Overview

Prediction of a random field based on observations of the random field at some set of locations arises in mining, hydrology, atmospheric sciences, and geography. Kriging, a prediction scheme defined as any prediction scheme that minimizes mean squared prediction error among some class of predictors under a particular model for the field, is commonly used in all these areas of prediction. This book summarizes past work and describes new approaches to thinking about kriging.

Full Product Details

Author:   Michael L. Stein
Publisher:   Springer-Verlag New York Inc.
Imprint:   Springer-Verlag New York Inc.
Edition:   Softcover reprint of the original 1st ed. 1999
Dimensions:   Width: 15.50cm , Height: 1.40cm , Length: 23.50cm
Weight:   0.421kg
ISBN:  

9781461271666

ISBN 10:   1461271665
Pages:   249
Publication Date:   17 October 2012
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   Manufactured on demand  
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Table of Contents

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 Matérn 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.- Matérn 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.- Matérn 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 Matérn model.- Exercise.- 6.6 A numerical study of the Fisher information matrix under the Matérn 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 Matérn 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 Matérn model.- Conclusions.- Exercises.- 6.10 Bayesian approach.- Application to simulated data.- Exercises.- A Multivariate Normal Distributions.- B Symbols.- References.

Reviews

From a review: GEODERMA the book is written with great care and dedication. Soil geostatisticians that are not easily scared off by mathematics will find this book to be a rich source of inspiration for many years to come.

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