This book provides an inter-disciplinary introduction to the theory of random fields and its applications. Spatial models and spatial data analysis are integral parts of many scientific and engineering disciplines. Random fields provide a general theoretical framework for the development of spatial models and their applications in data analysis.

The contents of the book include topics from classical statistics and random field theory (regression models, Gaussian random fields, stationarity, correlation functions) spatial statistics (variogram estimation, model inference, kriging-based prediction) and statistical physics (fractals, Ising model, simulated annealing, maximum entropy, functional integral representations, perturbation and variational methods). The book also explores links between random fields, Gaussian processes and neural networks used in machine learning. Connections with applied mathematics are highlighted by means of models based on stochastic partial differential equations. An interlude on autoregressive time series provides useful lower-dimensional analogies and a connection with the classical linear harmonic oscillator. Other chapters focus on non-Gaussian random fields and stochastic simulation methods. The book also presents results based on the author’s research on Spartan random fields that were inspired by statistical field theories originating in physics. The equivalence of the one-dimensional Spartan random field model with the classical, linear, damped harmonic oscillator driven by white noise is highlighted. Ideas with potentially significant computational gains for the processing of big spatial data are presented and discussed. The final chapter concludes with a description of the Karhunen-Loève expansion of the Spartan model.

The book will appeal to engineers, physicists, and geoscientists whose research involves spatial models or spatial data analysis. Anyone with background in probability and statistics can read at least parts of the book. Some chapters will be easier to understand by readers familiar with differential equations and Fourier transforms.

#### About the authors

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.