Local characteristics of functional marked point processes with applications to seismic data

We present a family of local inhomogeneous mark-weighted summary statistics for general marked point processes. These capture various types of local dependence structures depending on the specified involved weight function. We use them to propose a local random labeling test. This procedure enables us to identify points and thus regions where the random labeling assumption does not hold, for example, when the (functional) marks are spatially dependent. We further present an application to a seismic point pattern with functional marks provided by seismic waveforms. Indeed, despite the relatively long history of point process theory, few approaches to analyzing spatial point patterns where th…

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

To analyse interactions in marked spatio-temporal point processes (MSTPPs), we introduce marked second-order reduced moment measures and K-functions for inhomogeneous second-order intensity reweigh ...

Local inhomogeneous weighted summary statistics for marked point processes

We introduce a family of local inhomogeneous mark-weighted summary statistics, of order two and higher, for general marked point processes. Depending on how the involved weight function is specified, these summary statistics capture different kinds of local dependence structures. We first derive some basic properties and show how these new statistical tools can be used to construct most existing summary statistics for (marked) point processes. We then propose a local test of random labelling. This procedure allows us to identify points, and consequently regions, where the random labelling assumption does not hold, e.g.~when the (functional) marks are spatially dependent. Through a simulatio…

The second-order analysis of marked spatio-temporal point processes, with an application to earthquake data

To analyse interaction in marked spatio-temporal point processes (MSTPPs), we introduce marked (cross) second-order reduced moment measures and K-functions for general inhomogeneous second-order intensity reweighted stationary MSTPPs. These summary statistics, which allow us to quantify dependence between different mark categories of the points, are depending on the specific mark space and mark reference measure chosen. We also look closer at how the summary statistics reduce under assumptions such as the MSTPP being multivariate and/or stationary. A new test for independent marking is devised and unbiased minus-sampling estimators are derived for all statistics considered. In addition, we …

Local test of random labelling for functional marked point processes

We introduce the local t-weighted marked nth-order inhomogeneous K-function, in a Functional Marked Point Processes framework. We employ the proposed summary statistics to run a local test of random labelling, useful to identify points, and consequently regions, where this assumption does not hold, i.e. the functional marks are spatially dependent.

Seismic sequences identification in Italy by local test of random labelling

Local Inhomogeneous Weighted Summary Statistics for Marked Point Processes

We introduce a family of local inhomogeneous mark-weighted summary statistics, of order two and higher, for general marked point processes. Depending on how the involved weight function is specified, these summary statistics capture different kinds of local dependence structures. We first derive some basic properties and show how these new statistical tools can be used to construct most existing summary statistics for (marked) point processes. We then propose a local test of random labeling. This procedure allows us to identify points, and consequently regions, where the random labeling assumption does not hold, for example, when the (functional) marks are spatially dependent. Through a sim…