A gaussian kinematic formula

By Jonathan E. Taylor. 2006

76 citations recorded by the source corpus.

Our aim in this chapter is to show you how to prove the Gaussian kinematic formula, (1.3.3). That is, that for $ f:M\rightarrow\mathbb{R}^d, $with component random fields $f_1,\ldots,f_d$ which are smooth, zero mean, unit variance, and Gaussian, and where M and $D\subset\mathbb{R}^d$ are nice enough, $$\mathbb{E}\{{\mathcal{L}_i}(A(f,M,D))\}=\sum\limits_{j=0}^{{\rm dim}M-i }\left[\begin{array}{ll}i+j\cr j\end{array}\right](2\pi)^{-j/2}\mathcal{L}_{i+j}(M)\mathcal{M}_j^i(D).$$ (4.0.1)

Subfields: Applied Mathematics, Artificial Intelligence, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/430553156045617d855bd00ce04f142e2bc0360e

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Citations in the corpus

A quantitative central limit theorem for the Euler–Poincaré characteristic of random spherical eigenfunctions (2016, background): “The investigation of its behaviour for the excursion sets of Gaussian random fields has now a rather long history: seminal contributions were given by Robert Adler and his coauthors in the 1970s; the area was then very much revived by the discovery of the beautiful Gaussian kinematic formula [1, 30]”
Simultaneous Statistical Inference (2014, methodology): “7 of the textbook by Timm (2002). The case of potentially different correlation matrices ε1, . . . , εξ has been studied by Jensen (1970). From a practical perspective, it is remarkable that multivariate chi-square probabilities can exactly be computed, even for high dimensions, if the underlying co”
Random fields of multivariate test statistics, with applications to shape analysis (2008, methodology): “For many years this was thought to be a coincidence, but a new result of Taylor [20] on”
Topological complexity of smooth random functions (2011, background): “While special cases appeared as early as the mid 1970s, the elegant geometric form in which it can be found in these notes first appeared in Taylor’s 2001 McGill PhD thesis [79] and the paper [80].”
Random Fields and Geometry (2007, unknown): “[148]).”
Some new random field tools for spatial analysis (2008, background): “One is Jonathan Taylor, whose fundamental contributions to the theory have already been mentioned, and the second is Keith Worsley, who has made more contributions to the application of the Euler characteristic approach to random fields than anyone else I know.”