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Biometrika Advance Access originally published online on February 6, 2007
Biometrika 2007 94(1):1-18; doi:10.1093/biomet/asm004
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Copyright © 2007 Biometrika Trust

Articles

Maxima of discretely sampled random fields, with an application to ‘bubbles’

J. E. Taylor

Department of Statistics, Stanford University, Stanford, California 94305-4065, U.S.A

K. J. Worsley

Department of Mathematics & Statistics, McGill University, Montréal, Québec, Canada H3A 2K6

F. Gosselin

Département de Psychologie, Université de Montréal, Montréal, Québec, Canada H3 C 3 J7

jonathan.taylor{at}stanford.edu

keith.worsley{at}mcgill.ca

frederic.gosselin{at}umontreal.ca

Received for publication 1 March 2005. Revision received 1 June 2006.
  Abstract

A smooth Gaussian random field with zero mean and unit variance is sampled on a discrete lattice, and we are interested in the exceedance probability or P-value of the maximum in a finite region. If the random field is smooth relative to the mesh size, then the P-value can be well approximated by results for the continuously sampled smooth random field (Adler, 1981; Worsley, 1995a; Taylor & Adler, 2003; Adler & Taylor, 2007). If the random field is not smooth, so that adjacent lattice values are nearly independent, then the usual Bonferroni bound is very accurate. The purpose of this paper is to bridge the gap between the two, and derive a simple, accurate upper bound for intermediate mesh sizes. The result uses a new improved Bonferroni-type bound based on discrete local maxima. We give an application to the ‘bubbles’ technique for detecting areas of the face used to discriminate fear from happiness.

Key Words: Bonferroni • Bubbles • Euler characteristic • Random field

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