The interplay between trend and covariance structure 104

When the underlying mean function,µ(x), is not constant, empirical or sample variograms based on the observationsYiare potentially very misleading. In this situation, the empirical variogram wrongly attributes the variation induced by the non-constant mean,µ(x), to large-scale covariance structure in the unob-served processS(x). A solution is to estimateµ(x), typically by assuming either a trend surface model or, if covariate information is available, a more general regression model, and to convert the observations to residuals,Ri=Yi−µ(xˆ i), before calculating the empirical variogram. Of course, the properties of the observed residualsRi do not exactly match those of the theoretical but unob-served residuals,Yi−µ(xi). However, their covariance structure should not differ too much from that of the true residuals provided the number of parameters estimated in ˆµ(x) is small relative to n, the number of observations.

As an illustration, we consider a simulation in which the sample design mim-ics that of the surface elevation data, but the simulated observations Yi are

5.2. Variograms 105

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0.00.51.01.52.02.5

u

V(u)

Figure 5.4. The sample variogram for a simulated data-set with a non-constant mean function. The theoretical variogram is shown as a smooth curve. The dotted line is the sample variogram based on uncorrected observationsYi. The solid line is the sample variogram based on the true residuals,Yi−µi. The dashed line is the sample variogram based on the estimated residualsYi−µˆi.

generated by a modelYi =µ(xi) +S(xi) in whichµ(x) is a quadratic surface and S(x) is a stationary Gaussian process with mean zero, variance σ² = 1 and exponential correlation function,ρ(u) = exp(−u). Figure 5.4 compares the theoretical variogram with sample variograms based on the observed valuesYi, the observed residuals Ri = Yi −µ(xˆ i) with mean parameters estimated by ordinary least squares, and the true residualsR^∗_i =Yi−µi. The positive bias in the variogram based on the raw dataYiarises from the non-stationary variation induced by the quadratic trend surface. Using either observed or true residuals produces estimates which are closer to the theoretical variogram. Note, however, that the sample variogram based on observed residuals lies below that based on true residuals. Because the observed residuals are defined so as to minimise the variation about the estimated mean, we might generally expect the sample variogram of observed residuals to exhibit negative bias. The discrepancy be-tween observed and true residuals would be less marked in a larger data-set, and the negative bias in the sample variogram consequently smaller. This example illustrates how a decision on the data analyst’s part to ascribe part of the spa-tial variation in a real data-set to a deterministic trend model can materially affect the results obtained in any subsequent estimation of spatial correlation structure.

When analysing real data, we have to make a subjective judgment as to whether we should remove an empirically estimated trend before estimating spatial correlation structure. Figure 5.5 illustrates the point using the surface elevation data. It shows the sample variogram of observed residuals after fitting a quadratic trend surface to the observed elevation values using ordinary least squares. If we compare this with the sample variogram of the unadjusted data, shown as the right-hand panel of Figure 5.2, we see a number of qualitative sim-ilarities: an intercept close to zero, a smooth rising trend approaching a plateau

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u

V(u)

Figure 5.5. The sample variogram of observed residuals, after fitting a quadratic trend surface to the elevation data.

and erratic behaviour at large distances. However, the major differences are that the plateau is reached at smaller distances than before,u≈2 rather than u≈5, and its height is approximately 1000, whereas in Figure 5.2 the height of the plateau was approximately 6000. This shows that the fitted quadratic trend surface has accounted for approximately five-sixths of the total variation in the unadjusted elevation values, resulting in a weaker estimated spatial correlation structure for the residual variation than for the unadjusted elevations. For the time being, we regard these as alternative empirical descriptions of the pattern of spatial variation in the elevation data and make no attempt to say which, if either, is the better model in any scientific sense.

5.3 Curve-fitting methods for estimating covariance structure

In classical geostatistics, the variogram is used not only for exploratory pur-poses, but also for formal parameter estimation. In general we do not favour this approach, for reasons which we now discuss.

A possible rationale for using the variogram as the basis for parameter esti-mation is that the empirical variogram ordinates,vij, are unbiased estimates of the corresponding theoretical variogram ordinates,V(uij;θ), hence estimation ofθcan be considered as an exercise in fitting. In early work, the curve-fitting was often done “by eye,” in other words by trying different values for the model parameters and visually inspecting the fit to the sample variogram.

Although we do not advocate this as a method for parameter estimation, it can be a good way to find reasonable initial values for estimation methods involving numerical optimisation, which we discuss in the following sections. As discussed in Section 5.2.2, visual inspection of the empirical variogram is rarely helpful, and for curve-fitting by eye it is preferable to use the sample variogram.

5.3. Curve-fitting methods for estimating covariance structure 107 More objective curve-fitting methods include the use of non-linear regression analysis, treating the empirical or sample variogram ordinate as the response variable and inter-point distance as the corresponding explanatory variable. In the following discussion of this more objective approach, we use the notation (uk, vk, nk) : k = 1, . . . , m to denote a sample variogram. In this notation, vk represents the averaged empirical variogram ordinates over the distance-bin with mid-pointuk, andnk denotes the number of empirical variogram ordinates which contribute tovk. The unsmoothed empirical variogram is the special case in which allnk= 1. Rather than using the mid-point of the distance bin, a vari-ation is to defineukas the average of the inter-point distances which fall within the k^th bin. As pointed out by a reviewer, this may be particularly appro-priate when the empirical distribution of the inter-point distances is strongly multi-modal.