Assessing the covariance function in geostatistics

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1 Statistics & Probability Letters 52 (2001) Assessing the covariance function in geostatistics Ana F. Militino, M. Dolores Ugarte Departamento de Estadstica e Investigacion Operativa, Universidad Publica de Navarra, Campus Arrosada, Pamplona, Navarra, Spain Received October 1999; received in revised form October 2000 Abstract In geostatistics, one of the crucial problems is the choice of the covariance function. In this paper we show how to improve the cross-validation criterion, traditionally used for evaluating the t of a covariance function, in the case of unequally spaced data. c 2001 Elsevier Science B.V. All rights reserved Keywords: Dirichlet tessellation; Cross-validation; Mean integrated squared error; Spatial linear model 0. Introduction Interpolation based on spatial dependence of samples was rst used by Krige (1951) for estimation of the gold content of ore bodies in the mining industry of South Africa. However, classical interpolation procedures were considered inappropriate. They were biased and non-optimal due to the fact that they did not take into account local spatial dependence during estimation. Krige s practical methods were generalized by Matheron (1962) into the theory of regionalized variables. A regionalized variable Z(x) is a random variable that takes dierent values according to its location x within some region. Nowadays, kriging is a technique of making optimal, unbiased estimates of regionalized variables at unsampled locations using hypothesis of stationarity and structural properties of the covariance and the initial set of data values. The regionalized function, representing the variable under study, is split into a deterministic trend or drift plus a random error function with zero mean. The procedure called universal kriging assumes that the drift can be expressed locally as a linear combination of base functions (generally polynomials) with unknown location parameters. If the spatial variance covariance structure is known, universal kriging generally gives precise predictions, because it chooses optimal weights to be applied to the data. A basic tool in evaluating statistical models is cross-validation. In the geostatistical context, cross-validation is widely used to assess the t of a covariance function. Although the continuity of the parameters and Corresponding author. address: militino@unavarra.es (A.F. Militino) /01/$ - see front matter c 2001 Elsevier Science B.V. All rights reserved PII: S (00)

2 200 A.F. Militino, M.D. Ugarte / Statistics & Probability Letters 52 (2001) the great variety of covariance models make the task of choosing the right covariance model dicult, the cross-validation criterion plays an important role because it helps to choose the best model among a certain number of potential candidates. Besides the procedure is much used in practice by geologists. The purpose of this paper is two-fold. Firstly, we propose an improvement of the cross-validation procedure in the case of unequally spaced data. A weighted cross-validate statistic is presented by using the tile areas of the Dirichlet tessellation. This statistic is optimum in the sense that it gives estimators minimizing the mean integrated squared error. Secondly, we evaluate the behaviour of the new statistic presenting some simulation results based on two well-known examples of the geostatistical literature. 1. The spatial linear model Let {Z(x); x D} denote a covariance-stationary Gaussian process dened on D, x being the location, where an observation is taken. The spatial linear model is dened by Z(x)=S(x)+(x); (1) where S(x) is the underlying spatial surface. Universal kriging assumes that this surface may be expressed as a polynomial surface p S(x)= f k (x) k = f (x) T ; k=1 (2) where f (x)=(f 1 (x);:::;f p (x)) T is a known base function of the spatial coordinates x and T indicates transpose. Let =( 1 ;:::; p ) T be the location parameters, deterministic but unknown drift coecients and (x) a zero-mean random error process with var[(x)] = 2. The procedure requires knowledge of the covariance function K(x i ; x j ) = Cov[Z(x i );Z(x j )] for all x i ; x j D, which is usually unknown. There are several parametric models such as exponential, Gaussian or spherical which are widely used in practice. Much eort has been devoted to estimate its unknown parameters. Some of these methods require considerable more computational eort than others. Generally, however, several estimators are applicable in any given study and it is of practical use by geologists to make approximations, sometimes tted by eye from the theoretical covariance function to the empirical one. Alternatively, we may characterize the second-order moment structure necessary for the estimation procedure through the variogram. The variogram or equivalently the semivariogram is written as (h)= 1 2 Var[Z(x + h) Z(x)] = 1 2 E[Z(x + h) Z(x)]2 ; where Z(x+h) is the value of the random function at point x+h; Z(x) is the value of the random function at the point x and h is the deviation vector between the points in space x + h and x. It can be replaced by ( h ) under isotropy. It is easy to verify that the covariance function and semivariogram of a covariance-stationary random function are related, because (h) = Var[Z(x)] Cov[Z(x + h);z(x)]: This relationship allows for the estimation of the covariance function or the variogram in an equivalent way. 2. Kriging To obtain a kriging predictor let us suppose that a stationary process is observed at each of the n distinct locations x 1 ;:::;x n, yielding the sample vector Z N =(z(x 1 );:::;z(x n )) T. The kriging method follows a

3 A.F. Militino, M.D. Ugarte / Statistics & Probability Letters 52 (2001) three-step procedure: an ordinary least-squares regression (OLS), an estimation of the covariance matrix using OLS residuals, and a generalized least-squares estimator (GLS) using the estimated covariance matrix. Then, the location parameters of model (1) are given by ˆ GLS = arg min MSE [Ẑ(x)]; expressed in the form ˆ GLS =[F T K 1 F] 1 F T K 1 Z N ; (3) where F is the design matrix f 1 (x 1 ) ::: f p (x 1 ) f (x 1 ) T F =. :::. =. f 1 (x n ) ::: f p (x n ) f (x n ) T K is the covariance matrix K(x 1 ; x 1 ) ::: K(x 1 ; x n ) K =. :::. K(x n ; x 1 ) ::: K(x n ; x n ) ; and K 1 is the inverse of K. The covariance matrix of ˆ is given by [F T K 1 F T ] 1. If the covariance function is known, kriging gives an estimate of Z at a new location x as Ẑ(x)=f (x) T ˆ + k(x) T K 1 (Z N F ˆ); (4) where k(x)=(k(x 1 ; x);:::;k(x n ; x)) T is the covariance vector between the new prediction at x and the whole sample. Eq. (4) corresponds to the BLUP or best linear unbiased predictor, with kriging variance or mean squared error k(x)=e[z(x) 2 Ẑ(x)] 2 = 2 k(x) T K 1 k(x) +[f (x) F T K 1 k(x)] T (F T K 1 F) 1 [ f (x) F T K 1 k(x)]: (5) Robust alternatives to kriging have been proposed by Militino and Ugarte (1997). 3. A weighted cross-validate statistic Usually, dierent variogram models can often be tted to an empirical variogram. One of the crucial problems is how to know which is the best. Cross-validation is often used for such aim. The procedure consists of eliminating one of the data points temporarily from the set and then predict its value by kriging using the remaining data points. Repeating the procedure for all the points, we are able to estimate a cross-validate (CV) statistic n ( ) 2 zi ẑ (i) CV = (6) i=1 (i) and choose the parameters of the covariance function that make CV close to n. However, several authors have pointed out that practical problems may arise when using this technique (Ripley, 1981; Amstrong, 1998). The error sum of squares may be dominated by a few data points that are hard to predict, for example, points at the border of the region of interest which are more isolated. In the presence of unequally spaced data this problem could be more relevant. Our proposal consists of improving the CV criterion by using the tile areas

4 202 A.F. Militino, M.D. Ugarte / Statistics & Probability Letters 52 (2001) of the Dirichlet tessellation. With these areas, it is possible to balance the inuence of isolated data points in the leaving-out process, because under spatial dependence when we leave a data point, we also eliminate its area of inuence. In addition, we show below that the statistic that we provide is optimum in the sense that it gives estimators minimizing the mean integrated squared error (MISE). The MISE was introduced in the context of density estimation and has been frequently used to measure the discrepancy of an estimator and the true density. It has two dierent though equivalent interpretations: it is a measure of both the average global error and the accumulated point-wise error. We express it as MISE(Ẑ)= E[Z(x) Ẑ(x)] 2 dx = MSE(Ẑ)dx; (7) D D corresponding to the volume generated by the mean square error (MSE) over D, the region of interest. Now, it is possible to make a partition of the region of interest through the Dirichlet tessellation and assume that inside each polygon D i, the mean squared error is the same. MISE(Ẑ)= k(x)dx 2 = k(x)dx: D D 2 (8) i i D Dirichlet tessellation is based on associating a polygon D i to each point of a sample inside D in such a way that all of the points in this polygon, also called Dirichlet cell or tile, are nearer to the referenced sampled point than do any other. To approximate MISE(Ẑ) we use cross-validation, deleting an observation from the data and retting the model using the reduced data set. Typically, each observation is deleted in turn. Its purpose is to estimate in a realistic way the mean squared error of the prediction. Therefore, the use of kriging variance dened without the ith observation k(i) 2 (x)=e[z i(x) Ẑ (i) (x)] 2 is proposed, and then k(x)dx 2 w i E[Z i (x) Ẑ (i) (x)] 2 ; (9) D i D i D i where w i =area(tile) i for i =1;:::;n are the weights calculated as the tile areas of the Dirichlet tessellation. In order to estimate it, let us introduce the required notation and some quick formulae based on Christensen et al. (1992). The subscript (i) indicates that the observation i has been removed. Let Z (i) =(z 1 ;:::;z i 1 ;z i+1 ;:::;z n ) T be the vector of observations excluding the ith component, and K the covariance matrix of the whole sample, given in block form by ( kii k T K = i k i K (i) ) ; where k i = Cov(z i ; Z (i) ) is the vector of covariances between z i and the rest of the sample, k ii =Cov(z i ;z i ) and K (i) is the covariance matrix of Z (i). Denote by K 1 (i) the inverse of K (i), F (i) the design matrix after removing the ith observation, f (x i )=f (x i ) F(i) T K 1 (i) k i and z i =z i Z(i) T K 1 (i) k i where var( z i )=si 2 and si 2 =k ii ki T K 1 (i) k i. We dene ẽ i = z i ẑ (i) and after some algebra we obtain ẽ i = z i f T i ˆ (i) =( z i f T i where h i = f T i (F T K 1 F) 1 f i and s 2 i ˆ) ; si 2 h i ˆ (i) = ˆ (F T K 1 F) 1 f i ( z i f T i ˆ) : si 2 h i

5 A.F. Militino, M.D. Ugarte / Statistics & Probability Letters 52 (2001) Thus, approximating E[Z i (x) Ẑ (i) (x)] 2 by [z i ẑ (i) ] 2, we conclude n n MISE(Ẑ) w i [z i ẑ (i) ] 2 = w i ẽ 2 i : (10) i=1 i=1 As usual, we proceed standardizing the residuals and then dividing by the square root of Var(ẽ i )= s4 i : si 2 h i Therefore, we call SMISE (standardized MISE) the statistic n w i i=1 z i f T ˆ i : si 2 h i 2 (11) One just needs to minimize (11) with respect to the parameters of the covariance function to obtain the estimates of such parameters. In order to evaluate the behaviour of the proposed SMISE statistic (11) versus CV (6) we conducted a simulation study based on two well-known examples of the literature Example 1 The example corresponds to the 52 topographic data cited in Davis (1986, Table 5:11). They are observations over a small area on the northern side of a hill. Davis was interested in the analysis of maps, and he used the survey to produce contours of the region. We assume a quadratic surface and an exponential variogram given by { (h; a; b; r)= 0 if h =0; otherwise; a + b(1 exp( h r )) where a is the nugget eect, b is the sill, r is the range parameter and h is the Euclidean distance. Here, there is no nugget eect and the sill is a common constant that we can drop it out. Table 1 shows both CV and SMISE estimations for dierent range values of the exponential variogram. The areas of the topographic data set are plotted in Fig. 1. The minimum value of the SMISE statistic is that corresponds to the range value 0.9. This is also the weighted least-squares estimate provided by Cressie (1985). Now, we proceed to simulate 100 data sets using the scenario as encountered in the analysis of the topographic data, which means that we are using the same locations and we xed the range parameter of the covariance function equal to 0.9. We count the number of times where the minimum has been reached at ranges 0.1, 0.5, 0.9 and 1.4 with statistics (11) and (6). The results on the right of Table 1 show that SMISE is more successful than CV identifying the correct range of the covariance function. The identication of the range 0.9 is correct for Table 1 Analysis of the topographic data set and simulations Topo data set Simulations Range CV SMISE CV SMISE :63 a 44:09 a a Values of CV and SMISE for the correct parameter.

6 204 A.F. Militino, M.D. Ugarte / Statistics & Probability Letters 52 (2001) Fig. 1. Areas of the Dirichlet tessellation in the topographic data set. SMISE in all of the simulations, whereas CV fails almost 30% of them. In fact, it reaches its minimum at range 0.9, 72% of the times and the rest identies the range 0.5 as optimum Example 2 The aquifer head data from the Saratoga Valley in Wyoming is a data set of 93 observations which have been published and analyzed by Jones (1989) using stochastic partial dierential equations. This example has also been studied by Christensen et al. (1992), who t a linear trend and a Gaussian covariance function. The Gaussian variogram is given by { 0 if h =0; (h; a; b; r)= a + b(1 exp(( h r )2 )) otherwise: They show the great sensitivity of the covariance parameter estimates to the sampled data. In fact, this data set presents several outliers and give as restricted maximum likelihood estimates: a = 25, b = 1200 and r =8. The tile areas of the Dirichlet tessellation are given in Fig. 2. Table 2 shows the CV and SMISE estimations with nugget = 25 and b = Both statistics reach the minimum at range 4. Then, we simulate 100 data sets with parameters of the covariance function a = 25, b = 1200 and range = 4 and at the same spatial locations. We have checked the simulations in these parameters and allowed for small oscillations in the parameter estimates. This is why the right of Table 2 shows the frequencies of identifying the ranges 0.1, 4, 6, 8, 10 and 12. Range 4 is identied by SMISE in 95 simulations, whereas only 47 are identied using CV. We also observe larger dierences between both statistics due to the fact that data are quite irregularly spaced and the areas are quite big. Here, the use of the tile areas improves considerably the identication of the correct range parameter.

7 A.F. Militino, M.D. Ugarte / Statistics & Probability Letters 52 (2001) Fig. 2. Areas of the Dirichlet tessellation in the aquifer head data set. Table 2 Analysis of the aquifer data set and simulations Aquifer data set Simulations Range Nugget Sill CV SMISE CV SMISE :80 a 1262:14 a a Values of CV and SMISE for the correct parameters. 4. Discussion In geostatistics, one of the main uses of the cross-validation criterion has been to support the selection of a particular semivariogram model. Although this technique should not be interpreted as a conrmatory tool, it helps to choose the right model among a discrete number of potential candidates. Traditional cross-validation is based on minimizing a point-wise measure of error. This means that for example, points at the border of the region of interest that are dicult to predict, contribute heavily to the sum of errors. The weighted criterion that it is presented here, is based on minimizing a global measure of error which means that it will not be highly aected by the presence of points that are far apart. On the other hand, the use of the tile areas of the Dirichlet tessellation helps to cope with the problem of adapting the traditional cross-validation from independent and identically distributed data to the geostatistical context. The areas permit to take into account

8 206 A.F. Militino, M.D. Ugarte / Statistics & Probability Letters 52 (2001) not just a particular data point but its area of inuence. The new statistic does not warranty a complete success in identifying the covariance function parameters, but the simulation results show that it presents a reasonable criterion to choose the appropriate one, improving at the same time the traditional simple cross-validation. References Amstrong, M., Basic Linear Geostatistics. Springer, Berlin. Christensen, R., Johnson, W., Pearson, L.M., Prediction diagnostics for spatial linear models. Biometrika 79, Cressie, N., Fitting variogram models by weighted least squares. Math. Geol. 17, Davis, J.C., Statistics and Data Analysis in Geology, 2nd Edition. Wiley, New York. Jones, R.H., Fitting a stochastic partial dierential equation to aquifer data. Stochastic Hydrol. Hydraul. 3, Krige, D.G., A statistical approach to some basic mine valuation problems on the Witwatersrand. J. Chem. Metall. Min. Soc. South Africa 52, Matheron, G., Traite de Geostatistique Apliquee. Tome I: Memoires du Bureau de Reserches Geologiques et Minieres, Technip, Paris. Militino, A.F., Ugarte, M.D., Bounded inuence estimation in a spatial linear mixed model. In: Gregoire, T.G. et al. (Eds.), Modelling Longitudinal and Spatially Correlated Data, Lecture notes in statistics, vol Springer, New York, pp Ripley, B.D., Spatial Statistics. Wiley, New York.