A novel family of non-parametric cumulative based divergences for point processes

Hypothesis testing on point processes has several applications such as model fitting, plasticity detection, and non-stationarity detection. Standard tools for hypothesis testing include tests on mean firing rate and time varying rate function. However, these statistics do not fully describe a point process, and therefore, the conclusions drawn by these tests can be misleading. In this paper, we introduce a family of non-parametric divergence measures for hypothesis testing. A divergence measure compares the full probability structure and, therefore, leads to a more robust test of hypothesis. We extend the traditional Kolmogorov-Smirnov and Cramer-von-Mises tests to the space of spike trains via stratification, and show that these statistics can be consistently estimated from data without any free parameter. We demonstrate an application of the proposed divergences as a cost function to find optimally matched point processes.