Multiscale Change-Point Analysis of Inhomogeneous Poisson Processes Using Unbalanced Wavelet Decompositions

Abstract

We present a continuous wavelet analysis of count data with time-varying intensities. The objective is to extract intervals with significant intensities from background intervals. This includes the precise starting point of the significant interval, its exact duration and the (average) level of intensity. We allow multiple change points in the intensity curve, without specifying the number of change points in advance. We extend the classical (discretised) continuous Haar wavelet analysis towards an unbalanced (i.e., asymmetric) version. This additional degree of freedom allows more powerful detection. Locations of intensity change points are identified as persistent local maxima in the wavelet analysis at the successive scales. We illustrate the approach with simulations on low intensity data. Although the method is presented here in the context of Poisson (count) data, most ideas (apart from the specific Poisson normalization) apply for the detection of multiple change points in other circumstances (such as additive Gaussian noise) as well.