First-order separability of a spatio-temporal point process plays a fundamental role in the analysis of spatio-temporal point pattern data. While it is often a convenient assumption that simplifies the analysis greatly, existing non-separable structures should be accounted for in the model construction. Three different tests are proposed to investigate this hypothesis as a step of preliminary data analysis. The first two tests are exact or asymptotically exact for Poisson processes. The first test based on permutations and global envelopes allows one to detect at which spatial and temporal locations or lags the data deviate from the null hypothesis. The second test is a simple and computationally cheap X²-test. The third test is based on stochastic reconstruction method and can be generally applied for non-Poisson processes. The performance of the first two tests is studied in a simulation study for Poisson and non-Poisson models. The third test is applied to the real data of the UK 2001 epidemic foot and mouth disease.

This paper treats functional marked point processes (FMPPs), which are defined as marked point processes where the marks are random elements in some (Polish) function space. Such marks may represent, for example, spatial paths or functions of time. To be able to consider, for example, multivariate FMPPs, we also attach an additional, Euclidean, mark to each point. We indicate how the FMPP framework quite naturally connects the point process framework with both the functional data analysis framework and the geostatistical framework. We further show that various existing stochastic models fit well into the FMPP framework. To be able to carry out nonparametric statistical analyses for FMPPs, we study characteristics such as product densities and Palm distributions, which are the building blocks for many summary statistics. We proceed to defining a new family of summary statistics, so-called weighted marked reduced moment measures, together with their nonparametric estimators, in order to study features of the functional marks. We further show how other summary statistics may be obtained as special cases of these summary statistics. We finally apply these tools to analyse population structures, such as demographic evolution and sex ratio over time, in Spanish provinces. 

The Johnson–Mehl germination-growth model is a spatio-temporal point process model which among other things have been used for the description of neurotransmitters datasets. However, for such datasets parametric Johnson–Mehl models fitted by maximum likelihood have yet not been evaluated by means of functional summary statistics. This paper therefore invents four functional summary statistics adapted to the Johnson–Mehl model, with two of them based on the second-order properties and the other two on the nuclei-boundary distances for the associated Johnson–Mehl tessellation. The functional summary statistics theoretical properties are investigated, non-parametric estimators are suggested, and their usefulness for model checking is examined in a simulation study. The functional summary statistics are also used for checking fitted parametric Johnson–Mehl models for a neurotransmitters dataset.

In this paper we introduce an instance of the well-know Neyman–Scott cluster process model with clusters having a long tail behaviour. In our model the offspring points are distributed around the parent points according to a circular Cauchy distribution. Using a modified Cramér-von Misses test statistic and the simulated pointwise envelopes it is shown that this model fits better than the Thomas process to the frequently analyzed long-leaf pine data-set.

A common assumption in analyzing spatial and spatio-temporal point processes is stationarity, while in many real applications because of the environmental effects the stationarity condition is not often met. We propose two types of test statistics to test stationarity for spatio-temporal point processes, by adapting, Palahi, Pukkala & Mateu (2009) and by considering the square difference between observed and expected (under stationarity) intensities. We study the efficiency of the new statistics by simulated data, and we apply them to test the stationarity of real data.

Statistical methodology for spatio-temporal point processes is in its infancy. We consider second-order analysis based on pair correlation functions and K-functions for general inhomogeneous spatio-temporal point processes and for inhomogeneous spatio-temporal Cox processes. Assuming spatio-temporal separability of the intensity function, we clarify different meanings of second-order spatio-temporal separability. One is second-order spatio-temporal independence and relates to log-Gaussian Cox processes with an additive covariance structure of the underlying spatio-temporal Gaussian process. Another concerns shot-noise Cox processes with a separable spatio-temporal covariance density. We propose diagnostic procedures for checking hypotheses of second-order spatio-temporal separability, which we apply on simulated and real data.