Rough Dependencies as a Particular Case of Correlation: Application to the Calculation of Approximative Reducts

摘要

Rough Sets Theory provides a sound basis for the extraction of qualitative knowledge (dependencies) from very large relational databases. Dependencies may be expressed by means of formulas (implications) in the following way: (x_1, …, x_n) =>_rho (y) where (x_1, …, x_n) are attributes that induce partitions into equivalence classes on the underlying population. Coefficient rho is the dependency degree, it establishes the percentage of objects that can be correctly assigned to classes of y, taking into account the classification induced by (x_1, …, x_n). Dealing with decision tables, it is important to determine rho and to eliminate from (x_1, …, x_n) redundant attributes, to obtain minimal reducts having the same classification power as the original set. The problem of reduct extraction is NP-hard. Thus, approximative reducts are often determined. Reducts have the same classification power of the original set of attributes but quite often contain redundant attributes. The main idea developed in this paper is that attributes considered as random variables related by means of a dependency, are also correlated (the opposite, in general, is not true). From this fact we try to find, making use of well stated and widely used statistical methods, only the most significant variables, that is to say, the variables that contribute the most (in a quantitative sense) to determine y. The set of attributes (in general a subset of (x_1, …, x_n)) obtained by means of well-founded sound statistical methods could be considered as a good approximation of a reduct.