Interpolation of Spatial Data: Some Theory for Kriging

1 Linear Prediction. - 1.1 Introduction. - 1.2 Best linear prediction. - Exercises. - 1.3 Hilbert spaces and prediction. - Exercises. - 1.4 An example of a poor BLP. - Exercises. - 1.5 Best linear unbiased prediction. - Exercises. - 1.6 Some recurring themes. - The Matern model. - BLPs and BLUPs. - Inference for differentiable random fields. - Nested models are not tenable. - 1.7 Summary of practical suggestions. - 2 Properties of Random Fields. - 2.1 Preliminaries. - Stationarity. - Isotropy. - Exercise. - 2.2 The turning bands method. - Exercise. - 2.3 Elementary properties of autocovariance functions. - Exercise. - 2.4 Mean square continuity and differentiability. - Exercises. - 2.5 Spectral methods. - Spectral representation of a random field. - Bochner's Theorem. - Exercises. - 2.6 Two corresponding Hilbert spaces. - An application to mean square differentiability. - Exercises. - 2.7 Examples of spectral densities on 112. - Rational spectral densities. - Principal irregular term. - Gaussian model. - Triangular autocovariance functions. - Matern class. - Exercises. - 2.8 Abelian and Tauberian theorems. - Exercises. - 2.9 Random fields with nonintegrable spectral densities. - Intrinsic random functions. - Semivariograms. - Generalized random fields. - Exercises. - 2.10 Isotropic autocovariance functions. - Characterization. - Lower bound on isotropic autocorrelation functions. - Inversion formula. - Smoothness properties. - Matern class. - Spherical model. - Exercises. - 2.11 Tensor product autocovariances. - Exercises. - 3 Asymptotic Properties of Linear Predictors. - 3.1 Introduction. - 3.2 Finite sample results. - Exercise. - 3.3 The role of asymptotics. - 3.4 Behavior of prediction errors in the frequency domain. - Some examples. - Relationship to filtering theory. - Exercises. - 3.5 Prediction with the wrong spectral density. - Examples of interpolation. - An example with a triangular autocovariance function. - More criticism of Gaussian autocovariance functions. - Examples of extrapolation. - Pseudo-BLPs with spectral densities misspecified at high frequencies. - Exercises. - 3.6 Theoretical comparison of extrapolation and ointerpolation. - An interpolation problem. - An extrapolation problem. - Asymptotics for BLPs. - Inefficiency of pseudo-BLPs with misspecified high frequency behavior. - Presumed mses for pseudo-BLPs with misspecified high frequency behavior. - Pseudo-BLPs with correctly specified high frequency behavior. - Exercises. - 3.7 Measurement errors. - Some asymptotic theory. - Exercises. - 3.8 Observations on an infinite lattice. - Characterizing the BLP. - Bound on fraction of mse of BLP attributable to a set of frequencies. - Asymptotic optimality of pseudo-BLPs. - Rates of convergence to optimality. - Pseudo-BLPs with a misspecified mean function. - Exercises. - 4 Equivalence of Gaussian Measures and Prediction. - 4.1 Introduction. - 4.2 Equivalence and orthogonality of Gaussian measures. - Conditions for orthogonality. - Gaussian measures are equivalent or orthogonal. - Determining equivalence or orthogonality for periodic random fields. - Determining equivalence or orthogonality for nonperiodic random fields. - Measurement errors and equivalence and orthogonality. - Proof of Theorem 1. - Exercises. - 4.3 Applications of equivalence of Gaussian measures to linear prediction. - Asymptotically optimal pseudo-BLPs. - Observations not part of a sequence. - A theorem of Blackwell and Dubins. - Weaker conditions for asymptotic optimality of pseudo-BLPs. - Rates of convergence to asymptotic optimality. - Asymptotic optimality of BLUPs. - Exercises. - 4.4 Jeffreys's law. - A Bayesian version. - Exercises. - 5 Integration of Random Fields. - 5.1 Introduction. - 5.2 Asymptotic properties of simple average. - Results for sufficiently smooth random fields. - Results for sufficiently rough random fields. - Exercises. - 5.3 Observations on an infinite lattice. - Asymptotic mse of BLP. - Asymptotic optimality of simple average. - Exercises. - 5.4 Improving on the sample mean. - Approximating $$\int_0^1 {\exp } (ivt)dt$$. - Approximating $$\int_{{{[0,1]}^d}} {\exp (i{\omega ^T}x)} dx$$ in more than one dimension. - Asymptotic properties of modified predictors. - Are centered systematic samples good designs? - Exercises. - 5.5 Numerical results. - Exercises. - 6 Predicting With Estimated Parameters. - 6.1 Introduction. - 6.2 Microergodicity and equivalence and orthogonality of Gaussian measures. - Observations with measurement error. - Exercises. - 6.3 Is statistical inference for differentiable processes possible? - An example where it is possible. - Exercises. - 6.4 Likelihood Methods. - Restricted maximum likelihood estimation. - Gaussian assumption. - Computational issues. - Some asymptotic theory. - Exercises. - 6.5 Matern model. - Exercise. - 6.6 A numerical study of the Fisher information matrix under the Matern model. - No measurement error and? unknown. - No measurement error and? known. - Observations with measurement error. - Conclusions. - Exercises. - 6.7 Maximum likelihood estimation for a periodic version of the Matern model. - Discrete Fourier transforms. - Periodic case. - Asymptotic results. - Exercises. - 6.8 Predicting with estimated parameters. - Jeffreys's law revisited. - Numerical results. - Some issues regarding asymptotic optimality. - Exercises. - 6.9 An instructive example of plug-in prediction. - Behavior of plug-in predictions. - Cross-validation. - Application of Matern model. - Conclusions. - Exercises. - 6.10 Bayesian approach. - Application to simulated data. - Exercises. - A Multivariate Normal Distributions. - B Symbols. - References.