Information Technology Journal

Year: 2013 | Volume: 12 | Issue: 3 | Page No.: 548-550
DOI: 10.3923/itj.2013.548.550

Interval-valued Information Systems Rough Set Model Based on Multi-granulations

Wei Yang

Abstract: This study extends the rough set in interval-valued information systems to the multi-granulation rough set in interval-valued information systems. The lower and upper approximations of a set in interval-valued information systems based on multi-granulations are defined and some basic properties are introduced. In order to substantiate the conceptual arguments numerical examples are given.

Keywords: rough set, Interval-valued information systems, attribution reduction and multi-granulations

INTRODUCTION

Pawlak (1982) proposed the rough set theory, which is used to analyze imprecise, vague and uncertain data sets in many fields. Because of the classical rough set is not to use in some cases, such as the incomplete information system and so on. Several extensions on rough set have been proposed, such as a novel rough set approach is proposed by Leung et al. (2008). A dominance relation to interval information systems is presented by Qian et al. (2008). The knowledge reduction of α-maximal consistent blocks is proposed in interval-valued information systems by Miao et al. (2009).

In this study, the rough set of interval-valued information system is extended to multi-granulations rough set based on multi-granulations (Qian and Liang, 2006, 2010). Some interesting properties are discussed based on the multi-granulations rough set and proved two properties. Based on the given maximal intersection rates the reduction is computed.

THE MULTI-GRANULATIONS ROUGH SET IN INTERVAL-VALUED INFORMATION SYSTEMS

The basic notion of an interval-valued information system is cited for facilitating our discussion.

The interval-valued information system: “[L]et ζ = (U, AT, V, f) denote an information system called an interval-valued information system (IvIS), where U = [u₁, u₂,…, u_n] is a non-empty finite set called the universe of discourse, AT = [a₁, a₂,…, a_m] is non-empty finite set of m attribute, such that a_k (u_i) = [l_i^k, u_i^k], l_i^k≤u_i^k for all i = 1, 2,…, n and k = 1, 2,…, m. V is a set of values. f is called the information function as f:UxAT→V.

Example 1: Table 1 is an IvIS ζ = (U, AT, V, f), where U = [u₁, u₂,…, u_n], AT = [a₁, a₂,…, a_m], the attribute value a_k (u_i) = [l_i^k, u_i^k] is an interval number (Miao et al., 2009).

Definition 1: [L]et ζ = (U, AT, V, f) be an IvIS, X⊆U, A⊆AT, card(A) = s. The lower and upper approximations to a subset X of U can be defined as follows (Miao and Yang, 2010):

In the view of multi-granulations, in the interval-valued information systems the is called single granulation rough set.

Definition 2: Let ζ = (U, AT, V, f) be an IvIS, X⊆U, A, B⊆AT, card(A) = s, card(B) = t. The lower and upper approximations to a subset X of U can be defined as follows:

In the view of multi-granulations, in the IvIS the is called double granulations rough set.

Definition 3: Let ζ = (U, AT, V, f) be an IvIS, X⊆U, A₁, A₂,…, A_m⊆AT. The lower and upper approximations to a subset X of U can be defined as follows:

In the view of multi-granulations, in the IvIS the is called multi-granulations rough set.

Proposition 1: Let ζ = (U, AT, V, f) be an IvIS, X⊆U, A₁, A₂,…, A_m⊆AT. Some properties can be discussed as follows:

Proof: If j>1, the properties can be proved as follows:

Proposition 2: Let ζ = (U, AT, V, f) be an IvIS, X⊆U, A₁, A₂,…, A_m⊆AT. Then:

Proposition 3: Let ζ = (U, AT, V, f) be an IvIS, X⊆U, A₁, A₂,…, A_m⊆AT, X₁⊆X₂⊆…X_n⊆U. Then:

Example 2: We can consider the interval-valued information system given in Table 1 and assume the maximal intersection rates (Miao and Yang, 2010) β₁, β₂,…, β_m = 0.9 are given with respect to all the attributes.

For X = {u₁, u₂, u₄, u₈, u₉}

ATTRIBUTE REDUCTION

Definition 4: Let ζ = (U, AT, V, f) be an IvIS, A⊆AT, B⊂A, card(A) = s, card(B) = t, card(AT) = m, A = {a₁, a₂,…, a_s}, B = {b₁, b₂,…, b_t}, AT = {c₁, c₂,…, c_m}, X⊆U, is a reduction with respect to the given a₁, a₂,…, a_m in ζ, if and only if:

Definition 5: Let ζ = (U, AT, V, f) be an IvIS,A⊆AT, B⊂A, card(A) = s, card(B) = t, card(AT) = m, A = {a₁, a₂,…, a_s}, B = {b₁, b₂,…, b_t}, AT = {c₁, c₂,…, c_m}, X⊆U, is a reduction of ζ for u with respect to the given a₁, a₂,…, a_m in ζ, if and only if:

Example 3: We can consider the interval-valued information system given in Table 1 and assume the maximal intersection rates β₁, β₂,…, β_m = 0.9 are given with respect to all the attributes. It is easy to obtain the reductions:

A^0.9 = (a₁, a₃, a₄, a₅)
A₁^0.9 (u₁) = {a₁, a₃}, A₂^0.9 (u₁) = {a₁, a₅}
A₁^0.9 (u₂) = {a₁}, A₂^0.9 (u₂) = {a₂}
A₁^0.9 (u₃) = {a₂}, A₂^0.9 (u₃) = {a₅}
A^0.9 (u₄) = {a₃}
A^0.9 (u₅) = {a₁}
A^0.9 (u₆) = {a₄}
A^0.9 (u₇) = {a₅}
A^0.9 (u₈) = {a₁}
A^0.9 (u₉) = {a₁}
A₁^0.9 (u₁₀) = {a₁}, A₂^0.9 (u₁₀) = {a₃}, A₃^0.9 (u₁₀) = {a₄}, A₄^0.9 (u₁₀) = {a₅}

CONCLUSION

In this study, we extend the multi-granulations rough set theory to interval-valued information system. The extension of classical rough set model is an important direction of research. We proposed the rough set model based on multi-granulation in interval-valued information system and discussed some properties on the model. The reduction is defined with the new model. In order to substantiate the conceptual arguments numerical examples are given.

ACKNOWLEDGMENT

The research is supported by the National Natural Science Foundation of China under grant No.: 60970061 and 61075056.

REFERENCES