Transiogram: A spatial relationship measure for categorical data

Categorical geographical variables are normally classified into multinomial classes which are mutually exclusive and visualized as area-class maps. Typical categorical variables such as soil types and land cover classes are multinomial and exhibit complex interclass relationships. Interclass relationships may include three situations: cross-correlation (i.e. interdependency), neighbouring situation (i.e. juxtaposition), and directional asymmetry of class patterns. In a space, some classes may be cross-correlated with apparent correlation ranges, but some classes may not be cross-correlated in the traditional sense. For example, if class A and class B occur at two separate subareas of a watershed, respectively, it may be difficult to say they are cross-correlated; but we still can define their interclass relationship as nonneighbouring. If this interclass relationship is effectively incorporated into a geostatistical model, class A and class B will not occur closely as neighbours in simulated results; but if this interclass relationship is ignored in a simulation conditioned on sparse samples, they may occur as neighbours in simulation results. This means that any class has a relationship with another class existing in the same space, and quantifying various spatial relationships of classes and incorporating them into simulation models are helpful in generating realistic realizations of the real spatial distribution of multinomial classes and decreasing spatial uncertainty associated with simulated results. To describe the auto-correlations within single classes and the relationships between different classes, we need practical spatial measures. So far, indicator variograms have been widely used as two-point spatial measures for characterizing the spatial correlations of discrete geographic data in the geosciences (Chiles and Delfiner 1999). However, the physical meanings of indicator variograms, particularly indicator cross-variograms, are difficult to interpret. Variograms are widely used mainly because of the wide application and acceptance of kriging-based (or variogram-based) geostatistics as interpolation and simulation techniques for spatial variables, which normally use variograms as input parameters (Deutsch and Journel 1998). Recent studies (Li et al. 2004, 2005, Zhang and Li 2005) and further progress in the development of practical multidimensional (multi-D) Markov chain conditional simulation models and algorithms will suggest a Markov chain-based geostatistics for simulating categorical variables. As the accompanying spatial measure with this new geostatistics, the author proposes the concept of the transiogram (i.e. 1-D transition probability diagram) and suggests using transiograms to replace Markov transition probability matrices (TPMs) as parameter