Independent component models for replicated point processes

Abstract

We propose a semiparametric independent-component model for the intensity functions of a point process. When independent replications of the process are available, we show that the estimators are consistent and asymptotically normal. We study the finite-sample behavior of the estimators by simulation, and as an example of application we analyze the spatial distribution of street robberies in the city of Chicago.

Point process models for COVID-19 cases and deaths

“…The optimal transport result predicts that the death arrival times should be much more pushed to the right relative to the case arrival times for Norway as compared to Chile. That this is indeed the case is seen in Figure 5, where for Norway the cases and death arrival times are both unimodal with peaks around 19 and 30 days, respectively, while for Chile the deaths are pushed to the right with a higher density in the time interval t ∈ [14,38] days as compared to the cases density.…”

“…, T m iid ∼ f , where f (t) = λ(t)/τ is a density function and τ = T 0 λ(s) ds is a scalar [10], also referred to as intensity factor. Point process techniques have recently also been successfully applied for bike rentals as events, aiming at the analysis of repeated observations of the bike rental point process [15], and the spatial distribution of street robberies [14].…”

Joint models for grid point and response processes in longitudinal and functional data

“…1.4), but since we are interested in smooth processes in this paper, for practical purposes it is sufficient to consider only finite p 2 s. A similar decomposition for Λ would be problematic due to the non-negativity constraint. A nonnegative decomposition was proposed by Gervini (2016). However, for simplicity we will use an alternative approach in this paper, and decompose instead the logarithm of Λ, which is unconstrained:…”

Exploring Patterns of Demand in Bike Sharing Systems Via Replicated Point Process Models

“…From a statistical methodology perspective, we can mention early work on replicated point processes by Diggle et al (1991Diggle et al ( , 2000, Baddeley et al (1993), Mateu (2001) and Landau et al (2004), but they proposed tests for various hypotheses using summary statistics of the process (see also Baddeley et al (2015), chapter 16, andDiggle (2013), chapter 5.4), rather than explicitly estimating the intensity functions of the processes, as we do here. More recent work that does address the intensity function estimation problem was done by Wu et al (2013), Bouzas and Ruiz-Fuentes (2015) and Gervini (2016), but only in the context of univariate, not multivariate, processes as in this paper. Spatiotemporal processes have been widely studied in the literature, but mostly in the singlereplication context (see for example Li and Guan (2014), Shirota and Gelfand (2017), Diggle (2013) and references therein), not in the many-replication context of this paper.…”

“…More recent work that does address the intensity function estimation problem was done by Wu et al . (), Bouzas and Ruiz‐Fuentes () and Gervini (), but only in the context of univariate, not multivariate, processes as in this paper. Spatiotemporal processes have been widely studied in the literature, but mostly in the single‐replication context (see for example Li and Guan (), Shirota and Gelfand (), Diggle () and references therein), not in the many‐replication context of this paper.…”