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Inhomogeneous spatial point processes with hidden second-order stationarity

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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 −˘ v, where −˘ is a generalized subtraction operator. For the reweighted second-order stationary processes, the subtraction operator is simply u −˘ 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.
OriginalsprogEngelsk
UdgiverCentre for Stochastic Geometry and advanced Bioimaging, Aarhus University
Antal sider33
StatusUdgivet - 2013
SerietitelCSGB Research Reports
Nummer07
Vol/bind2013

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