A model for time series of spatial point patterns
Date:
A spatial point pattern $\mathbf{x}$ on $S\subset\mathbb{R}^{d}$, $d\geq2$, is a locally finite subset of $S$; i.e. for any bounded Borel set $B\subset S$, $\mathbf{x}\cap B$ if finite. Let $\mathcal{X}$ be the set of all point patterns on $S$ and $\mathcal{N}$ denotes the Borel $\sigma$-algebra of subsets of $\mathcal{X}$. Then a random object $X:(\Omega,\mathcal{F},\mathbb{P})\to(\mathcal{X},\mathcal{N},P_{X})$ is called a spatial point process on $S$. Assume that for each $t\in\mathbb{Z}$, $X_{t}$ is a spatial point process on $S$. Then ${X_t: t\in\mathbb{Z}}$ is a $\mathcal{X}$-valued time series that can not be explained in the classical time series framework. Such time series are encountered in various applications. In the present work, a framework for analyzing time series of spatial point patterns is developed and a simple model for the population dynamic in the BCI plot is introduced. In addition, parameter estimation for the proposed model is discussed.