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This article is intended to solve the systems of inverse fuzzy relational equations with max–min composition, which is beneficial for solving the well-known problems of fuzzy abductive/backward reasoning. Almost all the existing methods are based on numerical algorithms, and either cannot find the globally optimal solutions or consume significant computational cost to acquire the most accurate result. Due to these drawbacks, the existing methods seem not to render the retroduction applying inverse fuzzy relation popular in many real-time expert systems such as intelligent diagnosis. It is well known that the system of inverse fuzzy relational equations is not always consistent. In this case, according to the preselected weights, we employ weighted L 1 norm distances to define a variety of best approximate solutions. Then we show that there exist very straightforward algebraic formulae for finding the best approximate solutions. The proposed approach not only finds the globally optimal solutions, but also has the advantage of being computationally very simple and efficient, and hence it has a lot of potentials to perform real-time abductive reasoning in many expert systems. [ABSTRACT FROM AUTHOR]

This article is intended to solve the systems of inverse fuzzy relational equations with max–min composition, which is beneficial for solving the well-known problems of fuzzy abductive/backward reasoning. Almost all the existing methods are based on numerical algorithms, and either cannot find the globally optimal solutions or consume significant computational cost to acquire the most accurate result. Due to these drawbacks, the existing methods seem not to render the retroduction applying inverse fuzzy relation popular in many real-time expert systems such as intelligent diagnosis. It is well known that the system of inverse fuzzy relational equations is not always consistent. In this case, according to the preselected weights, we employ weighted L 1 norm distances to define a variety of best approximate solutions. Then we show that there exist very straightforward algebraic formulae for finding the best approximate solutions. The proposed approach not only finds the globally optimal solutions, but also has the advantage of being computationally very simple and efficient, and hence it has a lot of potentials to perform real-time abductive reasoning in many expert systems. [ABSTRACT FROM AUTHOR]

Information system is a database that shows relationships between objects and attributes. A real-valued information system is an information system whose information values are real numbers. A real-valued information system with decision attributes is referred to as a real-valued decision information system. If objects, conditional attributes and information values in a real-valued decision information system are cells, genes and gene expression values, respectively, then this information system is said to be a gene decision space. In a gene decision space, people are faced with gene expression data. If gene expression data in a gene decision space changes to single cell RNA-seq data, then this space is referred to as a single cell gene decision space. This paper explores gene selection in a single cell gene decision space in terms of Gaussian kernel. In the first place, the distance between two cells in each subspace of a single cell gene decision space is constructed. Next, the fuzzy T cos -equivalence relation on the cell set is obtained in terms of Gaussian kernel. After that, measures of uncertainty for a single cell gene decision space are investigated. Lastly, gene selection algorithms in a single cell gene decision space are presented in terms of the proposed information entropy and information granularity. The presented algorithms are testified in several publicly open single cell RNA-seq data sets. Experimental results reveal that the presented algorithms can select appropriate genes related to classification, and significantly improve classification performances. [ABSTRACT FROM AUTHOR]

Information system is a database that shows relationships between objects and attributes. A real-valued information system is an information system whose information values are real numbers. A real-valued information system with decision attributes is referred to as a real-valued decision information system. If objects, conditional attributes and information values in a real-valued decision information system are cells, genes and gene expression values, respectively, then this information system is said to be a gene decision space. In a gene decision space, people are faced with gene expression data. If gene expression data in a gene decision space changes to single cell RNA-seq data, then this space is referred to as a single cell gene decision space. This paper explores gene selection in a single cell gene decision space in terms of Gaussian kernel. In the first place, the distance between two cells in each subspace of a single cell gene decision space is constructed. Next, the fuzzy T cos -equivalence relation on the cell set is obtained in terms of Gaussian kernel. After that, measures of uncertainty for a single cell gene decision space are investigated. Lastly, gene selection algorithms in a single cell gene decision space are presented in terms of the proposed information entropy and information granularity. The presented algorithms are testified in several publicly open single cell RNA-seq data sets. Experimental results reveal that the presented algorithms can select appropriate genes related to classification, and significantly improve classification performances. [ABSTRACT FROM AUTHOR]

Due to limitations in data acquisition, data in real life often contains a wealth of uncertain information. Uncertainty measurement (UM) constructed within the framework of rough set theory (RST) is an important tool for processing uncertain information. Some basic UMs in RST such as classification precision, rough membership degree, dependence degree, and attribute importance cannot accurately measure the uncertainty of a hybrid information system (HIS). For example, dependence degree only considers the information provided by the lower approximation of the decision and ignores the upper approximation, which may lead to some information loss. In addition to these basic UMs, some extended entropy-based UMs such as rough entropy, information entropy and conditional entropy are also frequently used to measure the uncertainty of a HIS. However, these three UMs also have their own drawbacks. For instance, rough entropy is sensitive to the distribution of hybrid data. When the distribution of hybrid data is uneven, the calculation results of rough entropy may be greatly affected, leading to a decrease in measurement accuracy. This paper proposes four new UMs in a HIS and provides an application in attribute reduction. First of all, a distance function is defined to deal with each type of attribute in a HIS and construct a tolerance relation. On this basis, four UMs are listed to measure the uncertainty of a HIS. Next, the strength and weakness of the proposed UMs are verified by statistical analysis. Subsequently, the UM with the best performance is selected to design an attribute reduction algorithm. Finally, the designed algorithm is compared with other five attribute reduction algorithms to show its superior performance. [ABSTRACT FROM AUTHOR]

In the era of explosive data growth, data sources and volumes are rapidly increasing. A multi-source data refers to information from multi-sources. However, not every source of information is equally important; some sources are more important and some are essentially worthless. Therefore, it is very meaningful to study how to select the most valuable sources and to efficiently fuse information. Multi-source incomplete interval-valued data (MSIIV-data) is an important kind of multi-source data. This paper proposes a novel method to information fusion in MSIIV-data via conditional information entropy (CIE) and considers its application to attribute reduction based on mutual information entropy. First, the distance between two information values for each incomplete interval-valued data is defined, the neighborhood classes with a tunable parameter are obtained, and the neighborhood granularity structure is established. Then, a source selection method is given via CIE, which is used to fuse MSIIV-data into single-source incomplete interval-valued data (SSIIV-data). Based on the minimization of CIE, this method allows worthy and reliable information sources to be chosen. Moreover, an attribute reduction algorithm (denoted as MMQPSO) for the fused SSIIV-data is proposed by means of combining mutual information entropy and QPSO-algorithm. Finally, experiments are done to validate the effectiveness of the proposed algorithms. The results of experiment and statistical test on 12 datasets show that the proposed algorithms have certain feasibility and advancement than 6 other advanced algorithms. [ABSTRACT FROM AUTHOR]

Categorical data is a significant kind of data in machine learning.Generally, rough set theory (RS -theory) deals with categorical data in the following way.First, an equivalence relation based on the equality of attribute values of categorical data is established.Then, information granules (I -granules) based on equivalence classes are obtained.Finally, information structures (I -structures) consisting of I -granules are formed.However, an equivalence relation is too strict, and there are some limitations in the I -structure of a categorical information system (CIS) that may result in filtering out potentially useful information.This paper investigates fuzzy information structures (FI -structures) and new uncertainty measurements for categorical data from the perspective that "the equality of attribute values is fed back to the attribute set".First, a fuzzy symmetry relation based on the number of attributes with equal attribute values is established. Then, fuzzy information granules (FI -granules) based on the fuzzy symmetry relation are obtained. Next, FI -structures consisting of FI -granules are formed.Finally, some concepts related to FI -structures in a CIS are given.The set vector is used to denote FI -structures, and the inclusion degree is used to study the dependence between FI -structures.In addition, four new uncertainty measurements based on FI -structures in a CIS are proposed, including fuzzy information granulation ( G f ), fuzzy information entropy ( H f ), fuzzy rough entropy ( E r f ) and fuzzy information amount ( E f ).Moreover, numerical experiments and statistical tests to evaluate the performance of the proposed new measurements are carried out.The results of the paired t -test show that the performance of the four new measurements based on FI -structures is better than that of the corresponding four measurements based on I -structures.Finally, attribute reduction algorithms based on G f and H f are presented, and clustering analysis is conducted on the reduced CIS. The experimental results show that the proposed algorithms are effective and perform well on attribute reduction according to three evaluation indicators of clustering performance. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]