Large deviations for template matching between point processes

Details Large deviations for template matching between point processes Large deviations for template matching between point processes

2005-03-22

arxiv.org/abs/math/0503463v1

Annals of Applied Probability 2005, Vol. 15, No. 1A, 153-174

Mathematics

Probability

Published at http://dx.doi.org/10.1214/105051604000000576 in the Annals of Applied Probability (http://www.imstat.org/aap/) by

Scientific paper

10.1214/105051604000000576

We study the asymptotics related to the following matching criteria for two independent realizations of point processes X\sim X and Y\sim Y. Given l>0, X\cap [0,l) serves as a template. For each t>0, the matching score between the template and Y\cap [t,t+l) is a weighted sum of the Euclidean distances from y-t to the template over all y\in Y\cap [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, (1/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\to\infty, 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 \sqrtl, 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.

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