Disentangling mark/point interaction in marked-point processes

In many spatial situations, not only do the point locations of mark variables (e.g. tree heights) play a key role in the underlying process generating mechanism, but there can be interdependence between the marks and points themselves. Although Monte Carlo frequency-domain analyses can separate mark and point structure, theoretical advances for marks have so far related to the conditional mark spectrum based on a given point structure. A ‘discrepancy function’ is therefore developed which isolates the spatial structure of the marks alone, and involves a harmonic decomposition of the mark frequencies. The concept is introduced via various simulated examples based on mark cosine waves and thinned point processes, with particular attention given to the construction of sequential and simultaneous search procedures for developing parameter estimates. The procedure is then applied to Spanish daily ozone data with missing values, a spatial growth-interaction process, and a classic longleaf pine data set from the Wade Tract in Georgia, USA.