DOCSLIB.ORG

Even If Google Sent You Here, What Follows Is NOT the Book RANDOM

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text

Load more

i Note: Even if Google sent you here, what follows is NOT the book RANDOM FIELDS AND GEOMETRY published with Springer in 2007, but rather a companion volume, still under production, that gives a simpler version of the theory of the first book as well as lots of applications. You can find the original Random Fields and Geometry on the Springer site. Meanwhile, enjoy what is available of the second volume, and keep in mind that we are happy to receive all comments, suggestions and, most of all, cor- rections. ii Applications of RANDOM FIELDS AND GEOMETRY Foundations and Case Studies Robert J. Adler Faculty of Industrial Engineering and Management and Faculty of Electrical Engineering Technion { Israel Institute of Technology Haifa, Israel e-mail: [email protected] Jonathan E. Taylor Department of Statistics Stanford University Stanford, California. e-mail: [email protected] Keith J. Worsley Department of Mathematics and Statistics McGill University Montr´eal,Canada and Department of Statistics University of Chicago Chicago, Illinois. e-mail: [email protected] May 20, 2015 0 Preface Robert J. Adler Jonathan E. Taylor1 Keith Worsley Haifa, Israel Stanford, California Montreal, Canada ie.technion.ac.il/Adler.phtml www-stat.stanford.edu/∼jtaylo www.math.mcgill.ca/keith 1Research supported in part by NSF grant DMS-0405970 Contents 0 Preface .................................................... iii 1 Introduction ............................................... 1 1.1 An Initial Example......................................1 1.2 Gaussian and Related Random Fields......................3 1.3 Shape and the Euler Characteristic........................4 1.4 The Large-Scale Structure of the Universe..................7 1.4.1 Survey Data......................................7 1.4.2 Designing and Testing a Model......................8 1.5 Brain Imaging.......................................... 12 1.5.1 The Data........................................ 12 1.5.2 Statistical Testing................................. 13 1.6 Beyond the Euler Characteristic........................... 17 Part I The Underlying Theory 2 Random Fields ............................................ 23 2.1 Stochastic Processes and Random Fields................... 23 2.2 Gaussian and Gaussian Related Random Fields............. 25 2.2.1 Gaussian Variables................................ 25 2.2.2 Gaussian Fields................................... 27 2.2.3 Gaussian Related Fields............................ 28 2.3 Examples of Random Fields.............................. 30 2.3.1 Stationary and Isotropic Fields...................... 30 2.3.2 Random Polynomials.............................. 31 2.3.3 The Cosine Process................................ 32 2.3.4 The Cosine Field.................................. 34 2.3.5 Orthogonal Expansions............................ 36 2.4 Stationarity and Isotropy................................. 39 2.4.1 Basic Definitions.................................. 39 vi Contents 2.4.2 Spectral Distribution Theorem...................... 41 2.4.3 Spectral Moments and Derivatives of Random Fields... 42 2.4.4 Constant Variance................................. 44 2.4.5 White Noise and Integration........................ 45 2.4.6 Spectral Representation Theorem.................... 47 2.4.7 Isotropy.......................................... 52 2.4.8 Isotropic Fields on the Sphere and Other Homogenous Spaces........................................... 57 2.4.9 Transforming to Stationarity and Isotropy............ 58 2.5 Smoothness of Random Fields............................. 60 2.5.1 The General Gaussian Theory...................... 62 2.5.2 Gaussian Fields on RN ............................. 65 2.5.3 Non-Gaussian Processes............................ 69 2.6 Gaussian Exceedence Probabilities......................... 70 2.6.1 Borell-TIS Inequality.............................. 71 2.6.2 Comparison Inequalities............................ 74 2.6.3 Exceedence Probabilities for Smooth Processes........ 77 2.7 An Expectation Meta-Theorem: The Rice-Kac Formula...... 78 2.8 Exercises............................................... 82 3 Geometry ................................................. 89 3.1 Basic Complexes and the Euler Characteristic............... 89 3.2 Excursion Sets and Morse Theory......................... 93 3.3 Volume of Tubes and Intrinsic Volumes.................... 99 3.3.1 Euclidean Sets.................................... 99 3.3.2 Intrinsic Volumes for Simplicial Complexes........... 104 3.3.3 Volumes of Tubes on Spheres....................... 105 3.3.4 Probabilities of Tubes.............................. 107 3.4 Some Integral Geometric Formulae........................ 109 3.4.1 The Euclidean KFF............................... 109 3.4.2 The KFF on Spheres.............................. 110 3.4.3 Crofton's Formula................................. 111 3.5 Stratified Riemannian Manifolds........................... 112 3.6 Exercises............................................... 115 Part II Quantifiable Properties 4 The Expected Euler Characteristic . 121 4.1 Regularity Conditions.................................... 122 4.2 Stationary Gaussian Fields............................... 122 4.3 Non-Stationary Gaussian Fields........................... 127 4.4 The Gaussian Kinematic Formula, I....................... 131 4.5 Euler Characteristic Densities............................. 134 4.5.1 Gaussian Fields................................... 135 Contents vii 2 4.5.2 χk Fields......................................... 135 2 4.5.3χ ¯K Fields........................................ 138 4.5.4 F Fields......................................... 141 4.5.5 Student T Fields.................................. 146 4.5.6 Hotelling T 2 Fields................................ 147 4.5.7 Roy's Maximum Root Field......................... 149 4.5.8 Conjunctions/Correlated........................... 149 4.5.9 Correlation Fields................................. 149 4.5.10 Another Look at the F Field....................... 149 4.6 Some Lipschitz-Killing Curvatures......................... 149 4.6.1 Time-Space Fields................................. 150 4.6.2 Scale Space Fields................................. 150 4.6.3 Rotation Space Fields.............................. 163 4.7 Mean Lipschitz-Killing Curvatures under Isotropy........... 163 4.8 The Gaussian Kinematic Formula, II....................... 165 4.9 Exercises............................................... 167 5 Exceedence Probabilities ..................................169 5.1 The Mean Number of Maxima............................ 171 5.2 The Mean Number of Maxima and the Mean Euler Characteristic........................................... 174 5.3 Expected Euler Characteristic Heuristic.................... 178 5.3.1 About the proof................................... 180 2 2 5.3.2 χ andχ ¯K Fields.................................. 181 5.3.3 Numerical Examples............................... 182 5.4 Alternative Approaches to Exceedence Probabilities.......... 182 5.4.1 Volume of Tubes and Gaussian Exceedence Probabilities182 5.4.2 The Double Sum Method........................... 185 5.4.3 Double sums of moments........................... 187 5.5 Aproximately Gaussian Random Fields..................... 189 5.6 Exercises............................................... 189 6 The Structure of Excursion Sets . 193 6.1 Palm Distributions and Conditioning...................... 194 6.2 The Local Structure of Gaussian Maxima................... 196 6.2.1 The Exact Slepian Model........................... 197 6.2.2 Excursions Above High Levels...................... 202 6.3 More on the Local Structure of Gaussian Fields............. 203 6.3.1 Low Maxima and High Minima..................... 204 6.3.2 High Excursions................................... 205 6.3.3 Overshoot Distributions............................ 206 6.3.4 Numerical Approaches............................. 208 6.4 The Local Structure of Some Non-Gaussian Fields........... 209 2 6.4.1 χk fields.......................................... 209 6.4.2 Student T fields................................... 214 viii Contents 6.4.3 F fields.......................................... 215 6.4.4 On Computing the Mean Euler Characteristic......... 216 6.5 The Size of an Excursion Set Component................... 217 6.5.1 The Gaussian Case................................ 217 6.5.2 The Non-Gaussian Case............................ 220 6.6 The Poisson Clumping Heuristic and Excursions............. 221 6.6.1 Exceedence Probabilities, Again..................... 222 6.6.2 The Largest Component of an Excursion Set.......... 223 6.6.3 Numerical Experiments............................ 224 6.7 Exercises............................................... 227 Part III Numerics 7 Simulating Random Fields .................................231 7.1 Fields on Euclidean Spaces............................... 231 7.2 Fields on Spheres........................................ 231 7.3 Fields on Manifolds...................................... 231 8 Discrete approximation ....................................233 8.1 Estimating Intrinsic Volumes............................. 233 8.2 Data on a Lattice/Triangulated Surface.................... 233 8.3 Computing the Euler Characteristic....................... 233 8.4 Continuity Correction for Supremum Distribution........... 233 8.5 Improved Bonferroni Inequality........................... 233 Part IV Applications 9 Neuroimaging .............................................237
Recommended publications
  • An Infinite Dimensional Central Limit Theorem for Correlated Martingales

    An Infinite Dimensional Central Limit Theorem for Correlated Martingales

    Ann. I. H. Poincaré – PR 40 (2004) 167–196 www.elsevier.com/locate/anihpb An infinite dimensional central limit theorem for correlated martingales Ilie Grigorescu 1 Department of Mathematics, University of Miami, 1365 Memorial Drive, Ungar Building, Room 525, Coral Gables, FL 33146, USA Received 6 March 2003; accepted 27 April 2003 Abstract The paper derives a functional central limit theorem for the empirical distributions of a system of strongly correlated continuous martingales at the level of the full trajectory space. We provide a general class of functionals for which the weak convergence to a centered Gaussian random field takes place. An explicit formula for the covariance is established and a characterization of the limit is given in terms of an inductive system of SPDEs. We also show a density theorem for a Sobolev- type class of functionals on the space of continuous functions. 2003 Elsevier SAS. All rights reserved. Résumé L’article présent dérive d’un théorème limite centrale fonctionnelle au niveau de l’espace de toutes les trajectoires continues pour les distributions empiriques d’un système de martingales fortement corrélées. Nous fournissons une classe générale de fonctions pour lesquelles est établie la convergence faible vers un champ aléatoire gaussien centré. Une formule explicite pour la covariance est determinée et on offre une charactérisation de la limite à l’aide d’un système inductif d’équations aux dérivées partielles stochastiques. On démontre également que l’espace de fonctions pour lesquelles le champ des fluctuations converge est dense dans une classe de fonctionnelles de type Sobolev sur l’espace des trajectoires continues.
  • Stochastic Differential Equations with Variational Wishart Diffusions

    Stochastic Differential Equations with Variational Wishart Diffusions

    Stochastic Differential Equations with Variational Wishart Diffusions Martin Jørgensen 1 Marc Peter Deisenroth 2 Hugh Salimbeni 3 Abstract Why model the process noise? Assume that in the example above, the two states represent meteorological measure- We present a Bayesian non-parametric way of in- ments: rainfall and wind speed. Both are influenced by ferring stochastic differential equations for both confounders, such as atmospheric pressure, which are not regression tasks and continuous-time dynamical measured directly. This effect can in the case of the model modelling. The work has high emphasis on the in (1) only be modelled through the diffusion . Moreover, stochastic part of the differential equation, also wind and rain may not correlate identically for all states of known as the diffusion, and modelling it by means the confounders. of Wishart processes. Further, we present a semi- parametric approach that allows the framework Dynamical modelling with focus in the noise-term is not to scale to high dimensions. This successfully a new area of research. The most prominent one is the lead us onto how to model both latent and auto- Auto-Regressive Conditional Heteroskedasticity (ARCH) regressive temporal systems with conditional het- model (Engle, 1982), which is central to scientific fields eroskedastic noise. We provide experimental ev- like econometrics, climate science and meteorology. The idence that modelling diffusion often improves approach in these models is to estimate large process noise performance and that this randomness in the dif- when the system is exposed to a shock, i.e. an unforeseen ferential equation can be essential to avoid over- significant change in states.
  • Integral Geometry and Algebraic Structures for Tensor Valuations

    Integral Geometry and Algebraic Structures for Tensor Valuations

    Integral Geometry and Algebraic Structures for Tensor Valuations Andreas Bernig and Daniel Hug Abstract In this survey, we consider various integral geometric formulas for tensor- valued valuations that have been obtained by different methods. Furthermore we explain in an informal way recently introduced algebraic structures on the space of translation invariant, smooth tensor valuations, including convolution, product, Poincare´ duality and Alesker-Fourier transform, and their relation to kinematic formulas for tensor valuations. In particular, we describe how the algebraic viewpoint leads to new intersectional kinematic formulas and substantially simplified Crofton formulas for translation invariant tensor valuations. We also highlight the connection to general integral geometric formulas for area measures. 1 Introduction An important part of integral geometry is devoted to the investigation of integrals (mean values) of the form Z j(K \ gL) m(dg); G n where K; L ⊂ R are sets from a suitable intersection stable class of sets, G is a group n acting on R and thus on its subsets, m is a Haar measure on G, and j is a functional with values in some vector space W. Common choices for W are the reals or the space of signed Radon measures. Instead of the intersection, Minkowski addition is another natural choice for a set operation which has been studied. The principle aim then is to express such integrals by means of basic geometric functionals of K Andreas Bernig Institut fur¨ Mathematik, Goethe-Universitat¨ Frankfurt, Robert-Mayer-Str. 10, 60054 Frankfurt, Germany, e-mail: [email protected], Daniel Hug Karlsruhe Institute of Technology (KIT), Department of Mathematics, D-76128 Karlsruhe, Germany, e-mail: [email protected] 59 60 Andreas Bernig and Daniel Hug and L.
  • Mean Field Methods for Classification with Gaussian Processes

    Mean Field Methods for Classification with Gaussian Processes

    Mean field methods for classification with Gaussian processes Manfred Opper Neural Computing Research Group Division of Electronic Engineering and Computer Science Aston University Birmingham B4 7ET, UK. opperm~aston.ac.uk Ole Winther Theoretical Physics II, Lund University, S6lvegatan 14 A S-223 62 Lund, Sweden CONNECT, The Niels Bohr Institute, University of Copenhagen Blegdamsvej 17, 2100 Copenhagen 0, Denmark winther~thep.lu.se Abstract We discuss the application of TAP mean field methods known from the Statistical Mechanics of disordered systems to Bayesian classifi­ cation models with Gaussian processes. In contrast to previous ap­ proaches, no knowledge about the distribution of inputs is needed. Simulation results for the Sonar data set are given. 1 Modeling with Gaussian Processes Bayesian models which are based on Gaussian prior distributions on function spaces are promising non-parametric statistical tools. They have been recently introduced into the Neural Computation community (Neal 1996, Williams & Rasmussen 1996, Mackay 1997). To give their basic definition, we assume that the likelihood of the output or target variable T for a given input s E RN can be written in the form p(Tlh(s)) where h : RN --+ R is a priori assumed to be a Gaussian random field. If we assume fields with zero prior mean, the statistics of h is entirely defined by the second order correlations C(s, S') == E[h(s)h(S')], where E denotes expectations 310 M Opper and 0. Winther with respect to the prior. Interesting examples are C(s, s') (1) C(s, s') (2) The choice (1) can be motivated as a limit of a two-layered neural network with infinitely many hidden units with factorizable input-hidden weight priors (Williams 1997).
  • Integral Geometry and Radon Transforms, by Sigurdur Helgason, Springer, New York, 2010, Xiv+301 Pp., ISBN 978-1-4419-6054-2, Hardcover

    Integral Geometry and Radon Transforms, by Sigurdur Helgason, Springer, New York, 2010, Xiv+301 Pp., ISBN 978-1-4419-6054-2, Hardcover

    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY S 0273-0979(2012)01391-5 Article electronically published on December 6, 2012 Integral geometry and Radon transforms, by Sigurdur Helgason, Springer, New York, 2010, xiv+301 pp., ISBN 978-1-4419-6054-2, hardcover 1. Integral geometry The book under review is an excellent introduction to the group theoretical and analytic aspects of the field by one of its pioneers. Before reviewing the book, we will provide an overview of the field. Integral geometry draws together analysis, geometry, and numerical mathemat- ics. It has direct applications in PDEs, group representations, and the applied mathematical field of tomography. The fundamental problem in integral geometry is to determine properties of a function f in the plane or three-dimensional space or other manifolds from knowing the integrals of f over lines, planes, hyperplanes, spheres, or other submanifolds. The history of integral geometry starts in the early 1900s with Radon, Funk, Lorenz, and others. In 1917, Johann Radon [60] proved an inversion formula for whathasbecomeknownastheclassical Radon line transform: if is a line in the plane and f is an integrable function, then ż (1.1) RLfpq“ fpxqds, xP where ds is the arc length measure on the line . This is the mathematical model behind X-ray tomography, which we discuss in Section 5. One can extend this n to the hyperplane transform, RH , which integrates over all hyperplanes H in R . 3 RadonalsoprovedaninversionformulaforRH in R . (According to Bockwinkel [5], the inversion for RH had already been obtained by H. A. Lorenz before 1906.) In 1936, Cramer and Wold [10] proved injectivity of the hyperplane transform for probability measures.
  • Particle Filter Based Monitoring and Prediction of Spatiotemporal Corrosion Using Successive Measurements of Structural Responses

    Particle Filter Based Monitoring and Prediction of Spatiotemporal Corrosion Using Successive Measurements of Structural Responses

    sensors Article Particle Filter Based Monitoring and Prediction of Spatiotemporal Corrosion Using Successive Measurements of Structural Responses Sang-ri Yi and Junho Song * Department of Civil and Environmental Engineering, Seoul National University, Seoul 08826, Korea; [email protected] * Correspondence: [email protected]; Tel.: +82-2-880-7369 Received: 11 October 2018; Accepted: 10 November 2018; Published: 13 November 2018 Abstract: Prediction of structural deterioration is a challenging task due to various uncertainties and temporal changes in the environmental conditions, measurement noises as well as errors of mathematical models used for predicting the deterioration progress. Monitoring of deterioration progress is also challenging even with successive measurements, especially when only indirect measurements such as structural responses are available. Recent developments of Bayesian filters and Bayesian inversion methods make it possible to address these challenges through probabilistic assimilation of successive measurement data and deterioration progress models. To this end, this paper proposes a new framework to monitor and predict the spatiotemporal progress of structural deterioration using successive, indirect and noisy measurements. The framework adopts particle filter for the purpose of real-time monitoring and prediction of corrosion states and probabilistic inference of uncertain and/or time-varying parameters in the corrosion progress model. In order to infer deterioration states from sparse indirect inspection data, for example structural responses at sensor locations, a Bayesian inversion method is integrated with the particle filter. The dimension of a continuous domain is reduced by the use of basis functions of truncated Karhunen-Loève expansion. The proposed framework is demonstrated and successfully tested by numerical experiments of reinforcement bar and steel plates subject to corrosion.
  • Stochastic Differential Equations with Variational Wishart Diffusions

    Stochastic Differential Equations with Variational Wishart Diffusions

    Stochastic Differential Equations with Variational Wishart Diffusions Martin Jørgensen 1 Marc Peter Deisenroth 2 Hugh Salimbeni 3 Abstract Why model the process noise? Assume that in the example above, the two states represent meteorological measure- We present a Bayesian non-parametric way of in- ments: rainfall and wind speed. Both are influenced by ferring stochastic differential equations for both confounders, such as atmospheric pressure, which are not regression tasks and continuous-time dynamical measured directly. This effect can in the case of the model modelling. The work has high emphasis on the in (1) only be modelled through the diffusion . Moreover, stochastic part of the differential equation, also wind and rain may not correlate identically for all states of known as the diffusion, and modelling it by means the confounders. of Wishart processes. Further, we present a semi- parametric approach that allows the framework Dynamical modelling with focus in the noise-term is not to scale to high dimensions. This successfully a new area of research. The most prominent one is the lead us onto how to model both latent and auto- Auto-Regressive Conditional Heteroskedasticity (ARCH) regressive temporal systems with conditional het- model (Engle, 1982), which is central to scientific fields eroskedastic noise. We provide experimental ev- like econometrics, climate science and meteorology. The idence that modelling diffusion often improves approach in these models is to estimate large process noise performance and that this randomness in the dif- when the system is exposed to a shock, i.e. an unforeseen ferential equation can be essential to avoid over- significant change in states.
  • Markov Random Fields and Stochastic Image Models

    Markov Random Fields and Stochastic Image Models

    Markov Random Fields and Stochastic Image Models Charles A. Bouman School of Electrical and Computer Engineering Purdue University Phone: (317) 494-0340 Fax: (317) 494-3358 email [email protected] Available from: http://dynamo.ecn.purdue.edu/»bouman/ Tutorial Presented at: 1995 IEEE International Conference on Image Processing 23-26 October 1995 Washington, D.C. Special thanks to: Ken Sauer Suhail Saquib Department of Electrical School of Electrical and Computer Engineering Engineering University of Notre Dame Purdue University 1 Overview of Topics 1. Introduction (b) Non-Gaussian MRF's 2. The Bayesian Approach i. Quadratic functions ii. Non-Convex functions 3. Discrete Models iii. Continuous MAP estimation (a) Markov Chains iv. Convex functions (b) Markov Random Fields (MRF) (c) Parameter Estimation (c) Simulation i. Estimation of σ (d) Parameter estimation ii. Estimation of T and p parameters 4. Application of MRF's to Segmentation 6. Application to Tomography (a) The Model (a) Tomographic system and data models (b) Bayesian Estimation (b) MAP Optimization (c) MAP Optimization (c) Parameter estimation (d) Parameter Estimation 7. Multiscale Stochastic Models (e) Other Approaches (a) Continuous models 5. Continuous Models (b) Discrete models (a) Gaussian Random Process Models 8. High Level Image Models i. Autoregressive (AR) models ii. Simultaneous AR (SAR) models iii. Gaussian MRF's iv. Generalization to 2-D 2 References in Statistical Image Modeling 1. Overview references [100, 89, 50, 54, 162, 4, 44] 4. Simulation and Stochastic Optimization Methods [118, 80, 129, 100, 68, 141, 61, 76, 62, 63] 2. Type of Random Field Model 5. Computational Methods used with MRF Models (a) Discrete Models i.
  • Gaussian Processes on Trees: from Spin Glasses to Branching Brownian Motion

    Gaussian Processes on Trees: from Spin Glasses to Branching Brownian Motion

    Anton Bovier Gaussian processes on trees: From spin glasses to branching Brownian motion Lecture Notes, Bonn, WS 2014/15 March 11, 2015 in construction in construction Preface These are notes were written for a graduate course at Bonn University I gave in 2014/15. It extends a shorter version of lecture notes for a series of lectures given at the 7th Prague Summer School on Mathematical Statistical Mechanics in August 2013. BBM was studied over the last 50 years as a subject of interest in its own right, with seminal contributions by McKean, Bramson, Lalley, Sellke, Chauvin, Rouault, and others. Recently, the field has experienced a revival with many re- markable contributions and repercussions into other areas. The construction of the extremal process in our work with Arguin and Kistler [6] as well as, in parallel, in that of Aïdékon, Beresticky, Brunet, and Shi [2], is one highlight. Many results on BBM were extended to branching random walk, see e.g. Aïdékon [1]. Other devel- opments concern the connection to extremes of the free Gaussian random field in d = 2 by Bramson and Zeitouni [25], Ding [32], Bramson, Ding, and Zeitouni [24], and Biskup and Louidor [13, 14]. My personal motivation to study this object came, however, from spin glass theory. About ten years ago, Irina Kurkova and I wrote two papers [19, 20] on Derrida’s Generalised Random Energy models (GREM). It turns out that these models are closely related to BBM, and with BBM corresponding to a critical case, that we could not at the time fully analyse.
  • Integral Geometry and Radon Transforms

    Integral Geometry and Radon Transforms

    TO ARTIE Preface This book deals with a special subject in the wide field of Geometric Anal- ysis. The subject has its origins in results by Funk [1913] and Radon [1917] determining, respectively, a symmetric function on the two-sphere S2 from its great circle integrals and an integrable function on R2 from its straight line integrals. (See References.) The first of these is related to a geometric theorem of Minkowski [1911] (see Ch. III, 1). While the above work of Funk and Radon§ lay dormant for a while, Fritz John revived the subject in important papers during the thirties and found significant applications to differential equations. More recent applications to X-ray technology and tomography have widened interest in the subject. This book originated with lectures given at MIT in the Fall of 1966, based mostly on my papers during 1959–1965 on the Radon transform and its generalizations. The viewpoint of these generalizations is the following. The set of points on S2 and the set of great circles on S2 are both acted on transitively by the group O(3). Similarly, the set of points in R2 and the set P2 of lines in R2 are both homogeneous spaces of the group M(2) of rigid motions of R2. This motivates our general Radon transform definition from [1965a] and [1966a], which forms the framework of Chapter II: Given two homogeneous spaces X = G/K and Ξ = G/H of the same group G, two elements x = gK and ξ = γH are said to be incident (denoted x#ξ) if gK γH = (as subsets of G).
  • Spatial Process Generation Arxiv:1308.0399V1 [Stat.CO] 2 Aug

    Spatial Process Generation Arxiv:1308.0399V1 [Stat.CO] 2 Aug

    Spatial Process Generation Dirk P. Kroese∗ Zdravko I. Botevy 1 Introduction Spatial processes are mathematical models for spatial data; that is, spatially ar- ranged measurements and patterns. The collection and analysis of such data is of interest to many scientific and engineering disciplines, including the earth sciences, materials design, urban planning, and astronomy. Examples of spatial data are geo-statistical measurements, such as groundwater contaminant concen- trations, temperature reports from different cities, maps of the locations of mete- orite impacts or geological faults, and satellite images or demographic maps. The availability of fast computers and advances in Monte Carlo simulation methods have greatly enhanced the understanding of spatial processes. The aim of this chapter is to give an overview of the main types of spatial processes, and show how to generate them using a computer. From a mathematical point of view, a spatial process is a collection of random variables fXt; t 2 T g where the index set T is some subset of the d-dimensional d Euclidean space R . Thus, Xt is a random quantity associated with a spacial position t rather than time. The set of possible values of Xt is called the state space of the spatial process. Thus, spatial processes can be classified into four types, based on whether the index set and state space are continuous or discrete. An example of a spatial process with a discrete index set and a discrete state space is the Ising model in statistical mechanics, where sites arranged on a grid are assigned either a positive or negative \spin"; see, for example, [33].
  • A Minicourse on Stochastic Partial Differential Equations

    A Minicourse on Stochastic Partial Differential Equations

    Robert C. Dalang Davar Khoshnevisan Carl Mueller David Nualart Yimin Xiao A Minicourse on Stochastic Partial Differential Equations Salt Lake City, Utah, 2006 Editors: D. Khoshnevisan and F. Rassoul-Agha Springer Berlin Heidelberg NewYork Hong Kong London Milan Paris Tokyo Contents A Primer on Stochastic Partial Differential Equations Davar Khoshnevisan:::::::::::::::::::::::::::::::::::::::::::::: 5 1 What is an SPDE? . 5 2 Gaussian random vectors . 6 3 Gaussian processes . 6 4 Regularity of random processes. 12 5 Martingale measures . 18 6 A nonlinear heat equation . 27 7 From chaos to order . 36 References . 40 The Stochastic Wave Equation Robert C. Dalang :::::::::::::::::::::::::::::::::::::::::::::::: 43 1 Introduction . 43 2 The stochastic wave equation . 45 3 Spatially homogeneous Gaussian noise . 51 4 The wave equation in spatial dimension 2. 53 5 A function-valued stochastic integral . 60 6 The wave equation in spatial dimension d ≥ 1.................. 62 7 Spatial regularity of the stochastic integral (d = 3). 65 8 H¨older-continuity in the 3-d wave equation . 74 References . 75 Application of Malliavin Calculus to Stochastic Partial Differential Equations David Nualart ::::::::::::::::::::::::::::::::::::::::::::::::::: 77 1 Introduction . 77 2 Malliavin calculus . 78 3 Application of Malliavin calculus to regularity of probability laws . 88 4 Stochastic heat equation . 96 5 Spatially homogeneous SPDEs . 104 VI Contents References . 113 Some Tools and Results for Parabolic Stochastic Partial Differential Equations Carl Mueller :::::::::::::::::::::::::::::::::::::::::::::::::::: 115 1 Introduction . 115 2 Basic framework . 117 3 Duality . 119 4 Large deviations for SPDEs . 129 5 A comparison theorem . 134 6 Applications . 136 References . 147 Sample Path Properties of Anisotropic Gaussian Random Fields Yimin Xiao ::::::::::::::::::::::::::::::::::::::::::::::::::::: 151 1 Introduction . 151 2 Examples and general assumptions .