Multi knowledge based rough approximations and applications

doi:10.1016/j.knosys.2011.06.010

Knowledge-Based Systems

Volume 26, February 2012, Pages 20-29

https://doi.org/10.1016/j.knosys.2011.06.010 Get rights and content

Abstract

Rough set theory is an important technique in knowledge discovery in databases. In covering based rough sets, many types of rough set models are established in recent years. This paper presents new types of rough set approximations using multi knowledge base, that is, family of finite number of (reflexive, tolerance, dominance, equivalence) relations by two ways. Properties and applications of these approximation operators are studied and many examples are given.

Highlights

► We used a special neighborhood to define lower and upper approximations of any set. ► Generalized these definitions in two different ways. ► The first way is based on the intersection of all these neighborhoods. ► The second generalized based on the intersection and union of approximations. ► We introduced new general definitions for some Pawlak’s definitions.

Introduction

Rough set theory which was introduced by Pawlak [28] in the early 80s is an extension of set theory for the study of intelligent systems characterized by insufficient and incomplete information.

Rough set theory is an important tool for conceptualizing and approximate reasoning, organizing and analyzing various types of data mining. Axiomatic system of rough sets is significant for using rough set theory in logical reasoning systems. This method is especially useful for dealing with uncertain and vague knowledge in information systems. Many application examples of real-life fields of rough set method can be cited such as in process control, economics, medical diagnosis, biochemistry, environmental science, biology, chemistry, psychology, conflict analysis, pharmacology, banking, market research, engineering, speech recognition, material science, information analysis, data analysis, data mining, control and linguistics and many other fields [3], [5], [9], [11], [14], [18], [27], [32], [40], [44], [47], [48]. The original definition of rough sets and approximation operators is done, by means of unions of some equivalence classes [28], [29], [30]. The essential steps are done by generalizations of approximations spaces and approximations operators to tolerance relations [26], [31], [34]. However the most general step was done by Lin. In [20] Lin points out the usefulness and applicability of Frechet spaces for analyzing relational data model. Many proposals have been put forward for generalizing and interpreting rough sets. For example, rough set model is extended to arbitrary binary relations [4], [17], [19], [21], [22], [23], [24], [34], [38], [39], [42], [50], [51] and coverings [7], [9], [25], [49], [51], [52]. Some researchers even extended classical rough sets to fuzzy sets [6], [10], [12], [13], [37], [46]. The classical rough set theory is based on equivalence relations, but it has been extend by using different kinds of relations, it implies the equivalence class represented by different kinds of neighborhoods. The core concepts of classical rough sets are lower and upper approximations based on equivalence relations. Feng et al. [8] extend the approaches of attribute reduction by combing rough sets and vague sets and introduced basic ideas of rough sets given in the form of the low and upper approximations in Vague Approximate Space (VAS). Yang et al. [41] researched the neighborhood system from the view point of knowledge engineering and considered each neighborhood as a basic unit with knowledge. Also, discussed the rough approximations and the corresponding properties by using these knowledge in neighborhood system. Wei and Qi [36] discusses the relation between concept lattice reduction and rough set reduction based on classical formal context, which be meaningful for the relation research between these two theories, and for their knowledge discovery. In [1] Abu-Donia discussed three types of lower and upper approximations of any set with respect to any relation based on right neighborhood and generalized these three types of approximations into two ways using a finite number of any binary relations. This paper studies some of fundamental concepts of rough set theory by using a finite family of any (reflexive, tolerance, dominance, equivalence) relations. In this setting, the finite family can generate a lower approximation operator and an upper approximation operator based on a spacial neighborhood (〈x〉R). However some of the common properties of classical lower and upper approximations operator are no longer satisfied.

Rough sets theory has been applied in many fields such as machine learning, knowledge discovery, and expert systems. It deals with the classificatory analysis of data tables. Volumes of data are generated constantly by several computer applications running across the globe. This trend of rapid increase in data generation is triggered by the ever-increasing usage of internet and database applications. In any particular application domain, having a large database that represents an equally large collection of facts about the domain, is of little use to experts trying to analyze facts, recognize trends or patterns and make decisions. Extracting useful information and ironing out inconsistencies from raw data is a challenging task and draws the research interests of computer scientists globally. Where, decision quality depends on data analysis tools and techniques that used for extracting useful information from vast amounts of data often.

The data can be acquired from measurements or from human experts. The main goal of the rough set analysis is to synthesize approximation of concepts from the acquired data and makes reduction of data to a minimal representation. Many rough sets models have been developed in the rough set community in the last decades. Some of them have been applied in the industry decision support systems projects such as stock market predication, patient symptom diagnosis, telecommunication churner predication and financial bank customer attrition analysis to solve challenging business problems. These rough set models focus on the extension of the original model proposed by Pawlak [28] and attempt to deal with its limitations such as handling statistical distribution or noisy data.

Section snippets

Preliminaries

Let U be a non-empty set called universe, and E be an equivalence relation on U. The pair (U, E) is called an approximation space. Let [x]_E denotes the equivalence class for an element x ∈ U. Let A be a subset of U. A rough set corresponding to A is the ordered pair $(\underset{̲}{E} (A), \bar{E} (A))$ , where E(A) and $\bar{E} (A)$ are defined as follows: $\begin{matrix} \underset{̲}{E} (A) = {x \in U : [x]_{E} \subseteq A} (called lower approximation of A) . \\ \bar{E} (A) = {x \in U : [x]_{E} \cap A \neq \emptyset} (called upper approximation of A) . \end{matrix}$

Obviously, we have $\underset{̲}{E} (A) \subseteq A \subseteq \bar{E} (A)$ . The lower approximation of A contains

New type of approximations

In this section, we use a finite number of different types of relations to introduce a new form of upper and lower approximations.

Definition 3.1

Suppose that U is a finite and a non-empty set called universe. Let {R_i : i = 1, 2, … , n} be a finite family of binary relations on U, x R_i is the right neighborhood of the point x ∈ Uwith respect to the relation R_i and $〈 p 〉 R_{i} = ⋂_{p \in {xR}_{i} {xR}_{i}}$ . We can define the n-lower and n-upper approximations of a subset A ⊆ U according to R_i, i = 1, 2, … , n. As the following: $\begin{matrix} _{⊏ n ⊐} \underset{̲}{apr} (A) = \{x \in U : ⋂_{i = 1}^{n} 〈 x 〉 R_{i} \subseteq A\}, \end{matrix}$

General approximations

In this section, we introduce new approximations for any non-empty set according to a finite number of any reflexive, tolerance, dominance and equivalence relations based on the intersection (union) of the upper (lower) approximations defined in Definition 3.2. (See Table 2, Table 3, Table 4, Table 5, Table 6, Table 7, Table 8, Table 9, Table 10)

Definition 4.1

Let {R_i : i = 1, 2, … , n} be a finite family of binary relations on a non-empty finite set U. We can define n-lower and n-upper approximations of A ⊂ U,

New generalization of some Pawlak’s concepts

By using the lower and upper approximations defined in Definition 4.1, we decreased the boundary region more than that in Definition 2.2. As the new lower approximation becomes greater than the lower approximation defined in Definition 2.2 and the new upper approximation becomes smaller than the upper approximation defined in Definition 2.2.

We can define accuracy measure of any set A according to any binary relations.

Definition 5.1

Let {R_i : 1, 2, … , n} be a family of binary relations, then $_{⊏ n ⊐} α_{any}^{*} (A) = | A \cap_{⊏ n ⊐} {\underset{̲}{apr}}^{*} (A$

Conclusion

In this paper, we used a special neighborhood (〈x〉R) to define lower and upper approximations of any set with respect to any, reflexive, tolerance, dominance, equivalence relation. We generalized these definitions by using a family of the previous types of relations in two different ways. The first way is based on the intersection of all these neighborhoods and the second generalized definition is based on the intersection and union of upper (lower) approximations with respect to every relation

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Multi knowledge based rough approximations and applications

Abstract

Highlights

Introduction

Section snippets

Preliminaries

New type of approximations

General approximations

New generalization of some Pawlak’s concepts

Conclusion

Knowledge-Based Systems

Information Sciences

Knowledge-Based Systems

Computer-Aided Design

Knowledge-Based Systems

Knowledge-Based Systems

Expert Systems with Applications

Expert Systems with Applications

Information Sciences

European Journal of Operational Research

International Journal of Approximate Reasoning

Knowledge-Based Systems

International Journal of Approximate Reasoning

Information Sciences

International Journal of Approximate Reasoning

Information Sciences

Information Sciences

Knowledge-Based Systems

Information Sciences

Knowledge-Based Systems

Information Sciences

Knowledge-Based Systems

Information Sciences

Fuzzy Sets and Systems

Information Sciences

Knowledge-Based Systems