Pub Date : 2024-10-29DOI: 10.1016/j.fss.2024.109170
Noemí Lubomirsky , Paula Menchón , Hernán Javier San Martín
In this paper we generalize the well known relation between Heyting algebras and Nelson algebras in the framework of subresiduated lattices. In order to make it possible, we introduce the variety of subresiduated Nelson algebras. The main tool for its study is the construction provided by Vakarelov. Using it, we characterize the lattice of congruences of a subresiduated Nelson algebra through some of its implicative filters. We use this characterization to describe simple and subdirectly irreducible algebras, as well as principal congruences. Moreover, we prove that the variety of subresiduated Nelson algebras has equationally definable principal congruences and also the congruence extension property. Additionally, we present an equational base for the variety generated by the totally ordered subresiduated Nelson algebras. Finally, we show that there exists an equivalence between the algebraic category of subresiduated lattices and the algebraic category of centedred subresiduated Nelson algebras.
{"title":"Subresiduated Nelson algebras","authors":"Noemí Lubomirsky , Paula Menchón , Hernán Javier San Martín","doi":"10.1016/j.fss.2024.109170","DOIUrl":"10.1016/j.fss.2024.109170","url":null,"abstract":"<div><div>In this paper we generalize the well known relation between Heyting algebras and Nelson algebras in the framework of subresiduated lattices. In order to make it possible, we introduce the variety of subresiduated Nelson algebras. The main tool for its study is the construction provided by Vakarelov. Using it, we characterize the lattice of congruences of a subresiduated Nelson algebra through some of its implicative filters. We use this characterization to describe simple and subdirectly irreducible algebras, as well as principal congruences. Moreover, we prove that the variety of subresiduated Nelson algebras has equationally definable principal congruences and also the congruence extension property. Additionally, we present an equational base for the variety generated by the totally ordered subresiduated Nelson algebras. Finally, we show that there exists an equivalence between the algebraic category of subresiduated lattices and the algebraic category of centedred subresiduated Nelson algebras.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109170"},"PeriodicalIF":3.2,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560573","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 : 2024-10-29DOI: 10.1016/j.fss.2024.109156
Yara Quilles-Marinho, Ricardo C.L.F. Oliveira, Pedro L.D. Peres
This paper proposes a new strategy based on linear matrix inequalities (LMIs) for the synthesis of state- and output-feedback controllers through parallel distributed compensation for continuous-time Takagi-Sugeno fuzzy systems. The main novelty of the proposed design technique is the use of homogeneous polynomial Lyapunov functions of degree larger than two on the states, generalizing the results based on quadratic functions. Parametrized in terms of a constant matrix, those homogeneous polynomial Lyapunov functions provide less conservative results in terms of stability and decay rate of trajectories when dealing with membership functions whose time-derivatives have unknown bounds. The synthesis procedure is formulated in terms of a locally convergent iterative algorithm, where a set of LMIs defined in an augmented parameter space is solved at each iteration. Numerical examples are used to compare the proposed method with quadratic stabilizability techniques available in the literature, illustrating the advantages of higher degree Lyapunov functions.
{"title":"Output-feedback control design for Takagi-Sugeno fuzzy systems through Lyapunov functions depending polynomially on the states","authors":"Yara Quilles-Marinho, Ricardo C.L.F. Oliveira, Pedro L.D. Peres","doi":"10.1016/j.fss.2024.109156","DOIUrl":"10.1016/j.fss.2024.109156","url":null,"abstract":"<div><div>This paper proposes a new strategy based on linear matrix inequalities (LMIs) for the synthesis of state- and output-feedback controllers through parallel distributed compensation for continuous-time Takagi-Sugeno fuzzy systems. The main novelty of the proposed design technique is the use of homogeneous polynomial Lyapunov functions of degree larger than two on the states, generalizing the results based on quadratic functions. Parametrized in terms of a constant matrix, those homogeneous polynomial Lyapunov functions provide less conservative results in terms of stability and decay rate of trajectories when dealing with membership functions whose time-derivatives have unknown bounds. The synthesis procedure is formulated in terms of a locally convergent iterative algorithm, where a set of LMIs defined in an augmented parameter space is solved at each iteration. Numerical examples are used to compare the proposed method with quadratic stabilizability techniques available in the literature, illustrating the advantages of higher degree Lyapunov functions.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"499 ","pages":"Article 109156"},"PeriodicalIF":3.2,"publicationDate":"2024-10-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142578750","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 : 2024-10-28DOI: 10.1016/j.fss.2024.109154
Yingying Wang, Jianyu Zhang, Xin Li, Qing Luo
This paper investigates how to design a sliding mode surface and construct a sliding controller to stabilize a uncertain T-S fuzzy system with multiple input matrices and matched disturbances. The method proposed in this paper can expand the effective scope of sliding mode control into uncertain fuzzy systems even if there are not assumptions of input matrices. In order to expand the effective scope of controlled systems, firstly, transform the form of input matrices of system plants into a new input matrix B. Secondly, design a new sliding mode surface according to all the input matrices. It contains q sub-sliding surfaces. The number q is the column number of input matrix B. Thirdly, according to sliding mode theorem, the equivalent controller can be deduced. But, it may be nonexistence. A judgment criterion for the existence of the controller is given as a lemma. By use of the proposed sliding surface, a new sliding controller is constructed. It can settle the confliction difficult between sliding surface and input matrices. And it can make the system reach the sliding surface and keep on it thereafter. At last, two examples are given to illustrate the effectiveness of this paper.
本文研究了如何设计滑动模态面和构建滑动控制器，以稳定具有多个输入矩阵和匹配干扰的不确定 T-S 模糊系统。即使没有输入矩阵假设，本文提出的方法也能将滑模控制的有效范围扩大到不确定模糊系统。为了扩大控制系统的有效范围，首先要将系统植物的输入矩阵形式转换为新的输入矩阵 B。其次，根据所有输入矩阵设计新的滑模面。它包含 q 个子滑动面。q 是输入矩阵 B 的列数。第三，根据滑模定理，可以推导出等效控制器。但是，它可能不存在。现给出一个控制器是否存在的判断标准，作为一个两难问题。利用所提出的滑动面，构建了一个新的滑动控制器。它能解决滑动面与输入矩阵之间的冲突难题。它还能使系统到达滑动面并保持在滑动面上。最后，本文举了两个例子来说明其有效性。
{"title":"A design of sliding mode control for uncertain T-S fuzzy systems with multiple input matrices","authors":"Yingying Wang, Jianyu Zhang, Xin Li, Qing Luo","doi":"10.1016/j.fss.2024.109154","DOIUrl":"10.1016/j.fss.2024.109154","url":null,"abstract":"<div><div>This paper investigates how to design a sliding mode surface and construct a sliding controller to stabilize a uncertain T-S fuzzy system with multiple input matrices and matched disturbances. The method proposed in this paper can expand the effective scope of sliding mode control into uncertain fuzzy systems even if there are not assumptions of input matrices. In order to expand the effective scope of controlled systems, firstly, transform the form of input matrices of system plants into a new input matrix <em>B</em>. Secondly, design a new sliding mode surface according to all the input matrices. It contains <em>q</em> sub-sliding surfaces. The number <em>q</em> is the column number of input matrix <em>B</em>. Thirdly, according to sliding mode theorem, the equivalent controller can be deduced. But, it may be nonexistence. A judgment criterion for the existence of the controller is given as a lemma. By use of the proposed sliding surface, a new sliding controller is constructed. It can settle the confliction difficult between sliding surface and input matrices. And it can make the system reach the sliding surface and keep on it thereafter. At last, two examples are given to illustrate the effectiveness of this paper.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109154"},"PeriodicalIF":3.2,"publicationDate":"2024-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552855","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 : 2024-10-24DOI: 10.1016/j.fss.2024.109168
Benoît Albert, Violaine Antoine, Jonas Koko
Evidential c-means (ECM) is a prototype-based clustering algorithm that generates a credal partition. Such a partition encompasses the notions that can be encountered with a hard, fuzzy or possibilistic partition, allowing the representation of various situations concerning the class membership of an object. The ECM method provides a prototype for each subset of the possible classes, calculated by averaging the prototypes of the classes included in the subsets. Although this definition perfectly suits ECM when employing a Euclidean distance, it becomes inappropriate when using a Mahalanobis distance. In this context, a new definition of prototypes for the subsets is proposed. The ECM objective function is then optimized using the new definition of prototypes. The subsequent algorithm, named ECM+, is finally tested on various synthetic and real data sets to demonstrate its interest compared to ECM.
{"title":"ECM+: An improved evidential c-means with adaptive distance","authors":"Benoît Albert, Violaine Antoine, Jonas Koko","doi":"10.1016/j.fss.2024.109168","DOIUrl":"10.1016/j.fss.2024.109168","url":null,"abstract":"<div><div>Evidential c-means (ECM) is a prototype-based clustering algorithm that generates a credal partition. Such a partition encompasses the notions that can be encountered with a hard, fuzzy or possibilistic partition, allowing the representation of various situations concerning the class membership of an object. The ECM method provides a prototype for each subset of the possible classes, calculated by averaging the prototypes of the classes included in the subsets. Although this definition perfectly suits ECM when employing a Euclidean distance, it becomes inappropriate when using a Mahalanobis distance. In this context, a new definition of prototypes for the subsets is proposed. The ECM objective function is then optimized using the new definition of prototypes. The subsequent algorithm, named ECM+, is finally tested on various synthetic and real data sets to demonstrate its interest compared to ECM.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109168"},"PeriodicalIF":3.2,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142552857","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 : 2024-10-22DOI: 10.1016/j.fss.2024.109164
Haoming Han , Shixu Zhao , Jing Zhang , Yan Liu
In this study, we consider the stability of high-order nonlinear Takagi–Sugeno fuzzy impulsive delayed coupled systems (TSFICSs). It should be noted that the linear growth condition is removed in the investigation of high-order nonlinear TSFICSs, which weakens the conditions compared with previous studies of impulsive systems. In addition, the existing research methods do not work for high-order nonlinear TSFICSs, so novel impulsive differential inequalities with high-order terms are established to investigate the stabilization of high-order nonlinear TSFICSs and develop the classic impulsive differential inequalities for high-order nonlinear situations. Next, sufficient conditions for the stability are obtained based on the new impulsive differential inequalities together with graph theory and the Lyapunov method. Finally, the theoretical results are applied to modified impulsive delayed coupled Fitzhugh–Nagumo models, and numerical simulations are provided to illustrate the practical value of the derived results.
{"title":"Stability of high-order nonlinear Takagi–Sugeno fuzzy impulsive delayed coupled systems","authors":"Haoming Han , Shixu Zhao , Jing Zhang , Yan Liu","doi":"10.1016/j.fss.2024.109164","DOIUrl":"10.1016/j.fss.2024.109164","url":null,"abstract":"<div><div>In this study, we consider the stability of high-order nonlinear Takagi–Sugeno fuzzy impulsive delayed coupled systems (TSFICSs). It should be noted that the linear growth condition is removed in the investigation of high-order nonlinear TSFICSs, which weakens the conditions compared with previous studies of impulsive systems. In addition, the existing research methods do not work for high-order nonlinear TSFICSs, so novel impulsive differential inequalities with high-order terms are established to investigate the stabilization of high-order nonlinear TSFICSs and develop the classic impulsive differential inequalities for high-order nonlinear situations. Next, sufficient conditions for the stability are obtained based on the new impulsive differential inequalities together with graph theory and the Lyapunov method. Finally, the theoretical results are applied to modified impulsive delayed coupled Fitzhugh–Nagumo models, and numerical simulations are provided to illustrate the practical value of the derived results.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109164"},"PeriodicalIF":3.2,"publicationDate":"2024-10-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533114","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 : 2024-10-18DOI: 10.1016/j.fss.2024.109145
Renato Pelessoni, Paolo Vicig
We explore how the classical Bhatia-Davis inequality bounding variances can be extended to uncertainty evaluations of gambles (bounded random numbers) by means of imprecise (lower or upper) previsions with different degrees of consistency. Firstly, a number of extensions are found with 2-coherent imprecise previsions. Subsequently, bounds with coherent lower and upper previsions are investigated, together with applications bounding lower and upper variances as well as p-boxes. Finally, bounds for covariances and for lower and upper covariances are obtained. Like the classical situation, imprecise Bhatia-Davis inequalities require a reduced amount of uncertainty information to be applied. When even less information is available, we show that various versions of Popoviciu's inequality obtain.
我们探讨了经典的巴蒂亚-戴维斯（Bhatia-Davis）不等式如何通过不同一致性程度的不精确（下限或上限）预设，扩展到赌局（有界随机数）的不确定性评估。首先，利用 2 一致性不精确预设找到了一些扩展。随后，研究了一致的下部和上部预设的边界，以及下部和上部方差和 p 方框的应用边界。最后，我们得到了协方差以及下协方差和上协方差的边界。与经典情况一样，不精确的巴蒂亚-戴维斯不等式需要更少的不确定性信息才能应用。当可用信息更少时，我们证明可以得到各种版本的波波维奇不等式。
{"title":"Bhatia-Davis inequalities for lower and upper previsions and covariances","authors":"Renato Pelessoni, Paolo Vicig","doi":"10.1016/j.fss.2024.109145","DOIUrl":"10.1016/j.fss.2024.109145","url":null,"abstract":"<div><div>We explore how the classical Bhatia-Davis inequality bounding variances can be extended to uncertainty evaluations of gambles (bounded random numbers) by means of imprecise (lower or upper) previsions with different degrees of consistency. Firstly, a number of extensions are found with 2-coherent imprecise previsions. Subsequently, bounds with coherent lower and upper previsions are investigated, together with applications bounding lower and upper variances as well as p-boxes. Finally, bounds for covariances and for lower and upper covariances are obtained. Like the classical situation, imprecise Bhatia-Davis inequalities require a reduced amount of uncertainty information to be applied. When even less information is available, we show that various versions of Popoviciu's inequality obtain.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109145"},"PeriodicalIF":3.2,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142533113","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 : 2024-10-18DOI: 10.1016/j.fss.2024.109153
LeSheng Jin , Yi Yang , Zhen-Song Chen , Muhammet Deveci , Radko Mesiar
Basic uncertain information is a recently introduced and significant type of uncertainty that proves particularly valuable in decision-making environments with inherent uncertainties. In this study, we propose the concept of uncertainty cognition merging, which effectively combines basic uncertain information granules with probability measures to generate new probability measures within the same probability space. Additionally, we present a degenerated method that merges basic uncertain information granules with unit intervals to create new subintervals. We introduce four distinct uncertainty cognition merging methods and thoroughly compare and analyze their respective properties, limitations, and advantages. To demonstrate the practical application potential of our proposals, we provide numerical examples alongside further mathematical results.
{"title":"Uncertainty merging with basic uncertain information in probability environment","authors":"LeSheng Jin , Yi Yang , Zhen-Song Chen , Muhammet Deveci , Radko Mesiar","doi":"10.1016/j.fss.2024.109153","DOIUrl":"10.1016/j.fss.2024.109153","url":null,"abstract":"<div><div>Basic uncertain information is a recently introduced and significant type of uncertainty that proves particularly valuable in decision-making environments with inherent uncertainties. In this study, we propose the concept of uncertainty cognition merging, which effectively combines basic uncertain information granules with probability measures to generate new probability measures within the same probability space. Additionally, we present a degenerated method that merges basic uncertain information granules with unit intervals to create new subintervals. We introduce four distinct uncertainty cognition merging methods and thoroughly compare and analyze their respective properties, limitations, and advantages. To demonstrate the practical application potential of our proposals, we provide numerical examples alongside further mathematical results.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109153"},"PeriodicalIF":3.2,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142532657","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 : 2024-10-18DOI: 10.1016/j.fss.2024.109155
Meng Han , Yongjie Huang , Ge Guo , H.K. Lam , Zhengsong Wang
In this paper, the domain of attraction (DOA) of the continuous-time positive polynomial fuzzy systems subject to input saturation is estimated by using the level set of the linear copositive Lyapunov function. To relax the estimation of DOA, the restriction on the level set is removed by embedding the expression of the level set into the stability conditions and positivity conditions. Referring to the nonconvex terms caused by above novel analysis strategy, some polynomial inequality lemmas are proposed to handle them; the nonconvex terms caused by imperfect premise matching (IPM) nonlinear membership functions are dealt with by sector nonlinear methods and advanced Chebyshev membership-function-dependent (MFD) methods. In this advanced MFD method, the state space segmentation and polynomial order selection of the Chebyshev approximation method are improved based on breakpoints of the first derivative and curvature, respectively, which is helpful to reduce the conservatism and computational burden of the result. Thus, this advanced Chebyshev MFD method not only optimizes the convexification strategy, but also can further be extended to estimate the DOA when it is used to introduce the membership functions information for convex stability and positivity conditions. Finally, a numerical example and the lipoprotein metabolism and potassium ion transfer nonlinear model are presented to validate the effectiveness and feasibility of the aforementioned analysis and convexification strategies in the expansion of DOA estimation.
本文利用线性共正 Lyapunov 函数的水平集来估计受输入饱和影响的连续时间正多项式模糊系统的吸引域（DOA）。为了放宽对 DOA 的估计，通过将水平集的表达式嵌入稳定性条件和正向性条件，消除了对水平集的限制。针对上述新颖分析策略引起的非凸项，提出了一些多项式不等式定理来处理它们；对于不完全前提匹配（IPM）非线性成员函数引起的非凸项，则采用扇形非线性方法和高级切比雪夫成员函数依赖（MFD）方法来处理。在这种先进的 MFD 方法中，切比雪夫近似方法的状态空间分割和多项式阶数选择分别根据一阶导数和曲率的断点进行了改进，这有助于减少结果的保守性和计算负担。因此，这种先进的切比雪夫 MFD 方法不仅优化了凸化策略，而且当它用于引入凸稳定性和正性条件的成员函数信息时，还能进一步扩展到估计 DOA。最后，通过一个数值实例和脂蛋白代谢与钾离子传递非线性模型，验证了上述分析和凸化策略在扩展 DOA 估计中的有效性和可行性。
{"title":"Estimation of the domain of attraction for continuous-time saturated positive polynomial fuzzy systems based on novel analysis and convexification strategies","authors":"Meng Han , Yongjie Huang , Ge Guo , H.K. Lam , Zhengsong Wang","doi":"10.1016/j.fss.2024.109155","DOIUrl":"10.1016/j.fss.2024.109155","url":null,"abstract":"<div><div>In this paper, the domain of attraction (DOA) of the continuous-time positive polynomial fuzzy systems subject to input saturation is estimated by using the level set of the linear copositive Lyapunov function. To relax the estimation of DOA, the restriction on the level set is removed by embedding the expression of the level set into the stability conditions and positivity conditions. Referring to the nonconvex terms caused by above novel analysis strategy, some polynomial inequality lemmas are proposed to handle them; the nonconvex terms caused by imperfect premise matching (IPM) nonlinear membership functions are dealt with by sector nonlinear methods and advanced Chebyshev membership-function-dependent (MFD) methods. In this advanced MFD method, the state space segmentation and polynomial order selection of the Chebyshev approximation method are improved based on breakpoints of the first derivative and curvature, respectively, which is helpful to reduce the conservatism and computational burden of the result. Thus, this advanced Chebyshev MFD method not only optimizes the convexification strategy, but also can further be extended to estimate the DOA when it is used to introduce the membership functions information for convex stability and positivity conditions. Finally, a numerical example and the lipoprotein metabolism and potassium ion transfer nonlinear model are presented to validate the effectiveness and feasibility of the aforementioned analysis and convexification strategies in the expansion of DOA estimation.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"498 ","pages":"Article 109155"},"PeriodicalIF":3.2,"publicationDate":"2024-10-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142560572","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 : 2024-10-17DOI: 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}
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