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It is well known that the normal distribution is the most pleasant, one can even say, an exemplary object in the probability theory. It combines almost all conceivable nice properties that a distribution may ever have: symmetry, stability, indecomposability, a regular tail behavior, etc. Gaussian measures (the distributions of Gaussian random functions), as infinite-dimensional analogues of tht
Gaussian Random Processes by I.A. Ibragimov,Y.A. Rozanov Pdf
The book deals mainly with three problems involving Gaussian stationary processes. The first problem consists of clarifying the conditions for mutual absolute continuity (equivalence) of probability distributions of a "random process segment" and of finding effective formulas for densities of the equiva lent distributions. Our second problem is to describe the classes of spectral measures corresponding in some sense to regular stationary processes (in par ticular, satisfying the well-known "strong mixing condition") as well as to describe the subclasses associated with "mixing rate". The third problem involves estimation of an unknown mean value of a random process, this random process being stationary except for its mean, i. e. , it is the problem of "distinguishing a signal from stationary noise". Furthermore, we give here auxiliary information (on distributions in Hilbert spaces, properties of sam ple functions, theorems on functions of a complex variable, etc. ). Since 1958 many mathematicians have studied the problem of equivalence of various infinite-dimensional Gaussian distributions (detailed and sys tematic presentation of the basic results can be found, for instance, in [23]). In this book we have considered Gaussian stationary processes and arrived, we believe, at rather definite solutions. The second problem mentioned above is closely related with problems involving ergodic theory of Gaussian dynamic systems as well as prediction theory of stationary processes.
Probability Distributions Involving Gaussian Random Variables by Marvin K. Simon Pdf
This handbook, now available in paperback, brings together a comprehensive collection of mathematical material in one location. It also offers a variety of new results interpreted in a form that is particularly useful to engineers, scientists, and applied mathematicians. The handbook is not specific to fixed research areas, but rather it has a generic flavor that can be applied by anyone working with probabilistic and stochastic analysis and modeling. Classic results are presented in their final form without derivation or discussion, allowing for much material to be condensed into one volume.
Geostatistical Simulation by Christian Lantuejoul Pdf
This book deals with the estimation of natural resources using the Monte Carlo methodology. It includes a set of tools to describe the morphological, statistical and stereological properties of spatial random models. Furthermore, the author presents a wide range of spatial models, including random sets and functions, point processes and object populations applicable to the geosciences. The text is based on a series of courses given in the USA and Latin America to civil, mining and petroleum engineers as well as graduate students in statistics. It is the first book to discuss the geostatistical simulation techniques in such a specific way.
Author : Carl Edward Rasmussen,Christopher K. I. Williams Publisher : MIT Press Page : 266 pages File Size : 47,7 Mb Release : 2005-11-23 Category : Computers ISBN : 9780262182539
Gaussian Processes for Machine Learning by Carl Edward Rasmussen,Christopher K. I. Williams Pdf
A comprehensive and self-contained introduction to Gaussian processes, which provide a principled, practical, probabilistic approach to learning in kernel machines. Gaussian processes (GPs) provide a principled, practical, probabilistic approach to learning in kernel machines. GPs have received increased attention in the machine-learning community over the past decade, and this book provides a long-needed systematic and unified treatment of theoretical and practical aspects of GPs in machine learning. The treatment is comprehensive and self-contained, targeted at researchers and students in machine learning and applied statistics. The book deals with the supervised-learning problem for both regression and classification, and includes detailed algorithms. A wide variety of covariance (kernel) functions are presented and their properties discussed. Model selection is discussed both from a Bayesian and a classical perspective. Many connections to other well-known techniques from machine learning and statistics are discussed, including support-vector machines, neural networks, splines, regularization networks, relevance vector machines and others. Theoretical issues including learning curves and the PAC-Bayesian framework are treated, and several approximation methods for learning with large datasets are discussed. The book contains illustrative examples and exercises, and code and datasets are available on the Web. Appendixes provide mathematical background and a discussion of Gaussian Markov processes.
International Series of Monographs in Natural Philosophy, Volume 32: Random Functions and Turbulence focuses on the use of random functions as mathematical methods. The manuscript first offers information on the elements of the theory of random functions. Topics include determination of statistical moments by characteristic functions; functional transformations of random variables; multidimensional random variables with spherical symmetry; and random variables and distribution functions. The book then discusses random processes and random fields, including stationarity and ergodicity of random processes; influence of finiteness of the interval of averaging; scalar and vector random fields; and statistical moments. The text takes a look at the statistical theory of turbulence. Topics include turbulence with very large Reynolds numbers; emergence of turbulent motion; and energy spectrum in isothermal turbulent shear flow. The book also discusses small-scale and large-scale atmospheric turbulence and applications to numerical weather analysis and prediction. The manuscript is a vital source of data for readers interested in random theory.
Correlation Theory of Stationary and Related Random Functions by A.M. Yaglom Pdf
Correlation Theory of Stationary and Related Random Functions is an elementary introduction to the most important part of the theory dealing only with the first and second moments of these functions. This theory is a significant part of modern probability theory and offers both intrinsic mathematical interest and many concrete and practical applications. Stationary random functions arise in connection with stationary time series which are so important in many areas of engineering and other applications. This book presents the theory in such a way that it can be understood by readers without specialized mathematical backgrounds, requiring only the knowledge of elementary calculus. The first volume in this two-volume exposition contains the main theory; the supplementary notes and references of the second volume consist of detailed discussions of more specialized questions, some more additional material (which assumes a more thorough mathematical background than the rest of the book) and numerous references to the extensive literature.
Topological Complexity of Smooth Random Functions by Robert Adler,Jonathan E. Taylor Pdf
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.
This is the first systematic exposition of random sets theory since Matheron (1975), with full proofs, exhaustive bibliographies and literature notes Interdisciplinary connections and applications of random sets are emphasized throughout the book An extensive bibliography in the book is available on the Web at http://liinwww.ira.uka.de/bibliography/math/random.closed.sets.html, and is accompanied by a search engine
The Geometry of Random Fields by Robert J. Adler Pdf
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.
Random Functions and Hydrology by Rafael L. Bras,Ignacio Rodríguez-Iturbe Pdf
Advanced-level view of the tools of random processes and field theory as applied to the analysis and synthesis of hydrologic phenomena. Topics include time-series analysis, optimal estimation, optimal interpolation (Kriging), frequency-domain analysis of signals, and linear systems theory. Techniques and examples chosen to illustrate the latest advances in hydrologic signal analysis. Useable as graduate-level text in water resource systems, stochastic hydrology, random processes and signal analysis. 202 illustrations.
The Inverse Gaussian Distribution by Raj Chhikara Pdf
This monograph is a compilation of research on the inverse Gaussian distribution. It emphasizes the presentation of the statistical properties, methods, and applications of the two-parameter inverse Gaussian family of distribution. It is useful to statisticians and users of statistical distribution.
Spatial Point Patterns by Adrian Baddeley,Ege Rubak,Rolf Turner Pdf
Modern Statistical Methodology and Software for Analyzing Spatial Point PatternsSpatial Point Patterns: Methodology and Applications with R shows scientific researchers and applied statisticians from a wide range of fields how to analyze their spatial point pattern data. Making the techniques accessible to non-mathematicians, the authors draw on th