Second‐order analysis of marked inhomogeneous spatiotemporal point processes: Applications to earthquake data

Abstract

To analyze interactions in marked spatiotemporal point processes (MSTPPs), we introduce marked second-order reduced moment measures and K-functions for inhomogeneous second-order intensity-reweighted stationary MSTPPs. These summary statistics, which allow us to quantify dependence between different mark-based classifications of points, depend on the specific mark space and mark reference measure chosen. Unbiased and consistent minus-sampling estimators are derived for all statistics considered, and a test for random labeling is indicated. In addition, we treat Voronoi intensity estimators for MSTPPs. These new statistics are finally employed to analyze an Andaman Sea earthquake data set. KEYWORDSadaptive intensity estimation, earthquakes, marked inhomogeneous spatiotemporal point process, marked (second-order) intensity-reweighted stationarity, marked spatiotemporal K-function, marked spatiotemporal second-order reduced moment measure IFTIMI ET AL. so-called second-order nonparametric analyses of marked spatiotemporal point patterns. In particular, we want to explicitly study the behavior of earthquakes when we treat the magnitudes as marks. With such a setup, we may then study in detail how shocks of different magnitudes interact spatiotemporally. Once such an analysis has been carried out, on the basis of the outcome, one may proceed to fitting appropriate models to the data.Classically, when analyzing earthquakes within the framework of (marked) spatiotemporal point processes (STPPs), the analysis has been based on conditional intensity (CI) functions , a CI function gives the expected number of further events in a coming infinitesimal period, given the history of events up to that point. The beauty and appeal of CI functions is that, when existing, they specify the whole distribution of the MSTPP. As pointed out by Diggle (2014), however, not all MSTPP models have available/tractable CI functions. Furthermore, much of the CI function-based analysis is carried out within the framework of a given class of models. We want to define general fully nonparametric analysis tools; hence, we will proceed by following a random set/random measure formalism (nonconditional) intensity function reflects the univariate infinitesimal probability of finding a point of the MSTPP at a given spatial location, at a given time, with a given mark. The most common candidate for the nonparametric estimation of the intensity function is kernel estimation (Cronie & Van Lieshout, 2018;Diggle, 2014). However, due to the abrupt changes in the activity of earthquakes, both spatially and temporally, we here make the choice of considering an adaptive intensity estimation approach (Silverman, 1986;Van Lieshout, 2012), namely, a Voronoi intensity estimation approach (Barr & Schoenberg, 2010).Turning to the higher-order moment properties, to study the inherent dependence structure of the data-generating mechanism, that is, the underlying MSTPP, we focus on second-order summary statistics, of which Ripley's (stationary) K-function is the most pr...

Graphical modelling and partial characteristics for multitype and multivariate-marked spatio-temporal point processes

“…To introduce spatio-temporal point processes, we follow González et al (2016) and references therein as well as Iftimi et al (2019). A spatio-temporal point process X is, rigorously speaking, a random countable measure defined in a subset…”

“…where ds 1 and ds 2 are two infinitesimal spatial areas separated by a distance r and dt 1 and dt 2 are two infinitesimal temporal lengths separated by a distance t. S(r, t) represents a spatio-temporal mark correlation function that has not yet been examined in the current literature and that deserves especial attention given its extremely usefulness for analysing complex point patterns. Finally, the marked spatio-temporal K-function was defined in Iftimi et al (2019) in its general version. We can take advantage of the pair correlation function in the case of SOIRS processes.…”

“…Different from the mechanistic modelling approach, Särkkä and Renshaw (2006), Renshaw and Comas (2008), Comas et al (2009), Renshaw et al (2009), Cronie and Särkkä (2011), Cronie et al (2012), Comas et al (2013), and Redenbach and Särkkä (2013) discussed so-called growthinteraction processes to model the evolution of a quantitatively marked spatial point pattern over equidistant steps in time such as the diameter at breast height (DBH) value for a set of trees recorded over consecutive times. While the above specifications consider the analysis of quantitatively marked spatiotemporal point patterns, the investigation of cross-characteristics through marked ver-sions of the spatio-temporal reduced second-order moment measure and Ripley's Kfunction has just recently been discussed by Iftimi et al (2019). Unlike the classical second-order summary characteristics such as the spatio-temporal K-function, the corresponding marked version allows to investigate the pairwise interrelation between subsets of points within one quantitatively marked spatio-temporal point pattern, e.g.…”

Functional marked point processes: a natural structure to unify spatio-temporal frameworks and to analyse dependent functional data

“…To be able to treat the summary statistics considered in this paper, we first have to introduce the notion of kth-order marked intensity reweighted stationarity (k-MIRS) (cf. Cronie and van Lieshout 2016;Iftimi et al 2019).…”

“…Ψ (•) ≡ 1). In particular, the case k = 2 is referred to as Ψ being second-order marked intensity reweighted stationary (SOMIRS) (Cronie and van Lieshout 2016;Iftimi et al 2019).…”