Pub Date : 2026-01-06DOI: 10.1016/j.fss.2026.109764
Yao Ouyang , Qiuyu Cao , Hua-Peng Zhang
Given a curvilinear section, there are three known copulas. We prove that two of them are always singular and the third one is also singular under some mild condition. With the aid of the two always singular copulas and by employing the rectangular patchwork of copulas, we explore when there exists an absolutely continuous copula with a given curvilinear section. Several sufficient conditions and one necessary condition are discovered for the existence problem. The Durante-Jaworski theorem is retrieved when the curvilinear section reduces to a diagonal section.
{"title":"Absolutely continuous copulas with a given curvilinear section","authors":"Yao Ouyang , Qiuyu Cao , Hua-Peng Zhang","doi":"10.1016/j.fss.2026.109764","DOIUrl":"10.1016/j.fss.2026.109764","url":null,"abstract":"<div><div>Given a curvilinear section, there are three known copulas. We prove that two of them are always singular and the third one is also singular under some mild condition. With the aid of the two always singular copulas and by employing the rectangular patchwork of copulas, we explore when there exists an absolutely continuous copula with a given curvilinear section. Several sufficient conditions and one necessary condition are discovered for the existence problem. The Durante-Jaworski theorem is retrieved when the curvilinear section reduces to a diagonal section.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"530 ","pages":"Article 109764"},"PeriodicalIF":2.7,"publicationDate":"2026-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940521","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 : 2026-01-05DOI: 10.1016/j.fss.2025.109751
Luís Miguel Magalhães Torres , Ginalber Luiz de Oliveira Serra
This paper presents an evolving state-space interval type-2 Takagi-Sugeno identification algorithm. The methodology integrates an online particle swarm optimization scheme, which tunes the footprint of uncertainty by minimizing a coverage-width objective, with a filtering-based recursive estimation of fuzzy Markov parameters that provides unbiased consequent parameters under non-white noise. The antecedent structure is updated online through density-based rule creation, adaptation, and pruning. The approach is validated on a nonlinear benchmark with abrupt and gradual parameter changes and on a two-degrees-of-freedom helicopter. Compared to state-of-the-art methods, the proposed algorithm achieves lower error and higher-quality prediction intervals.
{"title":"Evolving interval type-2 fuzzy state-space identification using PSO-tuned footprint of uncertainty and filtered markov parameters","authors":"Luís Miguel Magalhães Torres , Ginalber Luiz de Oliveira Serra","doi":"10.1016/j.fss.2025.109751","DOIUrl":"10.1016/j.fss.2025.109751","url":null,"abstract":"<div><div>This paper presents an evolving state-space interval type-2 Takagi-Sugeno identification algorithm. The methodology integrates an online particle swarm optimization scheme, which tunes the footprint of uncertainty by minimizing a coverage-width objective, with a filtering-based recursive estimation of fuzzy Markov parameters that provides unbiased consequent parameters under non-white noise. The antecedent structure is updated online through density-based rule creation, adaptation, and pruning. The approach is validated on a nonlinear benchmark with abrupt and gradual parameter changes and on a two-degrees-of-freedom helicopter. Compared to state-of-the-art methods, the proposed algorithm achieves lower error and higher-quality prediction intervals.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"530 ","pages":"Article 109751"},"PeriodicalIF":2.7,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940522","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 : 2026-01-04DOI: 10.1016/j.fss.2026.109763
Jianzhang Wu
This comment concerns the paper “Another view on the non-additivity index of capacity” (Fuzzy Sets and Systems, Vol. 520, 2025, 109569). The classical non-additivity index (NI), defined through proper bipartitions, quantifies deviations from additivity. The generalized NI (GNI) proposed in the target paper extends this definition by including both the empty set and the coalition itself. We clarify that singleton cases are to be interpreted as importance rather than interaction, and show that the generalized extension adds little useful information and meanwhile misrepresents strict non-additivity. We further demonstrate that Properties 6, 7, and 8, together with the induced monotonicity constraints presented in the target paper as major contributions, are invalid and unreliable. Finally, we compare the properties of NI and GNI and emphasize that the classical NI uniquely satisfies key structural properties such as uniform range, maximality, and minimality.
{"title":"On “another view on the non-additivity index of capacity”","authors":"Jianzhang Wu","doi":"10.1016/j.fss.2026.109763","DOIUrl":"10.1016/j.fss.2026.109763","url":null,"abstract":"<div><div>This comment concerns the paper “Another view on the non-additivity index of capacity” (Fuzzy Sets and Systems, Vol. 520, 2025, 109569). The classical non-additivity index (NI), defined through proper bipartitions, quantifies deviations from additivity. The generalized NI (GNI) proposed in the target paper extends this definition by including both the empty set and the coalition itself. We clarify that singleton cases are to be interpreted as importance rather than interaction, and show that the generalized extension adds little useful information and meanwhile misrepresents strict non-additivity. We further demonstrate that Properties 6, 7, and 8, together with the induced monotonicity constraints presented in the target paper as major contributions, are invalid and unreliable. Finally, we compare the properties of NI and GNI and emphasize that the classical NI uniquely satisfies key structural properties such as uniform range, maximality, and minimality.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"530 ","pages":"Article 109763"},"PeriodicalIF":2.7,"publicationDate":"2026-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940518","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 : 2026-01-01DOI: 10.1016/j.fss.2025.109747
Francisco David Camacho-Gonzalez , Jesús Ariel Carrasco-Ochoa , José Francisco Martínez-Trinidad
Goldman’s fuzzy reducts are subsets of attributes that allows for preserving the discernibility capacity of the whole set of attributes, in these subsets each attribute has an associated discernibility value, which can be interpreted as the capacity the correspeponing attribute has to discern objects of different class. Computing all Goldman’s fuzzy reducts is time-consuming, and few algorithms have been proposed in the literature. For these reasons, this paper introduces a new algorithm for computing all Goldman’s fuzzy reducts. The proposed algorithm traverses the search space following a new ordering and applies pruning properties, introduced in this paper, that help avoid exhaustively evaluating the reduct definition and discarding subsets. Additionally, we introduced a concept of density for simplified non-Boolean discernibility matrices that allows a density-based characterization of the algorithms’ performance. The proposed algorithm is evaluated and compared against state-of-the-art algorithms on synthetic and real decision systems. From our experiments, we determine the matrix type regarding density, where our algorithm performs the best.
{"title":"Algorithm for computing all Goldman’s fuzzy reducts","authors":"Francisco David Camacho-Gonzalez , Jesús Ariel Carrasco-Ochoa , José Francisco Martínez-Trinidad","doi":"10.1016/j.fss.2025.109747","DOIUrl":"10.1016/j.fss.2025.109747","url":null,"abstract":"<div><div>Goldman’s fuzzy reducts are subsets of attributes that allows for preserving the discernibility capacity of the whole set of attributes, in these subsets each attribute has an associated discernibility value, which can be interpreted as the capacity the correspeponing attribute has to discern objects of different class. Computing all Goldman’s fuzzy reducts is time-consuming, and few algorithms have been proposed in the literature. For these reasons, this paper introduces a new algorithm for computing all Goldman’s fuzzy reducts. The proposed algorithm traverses the search space following a new ordering and applies pruning properties, introduced in this paper, that help avoid exhaustively evaluating the reduct definition and discarding subsets. Additionally, we introduced a concept of density for simplified non-Boolean discernibility matrices that allows a density-based characterization of the algorithms’ performance. The proposed algorithm is evaluated and compared against state-of-the-art algorithms on synthetic and real decision systems. From our experiments, we determine the matrix type regarding density, where our algorithm performs the best.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"530 ","pages":"Article 109747"},"PeriodicalIF":2.7,"publicationDate":"2026-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145978883","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 : 2025-12-31DOI: 10.1016/j.fss.2025.109750
Zhilong He , Chuandong Li , Cheng Hu , Zhiyong Yu , Haijun Jiang , Shiping Wen
This paper investigates the stability and stabilization of nonlinear time-delay systems via a novel Takagi-Sugeno (T-S) fuzzy hybrid impulsive controller that explicitly accounts for both input delay and saturation, enhancing its practical applicability. The main contributions are threefold. First, a new impulse-time-related Lyapunov function (ITRLF) is constructed, which synergistically integrates the impulsive Razumikhin technique with an improved convex hull representation to handle saturation nonlinearities effectively. Second, sufficient conditions in the form of linear matrix inequalities (LMIs) are established to ensure local exponential stability. A key advantage of these conditions is that they depend only on the bounds of impulse delays and intervals, eliminating the restrictive requirement of a specific relationship between them, thus reducing conservatism. Third, novel LMI-based optimization algorithms are proposed to maximize the estimation of the region of attraction (ROA), effectively trading off computational complexity for significantly reduced conservatism compared to conventional methods. The effectiveness and advantages of the proposed approach are validated through numerical simulations using the MATLAB LMI toolbox.
{"title":"Stability analysis of T-S fuzzy delayed impulsive systems with input saturation via an impulse-time-related function method","authors":"Zhilong He , Chuandong Li , Cheng Hu , Zhiyong Yu , Haijun Jiang , Shiping Wen","doi":"10.1016/j.fss.2025.109750","DOIUrl":"10.1016/j.fss.2025.109750","url":null,"abstract":"<div><div>This paper investigates the stability and stabilization of nonlinear time-delay systems via a novel Takagi-Sugeno (T-S) fuzzy hybrid impulsive controller that explicitly accounts for both input delay and saturation, enhancing its practical applicability. The main contributions are threefold. First, a new impulse-time-related Lyapunov function (ITRLF) is constructed, which synergistically integrates the impulsive Razumikhin technique with an improved convex hull representation to handle saturation nonlinearities effectively. Second, sufficient conditions in the form of linear matrix inequalities (LMIs) are established to ensure local exponential stability. A key advantage of these conditions is that they depend only on the bounds of impulse delays and intervals, eliminating the restrictive requirement of a specific relationship between them, thus reducing conservatism. Third, novel LMI-based optimization algorithms are proposed to maximize the estimation of the region of attraction (ROA), effectively trading off computational complexity for significantly reduced conservatism compared to conventional methods. The effectiveness and advantages of the proposed approach are validated through numerical simulations using the MATLAB LMI toolbox.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"530 ","pages":"Article 109750"},"PeriodicalIF":2.7,"publicationDate":"2025-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940524","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 : 2025-12-29DOI: 10.1016/j.fss.2025.109748
Ángel López Oriona, Ying Sun
The global approach has recently been shown to often outperform the local approach in time series forecasting. While the former fits a single model to all time series in a dataset, the latter fits a separate model to each series and therefore does not leverage potential similarities in the underlying structures across the time series collection. Empirical analyses have demonstrated that, in heterogeneous datasets, fitting per-cluster global models can further improve forecasting accuracy compared to the standard global approach. These methods obtain forecasts for a given time series using the global model associated with its corresponding cluster. In this paper, we show that combining the per-cluster approach with the fuzzy clustering paradigm can lead to even better predictive performance. Specifically, we propose a fuzzy clustering algorithm based on prediction accuracy of global models. The fuzzy partition is constructed by evaluating the prediction error of each series with respect to each global model. Forecasts for a given series are then obtained as a weighted average of the forecasts from all models, with weights determined by the corresponding membership degrees. The potential of the proposed approach is demonstrated using well-known time series datasets from several contexts. Improvements of up to 30% in forecasting error are achieved compared to the best of three strong benchmark methods.
{"title":"Forecasting time series collections via fuzzy clustering","authors":"Ángel López Oriona, Ying Sun","doi":"10.1016/j.fss.2025.109748","DOIUrl":"10.1016/j.fss.2025.109748","url":null,"abstract":"<div><div>The global approach has recently been shown to often outperform the local approach in time series forecasting. While the former fits a single model to all time series in a dataset, the latter fits a separate model to each series and therefore does not leverage potential similarities in the underlying structures across the time series collection. Empirical analyses have demonstrated that, in heterogeneous datasets, fitting per-cluster global models can further improve forecasting accuracy compared to the standard global approach. These methods obtain forecasts for a given time series using the global model associated with its corresponding cluster. In this paper, we show that combining the per-cluster approach with the fuzzy clustering paradigm can lead to even better predictive performance. Specifically, we propose a fuzzy clustering algorithm based on prediction accuracy of global models. The fuzzy partition is constructed by evaluating the prediction error of each series with respect to each global model. Forecasts for a given series are then obtained as a weighted average of the forecasts from all models, with weights determined by the corresponding membership degrees. The potential of the proposed approach is demonstrated using well-known time series datasets from several contexts. Improvements of up to 30% in forecasting error are achieved compared to the best of three strong benchmark methods.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"530 ","pages":"Article 109748"},"PeriodicalIF":2.7,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145877099","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 : 2025-12-29DOI: 10.1016/j.fss.2025.109749
Stefano Aguzzoli , Brunella Gerla , Sarah Nastasi
Fortresses constitute a class of language descriptors completely based only on two fundamental concepts of propositional logic: the notion of logical consequence and the notion of substitution. Fortresses based on classical propositional logic precisely recognise the class of regular languages.
In this paper, we characterise the formal languages obtained by replacing, in fortresses, the notion of logical consequence in classical logic with the one in Gödel infinitely-valued logic. We prove that fortresses in Gödel logic exactly recognise finite chains of inclusions of regular languages. To prove this, we make use of a Stone’s type categorical duality between the algebraic semantics of Gödel propositional logic and the category of finite forests and open maps.
{"title":"A Gödel logic descriptor for chains of regular languages","authors":"Stefano Aguzzoli , Brunella Gerla , Sarah Nastasi","doi":"10.1016/j.fss.2025.109749","DOIUrl":"10.1016/j.fss.2025.109749","url":null,"abstract":"<div><div><em>Fortresses</em> constitute a class of language descriptors completely based only on two fundamental concepts of propositional logic: the notion of logical consequence and the notion of substitution. Fortresses based on classical propositional logic precisely recognise the class of regular languages.</div><div>In this paper, we characterise the formal languages obtained by replacing, in fortresses, the notion of logical consequence in classical logic with the one in Gödel infinitely-valued logic. We prove that fortresses in Gödel logic exactly recognise finite chains of inclusions of regular languages. To prove this, we make use of a Stone’s type categorical duality between the algebraic semantics of Gödel propositional logic and the category of finite forests and open maps.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"530 ","pages":"Article 109749"},"PeriodicalIF":2.7,"publicationDate":"2025-12-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145940517","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 : 2025-12-22DOI: 10.1016/j.fss.2025.109745
Ignacio Montes
Robust statistics aims to develop methods that ensure the obtained results remain stable regardless of the quality of the data. A natural approach to enhancing the robustness of a probabilistic model is to consider a neighbourhood around a probability measure with a given radius. These models require two tools: a function for comparing probability measures and the radius measuring the level of imprecision or robustness to be added to the model. With these tools, a neighbourhood or distortion model is defined as the closed ball centred at the probability measure. Many well-known robust models, such as the ϵ-contamination, pari-mutuel model, or constant odds ratio, can be incorporated into the broader framework of neighbourhood models. Additionally, distances between probability measures, such as the total variation or the Kolmogorov distances, determine a neighbourhood model. This paper delves into the neighbourhood models induced by the Euclidean distance applied to the probability mass functions and the Kullback-Leibler divergence. For investigating these models from the perspective of imprecise probability theory, we make use of common optimisation tools with restrictions. We conclude this study providing a comparison study among different neighbourhood models.
{"title":"Neighbourhood models induced by the Euclidean distance and the Kullback-Leibler divergence","authors":"Ignacio Montes","doi":"10.1016/j.fss.2025.109745","DOIUrl":"10.1016/j.fss.2025.109745","url":null,"abstract":"<div><div>Robust statistics aims to develop methods that ensure the obtained results remain stable regardless of the quality of the data. A natural approach to enhancing the robustness of a probabilistic model is to consider a neighbourhood around a probability measure with a given radius. These models require two tools: a function for comparing probability measures and the radius measuring the level of imprecision or robustness to be added to the model. With these tools, a neighbourhood or distortion model is defined as the closed ball centred at the probability measure. Many well-known robust models, such as the ϵ-contamination, pari-mutuel model, or constant odds ratio, can be incorporated into the broader framework of neighbourhood models. Additionally, distances between probability measures, such as the total variation or the Kolmogorov distances, determine a neighbourhood model. This paper delves into the neighbourhood models induced by the Euclidean distance applied to the probability mass functions and the Kullback-Leibler divergence. For investigating these models from the perspective of imprecise probability theory, we make use of common optimisation tools with restrictions. We conclude this study providing a comparison study among different neighbourhood models.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"529 ","pages":"Article 109745"},"PeriodicalIF":2.7,"publicationDate":"2025-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145885601","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 : 2025-12-20DOI: 10.1016/j.fss.2025.109742
Wenqi Zhang , Xiang Wu , Hai Zhang , Xiaofeng Ye , Jinde Cao
This paper investigates finite-time projective synchronization (FTP-S) of Caputo fractional multilayer coupled Takagi-Sugeno (T-S) fuzzy delayed memristive neural networks with jumping mismatches. First, a new model combining multilayer coupling and jumping mismatches is proposed to address hierarchical interactions and synaptic weight shifts. Then, a novel Razumikhin finite-time stability lemma for Caputo fractional delayed systems is proposed. Leveraging this lemma and an event-triggered mechanism based on T-S fuzzy rules, sufficient conditions for FTP-S and its settling time are derived. Finally, the absence of Zeno behavior is demonstrated. Notably, the proposed lemma overcomes limitations of existing methods and applies broadly to other fractional delayed systems. Numerical simulations validate the effectiveness of the theory and control strategy.
{"title":"Innovative Razumikhin method for finite-time projective synchronization in fractional delay-coupled fuzzy memristive neural networks with jump mismatches","authors":"Wenqi Zhang , Xiang Wu , Hai Zhang , Xiaofeng Ye , Jinde Cao","doi":"10.1016/j.fss.2025.109742","DOIUrl":"10.1016/j.fss.2025.109742","url":null,"abstract":"<div><div>This paper investigates finite-time projective synchronization (FTP-S) of Caputo fractional multilayer coupled Takagi-Sugeno (T-S) fuzzy delayed memristive neural networks with jumping mismatches. First, a new model combining multilayer coupling and jumping mismatches is proposed to address hierarchical interactions and synaptic weight shifts. Then, a novel Razumikhin finite-time stability lemma for Caputo fractional delayed systems is proposed. Leveraging this lemma and an event-triggered mechanism based on T-S fuzzy rules, sufficient conditions for FTP-S and its settling time are derived. Finally, the absence of Zeno behavior is demonstrated. Notably, the proposed lemma overcomes limitations of existing methods and applies broadly to other fractional delayed systems. Numerical simulations validate the effectiveness of the theory and control strategy.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"529 ","pages":"Article 109742"},"PeriodicalIF":2.7,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842545","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}
The notion of spline linearity for aggregation functions is recalled. We then focus on linear spline 2-copulas and various constructions of 2-copulas to investigate the impact of these constructions on spline linearity. Moreover, associative copulas that are linear splines are discussed. Some other methods that yield linear spline copulas are also presented and illustrated.
{"title":"2-copulas and linear splines","authors":"Fateme Kouchakinejad , Radko Mesiar , Andrea Stupňanová , Radomír Halaš","doi":"10.1016/j.fss.2025.109746","DOIUrl":"10.1016/j.fss.2025.109746","url":null,"abstract":"<div><div>The notion of spline linearity for aggregation functions is recalled. We then focus on linear spline 2-copulas and various constructions of 2-copulas to investigate the impact of these constructions on spline linearity. Moreover, associative copulas that are linear splines are discussed. Some other methods that yield linear spline copulas are also presented and illustrated.</div></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":"529 ","pages":"Article 109746"},"PeriodicalIF":2.7,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842547","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}
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