A multiscale hierarchical Markov transition matrix model for generating and analyzing thematic raster maps

A model is described for generating hierarchically scaled spatial pattern as represented in a thematic raster map. The model involves a series of Markov transition matrices, one for each level in the scaling hierarchy. In full generality, the model allows the transition matrices to be different at each level, potentially making available a large number of parameters for landscape characterization. The model is self-similar when the transition matrices are all equal. A method is presented for fitting the model to data that take the form of a single-resolution thematic raster map. Explicit analytic solutions are obtained for the fitted parameters. The fitting method is based on a relationship between the hierarchical transitions in the model and spatial transitions at varying distance scales in the data map, a categorical analogy of the geostatistical variogram.