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