Framework for adaptive multiscale analysis of nonhomogeneous point processes

We develop the methodology for hypothesis testing and model selection in nonhomogeneous Poisson processes, with an eye toward the application of modeling and variability detection in heart beat data. Modeling the process´ non-constant rate function using templates of simple basis functions, we develop the generalized likelihood ratio statistic for a given template and a multiple testing scheme to model-select from a family of templates. A dynamic programming algorithm inspired by network flows is used to compute the maximum likelihood template in a multiscale manner. In a numerical example, the proposed procedure is nearly as powerful as the super-optimal procedures that know the true template size and true partition, respectively. Extensions to general history-dependent point processes is discussed.