Statistical Depth for Point Process via the Isometric Log-Ratio Transformation

Abstract

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 basic parametric representations such as Gaussian and Dirichlet densities were adopted in the definitions. However, these simplified forms ignore the underlying distribution of the process events, which makes the methods difficult to interpret and to apply to complicated patterns. To deal with these problems, we in this paper propose a distribution-based approach to the conditional depth via the well-known Isometric Log-Ratio (ILR) transformation on the inter-event times. The new depth, called the ILR depth, is at first defined for homogeneous Poisson process by using the density function on the transformed space. The definition is then extended to any general point process via a time-rescaling transformation. We illustrate the ILR depth using simulations of Poisson and non-Poisson processes and demonstrate its superiority over previous methods. We also thoroughly examine its mathematical properties and asymptotics in large samples. Finally, we apply the ILR depth in a real dataset and the result clearly shows the effectiveness of the new method.