Multiresolution analysis of spatial patterns to detect dominant directions

The assumption of direction invariance, i.e., isotropy, is often made in the practical analysis of spatial point processes due to simpler interpretation and ease of analysis. However, this assumption is many times hard to find in real applications. Many homogeneous point processes are indeed anisotropic. This paper concerns the analysis and detection of spatial anisotropies in terms of detection of linearities in spatial point processes, and even more generally, in terms of testing for spatial anisotropy.
We propose a wavelet-based approach to test for isotropy in spatial point processes based on the logarithm of the directional scalogram. Under the null hypothesis of isotropy, a random isotropic process should be expected to have the same value of the directional scalogram for any possible direction. Hence, Monte Carlo simulations of the logarithm of the directional scalograms over all directions are used to approximate the test distribution and the critical values. We demonstrate the efficacy of the approach through simulation studies and an application to a desert plant data set, where our approach confirms suspected directional effects in the spatial distribution of the desert plant species.