Statistical depth for point process via the isometric log-ratio transformation

Statistical depth, a useful tool to measure the center-outward rank of multivariate and functional data, is still under-explored in temporal point processes. Recent studies on point process depth proposed a weighted product of two terms - one indicates the depth of the cardinality of the process, and the other characterizes the conditional depth of the temporal events given the cardinality. The second term is of great challenge because of the apparent nonlinear structure of event times, and so far only parametric representations such as Gaussian and Dirichlet densities have been adopted in the definitions. However, these parametric forms ignore the underlying distribution of the process events and are difficult to apply to complicated patterns. To deal with these problems, a novel distribution-based approach is proposed to the conditional depth via the well-known Isometric Log-Ratio (ILR) transformation on the inter-event times. Motivated by a uniform distribution on simplex, the new method, called the ILR depth, is formally defined for general point process via a Time Rescaling. The mathematical properties of the ILR depth are thoroughly examined and the new method is well illustrated by using Poisson and non-Poisson processes to demonstrate its superiority over previous methods. Finally, the ILR depth is applied in a real dataset and the result clearly shows its effectiveness in ranking.