - Soi/ 2

28

Figure ,S: Anisotropic models for the direct indicator vanograms and median indi­ cator variogram.

Distance. km

Median Variogrsm {directions : 22.5, 67.5, 112.5, 157.5)

Distance. km .0 2 .0 1 .0 1 .0

0.50

~ -r 1.00 >

1.50

2.00 Soil 6 (directions: 0, 45, 90 1.A

...... -

Distance. km

0.50

E f 1.00

1.50

2.00 Soil 4 (directions: O, 45, 90, 135)

Distance. km

0.50

E ! 1.00

1.50

2.00

Soil 5 (directions : 01 45, 90, 135)

Distance, km

1.5_ E

~ 1.0 's > 0.5

2.0 Soil 3 {directions : 0, 45, 90, 135)

Figure 4: Median indicator variogram.

Distance, km

O.O+--~~~~~~--~~~ 0.0 5.0 1 .0 1 .0 2 .0 2 .0

02

~ .~ 0.6

~ 0-4

0.8

Median Variogram 1.0

29 COMPARISON OF Co!K, !K, mIK PERFORMANCES