Classical rough set theory is a technique of granular computing for handling the uncertainty, vagueness, and granularity in information systems. Covering-based rough sets are proposed to generalize this theory for dealing with covering data. By introducing a concept of misclassification rate functions, an extended variable precision covering-based rough set model is proposed in this paper. In addition, we define the

In the era of big data, it is difficult to obtain useful information in huge data. Many researchers have proposed lots of efficient means of dealing with the difficulty. As one of these efficient means, classical rough set theory based on equivalence relations is proposed by Pawlak [

In variable precision rough sets, the misclassification rate of equivalence classes of all elements in a universe is identical. Similarly, in variable precision covering-based rough sets, the misclassification rate of neighborhoods of all elements in a universe is identical too. However, in practical applications, since there are different understanding or demands about equivalence classes or neighborhoods of different elements, the misclassification rate usually varies. Hence, it is necessary to propose misclassification rate functions.

To address the above issue, we propose a variable precision covering-based rough set model based on functions by introducing misclassification rate functions in this paper. We present the concepts of the

The rest of this paper is arranged as follows. Section

In this section, we present some fundamental concepts and existing results of classical rough sets, covering-based rough sets, and variable precision covering-based rough sets. Throughout this paper, the universe

Let

They are called the lower and upper approximations of

Covering-based rough sets are presented as the extension of classical rough sets by extending partitions to coverings on a universe.

Let

Neighborhoods are important concepts in covering-based rough sets.

Let

Let

Clearly, if

Let

By introducing the parameter

Let

Let

Let

By the definition, it is clear that

If

Let

When

Let

Clearly, if

In variable precision rough sets, the misclassification rate of equivalence classes of all elements in a universe is identical. Similarly, in variable precision covering-based rough sets, the misclassification rate of neighborhoods of all elements in a universe is identical too. However, in practical applications, we have different understanding or demands about equivalence classes or neighborhoods of different elements. That means the misclassification rate usually varies. Therefore, we present variable precision covering-based rough sets based on functions by introducing a concept of misclassification rate functions.

Let

Let

Let

Let

In practical applications, according to various needs, the different misclassification rate functions can be given by workers or researchers.

Suppose that

Similar to classical rough sets or variable precision rough sets, we present the concepts of

Let

Let

(1)

(2)

(3) For any

In the following definition, we present the

Let

In this subsection, we present the properties and some significant results concerning the new model.

Let

If

(1) For any

(2) For any

(3) If

(4) For all

(5) Similar to the proof of

(6) For all

(7) Similar to the proof of (6).

(8) For any

Let

According to Proposition

Let

Let

From Definition

Let

However, the converse proposition of Theorem

Let

In this subsection, we will use a figure (Figure

Relationships among five types of rough sets.

In this paper, we proposed the variable precision covering-based rough set model based on functions as a generalization of a variable precision covering-based rough set model and studied its properties. Through the concept of reductions, we obtained that two coverings with the same reductions generate the same

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is in part supported by National Nature Science Foundation of China under Grant Nos. 61170128 and 61379049, the Key Project of Education Department of Fujian Province under Grant No. JA13192, and the Zhangzhou Municipal Natural Science Foundation under Grant No. ZZ2013J03.