Strata decision tree: a symbolic data analysis technique

Symbolic Data Analysis allow for the extension of Data Analysis to symbolic data, that is, more complex data structures than classical or monovalued data. Symbolic data are the symbolic descriptions given to individuals ω ∈ Ω associated to symbolic variables. For example, symbolic descriptions may be given by probability distributions, sets of categories or intervals. Strata Decision Tree is a generalized recursive tree-building algorithm (Breiman et al., 1984) for populations partitioned into strata and described by symbolic data. Common predictor and class variables describe population in all strata. Tree-growing methods are in general recursive non-parametric classification methods that recursively obtain the binary partition in the population which maximizes an information content measure of the new splits with respect to predefined classes. Our segmentation method is applied when the underlying population is not only subdivided into the classes to be investigated, but it is simultaneously composed of groups of individuals, which will be called strata such as when individuals of a country are partitioned into regions, individuals of a region are partitioned into towns, companies are partitioned into economic sectors, etc... Then, it is interesting to explain how strata influence the classification rules obtained by a tree-segmentation method. Some rules are applicable to some strata, and others are not. Our aim is: to predict or to explain the value of the class variable for individuals by the predictors, conditioned on the stratum; to explain how these predictions are affected by stratum membership; to detect sets of strata for which this explanation is the same; to describe a stratum symbolically by the set of prediction rules which apply to it with their importance. The algorithm is applied to classical data and to probabilistic symbolic data for predictors. A probabilistic-modal variable can be the variable Employment, which associates probability distributions to individuals ω. For example, the distribution given by (yes(0.8),no(0.2)) can represent either a subpopulation with 20% of them unemployed, or an individual with 20% of his/her working life unemployed. The algorithm also considers other kinds of symbolic data, as hierarchical dependencies between variables. Hierarchical dependence can be represented by non-applicable rules, as for example, if (smoke=no), then cigarrete mark=Non applicable. As output of the algorithm, symbolic objects describe decisional nodes and strata. Tree nodes are represented by assertions and strata are generalized by sets of assertions. A symbolic object provides a (symbolic) description of the properties of individuals in terms of (symbolic) variables, combined with an operator for selecting individu-