Cross-spectral analysis for spatial point-lattice processes
In this study, we explore the relationship between the components of a hybrid process consisting of a spatial point process arid a lattice process using two-dimensional spectral techniques. Simulated spatial point-lattice patterns are used to demonstrate how the different cross-spectral statistics can reveal correlation between the two components. A method to adjust for jumps that normally occur in the cross-spectral phase statistic is then proposed. Such adjustment is needed to enable us to calculate the slope of the phase spectrum which measures the shift between the two components. Several methods to calculate the slope are investigated. Asymptotic properties of the cross-spectral statistics are derived and their confidence intervals estimated. .A test that the components are independent is described. In a study region, lattice processes are observed at regular grids whereas point, processes can be observed anywhere. In order to account for discrepancies that. might arise due to this, methods to discretise the point pattern are suggested. Cross-spectral techniques are then applied to analyse the joint process of the discretised point pattern and the lattice pattern. Finally, we apply the techniques suggested above to study the joint properties of two data sets. The first consists of altitude data of a region in a rain forest in French Guyana together with the locations of a number of tree species in that region. The second set consists of altitude data of the Sahel region of Africa together with location of storms and some of their characteristics. In order to incorporate the storm characteristics in the analysis, cross-spectral tools used to analyse two components are extended to three components.