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

• 成果类型: Article
• 署名作者: Taylor, J. E.; Worsley, K. J.; Gosselin, F.
• 署名单位: Stanford University; McGill University; Universite de Montreal
• 刊物名称: BIOMETRIKA
• ISSN/ISSBN: 0006-3444
• DOI: 10.1093/biomet/asm004
• 发表日期: 2007
• 页码: 118
• 关键词: inclusion-exclusion identities; bonferroni inequalities; excursion sets; morse-theory; probability; RECOGNITION; bounds; UNION

摘要

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.