Probabilistic Rough Set Research Articles

Variable Precision Rough Set Model Based on Covering Relation and Uncertainty Measure

The variable precision probabilistic rough set model is based on equivalent relation and probabilistic measure. However, the requirements of equivalent relation and probabilistic measure are too strict to satisfy in some practical applications. In order to solve the above problem, a variable precision rough set model based on covering relation and uncertainty measure is proposed. Moreover, the upper and lower approximation operators of the proposed model are given, while the properties of the operators are discussed.

Analyzing uncertainties of probabilistic rough set regions with game-theoretic rough sets

Probabilistic rough set approach defines the positive, negative and boundary regions, each associated with a certain level of uncertainty. A pair of threshold values determines the uncertainty levels in these regions. A critical issue in the community is the determination of optimal values of these thresholds. This problem may be investigated by considering a possible relationship between changes in probabilistic thresholds and their impacts on uncertainty levels of different regions. We investigate the use of game-theoretic rough set (GTRS) model in exploring such a relationship. A threshold configuration mechanism is defined with the GTRS model in order to minimize the overall uncertainty level of rough set based classification. By realizing probabilistic regions as players in a game, a mechanism is introduced that repeatedly tunes the parameters in order to calculate effective threshold parameter values. Experimental results on text categorization suggest that the overall uncertainty of probabilistic regions may be reduced with the threshold configuration mechanism.

Rough set based attribute reduction with consistent confidence

In order to solve the problem of reduction anomaly in the existing probabilistic rough set models,non-parameterized and parameterized maximum decision entropy measures for attribute reduction were proposed by using the concept of maximum confidence of uncertain object.The monotonicity of the parameterized maximum decision entropy was explained and the relationship between its attribute reduction and other ones was analyzed.The definitions for core and relatively dispensable attributes in the proposed model were also given.Moreover,non-parameterized and parameterized confidence discernibility matrixes were put forward and the difference of classical discernibility matrix and the proposed ones in charactering the uncertain object were discussed.Finally,a case study was given to show the validity of the proposed model.

Optimal scale selection for multi-scale decision tables

Human beings often observe objects or deal with data hierarchically structured at different levels of granulations. In this paper, we study optimal scale selection in multi-scale decision tables from the perspective of granular computation. A multi-scale information table is an attribute-value system in which each object under each attribute is represented by different scales at different levels of granulations having a granular information transformation from a finer to a coarser labelled value. The concept of multi-scale information tables in the context of rough sets is introduced. Lower and upper approximations with reference to different levels of granulations in multi-scale information tables are defined and their properties are examined. Optimal scale selection with various requirements in multi-scale decision tables with the standard rough set model and a dual probabilistic rough set model are discussed respectively. Relationships among different notions of optimal scales in multi-scale decision tables are further analyzed.

An axiomatic characterization of probabilistic rough sets

Probabilistic approaches were successfully applied to the theory of rough sets in recent decades. As a result, various types of probabilistic rough set models have been proposed in constructive approaches. This paper focuses on axiomatic approaches of probabilistic rough sets. Some new properties of probabilistic rough set approximation operators are examined in detail. By investigating the dependence among these properties, the axiom sets characterizing two types of the probabilistic rough set approximation operators are given. Each set of axioms guarantees the existence of an equivalence relation reproducing the corresponding probabilistic rough set approximation operators.

On an optimization representation of decision-theoretic rough set model

Decision-theoretic rough set model can derive several probabilistic rough set models by providing proper cost functions. Learning cost functions from data automatically is the key to improving the applicability of decision-theoretic rough set model. Many region-related attribute reductions are not appropriate for probabilistic rough set models as the monotonic property of regions does not always hold. In this paper, we propose an optimization representation of decision-theoretic rough set model. An optimization problem is proposed by considering the minimization of the decision cost. Two significant inferences can be drawn from the solution of the optimization problem. Firstly, cost functions and thresholds used in decision-theoretic rough set model can be learned from the given data automatically. An adaptive learning algorithm and a genetic algorithm are designed. Secondly, a minimum cost attribute reduction can be defined. The attribute reduction is interpreted as finding the minimal attribute set to make the decision cost minimum. A heuristic approach and a particle swarm optimization approach are also proposed. The optimization representation can bring some new insights into the research on decision-theoretic rough set model.

Non-Monotonic Attribute Reduction in Decision-Theoretic Rough Sets

For most attribute reduction in Pawlak rough set model PRS, monotonicity is a basic property for the quantitative measure of an attribute set. Based on the monotonicity, a series of attribute reductions in Pawlak rough set model such as positive-region-preserved reductions and condition entropy-preserved reductions are defined and the corresponding heuristic algorithms are proposed in previous rough sets research. However, some quantitative measures of attribute set may be non-monotonic in probabilistic rough set model such as decision-theoretic rough set DTRS, and the non-monotonic definition of the attribute reduction should be reinvestigated and the heuristic algorithm should be reconsidered. In this paper, the monotonicity of the positive region in PRS and DTRS are comparatively discussed. Theoretic analysis shows that the positive region in DTRS model may be expanded with the decrease of the attributes, which is essentially different from that in PRS model. Hereby, a new non-monotonic attribute reduction is presented for the DTRS model in this paper, and a heuristic algorithm for searching the newly defined attribute reduction is proposed, in which the positive region is allowed to be expanded instead of remaining unchanged in the process of attribute reduction. Experimental analysis is included to validate the theoretic analysis and quantify the effectiveness of the proposed attribute reduction algorithm.

Matrix-based Approach for Calculation of the Approximations of a Concept under Tolerance Probabilistic Rough Sets Model

Matrix-based Approach for Calculation of the Approximations of a Concept under Tolerance Probabilistic Rough Sets Model

Probabilistic rough set over two universes and rough entropy

In this paper, we discuss the properties of the probabilistic rough set over two universes in detail. We present the parameter dependence or the continuous of the lower and upper approximations on parameters for probabilistic rough set over two universes. We also investigate some properties of the uncertainty measure, i.e., the rough degree and the precision, for probabilistic rough set over two universes. Meanwhile, we point out the limitation of the uncertainty measure for the traditional method and then define the general Shannon entropy of covering-based on universe. Then we discuss the uncertainty measure of the knowledge granularity and rough entropy for probabilistic rough set over two universes by the proposed concept. Finally, the validity of the methods and conclusions is tested by a numerical example.

On relationship between probabilistic rough set and Bayesian risk decision over two universes

We consider a problem of Bayesian risk decision based on probabilistic rough set over two universes. It is a new extension of classical probabilistic rough set on the same universe. We give four rough set models on probabilistic approximation space over two universes. Then we study the interrelationship between Bayesian risk decision and probabilistic rough set models over two universes. The results show that there must exist a kind of Bayesian minimum risk decision problem corresponding to one of the probabilistic rough set models over two universes. In fact, the conclusion also includes some generalized probabilistic rough set models on the same universe by other authors. And at the same time, the principal and validity of the Bayesian risk decision based on probabilistic rough set over two universes are tested by a numerical example of the medical diagnosis systems in detail. The probabilistic rough set approach over two universes gives an effective assistant for decision makers in the context of risk and uncertainty.

Two Semantic Issues in a Probabilistic Rough Set Model

Probabilistic rough set models are quantitative generalizations of the classical and qualitative Pawlak model by considering degrees of overlap between equivalence classes and a set to be approxima...

Game-Theoretic Rough Sets

This article investigates the Game-theoretic Rough Set (GTRS) model and its capability of analyzing a major decision problem evident in existing probabilistic rough set models. A major challenge in...

Credit scoring analysis using a fuzzy probabilistic rough set model

Credit scoring analysis is an important activity, especially nowadays after a huge number of defaults has been one of the main causes of the financial crisis. Among the many different tools used to model credit risk, the recent development of rough set models has proved effective. The original development of rough set theory has been widely generalized and combined with other approaches to uncertain reasoning, especially probability and fuzzy set theories. Since coherent conditional probability assessments cope well with the problem of unifying these different approaches, a merging of fuzzy rough set theory with this subjectivist approach is proposed. Specifically, expert partial probabilistic evaluations are encompassed inside a gradual decision rule structure, with coherence of the conclusion as a guideline. In line with Bayesian rough set models, credibility degrees of multiple premises are introduced through conditional probability assessments. Nonetheless, discernibility with this method remains too fine. Therefore, the basic partition is coarsened by equivalence classes based on the arity of positively, negatively and neutrally related criteria. A membership function, which grades the likelihood of default, is introduced by a peculiar choice of t-norms and t-conorms. To build and test the model, real data related to a sample of firms are used.

Probabilistic model criteria with decision-theoretic rough sets

In dealing with risk in real decision problems, decision-theoretic rough sets with loss functions aim to obtain optimization decisions by minimizing the overall risk with Bayesian decision procedures. Two parameters generated by loss functions divide the universe into three regions as the decision of acceptance, deferment and rejection. In this paper, we discuss the semantics of loss functions, and utilize the differences of losses replace actual losses to construct a new “four-level” approach of probabilistic rules choosing criteria. Ten types of probabilistic rough set models can be generated by the “four-level” approach and form two groups of models: two-way probabilistic decision models and three-way probabilistic decision models. A reasonable decision with these criteria is demonstrated by an illustration of oil investment.

Preface: A rough set approach to data mining

In recent years, we have observed rapid progress in research on data mining using rough sets. Rough set theory, invented by Zdzislaw Pawlak in 1982, is especially well-suited for research in data mining and related areas such as granular computing, intelligent information systems, nonclassical logics, web mining, and uncertainty reasoning. The scope of the International Conference on Rough Sets and Current Trends in Computing (RSCTC 2008) held in Akron, Ohio, included all of these aspects of innovative theories, methodologies, and applications of rough set theory. Five articles presented at the RSCTC 2008 were selected for publication in the International Journal of Intelligent Systems. An article on experiments with rough set approach to face recognition by X. Chen and W. Ziarko reports results of experiments on face recognition based on variable precision rough sets. The objective of this article was to explore a methodology to select the most discriminative features from facial photographs and to develop a decision-table classifier to achieve high face recognition rate. Another article on probabilistic rough sets (Probabilistic rule induction with the ERS data mining system) by J. W. Grzymala-Busse and Y. Yao presents a methodology for changing input data sets so that LERS (Learning from Examples based on Rough Sets), a standard rough set theory-based data mining system, can induce positive, boundary, and possible rule sets associated with two typical parameters, alpha and beta. A class of dynamic rough partitive algorithms by G. Peters and R. Weber discusses three rough partitive algorithms and extends these algorithms by dynamic elements. A new class of dynamic rough partitive algorithms is applicable to data sets with a variable data structure. The article (Granular computing in the frame of rough mereology. A case study on the classification of data into decision categories by means of granular reflections of data) by L. Polkowski and P. Artiemjew presents applications of a granular reflection of data using rough mereology. The authors illustrate their ideas by reporting results of experiments on a well-known data set on breast cancer from Wisconsin. Finally, the article entitled from data to classification rules and actions by Z. W. Ras and A. Dardzinska discusses action rules, defined as describing possible transitions of objects from one state to another with respect to some given attribute. Such rules

Two Semantic Issues in a Probabilistic Rough Set Model

Probabilistic rough set models are quantitative generalizations of the classical and qualitative Pawlak model by considering degrees of overlap between equivalence classes and a set to be approximated. The extensive studies, however, have not sufficiently addressed some semantic issues in a probabilistic rough set model. This paper examines two fundamental semantics-related questions. One is the interpretation and determination of the required parameters, i.e., thresholds on probabilities, for defining the probabilistic lower and upper approximations. The other is the interpretation of rules derived from the probabilistic positive, boundary and negative regions. We show that the two questions can be answered within the framework of a decision-theoretic rough set model. Parameters for defining probabilistic rough sets are interpreted and determined in terms of loss functions based on the well established Bayesian decision procedure. Rules constructed from the three regions are associated with different actions and decisions, which immediately leads to the notion of three-way decision rules. A positive rule makes a decision of acceptance, a negative rule makes a decision of rejection, and a boundary rules makes a decision of deferment. The three-way decisions are, again, interpreted based on the loss functions. (This work is partially supported by a Discovery Grant from NSERC Canada. The author thanks the reviewers for their constructive comments.)

Risk Decision Making Based on Decision-theoretic Rough Set: A Three-way View Decision Model

Rough set theory has witnessed great success in data mining and knowledge discovery, which provides a good support for decision making on a certain data. However, a practical decision problem always shows diversity under the same circumstance according to different personality of the decision makers. A simplex decision model can not provide a full description on such diverse decisions. In this article, a review of Pawlak rough set models and probabilistic rough set models is presented, and a three-way view decision model based on decision-theoretic rough set model is proposed, in which optimistic decision, pessimistic decision, and equable decision are provided according to the cost of misclassification. The thresholds of probabilistic inclusion are calculated based on minimization of risk cost under respective decision bias. The study not only presents a new theoretic decision model considering the different personality of the decision makers, but also provides a practical explanation and an illustrative example on diverse risk bias decision.

Research on Rough Set Model Based on Golden Ratio

How to make decision with pre-defined preference-ordered criteria also depends on the environment of the problem. Dominance rough set model is suitable for preference analysis and probabilistic rough set introduces probabilistic approaches to rough sets. In this paper, new dominance rough set rough set models are given by taking golden ratio into account. Also, we present steps to make decision using new dominance rough set models.

Game-Theoretic Rough Sets

This article investigates the Game-theoretic Rough Set (GTRS) model and its capability of analyzing a major decision problem evident in existing probabilistic rough set models. A major challenge in the application of probabilistic rough set models is their inability to formulate a method of decreasing the size of the boundary region through further explorations of the data. To decrease the size of this region, objects must be moved to either the positive or negative regions. Game theory allows a solution to this decision problem by having the regions compete or cooperate with each other in order to find which is best fit to be selected for the move. There are two approaches discussed in this article. First, the region parameters that define the minimum conditional probabilities for region inclusion can either compete or cooperate in order to increase their size. The second approach is formulated by having classification approximation measures compete against each other. We formulate a learning method using the GTRS model that repeatedly analyzes payoff tables created from approximation measures and modified conditional risk strategies to calculate parameter values.

The superiority of three-way decisions in probabilistic rough set models

Three-way decisions provide a means for trading off different types of classification error in order to obtain a minimum cost ternary classifier. This paper compares probabilistic three-way decisions, probabilistic two-way decisions, and qualitative three-way decisions of the standard rough set model. It is shown that, under certain conditions when considering the costs of different types of miss-classifications, probabilistic three-way decisions are superior to the other two.