Random Fields and Geometry
Since the term “random ?eld’’ has a variety of different connotations, ranging from agriculture to statistical mechanics, let us start by clarifying that, in this book, a random ?eld is a stochastic process, usually taking values in a Euclidean space, and de?ned over a parameter space of dimensionality at least 1. Consequently, random processes de?ned on countable parameter spaces will not 1 appear here. Indeed, even processes on R will make only rare appearances and, from the point of view of this book, are almost trivial. The parameter spaces we like best are manifolds, although for much of the time we shall require no more than that they be pseudometric spaces. With this clari?cation in hand, the next thing that you should know is that this book will have a sequel dealing primarily with applications. In fact, as we complete this book, we have already started, together with KW (Keith Worsley), on a companion volume [8] tentatively entitled RFG-A,or Random Fields and Geometry: Applications. The current volume—RFG—concentrates on the theory and mathematical background of random ?elds, while RFG-A is intended to do precisely what its title promises. Once the companion volume is published, you will ?nd there not only applications of the theory of this book, but of (smooth) random ?elds in general.
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Contents
Gaussian Fields
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7 |
Gaussian Inequalities
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49 |
Orthogonal Expansions
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65 |
Stationary Fields
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101 |
Integral Geometry
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127 |
Differential Geometry 149
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147 |
Piecewise Smooth Manifolds
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183 |
Critical Point Theory 193
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192 |
Random Fields on Euclidean Spaces
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263 |
Random Fields on Manifolds
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301 |
Mean Intrinsic Volumes
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331 |
Excursion Probabilities for Smooth Fields
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349 |
NonGaussian Geometry
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387 |
434 | |
443 | |
Volume of Tubes
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213 |
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Common terms and phrases
argument assume assumption basic bound boundary centered Gaussian Chapter compute constant convergence Corollary covariance function critical points Crofton's formula curvature measures defined definition denote density derivatives differential dimension embedded entropy Euclidean Euler characteristic example excursion probabilities excursion sets fact finite Furthermore Gaussian fields Gaussian process Gaussian random field geometry given gives implies independent inequality integral intrinsic volumes isotropic Lemma Lipschitz-Killing curvatures locally convex matrix Morse function Morse index N-dimensional notation Note orthonormal basis parameter space piecewise smooth proof of Theorem random fields random variables real-valued result Riemannian manifold Riemannian metric satisfying second fundamental form Section simple spectral stationary subset support cones tangent term theory tube formula variance vector fields Whitney stratified manifolds Whitney stratified space X₁ zero
References to this book
Statistical Parametric Mapping: The Analysis of Functional Brain Images Karl J. Friston No preview available - 2007 |