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• Estimates of functionals of discrete time homogeneous random fields
• Estimates of functionals of discrete time periodically correlated isotropic random fields
• Estimates of functionals of continuous time homogeneous random fields
• Estimates of functionals of continuous time periodically correlated isotropic random fields.

• Gaussian processes
• Some geometry and some topology
• The Gaussian kinematic formula
• On applications: topological inference
• Algebraic topology of excursions sets: a new challenge.

These notes, based on lectures delivered in Saint Flour, provide an easy introduction to the authors' 2007 Springer monograph "Random Fields and Geometry." While not as exhaustive as the full monograph, they are also less exhausting, while still covering the basic material, typically at a more intuitive and less technical level. They also cover some more recent material relating to random algebraic topology and statistical applications. The notes include an introduction to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness. This is followed by a quick review of geometry, both integral and Riemannian, with an emphasis on tube formulae, to provide the reader with the material needed to understand and use the Gaussian kinematic formula, the main result of the notes. This is followed by chapters on topological inference and random algebraic topology, both of which provide applications of the main results.

• Review of Basic Probability Theory in a Random Field Context
• New Quantum-Physics-Based Probability Distributions
• Correlation and Spectral-Density Functions
• Efficient Characterization in Term of the Scale of Fluctuation
• Spectral Parameters, Level Excursions and Extremes
• Local Average Processes
• Robust Stochastic Finite Element Modeling
• Stochastic Growth Processes
• Multi-scale Random Fields and Fractal Variation
• Estimation, Prediction and Optimal Sampling--.
• (source: Nielsen Book Data)

Random variation is a fact of life that provides substance to a wide range of problems in the sciences, engineering, and economics. There is a growing need in diverse disciplines to model complex patterns of variation and interdependence using random fields, as both deterministic treatment and conventional statistics are often insufficient. An ideal random field model will capture key features of complex random phenomena in terms of a minimum number of physically meaningful and experimentally accessible parameters. This volume, a revised and expanded edition of an acclaimed book first published by the M I T Press, offers a synthesis of methods to describe and analyze and, where appropriate, predict and control random fields. There is much new material, covering both theory and applications, notably on a class of probability distributions derived from quantum mechanics, relevant to stochastic modeling in fields such as cosmology, biology and system reliability, and on discrete-unit or agent-based random processes. "Random Fields" is self-contained and unified in presentation. The first edition was found, in a review in EOS (American Geophysical Union) to be both technically interesting and a pleasure to read - the presentation is clear and the book should be useful to almost anyone who uses random processes to solve problems in engineering or science - and (there is) continued emphasis on describing the mathematics in physical terms.
(source: Nielsen Book Data)

The theory of spatial models over lattices, or random fields as they are known, has developed significantly over recent years. This book provides a graduate-level introduction to the subject which assumes only a basic knowledge of probability and statistics, finite Markov chains, and the spectral theory of second-order processes. A particular strength of this book is its emphasis on examples - both to motivate the theory which is being developed, and to demonstrate the applications which range from statistical mechanics to image analysis and from statistics to stochastic algorithms.
(source: Nielsen Book Data)

• 1 Random Polymers.- 2 Spontaneous replica symmetry breaking and interpolation methods for complex statistical mechanics systems.- 3 Derrida's random energy models: from spin glasses to the extremes of correlated random fields.- 4 Isomorphism Theorems: Markov processes, Gaussian processes and beyond.- 5 Spectral properties of Wigner matrices.
• (source: Nielsen Book Data)

This volume presents five different methods recently developed to tackle the large scale behavior of highly correlated random systems, such as spin glasses, random polymers, local times and loop soups and random matrices. These methods, presented in a series of lectures delivered within the Jean-Morlet initiative (Spring 2013), play a fundamental role in the current development of probability theory and statistical mechanics. The lectures were: Random Polymers by E. Bolthausen, Spontaneous Replica Symmetry Breaking and Interpolation Methods by F. Guerra, Derrida's Random Energy Models by N. Kistler, Isomorphism Theorems by J. Rosen and Spectral Properties of Wigner Matrices by B. Schlein. This book is the first in a co-edition between the Jean-Morlet Chair at CIRM and the Springer Lecture Notes in Mathematics which aims to collect together courses and lectures on cutting-edge subjects given during the term of the Jean-Morlet Chair, as well as new material produced in its wake. It is targeted at researchers, in particular PhD students and postdocs, working in probability theory and statistical physics.
(source: Nielsen Book Data)

• 1.Introduction.- 2.Spectral Expansions.- 3.L2 Theory of Invariant Random Fields.- 4.Sample Path Properties of Gaussian Invariant Random Fields.- 5.Applications.- A.Mathematical Background.- References.- Index.
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The author describes the current state of the art in the theory of invariant random fields. This theory is based on several different areas of mathematics, including probability theory, differential geometry, harmonic analysis, and special functions. The present volume unifies many results scattered throughout the mathematical, physical, and engineering literature, as well as it introduces new results from this area first proved by the author. The book also presents many practical applications, in particular in such highly interesting areas as approximation theory, cosmology and earthquake engineering. It is intended for researchers and specialists working in the fields of stochastic processes, statistics, functional analysis, astronomy, and engineering.
(source: Nielsen Book Data)

• Random variables
• random vectors
• sampling random variables
• second order properties
• the Fourier transform
• second order random fields
• spectral representation of random fields
• sampling and modelling random fields
• the sources.
• (source: Nielsen Book Data)

Intended to be of use to upper level engineering students, this book of lecture notes contains theoretical background material required for computer generation of random fields, which is of interest in various fields of applied mathematics. The necessary probabilistic background suitable for applied work in water resources engineering as well as signal and image processing is also covered.
(source: Nielsen Book Data)

In recent years many researchers have investigated the use of Markov random fields (MRFs) for computer vision. The computational complexity of the implementation has been a drawback of MRFs. In this paper we derive deterministic approximations to MRFs models. All the theoretical results are obtained in the framework of the mean field theory from statistical mechanics. Because we use MRFs models the mean field equations lead to parallel and iterative algorithms. One of the considered models for image reconstruction is shown to give in a natural way the graduate non-convexity algorithm proposed by Blake and Zisserman.

This book presents the language of random field theory and the principles of stochastic geometry in order to give the systematic and concise knowledge necessary for modeling real random heterogeneous media. This book is ideal for a general scientific audience needing an in-depth guide to current ideas, methods and models for the stochastic modeling of microstructures. It is an essential reference for practitioners, researchers and professionals in material modeling, mechanical engineering, materials science, chemical, civil, environmental engineering and applied mathematics.
(source: Nielsen Book Data)

• Introduction Weakly Stationary Sequences Preliminaries Examples of Weakly Stationary Sequences Spectral Representation of the Covariance Function Examples of Spectral Measures Canonical Isomorphism between L2(T
• FX) and L(X) Spectral Representation of a Weakly Stationary Sequence The Shift Operator on L(X) Moving Averages and Densities Regular and Singular Sequences Examples of Regular and Singular Sequences The Wold Decomposition The Theory of Regular Sequences The Concordance Theorem Maximal Factors and Innovation Processes The Szeg o-Krein-Kolmogorov Theorem Weakly Stationary Random Fields Preliminaries Examples Random Field with Discrete Spectral Measure Product Random Field White Noise Random Field Moving Average Random Field Regularity and Singularity Examples Horizontally and Vertically Singular Horizontally Regular and Vertically Singular Horizontally and Vertically Regular Horizontal and Vertical Wold Decomposition Regularity and the Spectral Measure Spectral Measures and Spectral-type Wold Decompositions The Fourfold Wold Decomposition Quarter-plane Moving Average Representations Helson-Lowdenslager Theory Semi-group Moving Average Representations Wold-Type Decompositions Invariant Subspaces The Halmos Decomposition Invariant Subspaces of L2(T) Invariant Subspaces of H2(T) The Halmos Fourfold Decomposition The Doubly Commuting Condition Invariant Subspaces of L2(T2) Invariant Subspaces of H2(T2) Applications and Generalizations Texture Identi_cation Invariant Subspaces of Lp(T) Harmonizable S_S Sequences Invariant Subspaces of Lp(T2) Harmonizable S_S Fields Proper MA Representations and Outer Functions Background Material Projections Orthogonally Scattered Set Functions Representation Theorems Bibliography.
• (source: Nielsen Book Data)

The first book to examine weakly stationary random fields and their connections with invariant subspaces (an area associated with functional analysis). It reviews current literature, presents central issues and most important results within the area. For advanced Ph.D. students, researchers, especially those conducting research on Gaussian theory.
(source: Nielsen Book Data)

• Introduction Weakly Stationary Sequences Preliminaries Examples of Weakly Stationary Sequences Spectral Representation of the Covariance Function Examples of Spectral Measures Canonical Isomorphism between L2(T
• FX) and L(X) Spectral Representation of a Weakly Stationary Sequence The Shift Operator on L(X) Moving Averages and Densities Regular and Singular Sequences Examples of Regular and Singular Sequences The Wold Decomposition The Theory of Regular Sequences The Concordance Theorem Maximal Factors and Innovation Processes The Szeg o-Krein-Kolmogorov Theorem Weakly Stationary Random Fields Preliminaries Examples Random Field with Discrete Spectral Measure Product Random Field White Noise Random Field Moving Average Random Field Regularity and Singularity Examples Horizontally and Vertically Singular Horizontally Regular and Vertically Singular Horizontally and Vertically Regular Horizontal and Vertical Wold Decomposition Regularity and the Spectral Measure Spectral Measures and Spectral-type Wold Decompositions The Fourfold Wold Decomposition Quarter-plane Moving Average Representations Helson-Lowdenslager Theory Semi-group Moving Average Representations Wold-Type Decompositions Invariant Subspaces The Halmos Decomposition Invariant Subspaces of L2(T) Invariant Subspaces of H2(T) The Halmos Fourfold Decomposition The Doubly Commuting Condition Invariant Subspaces of L2(T2) Invariant Subspaces of H2(T2) Applications and Generalizations Texture Identi_cation Invariant Subspaces of Lp(T) Harmonizable S_S Sequences Invariant Subspaces of Lp(T2) Harmonizable S_S Fields Proper MA Representations and Outer Functions Background Material Projections Orthogonally Scattered Set Functions Representation Theorems Bibliography.
• (source: Nielsen Book Data)

Thefirst book to examine weakly stationary random fields and their connections with invariant subspaces (an area associated with functional analysis). It reviews current literature, presents central issues and most important results within the area.For advanced Ph.D. students, researchers, especially those conducting research on Gaussian theory.
(source: Nielsen Book Data)

• Fractional Levy Fields.- The Theory of Scale Functions for Spectrally Negative Levy Processes.
• (source: Nielsen Book Data)

This is the second volume in a subseries of the Lecture Notes in Mathematics called Levy Matters, which is published at irregular intervals over the years. Each volume examines a number of key topics in the theory or applications of Levy processes and pays tribute to the state of the art of this rapidly evolving subject with special emphasis on the non-Brownian world. The expository articles in this second volume cover two important topics in the area of Levy processes. The first article by Serge Cohen reviews the most important findings on fractional Levy fields to date in a self-contained piece, offering a theoretical introduction as well as possible applications and simulation techniques. The second article, by Alexey Kuznetsov, Andreas E. Kyprianou, and Victor Rivero, presents an up to date account of the theory and application of scale functions for spectrally negative Levy processes, including an extensive numerical overview.
(source: Nielsen Book Data)

• 1 Introduction.- 2 Gaussian Processes.- 3 Some Geometry and Some Topology.- 4 The Gaussian Kinematic Formula.- 5 On Applications: Topological Inference.- 6 Algebraic Topology of Excursion Sets: A New Challenge.
• (source: Nielsen Book Data)

These notes, based on lectures delivered in Saint Flour, provide an easy introduction to the authors' 2007 Springer monograph "Random Fields and Geometry." While not as exhaustive as the full monograph, they are also less exhausting, while still covering the basic material, typically at a more intuitive and less technical level. They also cover some more recent material relating to random algebraic topology and statistical applications. The notes include an introduction to the general theory of Gaussian random fields, treating classical topics such as continuity and boundedness. This is followed by a quick review of geometry, both integral and Riemannian, with an emphasis on tube formulae, to provide the reader with the material needed to understand and use the Gaussian kinematic formula, the main result of the notes. This is followed by chapters on topological inference and random algebraic topology, both of which provide applications of the main results.
(source: Nielsen Book Data)

• Fundamentals of Spatial Statistics.- Fundamentals of Experimental Design.- Exploratory Designs.- Designs for Spatial Trend Estimation.- Design and Dependence.- Multipurpose Designs.
• (source: Nielsen Book Data)

The book is concerned with the statistical theory for locating spatial sensors. It bridges the gap between spatial statistics and optimum design theory. After introductions to those two fields the topics of exploratory designs and designs for spatial trend and variogram estimation are treated. Special attention is devoted to describing new methodologies to cope with the problem of correlated observations.
(source: Nielsen Book Data)

• Formulation of Basic Results Numerical Solution of the Basic Integral Equation in Distributions Proofs Singular Perturbation Theory for a Class of Fredholm Integral Equations Arising in Random Fields Estimation Theory Estimation and Scattering Theory Applications Auxiliary Results Appendices: Analytical Solution of the Basic Integral Equation for a Class of One-Dimensional Problems Integral Operators Basic in Random Fields Estimation Theory.
• (source: Nielsen Book Data)

This book contains a novel theory of random fields estimation of Wiener type, developed originally by the author and presented here. No assumption about the Gaussian or Markovian nature of the fields are made. The theory, constructed entirely within the framework of covariance theory, is based on a detailed analytical study of a new class of multidimensional integral equations basic in estimation theory.This book is suitable for graduate courses in random fields estimation. It can also be used in courses in functional analysis, numerical analysis, integral equations, and scattering theory.
(source: Nielsen Book Data)