Feature Reduction with Inconsistency

1. Related Deffinitions And Concepts

Some related definitions and concepts are presented as follow:

Definition 1 Positive region, P and Q are two sets in the information system U(C, D),, then the positive region of Q in P, denoted as, can be calculated as:

Definition 2 Attribute dependency, P and Q are two sets in the information system U(C, D), , then the attribute dependency of attribute set Q on attribute set P, denoted as ,can be calculated as:

The attribute dependency can describe which variables are strongly related to which other variables, for example, if, then can be viewed as the measure between the decision attributes and the condition attributes, which can be implemented in further predictive modeling.

With the definition of attribute dependency, the attribute reduct can be defined as follow:

Definition 3 Attribute reduct, In information system U(C, D), , R is the reduct of C if and only if and or equivalently and

The essence of attribute reduct is to find a subset P from condition set, and the subset P can maintain the same discriminability under the instance space. So we can judge whether the set is a reduct by its discriminability under the instance space. So the positive region, which calculates the number of instances that can be discriminable with the attribute set, can be used to find the reduct.

From the definition of attribute reduct, we can see the reduct could keep the internal correlation of the attributes. Here we introduce the reduction into the feature selection, as the reduct can maintain the same discriminability as the original data set (Jensen & Shen, 2004).

Definition 4 Inconsistent condition, in information system U(C, D), C is the condition attribute set, D is the decision attribute set,, if and , then there are inconsistent condition between instance and instance .

Definition 5 Inconsistent instance number, in information system U(C, D), C is the condition attribute set, D is the decision attribute set, if , , , the inconsistent instance number of set P is denoted as , and calculated as follow: