Second order analysis of geometric anisotropic point processes revisited

Authors

Affiliations

Martina Sormani

University of Kaiserslautern

Claudia Redenbach

University of Kaiserslautern

Aila Särkkä

Chalmers University of Technology and the University of Gothenburg

Tuomas Rajala

Natural Resources Institute Finland

Abstract

Various methods for directional analysis of spatial point patterns have been introduced in the literature. In this paper, we formulate a unifying framework for methods based on integral transforms of the second order product density. Examples include directional versions of Ripley’s K-function, wavelet transforms, and spectral analysis. Furthermore, we propose an additional method based on the projection of the Fry points of the point pattern on the unit sphere. This method solves some of the practical problems appearing in the application of the integral transform. In particular, it can be easily applied in 3D and provides a definition of an isotropy test without the need of replicated data. The method is evaluated in a simulation study.

Fig. 3. Some template functions f . From left to right: indicator function of a cone, indicator function of an ellipse, and elliptic Gaussian kernel overlaid with the Fry points of the regular example pattern from Fig. 1.