Large deviations for template matching between point processes

• 成果类型: Article
• 署名作者: Chi, ZY
• 署名单位: University of Chicago
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/105051604000000576
• 发表日期: 2005
• 页码: 153-174
• 关键词: compression; replay

摘要

We study the asymptotics related to the following matching criteria for two independent realizations of point processes X similar to X and Y similar to Y. Given l > 0, X boolean AND [0, l) serves as a template. For each t > 0, the matching score between the template and Y boolean AND [t, t + l) is a weighted sum of the Euclidean distances from y - t to the template over all y is an element of Y boolean AND [t, t + l). The template matching criteria are used in neuroscience to detect neural activity with certain patterns. We first consider W-l (theta), the waiting time until the matching score is above a given threshold theta. We show that whether the score is scalar- or vector-valued, (l/l) log W-l (theta) converges almost surely to a constant whose explicit form is available, when X is a stationary ergodic process and Y is a homogeneous Poisson point process. Second, as l --> infinity, a strong approximation for -log[Pr{W-l(theta) = 0}] by its rate function is established, and in the case where X is sufficiently mixing, the rates, after being centered and normalized by rootl, satisfy a central limit theorem and almost sure invariance principle. The explicit form of the variance of the normal distribution is given for the case where X is a homogeneous Poisson process as well.