Abstract
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.
Highlights
Many types of functional data, such as financial time series, animal movements, growth functions for trees in a forest stand, the spatial extensions of outbreaks of a disease over time with respect to the outbreak centres, population growth functions of towns/cities in a country, and different functions describing spatial dependence (e.g. LISA functions; see Section 11 in Supplementary Materials and the references therein), are represented as collections { f1(t), . . . , fn(t)}, t ∈ T ⊂ [0, ∞), n ≥ 1, of functions/paths in some k-dimensional Euclidean space Rk, k ≥ 1; note that the argument t need not represent time, it could, for example, represent spatial distance
The common approach to deal with such data within the field of functional data analysis (FDA) (Ramsay and Silverman 2005) is to assume that the functions fi, i = 1, . . . , n, belong to some suitable family of functions and are realisations/sample paths of some collection of independent and identically distributed random functions/stochastic processes {F1(t), . . . , Fn(t)}, t ∈ T, with sample paths belonging to the family of functions in question
We argue that the natural setting to do this is through functional marked point processes (FMPPs)
Summary
Many types of functional data, such as financial time series, animal movements, growth functions for trees in a forest stand, the spatial extensions of outbreaks of a disease over time with respect to the outbreak centres, population growth functions of towns/cities in a country, and different functions describing spatial dependence (e.g. LISA functions; see Section 11 in Supplementary Materials and the references therein), are represented as collections { f1(t), . . . , fn(t)}, t ∈ T ⊂ [0, ∞), n ≥ 1, of functions/paths in some k-dimensional Euclidean space Rk, k ≥ 1; note that the argument t need not represent time, it could, for example, represent spatial distance. N, belong to some suitable family of functions (usually an L2-space) and are realisations/sample paths of some collection of independent and identically distributed (iid) random functions/stochastic processes {F1(t), . Fn(t)}, t ∈ T , with sample paths belonging to the family of functions in question. Functional data sets (believed to be) generated in accordance with the above remarks will be referred to as functional marked point patterns, and Fig. 1 provides illustrative examples of such data sets. In the top right panel, for each of the 47 functions/provinces, the horizontal red dashed line corresponds to y = 1, which illustrates the case where we have the same size of genders in the province in question. The lower left panel shows the movement tracks of two Mongolian wolves, starting from random initial monitoring locations (red squares); the data are taken from the Movebank website. The lower right panel shows the movement tracks of 15
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