Bayesian Transgaussian Kriging

Transcription

1 1 Bayesian Transgaussian Kriging Hannes Müller Institut für Statistik University of Klagenfurt 9020 Austria Keywords: Kriging, Bayesian statistics AMS: 62H11,60G90 Abstract In geostatistics a widely used method for prediction is kriging. It is well known and already used for many years. But some limits are inherent on the traditional ways of kriging, i.e. simple, ordinary and universal kriging. Kriging is based on the assumptions that the covariance function is exactly known and the underlying random field is a gaussian field. In practice, neither the trend or the variogram are exactly known, but on the other hand there may be expert knowledge for the trend or the variogram that should be used for prediction in geostatistics. So a mixture of kriging and Bayesian statistics can be useful. In theory, Baysian transgaussian kriging can handle random fields with non-gaussian behavior and with various trends and different link functions, but the problem is the computational effort. Many applications need near real-time evaluations of the random process, so the CPU-time is limited. Further work and increasing computational power is needed to face this problem. 1. Spatial statistics and geostatistics Under spatial statistics one understands a wide range of statistical models and methods used for the analysis of spatial data. The term geostatistics, that describes a special field of spatial statistics, is used when one has the following assumptions [2]: Values Y i are observed at some locations x i, i = 0,..., n. These points belong to some special spatial region D, where the values of Y are of interest. The values Y are either measurements of a continuous spatial phenomena or they are correlated to another spatial process, that cannot be measured directly, but is used for the prediction of Y. As first geostatistical model we consider Y (x) = m(x) + ε(x); x D, (1)

2 Bayesian Transgaussian Kriging 2 m denoting the mean or the spatial trend and ε is the error term with expectation zero, i.e.: ε(x) N (0, σ 2 ); x D. (2) If the expectation of Y should be constant, then we set m(x) = µ, otherwise we could use a linear model: m(x) = k β k f k (x) (3) j=0 The functions f k (x) are deterministic variates, with f 0 (x) 1 only we can include the constant case. Until now we do not really have an interesting model, we just could do something like a linear regression to fit the unknown parameters β to some given data Y i. We need some connection between the errors at different points x i and x j. Instead of the correlation the semivariogram γ is often used γ(h) = var(y (x + h) Y ). (4) Note that in this definition we did not write γ(x, h), so we assume stationarity for the semivariogram. If γ is just a function of the length of h, but not of the direction, then we call our model isotropic. γ(u) = var(y (x 1 ) Y (x 2 )), u = x 1 x 2 (5) If this assumption does not hold, a common way to cope with anisotropy is to try to transform the coordinates to get an isotropic model. This is done with an matrix A via x = Ax, in consequence the new isotropic distance is u = h A Ah. (6) 2. Kriging Kriging is the standard way for prediction in geostatistics [1] [6]. Assuming gaussian errors, with kriging one attempts to get the best linear predictor for Y (x) using the given data Y i : Ŷ (x) = n λ i Y (x i ). (7) i=0 To calculate the weights λ i we have to solve the following minimization problem for the prediction error σ 2 pred :

3 Bayesian Transgaussian Kriging 3 min(σ 2 pred) = E(Ŷ (x) Y (x))2 = E( n λ i Y (x i ) Y (x)) 2 (8) According to the deterministic part (3) of our model we can distinguish between the following three types of kriging: Simple Kriging In the minimum version of kriging we have no trend in the process. The constant mean is also assumed to be known. Ordinary Kriging The process is still stationary, but the mean is unknown and can only be estimated from the data. Universal Kriging Universal kriging is used in non-stationary geostatistics, now the trend parameters β also have to be estimated. Note that in case of a known trend, one can subtract the trend from the data and so one can still use simple kriging. A very important assumption is common to all kinds of kriging: the covariance function (or the semivariogram) of the data is assumed to be known. This is a very tight assumption, because normally the parameters of some class of correlation functions as well as the total variance have to be estimated from an empirical variogram. Afterwards in kriging the parameters are treated as given constants, this leads to an underestimation of the prediction error. 3. Transgaussian kriging Often the assumption of gaussian data does not hold, i.e. there are natural bounds for the values. For example, measurements like radiation levels can not fall below zero. A way to model such departures from Gaussianity is to assume that the random field of interest can be transformed [4] by a rather easy transformation to a near-gaussian distribution. A well known family of transformations is the Box-Cox family that is often used for normalizing positive data. The original data x are transformed to the gaussian data y with y = g λ (x) via i=0 y = xλ 1 λ if λ 0 (9) y = log(x) if λ = 0 For λ = 0 one has the famous lognormal-distribution, that is often used in finance. If there can not be found a reasonable Box-Cox parameter, on can try to use more complicate link functions [2] to find a transformation to normality, but the transformation should be monotonic, bijective and differentiable to avoid a strange behavior after doing the transformation or backtransformation.

4 Bayesian Transgaussian Kriging 4 4. Baysian framework As it was already mentioned, the two-phase approach in classical kriging leads to optimistic assessments of predicitive accuracy. In the Bayesian approach the distinction between estimation and prediction vanishes. Not only the realisation of the spatial process Y, but also the parameters are unobserved random variables. The spatial model is extended by the a priori distribution of the parameters. Via Bayes Theorem one gets the a posteriori distribution of the parameters from the measurements and the a priori distribution. Afterwards the a posteriori distribution is used to calculate the prediction, so in the Bayesian approach one not only has one exact model like in kriging, but on makes a model-averaging over the unknown model parameters. The a posteriori distribution should reflect expert knowledge, but often there is no real expert knowledge for the spatial data, so specifying a prior is hardly possible. In an empirical Bayes version of kriging [5] the a posteriori distribution is specified by means of simulations, reflecting the uncertainty of the transformation and the covariance parameters. The computation is started with an estimation of the transformation parameter λ of the transformed field and afterwards an estimation of the covariance parameters with classical methods like variogram fitting. In a non- Bayesian world we would now have plug-in estimates for all parameters and we could continue with classical kriging. In the empirical Bayesian approach this estimates are used as starting values for parametric bootstrap. By simulations using the estimates, a large number of random fields are produced. Afterwards, the simulated values at the measurement points are treated as new data-sets and estimations of the model parameters are made. Doing this simulation and estimation several times, one assumes to get the a posteriori function that is used for prediction. 5. Conclusion After this short article one just has a little overview about the wide field of geostatistics. The main objective of INTAMAP is to develop an interoperable framework for real time automatic mapping of critical environmental variables. In theory, Bayesian transgaussian kriging can handle random fields with nongaussian behaviour and with various trends and different link functions, but the problem is the computational effort. In INTAMAP real-time evaluations of the random process are needed, so the CPU-time is limited, with can be in conflict to the time-consuming calculation of the posterior distribution. The project will last for two additional years, so there still is a lot of work that is

5 Bayesian Transgaussian Kriging 5 waiting to be done by the project partners. Acknowledgements: The research is supported by INTAMAP [3]. 6. Bibliography [1] Cressie, Noel A. (1993). Statistics for spatial data. Rev. ed., New York, NY, Wiley. [2] Diggle, Peter J. and Ribeiro, Paulo J. (2007). Model-based geostatistics. New York, NY, Springer. [3] INTAMAP: [4] Oliveira, V. (1997). Bayesian Prediction of Transformed Gaussian Random Field, In Journal of the American Statistical Association; Dec 1997; 92, 440. [5] Spöck, G. (2005). Bayesian spatial prediction and sampling design. Ph.D. Thesis, Universität Klagenfurt. [6] Wackernagel, H. (2003). Multivariate geostatistics. 3. compl. rev. ed.,berlin, Springer.