Rough Membership Function Research Articles

Employing a Generalization of Open Sets Defined by Ideals to Initiate Novel Rough Approximation Spaces with a Chemical Application

A close similarity and analogy between rough set theory and topology is attributed to the corresponding behavior of lower and upper rough approximations with interior and closure topological operators, respectively. This relation motivates joint studies between topology and this theory. We endeavor by rough set theory to enlarge the knowledge we obtain from the information systems, for this reason, we apply the abstract concept of ideal structures to build new generalized approximation spaces with less vagueness. In the present work, we employ a novel type of nearly open sets in topology so-called ``${\mathcal{L}}$-${\theta\beta}_{\lambda}$-open" with an ideal structure to introduce novel approximation spaces satisfying the desired properties concerning shrinking the boundary region of uncertainty and expanding the domain of confirmed information. We set up the fundamentals of the proposed rough paradigms and demonstrate their superiority over the preceding paradigms induced by some nearly open sets. Two algorithms are furnished to illustrate the way of specifying the family of ${\mathcal{L}}$-${\theta\beta}_{\lambda}$-open sets and exploring whether a subset is ${\mathcal{L}}$-${\theta\beta}_{\lambda}$-definable or ${\mathcal{L}}$-${\theta\beta}_{\lambda}$-rough. Then, we put forward the concepts of rough membership relations and functions and uncover their core characterizations. Finally, we examine the proposed models to model a real situation in the Chemistry field and clarify how our models improve the outcomes of generalized approximation spaces over the previous models.

Granular computing in zero-divisor graphs of [formula omitted]

In this article, we study the zero-divisor graphs Γ(Zn) of rings of integers modulo n as information systems I(Γ(Zn)) using equivalence classes and rough sets. Equivalence classes are referred as granules and partitions are referred as indiscernible partitions. We define an indiscernibility relation on the vertex set and identify different sets of attributes that induce the same indiscernibility partition. A reduct is a minimal subset of attributes which yields the same partition as the original attribute set. We compute all reducts of the defined information system and classify them in to two types including: (i) the set P of all prime divisors of n and (ii) the set consisting of P∖pi, prime powers of pi in the prime factorization of n and the elements of the form pipj, where pj∈P∖pi. Moreover, we give the structures and the cardinalities of the attribute subsets whose removal yields a different indiscernibility partition than that of the set of all attributes, referred as essential sets. We prove that the essential sets of I(Γ(Zn)) either consist of two prime divisors of n or one prime divisor pi combined with prime powers of pi. Further, we determine the lower and upper approximations of various vertex subsets to study the properties of the zero-divisor graphs. We also study properties of the rough membership function for Γ(Zn). Furthermore, we introduce the class-based discernibility matrix induced by indiscernibility classes of zero divisors and determine general form of its entries. We also prove that the minimal entries of the class-based discernibility matrix coincide with the essential sets of Γ(Zn). Based on these results, we determine information-granularity measures corresponding to the notable partitions of Γ(Zn) and provide an example to establish consistency of the proved results with these well-known measures. Thus, starting from introducing an information system, we investigate the components of granular computing and utilize our findings to compute information granularity measures, contributing to a deeper understanding of the zero divisor graphs via rough set theory.

Rough Set Techniques in Wireless Sensor Networks using Membership Function and Rough Labelling Graphs for Energy Aware Routing

Rough set theory and rough graphs are employed for data analysis. Rough graphs utilize approximations, and this paper presents a method for implementing rough graphs using rough membership functions and graph labeling for data structures and reduction. This paper proposes a rough graph labeling method, termed rough -labeling similarity graph, that utilizes a similarity measure for vertex and edge labeling. This method aims to minimize boundary regions in rough graphs and is applicable to wireless sensor networks (WSNs). In WSN, the proposed algorithms such as PSO-LSTM, COA-LSTM, LOA-LSTM integrates with rough set theory based rough -labeling for boundary region identification. The proposed method incorporates the membership functions encapsulates the rough labeling graphs for cluster boundary region identification for WSN. The cluster boundary for WSN is based on rough set membership and rough labelling is termed as rough set membership boundary region (RMB). RMB in WSN and implementation of proposed routing algorithm such as LOA-LSTM provides high throughput and energy saving when compared to existing cluster boundary structure methods such as Voronoi, Spectrum and Chain.

Intelligent Height Adjustment Method of Shearer Drum Based on Rough Set Significance Reduction and Fuzzy Rough Radial Basis Function Neural Network

The intelligent adjustment method of the shearer drum is the key technology to improve the intelligent level and safety degree of fully mechanized mining face. This paper proposes a shearer drum intelligent height adjustment model based on rough set significance attribute reduction (AR) and fuzzy rough radial basis function neural network (FRRBFNN) optimized by adaptive immune genetic algorithm (AIGA). The model first selects the parameters of shearer process monitoring based on the importance attribute reduction algorithm of rough set, and establishes the attribute reduction set of shearer operation characteristic parameters and the drum height decision rule base. Next, a fuzzy rough radial basis function neural network determined by the decision rule space is proposed. By introducing the fuzzy rough membership function as the connection weight, the network can accurately describe the complex nonlinear relationship between the working characteristic parameters of the attribute shearer and the drum height, and measure the uncertainty of the coal seam distribution. Finally, to further optimize the performance of FRRBFNN, the adaptive immune genetic algorithm is introduced to optimize its parameters, to build a high-precision shearer drum height prediction system. For the evaluation method of the model, we use three indicators: mean absolute error, mean absolute percentage error, and root mean square error. Based on the measured data in Yujialiang area, Shaanxi Province, the experimental results show that—compared with the FRRBFNN and support vector regression (SVR) models, a gated current neural network (GRU), a radial basis function neural network (RBF), the memory strengthen long short-term memory (MSLSTM) model, and the adaptive fuzzy reasoning Petri net (AFRPN)—the MAE of the AR-AIGA-FRBFNN model for predicting the height of the left and right rollers are 18.3 mm and 17.2 mm, respectively; the MAPE is 0.96% and 0.93%, respectively; and the RMSE is 21.2 mm and 22.4 mm, respectively. The AR-AIGA-FRBFNN model is therefore more effective than the other considered methods.

Accuracy measure of rough membership multiset functions with application

In this paper, the notion of rough membership function has been introduced in the multiset context. As a consequence, we introduced the lower and upper approximations in terms of the rough membership multiset functions. Finally, an application about how to select the best possible suppliers of chemicals has been presented.

Construction of Rough Graph through Rough Membership Function

Rough membership function defines the degree of relationship between conditional and decision attributes of an information system. It is defined by <img src=image/13429145_01.gif> where <img src=image/13429145_07.gif> is the subset of <img src=image/13429145_08.gif> under the relation <img src=image/13429145_09.gif> where <img src=image/13429145_08.gif> is the universe of discourse. It can be expressed in different forms like cardinality form, probabilistic form etc. In cardinality form, it is expressed as <img src=image/13429145_02.gif> where as in probabilistic form it can be denoted as <img src=image/13429145_03.gif> where <img src=image/13429145_04.gif> is the equivalence class of <img src=image/13429145_10.gif> with respect to <img src=image/13429145_09.gif>. This membership function is used to measure the value of uncertainty. In this paper we have introduced the concept of graphical representation of rough sets. Rough graph was introduced by He Tong in 2006. In this paper, we propose a novel method for the construction of rough graph through rough membership function <img src=image/13429145_05.gif>. We propose that there is an edge between vertices if <img src=image/13429145_06.gif>. The rough graph is being constructed for an information system; here objects are considered as vertices. Rough path, rough cycle, rough ladder graph are introduced in this paper. We develop the operations on rough graph and also extend the properties of rough graph.

Rough sets in graphs using similarity relations

&lt;abstract&gt;&lt;p&gt;In this paper, we investigate the theory of rough set to study graphs using the concept of orbits. Rough sets are based on a clustering criterion and we use the idea of similarity of vertices under automorphism as a criterion. We introduce indiscernibility relation in terms of orbits and prove necessary and sufficient conditions under which the indiscernibility partitions remain the same when associated with different attribute sets. We show that automorphisms of the graph $ \mathcal{G} $ preserve the indiscernibility partitions. Further, we prove that for any graph $ \mathcal{G} $ with $ k $ orbits, any reduct $ \mathcal{R} $ consists of one element from $ k-1 $ orbits of the graph. We also study the rough membership functions for paths, cycles, complete and complete bipartite graphs. Moreover, we introduce essential sets and discernibility matrices induced by orbits of graphs and study their relationship. We also prove that every essential set consists of union of any two orbits of the graph.&lt;/p&gt;&lt;/abstract&gt;

Approximation Of Continuous Set Valued Mapping And Membership Function Rough Topology

In this paper, we introduce the new definition of rough membership function using continues function and we discuss several concepts and properties of rough continuous set value functions as new results on rough the continues function and membership continues function . Moreover, we extend the definition of rough membership function to topology spaces by substituted an equivalence class by continuous functions and prove some theorems on certain types of set value functions and some more general and fundamental properties of the generalized rough sets. Our result generalized the concept of the set valued function by using rough set theory.

Roughness in Membership Continuous Function

In this paper, we introduce the new definition of rough membership function using continuous function and we discuss several concepts and properties of rough continuous set value functions as new results on rough continuous function and membership continuous function. Moreover, we extend the definition of rough membership function to topology spaces by substituting an equivalence class by continuous functions and prove some theorems on certain types of set value functions and some more general and fundamental properties of the generalized rough sets. Our result generalized the concept of the set valued function by using rough set theory.

Generating Fuzzy Sets and Fuzzy Relations Based on Information

Fuzzy set theory and fuzzy relation are important techniques in knowledge discovery in databases. In this work, we presented fuzzy sets and fuzzy relations according to some giving Information by using rough membership function as a new way to get fuzzy set and fuzzy relation to help the decision in any topic . Some properties have been studied. And application of my life on the fuzzy set was introduced

Rough sets theory via new topological notions based on ideals and applications

&lt;abstract&gt;&lt;p&gt;There is a close analogy and similarity between topology and rough set theory. As, the leading idea of this theory is depended on two approximations, namely lower and upper approximations, which correspond to the interior and closure operators in topology, respectively. So, the joined study of this theory and topology becomes fundamental. This theory mainly propose to enlarge the lower approximations by adding new elements to it, which is an equivalent goal for canceling elements from the upper approximations. For this intention, one of the primary motivation of this paper is the desire of improving the accuracy measure and reducing the boundary region. This aim can be achieved easily by utilizing ideal in the construction of the approximations as it plays an important role in removing the vagueness of concept. The emergence of ideal in this theory leads to increase the lower approximations and decrease the upper approximations. Consequently, it minimizes the boundary and makes the accuracy higher than the previous. Therefore, this work expresses the set of approximations by using new topological notions relies on ideals namely $ \mathcal{I} $-$ {\delta_{\beta}}_{J} $-open sets and $ \mathcal{I} $-$ {\bigwedge_{\beta}}_{J} $-sets. Moreover, these notions are also utilized to extend the definitions of the rough membership relations and functions. The essential properties of the suggested approximations, relations and functions are studied. Comparisons between the current and previous studies are presented and turned out to be more precise and general. The brilliant idea of these results is increased in importance by applying it in the chemical field as it is shown in the end of this paper. Additionally, a practical example induced from an information system is introduced to elucidate that the current rough membership functions is better than the former ones in the other studies.&lt;/p&gt;&lt;/abstract&gt;

INCREMENTALLY UPDATING APPROXIMATION IN INCOMPLETE INFORMATION SYSTEMS UNDER THE VARIATION OF OBJECTS

Rough membership functions in covering approximation space not only give numerical characterizations of covering-based rough set approximations, but also establish the relationship between covering-based rough sets and fuzzy covering-based rough sets. In this paper, we introduce a new method to update the approximation sets with rough membership functions in covering approximation space. Firstly, we present the third types of rough membership functions and study their properties. And then, we consider the change of them while simultaneously adding and removing objects in the information system. Based on that change, we propose a method for updating the approximation sets when the objects vary over time.

Three-way Decisions with Rough Membership Functions in Covering Approximation Space

Three-way Decisions with Rough Membership Functions in Covering Approximation Space

Roughness measures of locally finite covering rough sets

The present paper defines four new kinds of measures of roughness of covering rough sets induced by locally finite covering approximation (LFC-, for brevity) spaces which are generalizations of finite covering approximation spaces. More precisely, consider an LFC-space (U,C) and a nonempty set X(⊆U). Using a reduction of a given LFC-space (U,C), the present paper firstly establishes two types of rough membership functions of the set X with respect to LFC-spaces (U,C), where each of the cardinalities of the sets U, C, and X need not be finite. Next, it also develops another two kinds of notions of roughness of digital topological rough sets. Indeed, these notions are based on the concepts of accuracy of rough sets derived from LFC-spaces. Furthermore, we use them to the estimation of roughness of a covering rough set of X(⊆U). Besides, we estimate roughness of a digital topological rough set, such as measures of roughness of the Khalimsky and Marcus–Wyse topological rough sets. Moreover, we compare between the measures of the Khalimsky topological and the Marcus–Wyse topological roughness.

The rough membership functions on four types of covering-based rough sets and their applications

Pawlak’s rough membership functions not only give numerical characterizations of Pawlak’s rough set approximations, but also establishes the relationships between Pawlak’s rough sets and fuzzy sets or probabilistic rough sets. However, it is noteworthy that Pawlak’s rough membership functions have limitations when handling incomplete data that exist widely in the real world. As will be shown in this paper, one way to overcome this is to construct rough membership functions for covering-based rough sets. In this paper, we first use an example in evidence-based medicine to illustrate how to use Pawlak’s rough membership function on numerically characterizing decisions under the circumstances where data are complete. Then, we construct covering-based rough membership functions for four types of covering-based rough sets which were examined by Zhu and Wang (in IEEE Transactions on Knowledge and Data Engineering 19(8)(2007) 1131-1144 and Information Sciences 201(2012) 80-92), and use them to characterize these covering-based rough sets numerically. Finally, we present theoretical backgrounds for these covering-based rough membership functions, and illustrate how to apply them on numerically characterizing decisions under the circumstances where data are incomplete.

The rough membership function based on C 10 ¯ and its applications

The rough membership function based on C 10 ¯ and its applications

Probabilities, dependence and rough membership functions

The goal of this research is to study the reflection of probability theories through rough membership functions in rough sets. Towards this, philosophy and variants of probability theories and their theorized connections with rough memberships functions are critically analyzed. The concept of rough membership functions and rough dependence are also generalized to granular operator spaces, characterized and used for the same purpose in more general contexts. A new theory of dependence based deviant probability is developed as a severe extension of the axiomatic approach to a dependence-based probability introduced recently by the present author. This permits clearer comparison of the concept with those of rough dependence. It is shown that the theories are very distinct semantically and similarities are poorly justified. These are relevant for rethinking the various probabilistic rough theories and related methodologies. The problem of contamination reduction was proposed recently across many papers by the present author. In this study, the scope of the problem within rough membership functions, probabilistic rough sets (PSTs) and three-way decision-making is also clarified by her. A new definition of artificial intelligence applicable in rough perspectives is also proposed on the basis of recent advances in algebraic semantics related to rough membership functions.

New rough approximations for n-cycles and n-paths

In this paper, we continue the study of a finite simple undirected graph through the rough set techniques introduced by Pawlak in the early 80’s. In this sense, we focus on the Boolean information system induced from a circle or a line graph with n vertices and, for such graphs, we provide a complete description of the rough membership function and some results on the positive regions and the attribute dependency function.

Uncertainty measures of Neighborhood System-based rough sets

Neighborhood System-based (NS-based) rough set theory is an extension of classical rough set theory. This paper aims to investigate the uncertainty measures of rough sets in a Neighborhood System-space (NS-space). We firstly develop a rough membership function and a rough intuitionistic membership function based on binary relationships. Then we propose a fuzzy entropy and an intuitionistic fuzzy entropy of rough sets, and further explore their corresponding properties. Examples show that the methods not only accord with people’s recognition rules of rough sets, but also can be used to depict the case that the NS-based rough sets degenerate into the crisp sets.

Generalized covering approximation space and near concepts with some applications

In this paper, we shall integrate some ideas in terms of concepts in topology. First, we introduce some new concepts of rough membership relations and functions in the generalized covering approximation space. Second, we introduce some topological applications namely “near concepts” in the generalized covering approximation space. Accordingly, several types of fuzzy sets are constructed. The basic notions of near approximations are introduced and sufficiently illustrated. Near concepts are provided to be easy tools to classify the sets and to help for measuring exactness and roughness of sets. Many proved results, examples and counter examples are provided. Finally, we give two practical examples to illustrate our approaches.