  Statistical depths have been well studied for multivariate and functional
data over the past few decades, but remain under-explored for point processes.
A first attempt on the notion of point process depth was conducted recently
where the depth was defined as a weighted product of two terms: (1) the
probability of the number of events in each process and (2) the depth of the
event times conditioned on the number of events by using a Mahalanobis depth.
We point out that multivariate depths such as the Mahalanobis depth cannot be
directly used because they often neglect the important ordered property in the
point process events. To deal with this problem, we propose a model-based
approach for point processes systematically. In particular, we develop a
Dirichlet-distribution-based framework on the conditional depth term, where the
new methods are referred to as Dirichlet depths. We examine the mathematical
properties of the new depths and conduct the asymptotic analysis. In addition,
we illustrate the new methods using various simulated and real experiment data.
It is found that the proposed framework provides a proper center-outward rank
and the new methods have superior decoding performance to previous methods in
two neural spike train datasets.
