  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.
