and thus, for all However, at the low (binary) variable resolution, two bilateral implications become detectable: and , since every occurs iff and occurs iff Thus, a pattern recognition system scanning for implications of this kind would find them at the binary variable resolution, but would fail to find them at the higher quaternary variable resolution. Issues and methods It is not feasible to exhaustively test all possible discretization resolutions on all variables in order to see which combination of resolutions yields interesting or significant results. Instead, the feature space must be preprocessed (often by an entropy analysis of some kind) so that some guidance can be given as to how the discretization process should proceed. Moreover, one cannot generally achieve good results by naively analyzing and discretizing each variable independently, since this may obliterate the very interactions that we had hoped to discover. A sample of papers that address the problem of variable discretization in general, and multiple-variable discretization in particular, is as follows: , , , , , , , , , , , , , , , , , , , , . Variable granulation (clustering/aggregation/transformation) Variable granulation is a term that could describe a variety of techniques, most of which are aimed at reducing dimensionality, redundancy, and storage requirements. We briefly describe some of the ideas here, and present pointers to the literature. Variable transformation A number of classical methods, such as principal component analysis, multidimensional scaling, factor analysis, and structural equation modeling, and their relatives, fall under the genus of "variable transformation." Also in this category are more modern areas of study such as dimensionality reduction, projection pursuit, and independent component analysis. The common goal of these methods in general is to find a representation of the data in terms of new variables, which are a linear or nonlinear transformation of the original variables, and in which important statistical relationships emerge. The resulting variable sets are almost always smaller than the original variable set, and hence these methods can be loosely said to impose a granulation on the feature space. These dimensionality reduction methods are all reviewed in the standard texts, such as , , and . Variable aggregation A different class of variable granulation methods derive more from data clustering methodologies than from the linear systems theory informing the above methods. It was noted fairly early that one may consider "clustering" related variables in