Set-valued Information Systems Research Articles

Uncertainty Measures in Fuzzy Set-Valued Information Systems Based on Fuzzy β-Neighborhood Similarity Relations

Uncertainty measures are instrumental in describing the classification abilities in information systems, and uncertain information has been measured and processed with granular computing theory. While the fuzzy set-valued information system is a generalization of fuzzy information systems, the relationship between the information granulation and the uncertainty in fuzzy set-valued information systems remains to be studied. This paper probes into uncertainty measures in fuzzy set-valued information systems based on the fuzzy [Formula: see text]-neighborhood and the idea of granulation. Specifically, the fuzzy [Formula: see text]-neighborhood similarity relation that reflects the similarity between two objects is defined in terms of the nearness degree. We propose the concepts of information granules and granular structures induced by fuzzy [Formula: see text]-neighborhood similarity relations, based on which we introduce the granularity measures and rough approximation measures of granular structures in fuzzy set-valued information systems. Given the situation of decision information systems, we propose the granularity-based rough approximation measures by combining granularity measures with rough approximation measures. Experiment results and effectiveness analysis show that the measures we proposed are reasonable and feasible.

Rough Pythagorean fuzzy approximations with neighborhood systems and information granulation

A rough set approximates a subset of a universal set on the basis of some binary relation and is significant for the reduction of attributes of an information system. On the other hand, a Pythagorean fuzzy set provides information about the extent of truthness and falsity of a statement. Both these theories deal with different forms of uncertainty and can be united to get their combined benefits. This paper contributes a new blend of rough sets and Pythagorean fuzzy sets namely, rough Pythagorean fuzzy sets. This model can encapsulate two distinct types of uncertainties that appear in imprecise available data through the approximation of Pythagorean fuzzy sets in crisp approximation space. We define rough Pythagorean fuzzy sets on the basis of equivalence relation and generalize it for arbitrary binary relations. The manuscript also provides a general framework to study rough Pythagorean fuzzy approximations of different k-step neighborhood systems. The identities and properties of upper and lower rough Pythagorean fuzzy approximation operators are discussed for the neighborhood systems induced from different types of binary relations. Further, we develop algorithms that compute reduct family, core and rough Pythagorean fuzzy approximations of single-valued and set-valued information systems using indiscernibility relation and similarity relation, respectively. These algorithms are subjected to simple yet interesting applications.

Outlier detection for set-valued data based on rough set theory and granular computing

Outlier detection has been broadly used in industrial practices such as public security and fraud detection, etc. Outlier detection from various perspectives against different backgrounds has been proposed. However, most of outlier detection consider categorical or numerical data. There are few researches on outlier detection for set-valued data, and a set-valued information system (SVIS) is a proper way of tackling the problem of missing values in data sets. This paper investigates outlier detection for set-valued data based on rough set theory (RST) and granular computing (GrC). First, the similarity between two information values in an SVIS is introduced and a variable parameter to control the similarity is given. Then, the tolerance relations on the object set are defined, and based on this tolerance relation, θ-lower and θ-upper approximations in an SVIS are put forward. Next, the outlier factor in an SVIS is presented and applied to various data sets. Finally, outlier detection method for set-valued data based on RST and GrC is proposed, and the corresponding algorithms are designed. Through numerical experiments based on UCI, the designed algorithm is compared with six other detection algorithms. The experimental results show the designed algorithm is arguably the best choice under the context of an SVIS. It is worth mentioning that for a comprehensive comparison, we use two criteria: AUC value and F-1 measure, to show the superiority of the designed algorithm.

Uncertainty measurement for probabilistic set-valued data: Gaussian kernel method

The uncertainty of information plays an important role in practical applications. Uncertainty measurement (UM) can help us in disclosing the substantive characteristics of information. Probabilistic set-valued data is an important class of data in machine learning. UM for probabilistic set-valued data is worth studying. This paper measures the uncertainty of a probability set-valued information system (PSVIS) by means of its information structures based on Gaussian kernel method. According to Bhattacharyya distance, the distance between objects in each subsystem of a PSVIS is first built. Then, the fuzzy Tcos-equivalence relations in a PSVIS by using Gaussian kernel method are obtained. Next, information structures in a PSVIS are defined. Moreover, dependence between information structures is investigated by using the inclusion degree. As an application for the information structures, UM in a PSVIS is investigated. Finally, to evaluate the performance of the investigated measures, effectiveness analysis is performed from dispersion analysis, correlation analysis, and analysis of variance and post-hoc test.

Fuzzy set-valued information systems and their homomorphisms based on data compression

A fuzzy set-valued information system (FSVIS) is a special information system (IS) where the value of an object under each attribute or each attribute value is a fuzzy set. Homomorphism is a powerful mathematical tool to deal with FSVISs, which can be used to study relationships among them. Based on data compression, we obtain some characterizations about FSVISs and their homomorphisms. First, some homomorphisms between FSVISs are introduced. After that, attribute reduction based on tolerance relation in a FSVIS is studied. Eventually, we get invariant characterizations of FSVISs based on some special homomorphisms under data compression.

Feature selection for a set-valued decision information system based on fuzzy rough iterative computation model

To deal with set-valued data, one usually exploits the information value similarity, which is fed back to the object set. Alternatively, it is feasible but scarcely considered to handle set-valued data from the perspective of the information-value similarity which is fed back to the feature set. Based on this perspective, this paper studies feature selection for a set-valued decision information system (SVDIS) by means of fuzzy rough iterative computation model. In order to describe the similarity between objects, the fuzzy symmetric relations in an SVDIS are first defined, and a variable parameter to control the similarity is introduced. Then, new fuzzy rough set model for set valued data is proposed, and this model employs the iterative computation strategy to define fuzzy rough approximations and dependency functions. Next, fuzzy rough iterative computation model is established by using the iteration of fuzzy positive region and fuzzy dependency. Moreover, fuzzy rough iterative computation model is applied to the feature selection for set-valued data, and the corresponding algorithm is given. Finally, experiments are carried out to evaluate the performance of the given algorithm. The experimental results indicate that the given algorithm is more effective than some existing algorithms.

Feature selection for incomplete set-valued data

Set-valued data is a significant kind of data, such as data obtained from different search engines, market data, patients’ symptoms and behaviours. An information system (IS) based on incomplete set-valued data is called an incomplete set-valued information system (ISVIS), which generalized model of a single-valued incomplete information system. This paper gives feature selection for an ISVIS by means of uncertainty measurement. Firstly, the similarity degree between two information values on a given feature of an ISVIS is proposed. Then, the tolerance relation on the object set with respect to a given feature subset in an ISVIS is obtained. Next, λ-reduction in an ISVIS is presented. What’s more, connections between the proposed feature selection and uncertainty measurement are exhibited. Lastly, feature selection algorithms based on λ-discernibility matrix, λ-information granulation, λ-information entropy and λ-significance in an ISVIS are provided. In order to better prove the practical significance of the provided algorithms, a numerical experiment is carried out, and experiment results show the number of features and average size of features by each feature selection algorithm.

Information structures in a fuzzy set-valued information system based on granular computing

A fuzzy set-valued information system (FSVIS) refers to an information system (IS) whose information values are fuzzy sets. This article investigates information structures (ISts) in a FSVIS based on granular computing (GrC). First, FSVISs and homomorphism between them are introduced. Next, ISts in a FSVIS are described. The dependence and information distance between ISts are discussed, and characterizations of ISts in a FSVIS are acquired. Additionally, as an application of ISts in a FSVIS, the entropy measure of uncertainty for a FSVIS is discussed, and on the basis of this uncertainty measure, the optimal selection of ISts is investigated. The obtained consequence will be conducive to establishing a framework for GrC in a FSVIS. Last, the difference between this paper and the references for ISts in this paper is compared and discussed.

Uncertainty measurement for a fuzzy set-valued information system

Uncertainty measurement (UM) can offer new visual angle for data analysis. A fuzzy set-valued information system (FSVIS) which means an information system (IS) where its information values are fuzzy sets. This article investigates UM for a FSVIS. First, a FSVIS is introduced. Then, the distance between two information values of each attribute in a FSVIS is founded. After that, the tolerance relation induced by a given subsystem is acquired by this distance. Moreover, the information structure of this subsystem is brought forward. Additionally, measures of uncertainty for a FSVIS are explored. Eventually, to verify the validity of these measures, statistical effectiveness analysis is carried out. The obtained results will help us understand the intrinsic properties of uncertainty in a FSVIS.

Local Generalized Multigranulation Variable Precision Tolerance Rough Sets and its Attribute Reduction

In the era of big data, as for an important granular computing model, rough set model is an important tool for us to deal with data. As a kind of extension of classical rough sets, multigranulation rough sets have two forms, including optimistic and pessimistic cases. However, these two models have their shortcomings, one is too loose, and the other is too strict. To overcome the above shortcomings, based on the concept of local multigranulation tolerance rough sets in set-valued information systems, the local generalized multigranulation variable precision tolerance rough sets model by introducing characteristic function is established. Then the related properties are studied and proved. In addition, we define the concepts of lower approximate quality, inner and outer importance of attribute according to different granularity structures in set-valued decision information systems because different granularity structures have different effectives on the decision classes. Finally, the local attribute reduction algorithm and the global attribute reduction algorithm of local generalized multigranulation variable precision tolerance rough sets in set-valued decision information systems are given, and the effectiveness of the algorithms is proved by using UCI data sets.

Semi-monolayer covering rough set on set-valued information systems and its efficient computation

For set-valued information systems, there are many original dot-based approximation models based on tolerance relations and other developed tolerance relations. Because of the lack of efficient algorithms, they are not to accommodate the bigger and bigger set-valued information table. Therefore, it is a real challenge on how to efficiently calculate a high-quality approximation set in set-valued information systems. To address the challenge, we propose reliable approximation operators based on semi-monolayer covering for set-valued information systems. Benefiting from considerable research about tolerance rough set models and covering rough set models, the proposed approximation operators used a piecewise design to effectively reduce the negative effects of the set-valued records and provided high-quality approximation sets for set-valued information systems. Furthermore, the reliable semi-monolayer covering approximation sets are more easily granulated and efficiently calculated than before. Based on the equivalent granule-based forms, the corresponding granular algorithms are designed for the improved approximation sets. The experiments on some UCI data sets show the improved approximation sets are high quality and efficient computational in set-valued information systems.

Measures of Uncertainty for an Incomplete Set-Valued Information System With the Optimal Selection of Subsystems: Gaussian Kernel Method

A set-valued information system (SVIS) with missing values is known as an incomplete set-valued information system (ISVIS). This article focuses on studying uncertainty measurement for an ISVIS and the optimal selection of subsystems by means of Gaussian kernel. First, the distance between two information values on each attribute in an ISVIS is put forward. Second, the fuzzy $T_{cos}$ -equivalence relation induced by a given subsystem is proposed based on Gaussian kernel. Next, some tools are used to measure the uncertainty of an ISVIS. Moreover, effectiveness analysis is done from a statistical point of view. In the end, the optimal selection of subsystems based on $\delta $ -information granulation and $\delta $ -information amount is given. These results will help us comprehend nature of uncertainty in an ISVIS.

Uncertainty Measurement for Fuzzy Set-Valued Data

Uncertainty measurement (UM) can offer new visual angle for data analysis. A fuzzy set-valued information system (FSVIS) indicates an information system (IS) where its information values are fuzzy sets. This article investigates UM for fuzzy set-valued data based on Chebyshev distance. First, the distance between information values is founded in a given subsystem. After that, the tolerance relation induced by this subsystem is obtained by means of this distance. Moreover, the information structure of this subsystem is proposed. Next, the uncertainty of a FSVIS are measured. Eventually, to show the feasibility of the proposed measures, effectiveness analysis is carried out from a statistical view. The obtained outcomes may be helpful for comprehending the nature of uncertainty in a FSVIS.

Fuzzy Set-Valued Information Systems and the Algorithm of Filling Missing Values for Incomplete Information Systems

How to effectively deal with missing values in incomplete information systems (IISs) according to the research target is still a key issue for investigating IISs. If the missing values in IISs are not handled properly, they will destroy the internal connection of data and reduce the efficiency of data usage. In this paper, in order to establish effective methods for filling missing values, we propose a new information system, namely, a fuzzy set-valued information system (FSvIS). By means of the similarity measures of fuzzy sets, we obtain several binary relations in FSvISs, and we investigate the relationship among them. This is a foundation for the researches on FSvISs in terms of rough set approach. Then, we provide an algorithm to fill the missing values in IISs with fuzzy set values. In fact, this algorithm can transform an IIS into an FSvIS. Furthermore, we also construct an algorithm to fill the missing values in IISs with set values (or real values). The effectiveness of these algorithms is analyzed. The results showed that the proposed algorithms achieve higher correct rate than traditional algorithms, and they have good stability. Finally, we discuss the importance of these algorithms for investigating IISs from the viewpoint of rough set theory.

A fuzzy similarity-based rough set approach for attribute selection in set-valued information systems

Databases obtained from different search engines, market data, patients’ symptoms and behaviours, etc., are some common examples of set-valued data, in which a set of values are correlated with a single entity. In real-world data deluge, various irrelevant attributes lower the ability of experts both in speed and in predictive accuracy due to high dimension and insignificant information, respectively. Attribute selection is the concept of selecting those attributes that ideally are necessary as well as sufficient to better describe the target knowledge. Rough set-based approaches can handle uncertainty available in the real-valued information systems after the discretization process. In this paper, we introduce a novel approach for attribute selection in set-valued information system based on tolerance rough set theory. The fuzzy tolerance relation between two objects using a similarity threshold is defined. We find reducts based on the degree of dependency method for selecting best subsets of attributes in order to obtain higher knowledge from the information system. Analogous results of rough set theory are established in case of the proposed method for validation. Moreover, we present a greedy algorithm along with some illustrative examples to clearly demonstrate our approach without checking for each pair of attributes in set-valued decision systems. Examples for calculating reduct of an incomplete information system are also given by using the proposed approach. Comparisons are performed between the proposed approach and fuzzy rough-assisted attribute selection on a real benchmark dataset as well as with three existing approaches for attribute selection on six real benchmark datasets to show the supremacy of proposed work.

Information structures in a set-valued information system based on granular computing1

Information structures in a set-valued information system based on granular computing1

A fuzzy rough set based fitting approach for fuzzy set-valued information system

Information systems comprising of multiple attribute values are known as set-valued information systems (SIS). This paper proposes a novel fuzzy set-valued information system (FSIS) as a generalized model of SIS. SIS is known as FSIS if some of its set-values are fuzzy set-values. Existing fuzzy rough set (FRS) model for SIS does not fit a dataset well and can lead to samples misclassification since sometimes there exist a large overlap between samples, so a fitting FRS model is introduced for handling FSIS. Fuzzy similarity relations obtained from each fuzzy set-valued attribute are parameterized using two parameter values to characterize the fuzzy information granules and then based on it, fuzzy lower and upper approximations are reconstructed. This approach can fit FSIS well and prevent samples misclassification. Later, the relative reduct of FSIS is computed using greedy forward algorithm. The effectiveness of proposed FSIS using fitting FRS model approach is proved by conducting a comparative study of proposed FSIS with existing information systems.

Uncertainty Measurement for a Set-Valued
 Information System: Gaussian Kernel Method

A set-valued information system (SIS) is the generalization of a single-valued informationsystem. This article explores uncertainty measurement for a SIS by using Gaussian kernel. The fuzzyTcos-equivalence relation lead by a SIS is first obtained by using Gaussian kernel. Then, informationstructures in this SIS are described by set vectors. Next, dependence between information structuresis presented and properties of information structures are investigated. Lastly, uncertainty measuresof a SIS are presented by using its information structures. Moreover, effectiveness analysis is doneto assess the feasibility of our presented measures. The consequence of this article will help usunderstand the intrinsic properties of uncertainty in a SIS.

Information Structures and Uncertainty Measures in an Incomplete Probabilistic Set-Valued Information System

An information system is a database that represents relationships between objects and attributes. A set-valued information system is the generalized model of a single-valued information system. A set-value information system that contains probability distributions and missing values is called an incomplete probability set-value information system (IPSIS). Uncertainty measure is an effective tool for evaluation. This paper explores information structures and uncertainty measures in an IPSIS. According to the Bhattacharyya distance, the distance between two objects in a given subsystem of an IPSIS is first proposed. Then, the tolerance relation on an object set, induced by a probability set-valued information system by using this distance, is obtained. Next, the information structure of this subsystem is introduced by a set vector. Moreover, the dependence between two information structures is studied by using the inclusion degree. Finally, as an application for information structures, measures of uncertainty for an IPSIS are investigated, and to evaluate the performance of the proposed measures, effectiveness analysis is given from the angle of statistics. These results will be helpful for establishing a framework of granular computing and understanding the essence of uncertainty in an IPSIS.

Entropy Based Feature Selection for Fuzzy Set-Valued Information Systems

In Set-valued Information Systems (SIS), several objects contain more than one value for some attributes. Tolerance relation used for handling SIS sometimes leads to loss of certain information. To surmount this problem, fuzzy rough model was introduced. However, in some cases, SIS may contain some real or continuous set-values. Therefore, the existing fuzzy rough model for handling Information system with fuzzy set-values needs some changes. In this paper, Fuzzy Set-valued Information System (FSIS) is proposed and fuzzy similarity relation for FSIS is defined. Yager's relative conditional entropy was studied to find the significance measure of a candidate attribute of FSIS. Later, using these significance values, three greedy forward algorithms are discussed for finding the reduct and relative reduct for the proposed FSIS. An experiment was conducted on a sample population of the real dataset and a comparison of classification accuracies of the proposed FSIS with the existing SIS and single-valued Fuzzy Information Systems was made, which demonstrated the effectiveness of proposed FSIS.