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

A model is described for generating hierarchically scaled spatial pattern a s represented in a thematic raster map. The model involves a series of Mark ov transition matrices, one for each level in the scaling hierarchy. In ful l generality, the model allows the transition matrices to be different at e ach level, potentially making available a large number of parameters for la ndscape characterization. The model is self-similar when the transition mat rices are all equal. A method is presented for fitting the model to data th at take the form of a single-resolution thematic raster map. Explicit analy tic 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 categ orical analogy of the geostatistical variogram.