Dependency Analysis and Attribute Reduction in the Probabilistic Approach to Rough Sets - Feature Selection for Data and Pattern Recognition - page 105

6.9 λ — Core Collection of Attributes

As in the original rough set approach [ 11 ], one can easily identify the set of most
essential condition attributes with respect to the ʻ — dependency . These attributes,

called the ʻ — core , are the ones which would never be eliminated in the process of

any ʻ — Reduct computation. They are included in all ʻ — reducts i.e. their collection

is equal to the intersection of all ʻ — reducts .

Any core attribute { a } satisfies the following inequality:

ʻ( X | C )>ʻ( X | C −{ a } ). (6.27)

The above inequality demonstrates that there is no need to compute all ʻ — reducts ,

which is NP-hard, to identify the ʻ — core as the core attributes can be found by simple

linear testing procedure.
As in the case of — core attributes, γ — core attributes can also be computed in

a probabilistic decision table with respect to the dependency γ (Region|C) by testing
the effect of removal of each condition attribute.

ʻ

6.10 Final Remarks

The chapter reviews results of our long-term research on data dependencies, within
the frameworks of the variable precision and Bayesian rough set models, occur-
ring in approximation spaces and in both, classification and decision tables. These
probabilistic dependencies are defined based on the degrees of overlap between
sets. The primary dependency measures discussed in the chapter are γ — dependency
and — dependency . They generalize and expand the attribute functional and partial

functional dependency measures introduced by Pawlak [ 10 , 11 ]. The applicability
of the measures to creation, analysis and optimization of classification and deci-
sion tables, via the concept of attribute reduct, was also discussed and two reduct
computation algorithms were presented. The variable precision rough set approach
was used inmany applications since its introduction in 1990s. To our best knowledge,
the most comprehensive application, involving the use of hierarchies of probabilistic
decision tables and the attribute dependency measures presented in this chapter, were
the experiments with face recognition [ 4 ]. It is our belief that the theory and meth-
ods presented in the chapter will find additional useful applications in areas dealing
with large amounts of data such as, for example, in medicine, pattern classification,
market analysis and prediction, machine learning and data mining in general, just to
mention a few areas where in our opinion this theory is applicable.

ʻ

Acknowledgments Thanks are due to anonymous referees for their detailed and inspiring
comments. The research reported in the chapter was supported by research grants from Natural
Sciences and Engineering Research Council of Canada.