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Inhomogeneous spatial point processes with hidden second-order stationarity

By Ute Hahn and Eva B. Vedel Jensen
CSGB Research Reports
No. 07, September 2013

Modelling of inhomogeneous spatial point patterns is a challenging research area with numerous applications in diverse areas of science. In recent years, the focus has mainly been on the class of reweighted second-order stationary point processes that is characterized by the mathematically attractive property of a translation invariant pair correlation function. Motivated by examples where this model class is not adequate, we extend the class of reweighted second-order stationary processes. The extended class consists of hidden second-order stationary point processes for which the pair correlation function $g(u,v)$ is a function of $ u\,\breve{\smash{-}}\, v$, where $\breve{\smash{-}}$ is a generalized subtraction operator. For the reweighted second-order stationary processes, the subtraction operator is simply $u \,\breve{\smash{-}}\, v= u-v$. The processes in the extended class are called hidden second-order stationary because, in many cases, they may be derived from second-order stationary template processes. We review and discuss different types of hidden second-order stationarity. Alternatives to reweighted second-order stationarity are retransformed and locally rescaled second-order stationarity. Permutation tests for the different types of hidden second-order stationarity are developed. A test for local anisotropy is also derived. We illustrate our approach by a detailed analysis of three point patterns.

Keywords: Inhomogeneity; Intensity reweighted stationarity; Local scaling; Second-order stationarity; Spatial point processes; Transformation

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