Random Fields for Spatial Data Modeling

D. T. Hristopulos holds a Dipl. Eng. in Electrical Engineering from the National Technical University of Athens (1985) and a PhD in Physics from Princeton University, USA (1991). His advisor at Princeton was Nobel laureate Prof. P.W. Anderson. D. T. Hristopulos worked for 7 years at the Dept. of Environmental Sciences and Engineering, University of North Carolina (Chapel Hill, USA), and for 2 years at the Pulp and Paper Research Institute of Canada – PAPRICAN (Pointe-Claire, Québec) before moving to the Technical University of Crete in 2002. For research conducted at PAPRICAN Hristopulos and Uesaka were awarded the 2003 Johannes A. Van den Akker Prize for Advances in Paper Physics. D. T. Hristopulos has more than 15 years of expertise in Geostatistics and mathematical modelling. His expertise includes the development of new geostatistical methods, algorithms for the simulation and interpolation of scattered data, analysis of mechanical properties and fracture of heterogeneous media, and applications of statistical physics in spatial analysis. In 2003 D. Hristopulos proposed a flexible and computationally efficient geostatistical model (Spartan Spatial Random Fields) with applications in automatic mapping of environmental processes and the simulation of geological spatial structures. D. T. Hristopulos is on the editorial board of the journal Stochastic Environmental Research and Risk Assessment, published by Springer. He also participates on organizing committees of international conferences on statistics, geographic information systems (GIS) and statistical physics (e.g., statGIS 2006, 2007, 2009, Sigma Phi 2008, 2011, Interpore 2011). D. T. Hristopulos actively pursues innovative research in the framework of European projects. Research results are presented by him and his group in various conferences and seminars in Europe (e.g. European Geophysical Union Assemblies, GeoENV, etc) and the USA (e.g. Univ. of North Carolina, Johns Hopkins Univ., etc).  The Marie Curie project SPATSTAT (2005-2008), coordinated by D. T. Hristopulos was selected by the European Commission as a success story and highlighted in the special edition “Marie Curie Actions: Inspiring Researchers”, European Commission, Luxembourg: Publications Office of the European Union, 2010 ISBN 978-92-79-14328-1.  D. T. Hristopulos has coauthored 75 scientific research papers in international journals (ISI Web of Knowledge database), 39 papers in proceedings of international conferences, 80 international conference abstracts, and the book Spatiotemporal Environmental Health Modelling (Kluwer, Boston, 1998). D. T. Hristopulos is on the editorial boards of the journals Stochastic Environmental Research and Risk Assessment, published by Springer and Computers and Geosciences, published by Elsevier.

Introduction.- Preliminary Remarks.- Why Random Fields?.- Notation and Definitions.- Noise and Errors.- Spatial Data and Basic Processing Procedures.- A Personal Selection of Relevant Books.- Trend Models and Estimation.- Empirical Trend Estimation.- Regression Analysis.- Global Trend Models.- Local Trend Models.- Trend Estimation based on Physical Information.- Trend Based on the Laplace Equation.- Basic Notions of Random Fields.- Introduction.- Single-Point Description.- Stationarity and Statistical Homogeneity.- Variogram versus Covariance.- Permissibility of Covariance Functions.- Permissibility of Variogram Functions.- Additional Topics of Random Field Modeling.- Ergodicity.- Statistical Isotropy.- Anisotropy.- Anisotropic Spectral Densities.- Multipoint Description of Random Fields.- Geometric Properties of Random Fields.- Local Properties.- Covariance Hessian Identity and Geometric Anisotropy.- Spectral Moments.- Length Scales of Random Fields.- Fractal Dimension.- Long-Range Dependence.- Intrinsic Random Fields.- Fractional Brownian Motion.- Classification of Random Fields.- Gaussian Random Fields.- Multivariate Normal Distribution.- Field Integral Formulation.- Useful Properties of Gaussian Random Fields.- Perturbation Theory for Non-Gaussian Probability Densities.- Non-stationary Covariance Functions.- Further Reading.- Random Fields based on Local Interactions.- Spartan Spatial Random Fields.- Two-point Functions and Realizations.- Statistical and Geometric Properties.- Bessel-Lommel Covariance Functions.- Lattice Representations of Spartan Random Fields.- Introduction to Gauss-Markov Random Fields.- From Spartan Random Fields to Gauss-Markov Random Fields.- Lattice Spectral Density.- SSRF Lattice Moments.- SSRF Inverse Covariance Operator on Lattices.- Spartan Random Fields and Langevin Equations.- Introduction to Stochastic Differential Equations.- Classical Harmonic Oscillator.- Stochastic Partial Differential Equations.- Spartan Random Fields and Stochastic Partial Differential Equations.- Covariance and Green’s functions.- Whittle-Matérn Stochastic Partial Differential Equation.- Diversion in Time Series.- Spatial Prediction Fundamentals.- General Principles of Linear Prediction.- Deterministic Interpolation.- Stochastic Methods.- Simple Kriging.- Ordinary Kriging.- Properties of the Kriging Predictor.- Topics Related to the Application of Kriging.- Evaluating Model Performance.- More on Spatial Prediction.- Linear Generalizations of Kriging.- Nonlinear Extensions of Kriging.- Connections with Gaussian Process Regression.- Bayesian Kriging.- Continuum Formulation of Linear Prediction.- The “Local-Interaction” Approach.- Big Spatial Data.- Basic Concepts and Methods of Estimation.- Estimator Properties.- Estimating the Mean with Ordinary Kriging.- Variogram Estimation.- Maximum Likelihood Estimation.- Cross Validation.- More on Estimation.- The Method of Normalized Correlations.- The Method of Maximum Entropy.- Stochastic Local Interactions.- Measuring Ergodicity.- Beyond the Gaussian Models.- Trans-Gaussian Random Fields.- Gaussian Anamorphosis.- Tukey g-h Random Fields.- Transformations based on Kappa Exponentials.- Hermite Polynomials.- Multivariate Student’s t-distribution.- Copula Models.- The Replica Method.- Binary Random Fields.- The Indicator Random Field.- Ising Model.- Generalized Linear Models.- Simulations.- Introduction.- Covariance Matrix Factorization.- Spectral Simulation Methods.- Fast-Fourier-Transform Simulation.- Randomized Spectral Sampling.- Conditional Simulation based on Polarization Method.- Conditional Simulation based on Covariance Matrix Factorization.- Monte Carlo Methods.- Sequential Simulation of Random Fields.- Simulated Annealing.- Karhunen-Loève Expansion.- Karhunen-Loève Expansion of Spartan Random Fields.- Epilogue.- A Jacobi’s Transformation Theorems.- B Tables of SSRF Properties.- C Linear Algebra Facts.- D Kolmogorov-Smirnov Test.- Glossary.- References.- Index.