The Geometry of Random FieldsOriginally published in 1981, The Geometry of Random Fields remains an important text for its coverage and exposition of the theory of both smooth and non-smooth random fields; closed form expressions for various geometric characteristics of the excursion sets of smooth, stationary, Gaussian random fields over N-dimensional rectangles; descriptions of the local behavior of random fields in the neighborhoods of high maxima; and a treatment of the Markov property for Gaussian fields. The Geometry of Random Fields is essential reading for researchers in probability and statistics, with no prior knowledge of geometry required. Since the book was originally published it has become a standard reference in areas of physical oceanography, cosmology, and neuroimaging. It is written at a level accessible to nonspecialists, including advanced undergraduates and early graduate students. |
Contents
CL62_ch1 | 1 |
CL62_ch2 | 22 |
CL62_ch3 | 39 |
CL62_ch4 | 66 |
CL62_ch5 | 93 |
CL62_ch6 | 122 |
CL62_ch7 | 168 |
CL62_ch8 | 184 |
CL62_appendixa | 253 |
| 263 | |
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Common terms and phrases
A₁ B₁ basic bounded Brownian motion Brownian sheet C₁ Chapter condition of order consider continuous convergence coordinate covariance function cube defined denote density differentiable distribution function ergodic erraticism example excursion characteristics excursion sets exists finite Firstly following lemma following result following theorem Furthermore Gaussian field Gaussian process Gaussian random field given Hausdorff dimension Hölder condition IG characteristic implies inequality integral geometry interval isotropic Lebesgue measure Let X(t level crossings matrix maxima mean number mean square mean value non-negative notation number of points obtain one-dimensional parameter partial derivatives probability problem properties random field X(t random variable real-valued sample functions sample paths satisfying second-order sequence splitting field stochastic stochastic process subset suitable regularity t₁ theory variance vector write X₁ X₁(t x² field zero zero-mean