The Geometry Of Random Fields

An important treatment of the geometric properties of sets generated by random fields, including a comprehensive treatment of the mathematical basics of random fields in general. It is a standard reference for all researchers with an interest in random fields, whether they be theoreticians or come from applied areas.

This monograph is devoted to a completely new approach to geometric problems arising in the study of random fields. The groundbreaking material in Part III, for which the background is carefully prepared in Parts I and II, is of both theoretical and practical importance, and striking in the way in which problems arising in geometry and probability are beautifully intertwined. "Random Fields and Geometry" will be useful for probabilists and statisticians, and for theoretical and applied mathematicians who wish to learn about new relationships between geometry and probability. It will be helpful for graduate students in a classroom setting, or for self-study. Finally, this text will serve as a basic reference for all those interested in the companion volume of the applications of the theory.

This self-contained monograph contains a number of topics that focus on the many developments in the area of mathematical probability/statistics known as random fields. Robert Adler has been one of the major contributors to this subject. In an earlier work (1981) he dealt with a wide variety of geometrical problems that are unique to stochastic processes in dimensions higher than one, fashioning an interesting blend of geometry and probability. Some old classical topics (non-geometric in nature) are presented in the first few chapters, but the material has been recast in the post-1980 setting of entropy. This is followed by a treatment of random geometry which takes into account the recent work by Worsley et al, and a significant update on our understanding of the fractal nature of erratic random fields. The geometric theme of the book is carried throughout in a user- friendly style. As in his previous work, Adler engages scientists outside of pure and applied mathematics, in particular, researchers involved in brain imaging, and astrophysicists. Applications of excursion set theory in astrophysics and brain imaging are detailed in a short chapter.It is hope that the approach employed in these two quite different areas will direct the reader to finding applications elsewhere. In recent years, the geometry of random fields has also found its way into the literature of materials (surface) science, oceanography, and geology. For applications in the medical community, a chapter discussing the tube formula approach involves the geometry on manifolds and an examination of the distribution of the maximum of Gaussian random

This volume is an attempt to provide a graduate level introduction to various aspects of stochastic geometry, spatial statistics and random fields, with special emphasis placed on fundamental classes of models and algorithms as well as on their applications, e.g. in materials science, biology and genetics. This book has a strong focus on simulations and includes extensive codes in Matlab and R which are widely used in the mathematical community. It can be seen as a continuation of the recent volume 2068 of Lecture Notes in Mathematics, where other issues of stochastic geometry, spatial statistics and random fields were considered with a focus on asymptotic methods.

This volume provides a modern introduction to stochastic geometry, random fields and spatial statistics at a (post)graduate level. It is focused on asymptotic methods in geometric probability including weak and strong limit theorems for random spatial structures (point processes, sets, graphs, fields) with applications to statistics. Written as a contributed volume of lecture notes, it will be useful not only for students but also for lecturers and researchers interested in geometric probability and related subjects.

Spatiotemporal Random Fields: Theory and Applications, Second Edition, provides readers with a new and updated edition of the text that explores the application of spatiotemporal random field models to problems in ocean, earth, and atmospheric sciences, spatiotemporal statistics, and geostatistics, among others. The new edition features considerable detail of spatiotemporal random field theory, including ordinary and generalized models, as well as space-time homostationary, isostationary and hetrogeneous approaches. Presenting new theoretical and applied results, with particular emphasis on space-time determination and interpretation, spatiotemporal analysis and modeling, random field geometry, random functionals, probability law, and covariance construction techniques, this book highlights the key role of space-time metrics, the physical interpretation of stochastic differential equations, higher-order space-time variability functions, the validity of major theoretical assumptions in real-world practice (covariance positive-definiteness, metric-adequacy etc.), and the emergence of interdisciplinary phenomena in conditions of multi-sourced real-world uncertainty. Contains applications in the form of examples and case studies, providing readers with first-hand experiences Presents an easy to follow narrative which progresses from simple concepts to more challenging ideas Includes significant updates from the previous edition, including a focus on new theoretical and applied results

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.

This volume offers a unique and accessible overview of the most active fields in Stochastic Geometry, up to the frontiers of recent research. Since 2014, the yearly meeting of the French research structure GDR GeoSto has been preceded by two introductory courses. This book contains five of these introductory lectures. The first chapter is a historically motivated introduction to Stochastic Geometry which relates four classical problems (the Buffon needle problem, the Bertrand paradox, the Sylvester four-point problem and the bicycle wheel problem) to current topics. The remaining chapters give an application motivated introduction to contemporary Stochastic Geometry, each one devoted to a particular branch of the subject: understanding spatial point patterns through intensity and conditional intensities; stochastic methods for image analysis; random fields and scale invariance; and the theory of Gibbs point processes. Exposing readers to a rich theory, this book will encourage further exploration of the subject and its wide applications.