Rongelap island

Our second case study is based on the data from Example 1.2. The data were collected as part of an investigation into the residual contamination arising from nuclear weapons testing during the 1950’s. This testing programme re-sulted in the deposition of large amounts of radioactive fallout on the pacific island of Rongelap. The island has been uninhabited since the mid-1980’s. A geostatistical analysis of residual contamination levels formed one component of a wide-ranging project undertaken to establish whether the island was safe for re-habitation. Earlier analyses of these data are reported by Diggle et al.

(1997) who used log-Gaussian kriging, and by Diggle et al. (1998) who used the model-based approach reported here, but with minor differences in the detailed implementation.

For our purposes, the data consist of nett photon emission counts Yi over time-periods ti at locations xi indicated by the map in Figure 1.2. The term

“nett” emission count refers to the fact that an estimate of natural background radiation has been subtracted from the raw count in such a way that the datum Yi can be attributed to the local level of radioactive caesium at or near the surface of the island. The background effect accounts for a very small fraction of the total radioactivity, and we shall ignore it from now on.

The gamma camera which records photon emissions integrates information received over a circular area centred on each locationxi. There is also a progres-sive “dilution” effect with increasing distance fromxi. Hence, if λ^∗(x) denotes

7.6. Case studies in generalized linear geostatistical modelling 181 the true rate of photon emissions per unit time at locationx, the raw count at locationxwill follow a Poisson distribution with mean

µ(x) =t(x) Z

w(x−u)λ^∗(u)du (7.23)

wheret(x) denotes the observation time corresponding to locationx. The func-tionw(·) decays to zero over a distance of approximately 10 metres, but we do not know its precise form. However, the minimum distance between any two locations in Figure 1.2 is 40 metres. Hence, rather than modelλ^∗(·) in (7.23) directly, we will modelλ(·), where

λ(x) = Z

w(x−u)λ^∗(u)du.

Our general objective is to describe the spatial variation in the spatial process λ(x). Note that any spatial correlation in λ(·) induced by the integration of the underlying processλ^∗(·) operates at a scale too small to be identified from the observed data. Hence, any empirically observed spatial correlation must be the result of genuine spatial variation in local levels of residual contamination, rather than an artefact of the sampling procedure.

The sampling design for the Rongelap island survey was a lattice plus in-fill design of the kind which we discuss in Chapter 8. This consists of a primary lattice overlaid by in-fill squares in selected lattice cells. In fact, the survey was conducted in two stages. The primary lattice, at 200m spacing, was used for the first visit to the island. The in-fill squares were added in a second visit, to en-able better estimation of the small-scale spatial variation. For the second-stage sample, two of the primary grid squares were selected randomly at either end of the island. As we discuss in Chapter 8, inclusion of pairs of closely spaced points in the sampling design can be important for identification of spatial covariance structure, and therefore for effective spatial prediction when the true model is unknown. In this application, we can also use the in-fill squares to make an admittedly incomplete assessment of the stationarity of the underlying signal process. For example, if we let y denote the nett count per second, then for 50 sample locations at the western end of the island including the two in-fill squares, the sample mean and standard deviation of log(y) are 2.17 and 0.29, whilst for 53 locations covering the eastern in-fill area the corresponding figures are 1.85 and 0.35. Hence, the western extremity is the more heavily contami-nated, but the variation over two areas of comparable size is quite similar; see also the exercises at the end of this chapter.

Taking all of the above into consideration, we adopt a Poisson log-linear model with log-observation time as an offset and a latent stationary spatial processS(·) in the linear predictor. Explicitly, ifY_i denotes the nett count over observation timeti at locationxi, then our modelling assumptions are

ˆ conditional on a latent spatial process S(·), the Yi are mutually independent Poisson variates with respective meansµi, where

logµi= logti+β+S(xi); (7.24)

ˆ S(·) is a stationary Gaussian process.

0 500 1000 1500

02468

distance

semivariance

Figure 7.11. Empirical variogram of the transformed Rongelap data.

In order to make a preliminary assessment of the covariance structure, we transform each datum (Y_i, t_i) toY_i^∗= log(Y_i/t_i). Under the assumed log-linear structure of the proposed model, we can think ofY_i^∗ as a noisy version of the unobserved S(xi). Hence, the sample variogram of the observed values of Y_i^∗ should give a qualitative pointer to the covariance structure of the latent process S(·). Figure 7.11 shows the resulting empirical variogram. The relatively large intercept suggests that measurement error, which in the model derives from the approximate Poisson sampling distribution of the nett counts, accounts for a substantial proportion of the total variation. However, there is also clear struc-ture to the empirical variogram, indicating that the residual spatial variation is also important. The convex shape of the empirical variogram suggests that this spatial variation is fairly rough in character.

The remaining results in this section are taken from the analysis reported in Diggle et al. (1998), who used the powered exponential family (3.7) to model the correlation structure ofS(·),

ρ(u) = exp{−(u/φ)^κ}.

Recall that for this model,κ≤2 and unlessk= 2 the model corresponds to a mean-square continuous but non-differentiable processS(·). Also, forκ≤1 the correlation functionρ(·) is convex, which would be consistent with our earlier comment on the shape of the sample variogram of theY_i^∗.

The priors for β, σ², φ and κ were independent uniforms, with respective ranges (−3,7), (0,15), (0,120) and (0.1,1.95). The corresponding marginal pos-teriors have means 1.7,0.89,22.8 and 0.7, and modes 1.7,0.65,4.7 and 0.7. Note in particular the strong positive skewness in the posterior forφ, and confirma-tion that the data favourκ <1 i.e., a convex correlation function for the process S(·).

For prediction, Diggle et al. (1998) ran their MCMC algorithm for 51,000 iterations, discarded the first 1000 and then sampled every 100 iterations to give a sample of 500 values from the posterior distributions of the model parameter, and from the predictive distribution of the surfaceS(x) at 960 locations forming

7.6. Case studies in generalized linear geostatistical modelling 183

-6000 -5000 -4000 -3000 -2000 -1000 0

-5000-4000-3000-2000-100001000

0 5 10 15

Figure 7.12. Point predictions of intensity (mean count per second) for the Rongelap data. Each value is the mean of a Monte Carlo sample of size 500.

a square lattice to cover the island at a spacing of 50 metres. By transforming each sampled S(x) to λ(x) = exp{β+S(x)}, they obtained a sample of 500 values from the predictive distribution of the spatially varying intensity, or mean emission count per second, over the island. Figure 7.12 shows the resulting point-wise mean surface. This map includes the southeast corner of the island as an enlarged inset, to show the nature of the predicted small-scale spatial variation in intensity. Note also the generally higher levels of the predictions at the western end of the island.

A question of particular practical importance in this example is the pattern of occurrence of relatively high levels of residual contamination. Maps of point predictions like the one shown in Figure 7.12 do not give a very satisfactory an-swer to questions of this kind because they do not convey predictive uncertainty.

One way round this is to define a specific targetT and to show the whole of the predictive distribution ofT, rather than just a summary. To illustrate this approach, the left-hand panel of Figure 7.13 shows the predictive distribution ofT = max{λ(x)}, where the maximum is computed over the same 960 predic-tion locapredic-tions as were used to construct Figure 7.12. Note that the predictive distribution extends far beyond the maximum of the point-wise predictions of λ(x) shown in Figure 7.12. The two versions of the predictive distribution refer to predictions with and without allowance for parameter uncertainty, showing that for this highly non-linear functional ofS(·) parameter uncertainty makes

Intensity level

Density

10 20 30 40 50 60

0.00.020.040.060.080.100.120.14

Intensity level

Survivor function

0 5 10 15 20

0.00.20.40.60.81.0

0 5 10 15 20

0.00.20.40.60.81.0

0 5 10 15 20

0.00.20.40.60.81.0

Figure 7.13. Predictive inference for the Rongelap island data. The left-hand panel shows the predictive distribution ofT = maxλ(x), computed from a grid to cover the island at a spacing of 50 metres, with (solid line) and without (dashed line) allowance for parameter uncertainty. The right-hand panel shows point predictions (solid line) and 95% credible limits (dashed lines) forT(z), the areal proportion of the island for which intensity exceedsz counts per second.

a material difference. Of course, by the same token so does the assumed para-metric model which underlies this predictive inference. By contrast, predictive inferences for the point-wise values ofλ(x) are much less sensitive to parameter uncertainty.

The right-hand panel of Figure 7.13 summarises the predictive distributions of a family of targets,T(z) equal to the areal proportion of the island for which λ(x)> z. The predictive distribution for each value ofz is summarised by its point prediction and associated 95% central quantile-based interval.