**Gaussian** Markov Random Fields: Throey and Applications (Chapman & Hall/CRC Monographs on Statistics & Applied Probability)

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Published by: CRC Press | Publication date: 04/16/2007

Kindle book details: Kindle Edition, 280 pages**Gaussian** Processes on Trees: From Spin Glasses to Branching Brownian Motion (Cambridge Studies in Advanced Mathematics)

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Branching Brownian motion (BBM) is a classical object in probability theory with deep connections to partial differential equations. This book highlights the connection to classical extreme value theory and to the theory of mean-field spin glasses in statistical mechanics. Starting with a concise review of classical extreme value statistics and a basic introduction to mean-field spin glasses, the author then focuses on branching Brownian motion. Here, the classical results of Bramson on the asymptotics of solutions of the F-KPP equation are reviewed in detail and applied to the recent construction of the extremal process of BBM. The extension of these results to branching Brownian motion with variable speed are then explained. As a self-contained exposition that is accessible to graduate students with some background in probability theory, this book makes a good introduction for anyone interested in accessing this exciting field of mathematics.

Published by: Cambridge University Press | Publication date: 10/20/2016

Kindle book details: Kindle Edition, 211 pages### Unsupervised Machine Learning in Python: Master Data Science and Machine Learning with Cluster Analysis, **Gaussian** Mixture Models, and Principal Components Analysis

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In a real-world environment, you can imagine that a robot or an artificial intelligence won’t always have access to the optimal answer, or maybe there isn’t an optimal correct answer. You’d want that robot to be able to explore the world on its own, and learn things just by looking for patterns.Think about the large amounts of data being collected today, by the likes of the NSA, Google, and other organizations. No human could possibly sift through all that data manually. It was reported recently in the Washington Post and Wall Street Journal that the National Security Agency collects so much surveillance data, it is no longer effective.Could automated pattern discovery solve this problem?Do you ever wonder how we get the data that we use in our supervised machine learning algorithms?Kaggle always seems to provide us with a nice CSV, complete with Xs and corresponding Ys.If you haven’t been involved in acquiring data yourself, you might not have thought about this, but someone has to make this data!A lot of the time this involves manual labor. Sometimes, you don’t have access to the correct information or it is infeasible or costly to acquire.You still want to have some idea of the structure of the data.This is where unsupervised machine learning comes into play.In this book we are first going to talk about clustering. This is where instead of training on labels, we try to create our own labels. We’ll do this by grouping together data that looks alike.The 2 methods of clustering we’ll talk about: k-means clustering and hierarchical clustering.Next, because in machine learning we like to talk about probability distributions, we’ll go into

**Gaussian**mixture models and kernel density estimation, where we talk about how to learn the probability distribution of a set of data.One interesting fact is that under certain conditions,**Gaussian**mixture models and k-means clustering are exactly the same! We’ll prove how this is the case.Lastly, we’ll look at the theory behind principal components analysis or PCA. PCA has many useful applications: visualization, dimensionality reduction, denoising, and de-correlation. You will see how it allows us to take a different perspective on latent variables, which first appear when we talk about k-means clustering and GMMs.All the algorithms we’ll talk about in this course are staples in machine learning and data science, so if you want to know how to automatically find patterns in your data with data mining and pattern extraction, without needing someone to put in manual work to label that data, then this book is for you.All of the materials required to follow along in this book are free: You just need to able to download and install Python, Numpy, Scipy, Matplotlib, and Sci-kit Learn. Publication date: 05/22/2016

Kindle book details: Kindle Edition, 38 pages**Gaussian** Process Regression Analysis for Functional Data

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**Gaussian**Process Regression Analysis for Functional Data presents nonparametric statistical methods for functional regression analysis, specifically the methods based on a

**Gaussian**process prior in a functional space. The authors focus on problems involving functional response variables and mixed covariates of functional and scalar variables.Covering the basics of

**Gaussian**process regression, the first several chapters discuss functional data analysis, theoretical aspects based on the asymptotic properties of

**Gaussian**process regression models, and new methodological developments for high dimensional data and variable selection. The remainder of the text explores advanced topics of functional regression analysis, including novel nonparametric statistical methods for curve prediction, curve clustering, functional ANOVA, and functional regression analysis of batch data, repeated curves, and non-

**Gaussian**data.Many flexible models based on

**Gaussian**processes provide efficient ways of model learning, interpreting model structure, and carrying out inference, particularly when dealing with large dimensional functional data. This book shows how to use these

**Gaussian**process regression models in the analysis of functional data. Some MATLAB® and C codes are available on the first author’s website.

Published by: Chapman and Hall/CRC | Publication date: 07/01/2011

Kindle book details: Kindle Edition, 216 pages### Lectures on **Gaussian** Processes (SpringerBriefs in Mathematics)

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**Gaussian**processes can be viewed as a far-reaching infinite-dimensional extension of classical normal random variables. Their theory presents a powerful range of tools for probabilistic modelling in various academic and technical domains such as Statistics, Forecasting, Finance, Information Transmission, Machine Learning - to mention just a few. The objective of these Briefs is to present a quick and condensed treatment of the core theory that a reader must understand in order to make his own independent contributions. The primary intended readership are PhD/Masters students and researchers working in pure or applied mathematics. The first chapters introduce essentials of the classical theory of

**Gaussian**processes and measures with the core notions of reproducing kernel, integral representation, isoperimetric property, large deviation principle. The brevity being a priority for teaching and learning purposes, certain technical details and proofs are omitted. The later chapters touch important recent issues not sufficiently reflected in the literature, such as small deviations, expansions, and quantization of processes. In university teaching, one can build a one-semester advanced course upon these Briefs.

Published by: Springer | Publication date: 01/11/2012

Kindle book details: Kindle Edition, 134 pages### Financial Modeling Under Non-**Gaussian** Distributions (Springer Finance)

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This book examines non-

**Gaussian**distributions. It addresses the causes and consequences of non-normality and time dependency in both asset returns and option prices. The book is written for non-mathematicians who want to model financial market prices so the emphasis throughout is on practice. There are abundant empirical illustrations of the models and techniques described, many of which could be equally applied to other financial time series.Published by: Springer | Publication date: 04/05/2007

Kindle book details: Kindle Edition, 541 pages### Analysis on **Gaussian** Spaces

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Analysis of functions on the finite dimensional Euclidean space with respect to the Lebesgue measure is fundamental in mathematics. The extension to infinite dimension is a great challenge due to the lack of Lebesgue measure on infinite dimensional space. Instead the most popular measure used in infinite dimensional space is the

**Gaussian**measure, which has been unified under the terminology of "abstract Wiener space".Out of the large amount of work on this topic, this book presents some fundamental results plus recent progress. We shall present some results on the**Gaussian**space itself such as the Brunn–Minkowski inequality, Small ball estimates, large tail estimates. The majority part of this book is devoted to the analysis of nonlinear functions on the**Gaussian**space. Derivative, Sobolev spaces are introduced, while the famous Poincaré inequality, logarithmic inequality, hypercontractive inequality, Meyer's inequality, Littlewood–Paley–Stein–Meyer theory are given in details.This book includes some basic material that cannot be found elsewhere that the author believes should be an integral part of the subject. For example, the book includes some interesting and important inequalities, the Littlewood–Paley–Stein–Meyer theory, and the Hörmander theorem. The book also includes some recent progress achieved by the author and collaborators on density convergence, numerical solutions, local times.Published by: WSPC | Publication date: 08/30/2016

Kindle book details: Kindle Edition, 484 pages### Molecular Physical Chemistry: A Computer-based Approach using Mathematica® and **Gaussian**

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This is the physical chemistry textbook for students with an affinity for computers! It offers basic and advanced knowledge for students in the second year of chemistry masters studies and beyond. In seven chapters, the book presents thermodynamics, chemical kinetics, quantum mechanics and molecular structure (including an introduction to quantum chemical calculations), molecular symmetry and crystals. The application of physical-chemical knowledge and problem solving is demonstrated in a chapter on water, treating both the water molecule as well as water in condensed phases.Instead of a traditional textbook top-down approach, this book presents the subjects on the basis of examples, exploring and running computer programs (Mathematica®), discussing the results of molecular orbital calculations (performed using

**Gaussian**) on small molecules and turning to suitable reference works to obtain thermodynamic data. Selected Mathematica® codes are explained at the end of each chapter and cross-referenced with the text, enabling students to plot functions, solve equations, fit data, normalize probability functions, manipulate matrices and test physical models. In addition, the book presents clear and step-by-step explanations and provides detailed and complete answers to all exercises. In this way, it creates an active learning environment that can prepare students for pursuing their own research projects further down the road.Students who are not yet familiar with Mathematica® or**Gaussian**will find a valuable introduction to computer-based problem solving in the molecular sciences. Other computer applications can alternatively be used. For every chapter learning goals are clearly listed in the beginning, so that readers can easily spot the highlights, and a glossary in the end of the chapter offers a quick look-up of important terms.Published by: Springer | Publication date: 01/16/2017

Kindle book details: Kindle Edition, 457 pages### Markov Processes, **Gaussian** Processes, and Local Times (Cambridge Studies in Advanced Mathematics)

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This book was first published in 2006. Written by two of the foremost researchers in the field, this book studies the local times of Markov processes by employing isomorphism theorems that relate them to certain associated

**Gaussian**processes. It builds to this material through self-contained but harmonized 'mini-courses' on the relevant ingredients, which assume only knowledge of measure-theoretic probability. The streamlined selection of topics creates an easy entrance for students and experts in related fields. The book starts by developing the fundamentals of Markov process theory and then of**Gaussian**process theory, including sample path properties. It then proceeds to more advanced results, bringing the reader to the heart of contemporary research. It presents the remarkable isomorphism theorems of Dynkin and Eisenbaum and then shows how they can be applied to obtain new properties of Markov processes by using well-established techniques in**Gaussian**process theory. This original, readable book will appeal to both researchers and advanced graduate students.Published by: Cambridge University Press | Publication date: 07/24/2006

Kindle book details: Kindle Edition, 630 pages### Non-**Gaussian** Statistical Communication Theory (IEEE Series on Digital & Mobile Communication)

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The book is based on the observation that communication is the central operation of discovery in all the sciences. In its "active mode" we use it to "interrogate" the physical world, sending appropriate "signals" and receiving nature's "reply". In the "passive mode" we receive nature's signals directly. Since we never know a prioriwhat particular return signal will be forthcoming, we must necessarily adopt a probabilistic model of communication. This has developed over the approximately seventy years since it's beginning, into a Statistical Communication Theory (or SCT). Here it is the set or ensemble of possible results which is meaningful. From this ensemble we attempt to construct in the appropriate model format, based on our understanding of the observed physical data and on the associated statistical mechanism, analytically represented by suitable probability measures. Since its inception in the late '30's of the last century, and in particular subsequent to World War II, SCT has grown into a major field of study. As we have noted above, SCT is applicable to all branches of science. The latter itself is inherently and ultimately probabilistic at all levels. Moreover, in the natural world there is always a random background "noise" as well as an inherent a priori uncertainty in the presentation of deterministic observations, i.e. those which are specifically obtained, a posteriori. The purpose of the book is to introduce Non-

**Gaussian**statistical communication theory and demonstrate how the theory improves probabilistic model. The book was originally planed to include 24 chapters as seen in the table of preface. Dr. Middleton completed first 10 chapters prior to his passing in 2008. Bibliography which represents remaining chapters are put together by the author's close colleagues; Drs. Vincent Poor, Leon Cohen and John Anderson. email pressbooks@ieee.org to request Ch.10Published by: Wiley-IEEE Press | Publication date: 05/11/2012

Kindle book details: Kindle Edition, 664 pages