Measure Identification for the Choquet Integral: A Python Module
Fuzzy integrals are common concepts which are used to aggregate input values in practical applications. Aggregation of inputs using fuzzy integrals opens up numerous possibilities for modeling interaction, redundancy, and synergy of inputs. However, fuzzy integrals need a fuzzy measure to start this...
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| Vydané v: | International journal of computational intelligence systems Ročník 15; číslo 1; s. 1 - 10 |
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| Jazyk: | English |
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Dordrecht
Springer Netherlands
21.10.2022
Springer Nature B.V Springer |
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| ISSN: | 1875-6883, 1875-6891, 1875-6883 |
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| Abstract | Fuzzy integrals are common concepts which are used to aggregate input values in practical applications. Aggregation of inputs using fuzzy integrals opens up numerous possibilities for modeling interaction, redundancy, and synergy of inputs. However, fuzzy integrals need a fuzzy measure to start this aggregation process. This situation pushes us into the fuzzy measure identification process. This process becomes difficult due to the monotony condition of the fuzzy measure and the exponential increase on the number of measure parameters. There are in the literature many ways to determine fuzzy measures. One of them is learning from data. In this paper, our aim is to introduce a new fuzzy measure identification tool to learn measures from empirical data. It is a Python module which finds the measure that minimizes the difference between the computed and expected outputs of the Choquet integral. In addition, we study some properties of the learning process. In particular, we consider
k
-additive fuzzy measures and belief functions as well as arbitrary fuzzy measures. Using these variety of measures we examine the effect of
k
and noisy data on the learning process. |
|---|---|
| AbstractList | Abstract Fuzzy integrals are common concepts which are used to aggregate input values in practical applications. Aggregation of inputs using fuzzy integrals opens up numerous possibilities for modeling interaction, redundancy, and synergy of inputs. However, fuzzy integrals need a fuzzy measure to start this aggregation process. This situation pushes us into the fuzzy measure identification process. This process becomes difficult due to the monotony condition of the fuzzy measure and the exponential increase on the number of measure parameters. There are in the literature many ways to determine fuzzy measures. One of them is learning from data. In this paper, our aim is to introduce a new fuzzy measure identification tool to learn measures from empirical data. It is a Python module which finds the measure that minimizes the difference between the computed and expected outputs of the Choquet integral. In addition, we study some properties of the learning process. In particular, we consider k-additive fuzzy measures and belief functions as well as arbitrary fuzzy measures. Using these variety of measures we examine the effect of k and noisy data on the learning process. Fuzzy integrals are common concepts which are used to aggregate input values in practical applications. Aggregation of inputs using fuzzy integrals opens up numerous possibilities for modeling interaction, redundancy, and synergy of inputs. However, fuzzy integrals need a fuzzy measure to start this aggregation process. This situation pushes us into the fuzzy measure identification process. This process becomes difficult due to the monotony condition of the fuzzy measure and the exponential increase on the number of measure parameters. There are in the literature many ways to determine fuzzy measures. One of them is learning from data. In this paper, our aim is to introduce a new fuzzy measure identification tool to learn measures from empirical data. It is a Python module which finds the measure that minimizes the difference between the computed and expected outputs of the Choquet integral. In addition, we study some properties of the learning process. In particular, we consider k -additive fuzzy measures and belief functions as well as arbitrary fuzzy measures. Using these variety of measures we examine the effect of k and noisy data on the learning process. Fuzzy integrals are common concepts which are used to aggregate input values in practical applications. Aggregation of inputs using fuzzy integrals opens up numerous possibilities for modeling interaction, redundancy, and synergy of inputs. However, fuzzy integrals need a fuzzy measure to start this aggregation process. This situation pushes us into the fuzzy measure identification process. This process becomes difficult due to the monotony condition of the fuzzy measure and the exponential increase on the number of measure parameters. There are in the literature many ways to determine fuzzy measures. One of them is learning from data. In this paper, our aim is to introduce a new fuzzy measure identification tool to learn measures from empirical data. It is a Python module which finds the measure that minimizes the difference between the computed and expected outputs of the Choquet integral. In addition, we study some properties of the learning process. In particular, we consider k-additive fuzzy measures and belief functions as well as arbitrary fuzzy measures. Using these variety of measures we examine the effect of k and noisy data on the learning process. |
| ArticleNumber | 89 |
| Author | Torra, Vicenç Türkarslan, Ezgi |
| Author_xml | – sequence: 1 givenname: Ezgi surname: Türkarslan fullname: Türkarslan, Ezgi email: ezgi.turkarslan@tedu.edu.tr organization: Department of Mathematics, Ankara University, Faculty of Science, Department of Mathematics, TED University, Faculty of Arts and Science – sequence: 2 givenname: Vicenç orcidid: 0000-0002-0368-8037 surname: Torra fullname: Torra, Vicenç organization: Department of Computing Sciences, Umeå University |
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| References | Grabisch, M.: New algorithm for identifying fuzzy measures and its application to pattern recognition. IEEE International Conference on Fuzzy Systems (1995) JavierMSergeGPilarBk-maxitive fuzzy measures: A scalable approach to model interactionsFuzzy Sets and Systems20173243348368550010.1016/j.fss.2017.04.0111382.28021 GrabischMk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-order additive discrete fuzzy measures and their representationFuzzy Sets and Systems199792167189148641710.1016/S0165-0114(97)00168-10927.28014 DimuroPGFernandezJBedregalBMesiarRSanzJALuccaGBustinceHThe state-of-art of the generalizations of the choquet integral: From aggregation and pre-aggregation to ordered directionally monotone functionsInf. Sci.202057274310.1016/j.inffus.2019.10.005 Marco-DetchartCLuccaGLopez-MolinaCDe MiguelLDimuroPGBustinceHNeuro-inspired edge feature fusion using choquet integralsInf. Sci.202158174075410.1016/j.ins.2021.10.016 Torra, V., Narukawa, Y., Sugeno, M.: Non-Additive Measures, Theory and Applications, Studies in Fuzziness and Soft Computing. Springer, Berlin (2013). https://doi.org/10.1007/978-3-319-03155-2 ChoquetGTheory of capacitiesAnnales de L’Institut Fourier195451312958076010.5802/aif.530064.35101 BeliakovGWuJZLearning fuzzy measures from data: Simplifications and optimisation strategiesInformation Sciences201949410011310.1016/j.ins.2019.04.0421451.68221 http://www.mdai.cat/ifao MesiarRGeneralizations of k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-order additive discrete fuzzy measuresFuzzy Sets and Systems1999102423428167690910.1142/S02184885990004890936.28014 Saleh, E., Valls, A., Moreno, A., Romero-Aroca, P., Bustince, H., Torra, V.: A hierarchically ⊥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\perp $$\end{document}-decomposable fuzzy measure-based approach for fuzzy rules aggregation. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 27, 59–76 (2019). https://doi.org/10.1142/S0218488519400038 KojadinovicIMinimum variance capacity identificationEuropean Journal of Operational Research2007177498514227862910.1016/j.ejor.2005.10.0591111.90055 Torra, V., Narukawa, Y.: Modeling Decisions: Information Fusion and Aggregation Operators. Springer, Berlin, Heidelberg (2007). https://doi.org/10.1007/978-3-540-68791-7 Rota, G.C.: On the foundations of combinatorial theory. i. the theory of möbius functions. Z. Wahrscheinlichkeitstheorie verw Gebiete 2, 340–368 (1964). https://doi.org/10.1007/BF00531932 Sugeno, M.: Theory of Fuzzy Integrals and Its Applications. Tokyo Institute of Technology,, Ph.D. Thesis (1974) Torra, V.: On a family of fuzzy measures for data fusion with reduced complexity. Proc. 3rd Int. Conf on Information Fusion, TuCa17-23 (2000). https://doi.org/10.1109/IFIC.2000.862689 BeliakovGConstruction of aggregation functions from data using linear programmingFuzzy Sets and Systems20091606575246943210.1016/j.fss.2008.07.0041183.62006 GrabischMKojadinovicIMeyerPA review of methods for capacity identification in choquet integral based multi-attribute utility theory: Applications of the kappalab r packageEuropean Journal of Operational Research2008186766785236371810.1016/j.ejor.2007.02.0251138.90407 Grabisch, M., Kojadinovic, I., Meyer, P.: Non-Additive Measure and Integral Manipulation Functions, (kappalab package in R). https://CRAN.R-project.org/package=kappalab (2015) Imai, D. H. Asano, Sato, Y.:An algorithm based on alternative projections for a fuzzy measure identification problem. In:Torra, V. (ed.) Information Fusion in Data Mining. Studies in Fuzziness and Soft Computing, pp. 149–158. Springer, Berlin, Heidelberg (2003). https://doi.org/10.1007/978-3-540-72434-6 Grabisch, M., Marichal, J.L., Mesiar, R.: Aggregation Functions, Encyclopedia of Mathematics and Its Applications. Cambridge University Pres, ??? (2009). https://doi.org/10.1017/CBO9781139644150 GrabischMThe application of fuzzy integrals in multi criteria decision makingEuropean Journal of Operational Research19968944545610.1016/0377-2217(95)00176-X0916.90164 CombarroEFIdentification of fuzzy measures from sample data with genetic algorithmsComputers & Operations Research2006333046306610.1016/j.cor.2005.02.0341086.90069 G Beliakov (146_CR9) 2019; 494 146_CR13 146_CR23 M Javier (146_CR8) 2017; 324 146_CR15 146_CR1 146_CR2 PG Dimuro (146_CR3) 2020; 57 C Marco-Detchart (146_CR6) 2021; 581 146_CR22 146_CR10 146_CR21 146_CR5 M Grabisch (146_CR14) 2008; 186 G Beliakov (146_CR7) 2009; 160 EF Combarro (146_CR12) 2006; 33 I Kojadinovic (146_CR11) 2007; 177 146_CR4 M Grabisch (146_CR19) 1997; 92 G Choquet (146_CR20) 1954; 5 R Mesiar (146_CR18) 1999; 102 M Grabisch (146_CR17) 1996; 89 146_CR16 |
| References_xml | – reference: Marco-DetchartCLuccaGLopez-MolinaCDe MiguelLDimuroPGBustinceHNeuro-inspired edge feature fusion using choquet integralsInf. Sci.202158174075410.1016/j.ins.2021.10.016 – reference: ChoquetGTheory of capacitiesAnnales de L’Institut Fourier195451312958076010.5802/aif.530064.35101 – reference: JavierMSergeGPilarBk-maxitive fuzzy measures: A scalable approach to model interactionsFuzzy Sets and Systems20173243348368550010.1016/j.fss.2017.04.0111382.28021 – reference: GrabischMk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-order additive discrete fuzzy measures and their representationFuzzy Sets and Systems199792167189148641710.1016/S0165-0114(97)00168-10927.28014 – reference: Grabisch, M., Kojadinovic, I., Meyer, P.: Non-Additive Measure and Integral Manipulation Functions, (kappalab package in R). https://CRAN.R-project.org/package=kappalab (2015) – reference: BeliakovGConstruction of aggregation functions from data using linear programmingFuzzy Sets and Systems20091606575246943210.1016/j.fss.2008.07.0041183.62006 – reference: DimuroPGFernandezJBedregalBMesiarRSanzJALuccaGBustinceHThe state-of-art of the generalizations of the choquet integral: From aggregation and pre-aggregation to ordered directionally monotone functionsInf. Sci.202057274310.1016/j.inffus.2019.10.005 – reference: Torra, V.: On a family of fuzzy measures for data fusion with reduced complexity. Proc. 3rd Int. Conf on Information Fusion, TuCa17-23 (2000). https://doi.org/10.1109/IFIC.2000.862689 – reference: Imai, D. H. Asano, Sato, Y.:An algorithm based on alternative projections for a fuzzy measure identification problem. In:Torra, V. (ed.) Information Fusion in Data Mining. Studies in Fuzziness and Soft Computing, pp. 149–158. Springer, Berlin, Heidelberg (2003). https://doi.org/10.1007/978-3-540-72434-6 – reference: Torra, V., Narukawa, Y.: Modeling Decisions: Information Fusion and Aggregation Operators. Springer, Berlin, Heidelberg (2007). https://doi.org/10.1007/978-3-540-68791-7 – reference: Sugeno, M.: Theory of Fuzzy Integrals and Its Applications. Tokyo Institute of Technology,, Ph.D. Thesis (1974) – reference: KojadinovicIMinimum variance capacity identificationEuropean Journal of Operational Research2007177498514227862910.1016/j.ejor.2005.10.0591111.90055 – reference: Saleh, E., Valls, A., Moreno, A., Romero-Aroca, P., Bustince, H., Torra, V.: A hierarchically ⊥\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\perp $$\end{document}-decomposable fuzzy measure-based approach for fuzzy rules aggregation. Int. J. Uncertain. Fuzziness Knowl. Based Syst. 27, 59–76 (2019). https://doi.org/10.1142/S0218488519400038 – reference: CombarroEFIdentification of fuzzy measures from sample data with genetic algorithmsComputers & Operations Research2006333046306610.1016/j.cor.2005.02.0341086.90069 – reference: BeliakovGWuJZLearning fuzzy measures from data: Simplifications and optimisation strategiesInformation Sciences201949410011310.1016/j.ins.2019.04.0421451.68221 – reference: Grabisch, M.: New algorithm for identifying fuzzy measures and its application to pattern recognition. IEEE International Conference on Fuzzy Systems (1995) – reference: Grabisch, M., Marichal, J.L., Mesiar, R.: Aggregation Functions, Encyclopedia of Mathematics and Its Applications. Cambridge University Pres, ??? (2009). https://doi.org/10.1017/CBO9781139644150 – reference: Rota, G.C.: On the foundations of combinatorial theory. i. the theory of möbius functions. Z. Wahrscheinlichkeitstheorie verw Gebiete 2, 340–368 (1964). https://doi.org/10.1007/BF00531932 – reference: GrabischMThe application of fuzzy integrals in multi criteria decision makingEuropean Journal of Operational Research19968944545610.1016/0377-2217(95)00176-X0916.90164 – reference: GrabischMKojadinovicIMeyerPA review of methods for capacity identification in choquet integral based multi-attribute utility theory: Applications of the kappalab r packageEuropean Journal of Operational Research2008186766785236371810.1016/j.ejor.2007.02.0251138.90407 – reference: http://www.mdai.cat/ifao/ – reference: Torra, V., Narukawa, Y., Sugeno, M.: Non-Additive Measures, Theory and Applications, Studies in Fuzziness and Soft Computing. Springer, Berlin (2013). https://doi.org/10.1007/978-3-319-03155-2 – reference: MesiarRGeneralizations of k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k$$\end{document}-order additive discrete fuzzy measuresFuzzy Sets and Systems1999102423428167690910.1142/S02184885990004890936.28014 – volume: 494 start-page: 100 year: 2019 ident: 146_CR9 publication-title: Information Sciences doi: 10.1016/j.ins.2019.04.042 – ident: 146_CR10 – volume: 5 start-page: 131 year: 1954 ident: 146_CR20 publication-title: Annales de L’Institut Fourier doi: 10.5802/aif.53 – volume: 57 start-page: 27 year: 2020 ident: 146_CR3 publication-title: Inf. Sci. doi: 10.1016/j.inffus.2019.10.005 – volume: 581 start-page: 740 year: 2021 ident: 146_CR6 publication-title: Inf. 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| Snippet | Fuzzy integrals are common concepts which are used to aggregate input values in practical applications. Aggregation of inputs using fuzzy integrals opens up... Abstract Fuzzy integrals are common concepts which are used to aggregate input values in practical applications. Aggregation of inputs using fuzzy integrals... |
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| SubjectTerms | Artificial Intelligence Belief functions Computational Intelligence Control Engineering Fuzzy measure identification Integrals k-additive fuzzy measure Learning Mathematical Logic and Foundations Mechatronics Modules Monotony Möbius transform Python Redundancy Research Article Robotics |
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| Title | Measure Identification for the Choquet Integral: A Python Module |
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