Pub Date : 2024-10-17DOI: 10.1016/j.fss.2024.109152
Wei Zhang , Bao Qing Hu
The α-cut (i.e., α-plane) of type-2 fuzzy sets is a very useful tool for computation. However, there are some theoretical mistakes in type-2 fuzzy sets literature discussing the topic of α-cuts. This paper will illustrate these mistakes through examples and specifically address the two new questions induced by them: (1) Taking the α-cut (resp. α-strong cut) of the result obtained by performing a T-extension operation of ⁎ (i.e., t-norm extension operation of a general binary operation) on two type-2 fuzzy sets is equal to what? (2) What conditions are required for taking the α-cut (resp. α-strong cut) of the result obtained by performing a T-extension operation of ⁎ on two type-2 fuzzy sets to be equal to performing the ⁎ operation on the α-cuts (resp. α-strong cuts) of these two type-2 fuzzy sets? Finally, we will get a comprehensive answer to these two questions.
{"title":"New results on α-cuts of type-2 fuzzy sets","authors":"Wei Zhang , Bao Qing Hu","doi":"10.1016/j.fss.2024.109152","DOIUrl":"10.1016/j.fss.2024.109152","url":null,"abstract":"<div><div>The <em>α</em>-cut (i.e., <em>α</em>-plane) of type-2 fuzzy sets is a very useful tool for computation. However, there are some theoretical mistakes in type-2 fuzzy sets literature discussing the topic of <em>α</em>-cuts. This paper will illustrate these mistakes through examples and specifically address the two new questions induced by them: (1) Taking the <em>α</em>-cut (resp. <em>α</em>-strong cut) of the result obtained by performing a <em>T</em>-extension operation of ⁎ (i.e., t-norm extension operation of a general binary operation) on two type-2 fuzzy sets is equal to what? (2) What conditions are required for taking the <em>α</em>-cut (resp. <em>α</em>-strong cut) of the result obtained by performing a <em>T</em>-extension operation of ⁎ on two type-2 fuzzy sets to be equal to performing the ⁎ operation on the <em>α</em>-cuts (resp. <em>α</em>-strong cuts) of these two type-2 fuzzy sets? Finally, we will get a comprehensive answer to these two questions.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109152"},"PeriodicalIF":3.2,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-16DOI: 10.1016/j.fss.2024.109150
Nicolas Pascal Dietrich, Wolfgang Trutschnig
Despite the fact that copulas are commonly considered as analytically smooth/regular objects, derivatives of copulas have to be handled with care. Triggered by a recently published result characterizing multivariate copulas via -increasingness of their partial derivative we study the bivariate setting in detail and show that the set of non-differentiability points of a copula may be quite large. We first construct examples of copulas C whose first partial derivative is pathological in the sense that for almost every it does not exist on a dense subset of , and then show that the family of these copulas is dense. Since in commonly considered subfamilies more regularity might be typical, we then focus on bivariate Extreme Value copulas (EVCs) and show that a topologically typical EVC is not absolutely continuous but has degenerated discrete component, implying that in this class typically exists in full .
Considering that regularity of copulas is closely related to their mass distributions we then study mass distributions of topologically typical copulas and prove the surprising fact that topologically typical bivariate copulas are mutually completely dependent with full support. Furthermore, we use the characterization of EVCs in terms of their associated Pickands dependence measures ϑ on , show that regularity of ϑ carries over to the corresponding EVC and prove that the subfamily of all EVCs whose absolutely continuous, discrete and singular component has full support is dense in the class of all EVCs.
{"title":"On differentiability and mass distributions of typical bivariate copulas","authors":"Nicolas Pascal Dietrich, Wolfgang Trutschnig","doi":"10.1016/j.fss.2024.109150","DOIUrl":"10.1016/j.fss.2024.109150","url":null,"abstract":"<div><div>Despite the fact that copulas are commonly considered as analytically smooth/regular objects, derivatives of copulas have to be handled with care. Triggered by a recently published result characterizing multivariate copulas via <span><math><mo>(</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-increasingness of their partial derivative we study the bivariate setting in detail and show that the set of non-differentiability points of a copula may be quite large. We first construct examples of copulas <em>C</em> whose first partial derivative <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mn>1</mn></mrow></msub><mi>C</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is pathological in the sense that for almost every <span><math><mi>x</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span> it does not exist on a dense subset of <span><math><mi>y</mi><mo>∈</mo><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></math></span>, and then show that the family of these copulas is dense. Since in commonly considered subfamilies more regularity might be typical, we then focus on bivariate Extreme Value copulas (EVCs) and show that a topologically typical EVC is not absolutely continuous but has degenerated discrete component, implying that in this class typically <span><math><msub><mrow><mo>∂</mo></mrow><mrow><mn>1</mn></mrow></msub><mi>C</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> exists in full <span><math><msup><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span>.</div><div>Considering that regularity of copulas is closely related to their mass distributions we then study mass distributions of topologically typical copulas and prove the surprising fact that topologically typical bivariate copulas are mutually completely dependent with full support. Furthermore, we use the characterization of EVCs in terms of their associated Pickands dependence measures <em>ϑ</em> on <span><math><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math></span>, show that regularity of <em>ϑ</em> carries over to the corresponding EVC and prove that the subfamily of all EVCs whose absolutely continuous, discrete and singular component has full support is dense in the class of all EVCs.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109150"},"PeriodicalIF":3.2,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445721","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
It is known that every continuous t-norm ⁎ generates a functor of the so-called ⁎-measures in the category of compact Hausdorff spaces. Similarly to the case of the hyperspace functor and the probability measure functors one can define the notion of invariant ⁎-measure for iterated function systems of contractions on compact metric spaces.
We provide a simple proof of existence and uniqueness of invariant ⁎-measures. Some examples of invariant ⁎-measures, for different t-norms ⁎, are presented.
{"title":"Invariant idempotent ⁎-measures generated by iterated function systems","authors":"Natalia Mazurenko , Khrystyna Sukhorukova , Mykhailo Zarichnyi","doi":"10.1016/j.fss.2024.109151","DOIUrl":"10.1016/j.fss.2024.109151","url":null,"abstract":"<div><div>It is known that every continuous t-norm ⁎ generates a functor of the so-called ⁎-measures in the category of compact Hausdorff spaces. Similarly to the case of the hyperspace functor and the probability measure functors one can define the notion of invariant ⁎-measure for iterated function systems of contractions on compact metric spaces.</div><div>We provide a simple proof of existence and uniqueness of invariant ⁎-measures. Some examples of invariant ⁎-measures, for different t-norms ⁎, are presented.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109151"},"PeriodicalIF":3.2,"publicationDate":"2024-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533115","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-10DOI: 10.1016/j.fss.2024.109147
Tadeusz Antczak
There is the growing use in practice of optimization models with uncertain data related to human activity in which hypotheses are not verified in a way specific for classical optimization. Fuzzy optimization problems have been introduced and developed for formulating and solving such real-world extremum problems which are usually not well defined. In most works devoted to fuzzy optimization problems, fuzzy numbers are characterized by their vertical membership functions which causes some difficulties in calculations and is the reason for arithmetic paradoxes. In the paper, therefore, fuzzy numbers are characterized by their horizontal membership functions and the concept of a gr-derivative of a fuzzy function is used which is based on the horizontal membership function and the granular difference. Although the convexity notion is a very important property of optimization models, there are real-world processes and systems with uncertainty that cannot be modeled with convex fuzzy optimization problems. Therefore, new concepts of granular generalized convexity notions, that is, the concepts of granular pre-invexity and gr-differentiable invexity are introduced to fuzzy analysis and some properties of the aforesaid granular generalized convexity concepts are investigated. Further, the class of nonconvex smooth optimization problems with gr-differentiable fuzzy-valued objective function and differentiable inequality constraint functions is considered as an application of the concept of gr-differentiable invexity. Then, the Karush-Kuhn-Tucker necessary optimality conditions are established for a global fuzzy minimizer with regard to the distinct fuzzy numbers in the analyzed fuzzy extremum problem. Further, the sufficiency of the aforesaid necessary optimality conditions of a Karush-Kuhn-Tucker type is also proved.
{"title":"Optimality results for a class of nonconvex fuzzy optimization problems with granular differentiable objective functions","authors":"Tadeusz Antczak","doi":"10.1016/j.fss.2024.109147","DOIUrl":"10.1016/j.fss.2024.109147","url":null,"abstract":"<div><div>There is the growing use in practice of optimization models with uncertain data related to human activity in which hypotheses are not verified in a way specific for classical optimization. Fuzzy optimization problems have been introduced and developed for formulating and solving such real-world extremum problems which are usually not well defined. In most works devoted to fuzzy optimization problems, fuzzy numbers are characterized by their vertical membership functions which causes some difficulties in calculations and is the reason for arithmetic paradoxes. In the paper, therefore, fuzzy numbers are characterized by their horizontal membership functions and the concept of a <em>gr</em>-derivative of a fuzzy function is used which is based on the horizontal membership function and the granular difference. Although the convexity notion is a very important property of optimization models, there are real-world processes and systems with uncertainty that cannot be modeled with convex fuzzy optimization problems. Therefore, new concepts of granular generalized convexity notions, that is, the concepts of granular pre-invexity and <em>gr</em>-differentiable invexity are introduced to fuzzy analysis and some properties of the aforesaid granular generalized convexity concepts are investigated. Further, the class of nonconvex smooth optimization problems with <em>gr</em>-differentiable fuzzy-valued objective function and differentiable inequality constraint functions is considered as an application of the concept of <em>gr</em>-differentiable invexity. Then, the Karush-Kuhn-Tucker necessary optimality conditions are established for a global fuzzy minimizer with regard to the distinct fuzzy numbers in the analyzed fuzzy extremum problem. Further, the sufficiency of the aforesaid necessary optimality conditions of a Karush-Kuhn-Tucker type is also proved.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109147"},"PeriodicalIF":3.2,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532658","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-10DOI: 10.1016/j.fss.2024.109146
Li Zhu , Qianli Zhou , Yong Deng , Witold Pedrycz
Order-2 information granules, as a representation of multi-layer structured information, have recently rekindled discussions. It is usually generated by abstracting order-1 information granules. When the dependence relationship and corresponding membership function of the order-1 information granules (reference information granules) are known, Pedrycz et al. used a method based on gradient optimization to complete the aggregation of the order-2 information granules. In this paper, we discuss a more specific scenario: not only the dependencies between reference information granules are captured, but also the order-1 information granules can be expressed as specific fuzzy information distributions. For this, we propose a fusion scheme of order-2 fuzzy sets based on matrix transformation called CQRP. To our knowledge, it is the first method that completely integrates the structural information of the order-2 environment into the fusion of order-2 fuzzy sets. In the process, we also creatively proposed the attraction and exclusion of structural information, deepening the understanding of structural information. Through sufficient comparison and analysis, we prove that it makes fuller use of information in the order-2 environment and is more reasonable and effective in tasks such as classification and identification.
{"title":"Information fusion in order-2 fuzzy environments: A matrix transformation perspective","authors":"Li Zhu , Qianli Zhou , Yong Deng , Witold Pedrycz","doi":"10.1016/j.fss.2024.109146","DOIUrl":"10.1016/j.fss.2024.109146","url":null,"abstract":"<div><div>Order-2 information granules, as a representation of multi-layer structured information, have recently rekindled discussions. It is usually generated by abstracting order-1 information granules. When the dependence relationship and corresponding membership function of the order-1 information granules (reference information granules) are known, Pedrycz et al. used a method based on gradient optimization to complete the aggregation of the order-2 information granules. In this paper, we discuss a more specific scenario: not only the dependencies between reference information granules are captured, but also the order-1 information granules can be expressed as specific fuzzy information distributions. For this, we propose a fusion scheme of order-2 fuzzy sets based on matrix transformation called CQRP. To our knowledge, it is the first method that completely integrates the structural information of the order-2 environment into the fusion of order-2 fuzzy sets. In the process, we also creatively proposed the attraction and exclusion of structural information, deepening the understanding of structural information. Through sufficient comparison and analysis, we prove that it makes fuller use of information in the order-2 environment and is more reasonable and effective in tasks such as classification and identification.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109146"},"PeriodicalIF":3.2,"publicationDate":"2024-10-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432498","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-09DOI: 10.1016/j.fss.2024.109149
Yifan Zhao, Hua-Wen Liu
Recently, Baczyński et al. introduced the axiomatic definition of fuzzy Sheffer strokes and incorporated the Sheffer stroke operation into the fuzzy logic framework [Fuzzy Sets Syst. 431 (2022) 110-128]. In this paper, we introduce a new class of directionally monotone functions, called -Sheffer strokes. Firstly, we propose the notion of -Sheffer strokes by relaxing the monotonicity of fuzzy Sheffer strokes to the directional monotonicity. And then, we discuss some vital properties of such functions as well as its relationship between fuzzy Sheffer strokes. Subsequently, we give a representation of -Sheffer strokes by means of -pre-conjunctions and fuzzy negations. Meanwhile, we give a characterization of -(constant) Sheffer strokes. Besides, we provide several construction methods of -Sheffer strokes. Interestingly, we show that -pre-conjunctions, -pre-disjunctions, (light) -pre-t-norms, (light) -pre-t-conorms, -(quasi-)overlap and grouping functions, and -implication functions can be obtained through adequate combinations of -Sheffer strokes. Finally, we present an example of a potential application of -Sheffer strokes in fire detectors.
{"title":"On r→-Sheffer strokes: A new class of directionally monotone functions","authors":"Yifan Zhao, Hua-Wen Liu","doi":"10.1016/j.fss.2024.109149","DOIUrl":"10.1016/j.fss.2024.109149","url":null,"abstract":"<div><div>Recently, Baczyński et al. introduced the axiomatic definition of fuzzy Sheffer strokes and incorporated the Sheffer stroke operation into the fuzzy logic framework [Fuzzy Sets Syst. 431 (2022) 110-128]. In this paper, we introduce a new class of directionally monotone functions, called <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-Sheffer strokes. Firstly, we propose the notion of <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-Sheffer strokes by relaxing the monotonicity of fuzzy Sheffer strokes to the directional monotonicity. And then, we discuss some vital properties of such functions as well as its relationship between fuzzy Sheffer strokes. Subsequently, we give a representation of <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-Sheffer strokes by means of <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-pre-conjunctions and fuzzy negations. Meanwhile, we give a characterization of <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-(constant) Sheffer strokes. Besides, we provide several construction methods of <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-Sheffer strokes. Interestingly, we show that <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-pre-conjunctions, <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-pre-disjunctions, (light) <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-pre-t-norms, (light) <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-pre-t-conorms, <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-(quasi-)overlap and grouping functions, and <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-implication functions can be obtained through adequate combinations of <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-Sheffer strokes. Finally, we present an example of a potential application of <span><math><mover><mrow><mi>r</mi></mrow><mrow><mo>→</mo></mrow></mover></math></span>-Sheffer strokes in fire detectors.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109149"},"PeriodicalIF":3.2,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142432615","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper mainly focuses on building the fuzzy prime filter theorem for multilattices. Firstly, we introduce the notion of a fuzzy filter generated by a fuzzy subset of a multilattice and we give a characterization. Also, we define four types of fuzzy prime filters and establish some relationships between them. Finally, we state and prove the fuzzy prime filter theorem in distributive multilattices.
{"title":"Fuzzy prime filter theorem in multilattices","authors":"Luc Éméry Diékouam Fotso , Carole Pierre Kengne , Daquin Cédric Awouafack","doi":"10.1016/j.fss.2024.109148","DOIUrl":"10.1016/j.fss.2024.109148","url":null,"abstract":"<div><div>This paper mainly focuses on building the fuzzy prime filter theorem for multilattices. Firstly, we introduce the notion of a fuzzy filter generated by a fuzzy subset of a multilattice and we give a characterization. Also, we define four types of fuzzy prime filters and establish some relationships between them. Finally, we state and prove the fuzzy prime filter theorem in distributive multilattices.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"497 ","pages":"Article 109148"},"PeriodicalIF":3.2,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142531222","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-03DOI: 10.1016/j.fss.2024.109144
Jian-Zhong Xiao , Chen-Ying Wang
In this paper a new definition for fuzzy inner product spaces is presented. Just as the set of real numbers can be embedded in a set of fuzzy numbers, a crisp inner product space can be considered as a special case of fuzzy inner product spaces. An example is given to demonstrate that, the new definition is a nontrivial generalization for the crisp inner product spaces. Under certain restrictions, a fuzzy inner product space can become a fuzzy normed space. Based on some elementary properties for the families of semi-inner products of endpoints, the linearly topological structure of new spaces is discussed. Moreover, the orthogonality between two vectors is considered and a fuzzy version of Pythagorean theorem is given.
{"title":"On fuzzy inner products constructed by fuzzy numbers in linear spaces","authors":"Jian-Zhong Xiao , Chen-Ying Wang","doi":"10.1016/j.fss.2024.109144","DOIUrl":"10.1016/j.fss.2024.109144","url":null,"abstract":"<div><div>In this paper a new definition for fuzzy inner product spaces is presented. Just as the set of real numbers can be embedded in a set of fuzzy numbers, a crisp inner product space can be considered as a special case of fuzzy inner product spaces. An example is given to demonstrate that, the new definition is a nontrivial generalization for the crisp inner product spaces. Under certain restrictions, a fuzzy inner product space can become a fuzzy normed space. Based on some elementary properties for the families of semi-inner products of endpoints, the linearly topological structure of new spaces is discussed. Moreover, the orthogonality between two vectors is considered and a fuzzy version of Pythagorean theorem is given.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109144"},"PeriodicalIF":3.2,"publicationDate":"2024-10-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142420975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.fss.2024.109143
Jiang Deng , Degang Chen , Hui Wang , Ruifeng Shi
In multi-label learning, samples of practical classification task may associated with multiple labels, it is challenging to acquire all labels of the training samples, the rapid expansion of the label space and the significant increase in annotation costs have exacerbated the issue of missing labels in multi-label learning. By utilizing the low rank structure of the label matrix, missing labels can be effectively recovered with matrix factorization technique. Nevertheless, current approaches disregard the potential correlation between the feature space and the multi-dimensional label data. In this paper, key features associated to different labels are extracted and then a non-negative matrix factorization algorithm is proposed for recover missing labels. Firstly, the fuzzy rough set theory is used to analyze the consistency between label matrix and feature space, potential feature information is employed to determine the latent variable dimension in non-negative matrix decomposition and the symbolic label matrix is also converted to a numerical one by means of the lower approximation operator. Then, feature-based manifold regularization and local label correlations are used to model the multi-label completion algorithm. In order to verify the effectiveness in dealing incomplete label data, comparison experiments with varying levels of missing values are designed, the experiments show that compared with the state-of-the-art algorithm, the proposed algorithm is effective in the completion of missing labels. In addition, the sensitivity analysis experiments also show that the proposed method has good stability.
{"title":"Matrix factorization algorithm for multi-label learning with missing labels based on fuzzy rough set","authors":"Jiang Deng , Degang Chen , Hui Wang , Ruifeng Shi","doi":"10.1016/j.fss.2024.109143","DOIUrl":"10.1016/j.fss.2024.109143","url":null,"abstract":"<div><div>In multi-label learning, samples of practical classification task may associated with multiple labels, it is challenging to acquire all labels of the training samples, the rapid expansion of the label space and the significant increase in annotation costs have exacerbated the issue of missing labels in multi-label learning. By utilizing the low rank structure of the label matrix, missing labels can be effectively recovered with matrix factorization technique. Nevertheless, current approaches disregard the potential correlation between the feature space and the multi-dimensional label data. In this paper, key features associated to different labels are extracted and then a non-negative matrix factorization algorithm is proposed for recover missing labels. Firstly, the fuzzy rough set theory is used to analyze the consistency between label matrix and feature space, potential feature information is employed to determine the latent variable dimension in non-negative matrix decomposition and the symbolic label matrix is also converted to a numerical one by means of the lower approximation operator. Then, feature-based manifold regularization and local label correlations are used to model the multi-label completion algorithm. In order to verify the effectiveness in dealing incomplete label data, comparison experiments with varying levels of missing values are designed, the experiments show that compared with the state-of-the-art algorithm, the proposed algorithm is effective in the completion of missing labels. In addition, the sensitivity analysis experiments also show that the proposed method has good stability.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109143"},"PeriodicalIF":3.2,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142442457","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-30DOI: 10.1016/j.fss.2024.109139
Francisco Javier Talavera , Sergio Ardanza-Trevijano , Jean Bragard , Jorge Elorza
This study builds upon results concerning the preservation of fuzzy structures by aggregation operators. We analyze when an aggregation operator preserves the structure of T-subgroup in the last cases remaining unresolved in the characterization of such operators. In order to characterize these operators for different groups, relaxed forms of domination are introduced and their properties and interconnections are studied. We also include a thorough review of the features that an aggregation operator must fulfill to preserve T-subgroups of an arbitrary group.
本研究以有关聚合算子保留模糊结构的结果为基础。我们分析了聚合算子在什么情况下会保留 T 子群的结构，这是此类算子表征中尚未解决的最后一种情况。为了描述不同群组的这些算子，我们引入了松弛的支配形式，并研究了它们的性质和相互联系。我们还全面回顾了聚合算子在保留任意群的 T 子群时必须满足的特征。
{"title":"New types of domination to characterize the preservation of T-subgroups under aggregation","authors":"Francisco Javier Talavera , Sergio Ardanza-Trevijano , Jean Bragard , Jorge Elorza","doi":"10.1016/j.fss.2024.109139","DOIUrl":"10.1016/j.fss.2024.109139","url":null,"abstract":"<div><div>This study builds upon results concerning the preservation of fuzzy structures by aggregation operators. We analyze when an aggregation operator preserves the structure of <em>T</em>-subgroup in the last cases remaining unresolved in the characterization of such operators. In order to characterize these operators for different groups, relaxed forms of domination are introduced and their properties and interconnections are studied. We also include a thorough review of the features that an aggregation operator must fulfill to preserve <em>T</em>-subgroups of an arbitrary group.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109139"},"PeriodicalIF":3.2,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142358021","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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