Probabilistic Rough Set Model Research Articles

Reduction Algorithm of Interval-Valued Intuitionistic Fuzzy Probability Rough Set Under Dominant Relation

Attribute reduction is a part of the most basic and significant research contents in rough set theory. The so-called attribute reduction is the smallest independent subset that keeps certain properties of information table unchanged. In this paper, four interval-valued intuitionistic fuzzy probabilistic rough set models and their natures are given on the basis of the dominant-, inferior- and interval-valued intuitionistic fuzzy probabilistic rough set models. At the same time, the interval-valued intuitionistic fuzzy numbers are transformed by fuzzy degree, and the approximate accuracy and approximate classification quality under the dominance relation are used for reduction, at last, the feasibility is verified by an example.

Multigranulation fuzzy probabilistic rough set model on two universes

Fuzzy probabilistic rough set is an important mathematical model in rough set theory and has been widely used in applications. Although many studies have been taken on this topic, the existing fuzzy probabilistic rough set models are insufficient for dealing with data noise and data discretization. This motivates us to develop a better method to handle complex data. In this paper, we propose a novel fuzzy probabilistic rough set model on two universes by using λ-fuzzy sets. To further extend application field, we define upper and lower approximation operators of fuzzy probabilistic rough set in a multigranulation context and propose a novel multigranulation fuzzy probabilistic rough set model on two universes. Moreover, some interesting properties of the models are studied. The effectiveness and superiority of them are illustrated by numerical examples and experiments compared with existing models. Experimental results show that the proposed models have better performance in classification and have potential applications in decision making problems.

Incremental updating probabilistic approximations under multi-level and multi-dimensional variations in hybrid incomplete decision systems

The data types of practical application systems are various (for example, Boolean, categorical, numerical, set-valued, interval-valued and incomplete, etc.), and such complex data systems exist widely in the real world; In addition, the data is dynamic in the process of collection and screening, not only the number of objects will change, but also the number of features will vary, which leads to the knowledge being constantly changed and needing to be updated with the collation process. In this paper, aiming at the dynamic change of data in the hybrid incomplete decision system (HIDS), we mainly focus on researching the incremental updating theory and method of probabilistic approximations under the multi-level and multi-dimensional variations of objects and attributes. Firstly, for the different binary relations of multiple data types in HIDS, a normalized combination relationship-based probabilistic rough set model is proposed. Next, multi-level and multi-dimensional variations (MLMDV) of objects and attributes are analyzed; for MLMDV of the object set and the attribute set in HIDS, dynamic knowledge updating mechanisms are researched, and a matrix-based incremental algorithm for updating probabilistic approximations is designed to avoid the repeated calculation of the static algorithm and improve efficiency. Finally, a series of experiments are conducted to evaluate the efficiency of the proposed method. The experimental results of 9 data sets show that the proposed incremental algorithm can effectively update the knowledge for the multi-level and multi-dimensional variants of objects and attributes, and is superior to the static knowledge acquisition method.

A novel three-way decision approach in decision information systems

Given that the three-way decision (3WD) theory provides a scientific solution and reasonable semantic interpretation for solving multi-criteria decision-making (MCDM) problems, this paper presents a new 3WD model and applies it to realistic MCDM problems. Since the fuzzy neighborhood operator is effective in handling uncertain data, we first define a pair of probabilistic approximation operators via the constructed α-fuzzy neigh borhood class. Then, a 3WD model is explored in decision information systems to address the related issues in the proposed fuzzy probabilistic rough set model. In light of the above works, we establish a novel approach to classify and rank applicants for enterprise talent recruitment problems. Instead of the rule of tie-breaking, a way is given to separate applicants into three parts by combining decision attributes, and the feasibility of the approach is confirmed as well. Finally, by using a data set in the UCI database, the results of comparative and experimental analyses demonstrate that the constructed MCDM approach owns better performances in terms of effectiveness and stability.

Sentence Similarity Calculation Based on Probabilistic Tolerance Rough Sets

Sentence similarity calculation is one of the important foundations of natural language processing. The existing sentence similarity calculation measurements are based on either shallow semantics with the limitation of inadequately capturing latent semantics information or deep learning algorithms with the limitation of supervision. In this paper, we improve the traditional tolerance rough set model, with the advantages of lower time complexity and becoming incremental compared to the traditional one. And then we propose a sentence similarity computation model from the perspective of uncertainty of text data based on the probabilistic tolerance rough set model. It has the ability of mining latent semantics information and is unsupervised. Experiments on SICK2014 task and STSbenchmark dataset to calculate sentence similarity identify a significant and efficient performance of our model.

Object-First Incremental Algorithms for Updating Approximations in Multi-Granulation Fuzzy Probabilistic Rough Sets Over Two Universes

Multi-granulation fuzzy probabilistic rough set (MFPRS) model over two universes has various applications in recent years. However, data are typically changing with time in real-life applications. When data changes, people have to spend a lot of unnecessary time to recalculate the approximations in MFPRS over two universes. To address this problem, we design incremental algorithms based on the object-first approach for updating approximations in MFPRS over two universes while the objects are added into or removed from the two universes. Firstly, considering a case where the equivalence class of the object with respect to the fuzzy relation could be empty, we introduce a model of type-empty MFPRS over two universes. Then, based on the matrix, a traditional algorithm is introduced to calculate approximations in MFPRS over two universes. In addition, by optimizing the traditional algorithm, we design the object-first algorithm which could calculate approximations more efficient in MFPRS over two universes. After that, based on the object-first algorithm, four object-first incremental algorithms are designed for updating approximations in MFPRS over two universes while the objects are added into or removed from two universes. Finally, experimental results show that the proposed object-first incremental algorithms for updating approximations in MFPRS over two universes are more efficient in comparison with the non-incremental algorithms and the traditional incremental algorithms.

A novel sequential three-way decision model with autonomous error correction

As an approach to granular computing, the sequential three-way decision (S3WD) model has been widely studied in practical applications. In terms of improving the accuracy of the S3WD model, existing studies have achieved fruitful results. However, the two types of classification errors and two types of uncertain classifications caused by a probabilistic rough set model have received less consideration, which will result in a higher error classification rate (ECR) in the decision process. In this paper, from the perspective of the subdivision of granules, a new sequential three-way decision model with autonomous error correction (S3WD-AEC) is proposed to reduce the ECR. First, two types of errors correction and two types of effective classifications in the S3WD model are defined. Next, according to the process of information granulation, four subdivisions of equivalence classes are discussed in detail. Subsequently, the total ECR composed of the positive and negative regions in each granularity layer is proved to gradually decrease with the subdivision of the equivalence classes. Then, during the S3WD process, four commonly used clustering algorithms are introduced to select a portion of the equivalence classes near the boundary region for further subdivision, implementing an error correction for some misclassified objects. Finally, the experimental results show that the S3WD-AEC model has a smaller ECR compared with the S3WD model.

Fuzzy entropies for class-specific and classification-based attribute reducts in three-way probabilistic rough set models

There exist two formulations of the theory of rough sets, consisting of the conceptual formulations and the computational formulations. Class-specific and classification-based attribute reducts are two crucial notions in three-way probabilistic rough set models. In terms of conceptual formulations, the two types of attribute reducts can be defined by considering probabilistic positive or negative region preservations of a decision class and a decision classification, respectively. However, in three-way probabilistic rough set models, there are few studies on the computational formulations of the two types of attribute reducts due to the non-monotonicity of probabilistic positive and negative regions. In this paper, we examine the computational formulations of the two types of attribute reducts in three-way probabilistic rough set models based on fuzzy entropies. We construct monotonic measures based on fuzzy entropies, from which we can obtain the computational formulations of the two types of attribute reducts. On this basis, we develop algorithms for finding the two types of attribute reducts based on addition-deletion method or deletion method. Finally, the experimental results verify the monotonicity of the proposed measures with respect to the set inclusion of attributes and show that class-specific attribute reducts provide a more effective way of attribute reduction with respect to a particular decision class compared with classification-based attribute reducts.

A novel decision-making approach based on three-way decisions in fuzzy information systems

The decision-making theory with three-way decisions (3WD) improves the either-or decision results of the traditional multi-criteria decision-making (MCDM) methods and provides a more reasonable decision basis. This paper presents a novel 3WD method to solve MCDM problems in fuzzy information systems. Considering the usefulness of fuzzy neighborhood operators when dealing with fuzzy numerical data, we define a reflexive fuzzy α-neighborhood operator for overcoming the weaknesses of the existing fuzzy β-neighborhood operators in some applications. A probabilistic rough fuzzy set model and MCDM-based 3WD model are established by means of the constructed fuzzy ε-neighborhood classes. We give a determination method of the conditional probability according to data tables of MCDM problems. Particularly, the steps and algorithm of 3WD method are proposed. Then the feasibility and validity of the method are illustrated by the validity test and the comparative analysis on an example of project investments. Finally, we demonstrate the stability of the method through the experimental analysis on different data sets.

Double-quantitative variable consistency dominance-based rough set approach

Rough set model with double quantification satisfies the requirement of quantitative information in practical applications, it has better fault tolerance than probabilistic rough set model considering only relative quantification and graded rough set model considering only absolute quantification. In this paper, two kinds of consistency levels are introduced from the perspective of double quantification in an ordered information system, namely relative quantitative consistency level and absolute quantitative consistency level. The single-quantitative variable consistency dominance-based rough set models based on these two kinds of quantitative consistency levels and their basic properties with the relevant three-way decision rules are discussed respectively in an ordered information system. Moreover, two kinds of double-quantitative variable consistency dominance-based rough set models and their basic properties with the relevant decision rules based on these two kinds of quantitative consistency levels are introduced. A consistency analysis of decision making in a practical case study is used to illustrate and interpret the double-quantitative variable consistency rough set models and the related decision rules in the ordered information system. The obvious shortcomings of dominance-based rough set approach (DRSA) without quantitative information are compared to explain the advantages of the quantitative variable consistency dominance-based rough sets with the two consistency levels in the practical case study.

A computational formulation of distribution reducts in probabilistic rough set models

Conceptual and computational formulations are the two sides of the theory of rough sets. Conceptual formulation focuses on the meaning and interpretation of the concepts. Computational formulation focuses on procedures and algorithms for constructing these notions. In probabilistic rough set models, a distribution reduct is defined as a minimal subset of attributes that preserves the probabilistic lower or upper approximations of all decision classes. The definition is a conceptual formulation that provides an essential understanding of distribution reducts, but it does not directly give a computationally efficient method. In this paper, we study the computational formulation of distribution reducts in probabilistic rough set models by constructing monotonic measures, resulting in a more efficient computational method. We first construct two monotonic measures called the probabilistic low and upper approximation distribution measures, respectively, from which the computational formulation of distribution reducts can be obtained. We then propose the granularity-based probabilistic low and upper approximation distribution measures to evaluate the significance of attributes more effectively. On this basis, we develop two algorithms for finding distribution reducts based on addition–deletion method and deletion method, respectively. Finally, the experimental results show the effectiveness of the proposed measures.

Double-quantitative multigranulation decision-theoretic rough fuzzy set model

Probabilistic rough set model and graded rough set model are used to measure relative quantitative information and absolute quantitative information between equivalence classes and basic concepts, respectively. Since fuzzy concepts are more common in real life than classical concepts, how to use relative and absolute quantitative information to determine fuzzy concepts is a extremely important research topic. In this study, we propose a double-quantitative decision theory rough fuzzy set frame based on the fusion of decision theory rough set and graded rough set, and the framework mainly studies the fuzzy concepts in multigranulation approximation spaces. Three pairs of double-quantitative multigranulation decision theory rough fuzzy set models are established. Some basic characteristics of these models are discussed. The decision rules including relative and absolute quantitative information are studied. The intrinsic relationship between the double-quantitative decision theory rough fuzzy set and the multigranulation rough set is analyzed. Finally, an illustrative case of medical diagnosis is conducted to explain and evaluate the dual quantitative decision theory approach.

Three-way decision perspectives on class-specific attribute reducts

In rough set theory, a decision class (i.e., a subset of objects) is approximated by three pair-wise disjoint positive, boundary, and negative regions. The concept of three-way decisions is introduced to provide a new interpretation of the three regions. We construct acceptance, non-commitment, and rejection rules, respectively, from the positive, boundary, and negative regions. The notion of class-specific attribute reducts concerns a minimal set of attributes used in constructing such rules. Existing studies on class-specific attribute reducts only consider the positive region and hence only the acceptance rules. In many situations such as medical diagnosis, we are also interested in negative rules or rule-out rules. This motivates the present study on three-way decision perspectives on class-specific attribute reducts. In addition to positive-region based attribute reducts, we study negative-region and positive-and-negative-region based attribute reducts. We investigate relationships among the three types of reducts. Although the three types of reducts are equivalent in consistent decision tables, they are not equivalent in inconsistent decision tables. By extending the framework, we study the three types of class-specific attribute reducts in probabilistic rough set models and their relationships. Finally, we give a general definition of class-specific attribute reducts.

Parameter Selection and Uncertainty Measurement for Variable Precision Probabilistic Rough Set

In this paper, we consider the problem of parameter selection and uncertainty measurement for a variable precision probabilistic rough set. Firstly, within the framework of the variable precision probabilistic rough set model, the relative discernibility of a variable precision rough set in probabilistic approximation space is discussed, and the conditions that make precision parameters α discernible in a variable precision probabilistic rough set are put forward. Concurrently, we consider the lack of predictability of precision parameters in a variable precision probabilistic rough set, and we propose a systematic threshold selection method based on relative discernibility of sets, using the concept of relative discernibility in probabilistic approximation space. Furthermore, a numerical example is applied to test the validity of the proposed method in this paper. Secondly, we discuss the problem of uncertainty measurement for the variable precision probabilistic rough set. The concept of classical fuzzy entropy is introduced into probabilistic approximation space, and the uncertain information that comes from approximation space and the approximated objects is fully considered. Then, an axiomatic approach is established for uncertainty measurement in a variable precision probabilistic rough set, and several related interesting properties are also discussed. Thirdly, we study the attribute reduction for the variable precision probabilistic rough set. The definition of reduction and its characteristic theorems are given for the variable precision probabilistic rough set. The main contribution of this paper is twofold. One is to propose a method of parameter selection for a variable precision probabilistic rough set. Another is to present a new approach to measurement uncertainty and the method of attribute reduction for a variable precision probabilistic rough set.

Three-way decisions approach to multiple attribute group decision making with linguistic information-based decision-theoretic rough fuzzy set

This paper investigates the decision-theoretic rough sets involved multiple decision-makers with linguistic evaluation information for the uncertainty and risk under the framework of three-way decisions theory. We first discuss the probabilistic rough fuzzy set model with linguistic information for the value of objects on the universe about the attribute set. Subsequently, we present the decision-theoretic rough fuzzy set with linguistic information for the loss functions. Then, we establish the three-way decisions based on linguistic information-based decision-theoretic rough fuzzy set as well as the corresponded decision rules that satisfy minimum risk are given. Furthermore, we consider a kind of multiple attribute group decision making problem with linguistic information and then present a new decision model using the decision-theoretic rough fuzzy set with linguistic information-based three-way decisions. Finally, the proposed decision model and method is applied to an emergency decision making problem of unconventional emergency events, the steps and the principle of the proposed method are illustrated by a numerical example with the background of emergency decision-making.

Measuring Uncertainty of Probabilistic Rough Set Model From Its Three Regions

The probabilistic rough set model is a generalized rough set model, which was proposed to improve the fault tolerance ability of the classical Pawlak’s rough set model. Compared with the classical Pawlak’s rough set model, the probabilistic rough set model and its several special forms (such as variable precision rough set model, Bayesian rough set model, and decision-theoretic rough set model) divide a domain into three regions by two parameters ( ${\alpha , \beta }$ ), and the objects belong to every region with a certain uncertainty. With the increase of information (attributes), the boundary region of the probabilistic rough set model may become smaller, bigger, or remain unchanged. This paper aims first to study some more complex uncertainty of the probabilistic rough set model. Then the uncertainty of the three regions is defined and analyzed. Besides, the changing regularities of uncertainty of a target concept in the probabilistic rough set model are discovered. Finally, three kinds of incremental information are defined, and their judging theorems are proposed. These results could enrich and improve rough set theory to deal with uncertain information systems.

Three-way decision model with two types of classification errors

For an uncertain target set, three disjoint regions, i.e., positive, negative, and boundary regions, are established in a rough approximation space. To improve the fault-tolerance ability of the rough set model, a probabilistic rough set model with a pair of thresholds is proposed, and two types of classification errors are introduced from the viewpoint of probability and statistics. A decision-theoretic rough set model is proposed to minimize the decision risk, and a pair of thresholds of the probabilistic rough set model can be calculated by the given cost parameters. In this paper, first, two types of classification errors and two types of uncertain classifications are defined in the probabilistic rough set model. Then, by considering four cost parameters that characterize the cost of two types of classification errors and two types of uncertain classifications, a pair of thresholds of the probabilistic rough set model can again be obtained. Finally, with the increase in new attributes, the change regularities of the total cost with changing rough approximation spaces are discussed in detail, and experimental results show that the proposed model for three-way decision is reasonable and effective. These results further enrich three-way decision theory to effectively deal with uncertain classification problems.

A three-way decisions model with probabilistic rough sets for stream computing

Stream computing paradigm, with the characteristics of real-time arrival and departure, has been admitted as a major computing paradigm in big data. Relevant theories are flourishing recently with the surge development of stream computing platforms such as Storm, Kafka and Spark. Rough set theory is an effective tool to extract knowledge with imperfect information, however, related discussions on synchronous immigration and emigration of objects have not been investigated. In this paper, stream computing learning method is proposed on the basis of existing incremental learning studies. This method aims at solving challenges resulted from simultaneous addition and deletion of objects. Based on novel learning method, a stream computing algorithm called single-object stream-computing-based three-way decisions (SS3WD) is developed. In this algorithm, the probabilistic rough set model is applied to approximate the dynamic variation of concepts. Three-way regions can be determined without multiple scans of existing information granular. Extensive experiments not only demonstrate better efficiency and robustness of SS3WD in the presence of objects streaming variation, but also illustrate that stream computing learning method is an effective computing strategy for big data.

Minimal decision cost reduct in fuzzy decision-theoretic rough set model

Decision-theoretic rough set model interprets the parameters of existing probabilistic rough sets by introducing Bayesian theory to minimize the risks of classification. It is a representative probabilistic rough set model. As an important extension of decision-theoretic rough set model, fuzzy decision-theoretic rough set model can deal with numerical data directly. To investigate the attribute reduction in fuzzy decision-theoretic rough set model, we propose two attribute reduction approaches: global reduction and local reduction. Global reduction can keep the cost of all the decision classes unchanged or decreased. Moreover, local reduction can make the cost of single decision class unchanged or decreased. In this paper, we use two different heuristic functions to implement heuristic algorithms. Heuristic algorithms are used to find the minimal subset of condition attributes which minimizes the (P) cost, (B) cost and (N) cost in fuzzy decision theoretic-rough set model. Six UCI datasets are employed to test our heuristic algorithms. The experimental results indicate that global reduction and local reduction can minimize the decision cost in fuzzy decision-theoretic rough set model effectively. After reduction, the numbers of condition attributes are reduced greatly. On the other hand, global reduction and local reduction both can decrease the numbers of (P) rule and (B) rule on the whole. The results show that when we reduce the decision cost, we will also reduce the rule numbers as well.

A three-way decision making approach to malware analysis using probabilistic rough sets

Malware analysis aims to identify malware by examining applications behaviour on the host operating system. A common issue in malware analysis is how to mitigate and handle the false decisions such as false positives. Existing approaches which are based on two-way decisions (such as acceptance and rejection) for classifying applications behaviour result in two shortcomings. Firstly, the two-way decisions are rigid and strict in the sense that they demand that a classification decision must be made irrespective of the quality of available information. This potentially leads to wrong classification decisions whenever we do not have sufficient and complete information. Secondly, two-way decisions do not involve any explicit mechanism for dealing with the false decisions at the model level. The existing approaches generally work like an add-on to learning models and are only exercised after incorrect decisions are being made by the learning models. This results in additional processing and increases the complexity of the task. In this paper, we investigate a three-way decision making approach based on decisions of acceptance, rejection or deferment. The added deferment decision option provides flexibility for delaying a certain decision whenever we do not have sufficient information. Moreover, it aims to mitigate the false decisions at the model level by determining a tradeoff between different properties of decision making such as accuracy, generality and uncertainty. We considered three-way decisions based on two probabilistic rough set models, namely, game-theoretic rough sets (GTRS) and information-theoretic rough sets (ITRS) in this study. An architecture of malware analysis realized with probabilistic rough sets based three-way decisions is proposed. A new algorithm termed as sequentially stackable linux security (SSLS) based on the proposed architecture is presented. Experimental results on the system call sequences from the UNM data set advocate for the use of three-way decisions in malware analysis.