Attribute reduction based on evidence theory in incomplete decision systems

doi:10.1016/j.ins.2007.10.006

Information Sciences

Volume 178, Issue 5, 1 March 2008, Pages 1355-1371

https://doi.org/10.1016/j.ins.2007.10.006 Get rights and content

Abstract

Attribute reduction is a basic issue in knowledge representation and data mining. This paper deals with attribute reduction in incomplete information systems and incomplete decision systems based on Dempster–Shafer theory of evidence. The concepts of plausibility reduct and belief reduct in incomplete information systems as well as relative plausibility reduct and relative belief reduct in incomplete decision systems are introduced. It is shown that in an incomplete information system an attribute set is a belief reduct if and only if it is a classical reduct and a plausibility consistent set must be a classical consistent set. In a consistent incomplete decision system, the concepts of relative reduct, relative plausibility reduct, and relative belief reduct are all equivalent. In an inconsistent incomplete decision system, an attribute set is a relative plausibility reduct if and only if it is a relative reduct, a plausibility consistent set must be a belief consistent set, and a belief consistent set is not a plausibility consistent set in general.

Introduction

The theory of rough sets, proposed by Pawlak [39], is an extension of classical set theory for the study of intelligent systems characterized by insufficient and incomplete information. With more than twenty years development, rough set theory has been found to have very successful applications in the fields of artificial intelligence such as expert systems, machine learning, pattern recognition, decision analysis, process control, and knowledge discovery in databases.

A basic concept related to rough set is information system (attribute-value system). Most applications based on rough set theory can fall into the attribute-value representation model. According to whether or not a system is deterministic, information systems can be classified into two categories: complete and incomplete. A complete information system is a system in which the values of all the attributes are deterministic. By an incomplete information system we mean a system that the values of some of the attributes are not known, i.e., missing or partially known, an incomplete information system is also called a nondeterministic information system [38].

The basic idea of rough set theory is knowledge acquisition in the sense of unravelling a set of decision rules from an information system via an objective knowledge induction process for decision making. Various approaches using rough set theory have been proposed to induce decision rules from data sets taking the form of complete decision systems [8], [10], [20], [39], [40], [41], [42], [43], [44], [53], [56], [66], [73]. Due to the rampant existence of incomplete information systems in real life, many authors employed extensions of Pawlak’s rough set model to reason in incomplete information systems [5], [7], [9], [11], [12], [22], [23], [25], [26], [28], [30], [31], [36], [37], [38], [46], [53], [54], [57], [68]. For example, Greco et al. [7], Grzymala-Busse [9], Kryszkiewicz [22], [23], used similarity relations in incomplete information systems with missing values. By analyzing similarity classes defined by Kryszkiewicz, Leung and Li [25] introduced the concept of maximal consistent block technique for rule acquisition in incomplete information systems. To unravel certain and possible decision rules in incomplete information systems, Leung et al. [26] developed a new rough set approximations by defining a new information structure called labelled blocks. Other researchers, such as Deng et al. [6], Hong et al. [13], [14], Jensen and Shen [18], Korvin et al. [21], Liu et al. [32], Slowinski and Stefanowski [53], Wang et al. [60] and Wu et al. [63], used rough set models to handle fuzzy and quantitative data.

It is well-known that not all conditional attributes in an information system are necessary to depict the decision attribute before decision rules are generated. Knowledge reduction in the sense of reducing attributes is thus an outstanding contribution made by rough set research to data analysis [40]. It is performed in information systems by means of the notion of a reduct based on a specialization of the general notion of independence due to Marczewski [33]. Many types of attribute reduction have been proposed in complete information systems and complete decision systems [1], [2], [10], [17], [19], [20], [24], [27], [34], [35], [40], [43], [50], [51], [52], [55], [56], [58], [59], [69], [71], each of them aimed at some basic requirement. In recent years, more attention has been paid to attribute reduction in incomplete information systems, incomplete decision systems, covering information systems, and fuzzy information systems in rough set research [3], [11], [15], [16], [22], [23], [25], [26], [32], [72].

Another important method used to deal with uncertainty in information systems is the Dempster–Shafer theory of evidence. It was originated by Dempster’s concepts of lower and upper probabilities [4], and extended by Shafer [45] as a theory. The basic representational structure in this theory is a belief structure which consists of a family of subsets, called focal elements, with associated individual positive weights summing to one. The primitive numeric measures derived from the belief structure are a dual pair of belief and plausibility functions.

There are strong connections between rough set theory and Dempster–Shafer theory of evidence. It has been demonstrated that various belief structures are associated with various rough approximation spaces such that the different dual pairs of lower and upper approximation operators induced by the rough approximation spaces may be used to interpret the corresponding dual pairs of belief and plausibility functions induced by the belief structures [29], [30], [47], [48], [49], [61], [67]. The Dempster–Shafer theory of evidence may be used to analyze knowledge acquisition in information systems. For example, Zhang et al. [70] proposed the concepts of belief reduct and plausibility reduct in complete information systems without decisions. Wu et al. [64] discussed knowledge reduction in complete decision systems via the Dempster–Shafer theory of evidence. Lingras and Yao [30] employed two different generalizations of rough set models to generate plausibilistic rules with incomplete databases instead of probabilistic rules generated by a Pawlak’s rough set model with complete decision tables. Wu and Mi [62] studied knowledge reduction in incomplete information systems without decisions within evidence theory. We attempt to investigate in this paper attribute reduction in incomplete decision systems within the Dempster–Shafer theory of evidence.

In the next section, we give some basic notions related to incomplete information systems and incomplete decision systems, we also review rough set approximations in incomplete information systems. Some basic notions of evidence theory are introduced in Section 3. The concepts of belief reducts and plausibility reducts in incomplete information systems are proposed in Section 4. In Section 5, we study relative belief reducts and relative plausibility reducts in consistent and inconsistent incomplete decision systems and discuss the relationships among the new concepts of reducts and some existing ones. We then conclude the paper with a summary and outlook for further research in Section 6.

Section snippets

Incomplete information systems and rough set approximations

The notion of information systems (sometimes called data tables, information tables, attribute-value systems, knowledge representation systems etc.) provides a convenient tool for the representation of objects in terms of their attribute values.

A complete information system (CIS) S is a pair (U, AT), where U = {x₁, x₂, … , x_n} is a nonempty finite set of objects called the universe of discourse and AT = {a₁, a₂, … , a_m} is a nonempty finite set of attributes such that a:U → V_a for any a ∈ AT, i.e., a(x) ∈ V_a, x ∈ U

Belief structures and belief functions

The Dempster–Shafer theory of evidence, also called the “evidence theory” or the “belief function theory”, is treated as a promising method of dealing with uncertainty in intelligence systems. The basic representational structure in the Dempster–Shafer theory of evidence is a belief structure [45].

Definition 1

Let U be a nonempty finite set, a set function $m : P (U) \to [0, 1]$ is referred to as a basic probability assignment if it satisfies axioms (M1) and (M2): $(M1) m (\emptyset) = 0, (M2) \sum_{A \subseteq U} m (A) = 1 .$

The value m(A) represents

Attribute reduction in incomplete information systems

In this section, we propose the concepts of belief and plausibility reducts in IISs and compare them with the concept of classical reduct.

Definition 3

Let S = (U, AT) be an IIS, then

(1)
an attribute subset A ⊆ AT is referred to as a classical consistent set of S if R_A = R_AT. If A ⊆ AT is a classical consistent set of S and no proper subset of A is a classical consistent set of S, then A is referred to as a classical reduct of S. Thus a classical reduct of S is a minimal subset A ⊆ AT satisfying R_A = R_AT.
(2)
an attribute subset A

Attribute reduction in incomplete decision systems

In this section, we introduce the concepts of relative belief reduct and relative plausibility reduct in an IDS and compare them with the existing concept of relative reduct.

Let S = (U, C ∪ {d}) be an IDS and B ⊆ C, denote $\partial_{B} (x) = {d (y) : y \in S_{B} (x)}, x \in U .$ ∂_B(x) is called the generalized decision of x w.r.t. B in S. S is said to be consistent if |∂_C(x)| = 1 for all x ∈ U, otherwise it is inconsistent.

Definition 4

Let S = (U, C ∪ {d}) be an IDS and B ⊆ C. Then

(1)
B is referred to as a relative consistent set of S if ∂_B(x) = ∂_C(x) for all x ∈ U

Conclusions

An incomplete information system is an attribute-value system in which some of the attribute values are non-deterministic, i.e., missing or partially known. We make an assumption in this paper that a non-deterministic value in the system is a set of possible values of the attribute for the object. Obviously such an incomplete information system can be transformed into a set-valued information system.

We have discussed in this paper attribute reduction via the Dempster–Shafer theory of evidence

Acknowledgements

The author would like to thank the anonymous referees for their valuable comments and suggestions. This work was supported by grants from the National Natural Science Foundation of China (No. 60373078, No. 60673096 and No. 60773174).

References (73)

M. Beynon
Reducts within the variable precision rough sets model: a further investigation
European Journal of Operational Research
(2001)
T.Q. Deng et al.
A novel approach to fuzzy rough sets based on a fuzzy covering
Information Sciences
(2007)
Y.-Y. Guan et al.
Set-valued information systems
Information Sciences
(2006)
T.P. Hong et al.
Learning rules from incomplete training examples by rough sets
Expert Systems with Applications
(2002)
T.P. Hong et al.
Mining fuzzy β-certain and β-possible rules from quantitative data based on the variable precision rough-set model
Expert Systems with Applications
(2007)
Q. Hu et al.
Hybrid attribute reduction based on a novel fuzzy-rough model and information granulation
Pattern Recognition
(2007)
Q. Hu et al.
Information-preserving hybrid data reduction based on fuzzy-rough techniques
Pattern Recognition Letters
(2006)
M. Kryszkiewicz
Rough set approach to incomplete information systems
Information Sciences
(1998)
M. Kryszkiewicz
Rules in incomplete information systems
Information Sciences
(1999)
Y. Leung et al.
Maximal consistent block technique for rule acquisition in incomplete information systems
Information Sciences
(2003)

W. Zhu et al.

Reduction and axiomization of covering generalized rough sets

Information Sciences

(2003)

W. Ziarko

Variable precision rough set model

Journal of Computer and System Sciences

(1993)

J.A. Bazan

Comparison of dynamic and non-dynamic rough set methods for extracting laws from decision tables

D. Chen et al.

A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets

Information Sciences

(2007)

A.P. Dempster

Upper and lower probabilities induced by a multivalued mapping

Annals of Mathematical Statistics

(1967)

S. Demri et al.

Incomplete Information: Structure, Inference, Complexity

(2002)

S. Greco et al.

Handling missing values in rough set analysis of multi-attribute and multi-criteria decision problems

S. Greco et al.

Rough approximation by dominance relation

International Journal of Intelligent Systems

(2002)

J. Grzymala-Busse

A rough set approach to data with missing attribute values

J. Grzymala-Busse et al.

Classification strategies using certain and possible rules

T.P. Hong et al.

Knowledge acquisition from quantitative data using the rough-set theory

Intelligent Data Analysis

(2000)

Cited by (197)

Lattices arising from fuzzy coverings
2023, Fuzzy Sets and Systems
A fuzzy covering is a natural generalization of coverings. This paper shows that the families of all definable fuzzy sets induced by a fuzzy covering-based rough set have lattice structures. There may be some superfluous elements in the universal set and the fuzzy covering for the families of all definable sets. We define class reduction, element reduction, and bireduction to remove these elements respectively. For the family of all lower definable fuzzy sets, it is shown that the class reduction coincides with covering reduction, and the element reduction can be transformed into a reduction in information systems. Bireduction considers class reduction and element reduction, and it can be achieved by class and element reductions. Moreover, our results also hold for coverings because a covering is a special fuzzy covering.
A bi-variable precision rough set model and its application to attribute reduction
2023, Information Sciences
Significant indicators for measuring algorithms for attribute reduction include their fault tolerance. The majority of existing algorithms rely on the incorporation of fault tolerance at the upper and lower approximation operators. However, information granules partitioned by these algorithms are not inherently fault-tolerant. To enhance the fault tolerance of attribute reduction algorithms, this study introduces a new model based on variable precision rough sets, comprising of three parameters: the inclusion of fault tolerance in the comparison of attribute values between two objects, the partition of information granules, and the calculation of upper and lower approximation operators. As a result, the model tolerates errors, enhances the fault-tolerance ability of information granules, and improves its inclusivity. Using this new model, we design a novel algorithm that deletes both redundant and noise attributes to improve the classification accuracy. Through analysis of classification accuracy and robustness, we confirm that the novel algorithm's fault tolerance and robustness are maintained even after adding random Gaussian noise. The parameters included as part of the model are also found to be mostly unchanged after the addition of random Gaussian noise.
TI-fuzzy neighborhood measures and generalized Choquet integrals for granular structure reduction and decision making
2023, Fuzzy Sets and Systems
Granular structures are mathematical representations of knowledge used in granular computing. Any fuzzy information table can be seen as a fuzzy granular structure, which is called a fuzzy β-covering approximation space. Hence, corresponding granular computing methods are widely used in data analysis, granular reduction and decision making. Currently, these methods mainly use rough approximate operators to process data where attributes are independent of each other in rough approximate operators. However, attributes are often associated with each other in real-life problems. As a parametric nonlinear aggregation function, the Choquet integral (CI) with respect to a fuzzy measure (FM) solves the problem of attribute association well. In this paper, we present TI-fuzzy β-neighborhood measures, which are FMs and generalized CIs, to deal with granularity reduction and decision making in a fuzzy β-covering approximation space. Firstly, four pairs of TI-fuzzy β-neighborhood measures under the t-norm “ $T$ ” and its residual implication “ $I_{T}$ ”, as FMs, are presented for use in granular computing instead of rough approximation operators. Then, a novel method with TI-fuzzy β-neighborhood measures is presented to deal with granularity reduction in the fuzzy β-covering approximation space. Thirdly, four pairs of generalized CIs based on the TI-fuzzy β-neighborhood measures are constructed. By combining the presented CIs with the fuzzy PROMETHEE method, we propose a new method to solve the problem of decision making. Finally, several numerical examples and UCI data sets are used to illustrate the feasibility and effectiveness of our proposed methods.
A novel variable precision rough set attribute reduction algorithm based on local attribute significance
2023, International Journal of Approximate Reasoning
In attribute reduction, the method of partitioning information granules is crucial; however, the existing attribute reduction algorithms have limitations in this respect. To solve this problem, this study proposes the concept of an $(α, β)$ -indiscernibility relation. This is a fresh partition approach for information granules, tolerating a certain degree of error information. On the basis of this concept, we propose a novel $(α, β)$ -rough set model. Then we use this model to design a novel attribute reduction algorithm based on local attribute significance. To maintain the objectivity of this research and ensure fairness of the experiments, we present an integrated classifier to achieve the greatest classification accuracy. Ultimately, the experimental results verify that the fault-tolerance ability of information granules is increased. Moreover, the novel attribute reduction algorithm deletes redundant attributes and noise attributes to increase classification accuracy. In the later part of the experiment, different proportions of random Gaussian noise are added to each data set. The experimental results demonstrate the robustness and superiority of the new algorithm.
Attribute reduction based on D-S evidence theory in a hybrid information system
2022, International Journal of Approximate Reasoning
As an important uncertainty reasoning method, Dempster-Shafer (D-S) evidence theory has been widely applied to expert system, comprehensive evaluation, information fusion and decision analysis. However, it has not been fully valued in uncertainty measurement. A hybrid information system (HIS) means an information system that contains many types of attributes (e.g., categorical attribute, real-valued attribute and attribute with missing values, etc.). It is more difficult to measure an HIS than an ordinary IS. This paper studies the use of evidence theory to measure the uncertainty of an HIS. Firstly, a novel distance between two objects in an HIS considering decision attributes is constructed, and secondly, the tolerance relation based on the constructed distance is established in an HIS. And then, belief and plausibility functions are defined by the tolerance relation. Furthermore, several algorithms for attribute reduction are designed on the basis of the defined belief and plausibility functions. In addition, we come to a series of conclusions on the relation among θ-reduction by using decision attributes, θ-belief reduction and θ-plausibility reduction, which further confirms the effectiveness of the designed attribute reduction algorithms. Finally, the experimental results and statistical test show that the defined belief and plausibility functions work well in measuring the uncertainty of an HIS and the designed reduction algorithm is superior to several state-of-the-art algorithms in classification accuracy. These results will provide a wider perspective on the uncertainty of an HIS.
Belief functions and rough sets: Survey and new insights
2022, International Journal of Approximate Reasoning
Rough set theory and belief function theory, two popular mathematical frameworks for uncertainty representation, have been widely applied in different settings and contexts. Despite different origins and mathematical foundations, the fundamental concepts of the two formalisms (i.e., approximations in rough set theory, belief and plausibility functions in belief function theory) are closely related. In this survey article, we review the most relevant contributions studying the links between these two uncertainty representation formalisms. In particular, we discuss the theoretical relationships connecting the two approaches, as well as their applications in knowledge representation and machine learning. Special attention is paid to the combined use of these formalisms as a way of dealing with imprecise and uncertain information. The aim of this work is, thus, to provide a focused picture of these two important fields, discuss some known results and point to relevant future research directions.

View all citing articles on Scopus

View full text

Attribute reduction based on evidence theory in incomplete decision systems

Abstract

Introduction

Section snippets

Incomplete information systems and rough set approximations

Belief structures and belief functions

Attribute reduction in incomplete information systems

Attribute reduction in incomplete decision systems

Conclusions

Acknowledgements

European Journal of Operational Research

Information Sciences

Information Sciences

Expert Systems with Applications

Expert Systems with Applications

Pattern Recognition

Pattern Recognition Letters

Information Sciences

Information Sciences

Information Sciences

European Journal of Operational Research

Computers and Mathematics with Applications

Information Sciences

Theoretical Computer Science

Theoretical Computer Science

Information Sciences

Information Sciences

Mathematical and Computer Modelling

Information Sciences

Information Sciences

Information Sciences

International Journal of Man–Machine Study

Information Sciences

Computers and Mathematics with Applications

Information Sciences

Information Sciences

Journal of Computer and System Sciences

Comparison of dynamic and non-dynamic rough set methods for extracting laws from decision tables

A new approach to attribute reduction of consistent and inconsistent covering decision systems with covering rough sets

Information Sciences

Upper and lower probabilities induced by a multivalued mapping

Annals of Mathematical Statistics

Incomplete Information: Structure, Inference, Complexity

Handling missing values in rough set analysis of multi-attribute and multi-criteria decision problems

Rough approximation by dominance relation

International Journal of Intelligent Systems

A rough set approach to data with missing attribute values

Classification strategies using certain and possible rules

Knowledge acquisition from quantitative data using the rough-set theory

Intelligent Data Analysis