On Spatial Transition Probabilities as Continuity Measures in Categorical Fields

TL;DR

This paper analyzes properties of transiogram models as measures of spatial continuity in categorical fields, revealing their behavior near the origin, limitations of common covariogram forms, and proposing a kernel regression method for joint modeling.

Contribution

It provides an analytical examination of transiogram models, identifies limitations of covariogram forms for Gaussian fields, and introduces a kernel regression approach for efficient joint modeling.

Findings

Auto-transiogram behavior relates to category distribution.

Most covariogram models are unsuitable for transiograms except exponential decay.

Kernel regression enables non-parametric joint modeling of transiograms.

Abstract

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.