  Models of spatial transition probabilities, or equivalently, transiogram
models have been recently proposed as spatial continuity measures in
categorical fields. In this paper, properties of transiogram models are
examined analytically, and three important findings are reported. Firstly,
connections between the behaviors of auto-transiogram models near the origin
and the spatial distribution of the corresponding category are carefully
investigated. Secondly, it is demonstrated that for the indicators of excursion
sets of Gaussian random fields, most of the commonly used basic mathematical
forms of covariogram models are not eligible for transiograms in most cases; an
exception is the exponential distance-decay function and models that are
constructed from it. Finally, a kernel regression method is proposed for
efficient, non-parametric joint modeling of auto- and cross-transiograms, which
is particularly useful for situations where the number of categories is large.
