Learning Fuzzy Measures

This chapter is a key contribution of this work in which various computational approaches to learning fuzzy measures are described. The learning problem is framed from the perspective of data fitting, where we aim to define a model that interpolates or approximates a set of observed or user-specified instances. Fitting is performed with respect to different metrics, and by solving different convex and non-convex optimisation problems. The computational complexity of fuzzy measures is addressed by using simplifying assumptions, in particular the notion of k-order fuzzy measures and the software packages implementing the presented fitting approaches are also described.

[1]  Mathieu Serrurier,et al.  Elicitation of Sugeno Integrals: A Version Space Learning Perspective , 2009, ISMIS.

[2]  S. Greco,et al.  Combining analytical hierarchy process and Choquet integral within non-additive robust ordinal regression ☆ , 2016 .

[3]  Gleb Beliakov,et al.  Learning k-maxitive fuzzy measures from data by mixed integer programming , 2021, Fuzzy Sets Syst..

[4]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[5]  Alain Chateauneuf,et al.  Some Characterizations of Lower Probabilities and Other Monotone Capacities through the use of Möbius Inversion , 1989, Classic Works of the Dempster-Shafer Theory of Belief Functions.

[6]  Michel Grabisch,et al.  Classification by fuzzy integral: performance and tests , 1994, CVPR 1994.

[7]  Stephen J. Wright,et al.  Object-oriented software for quadratic programming , 2003, TOMS.

[8]  Milosz Kadzinski,et al.  Robust ordinal regression in preference learning and ranking , 2013, Machine Learning.

[9]  Ivan Kojadinovic,et al.  Minimum variance capacity identification , 2007, Eur. J. Oper. Res..

[10]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[11]  Gang Li,et al.  Learning Choquet-Integral-Based Metrics for Semisupervised Clustering , 2011, IEEE Transactions on Fuzzy Systems.

[12]  Salvatore Greco,et al.  Non-additive robust ordinal regression: A multiple criteria decision model based on the Choquet integral , 2010, Eur. J. Oper. Res..

[13]  Qiang Zhang,et al.  2-order additive fuzzy measure identification method based on diamond pairwise comparison and maximum entropy principle , 2010, Fuzzy Optim. Decis. Mak..

[14]  Gleb Beliakov,et al.  Construction of aggregation functions from data using linear programming , 2009, Fuzzy Sets Syst..

[15]  Marc Roubens,et al.  Choice, Ranking and Sorting in Fuzzy Multiple Criteria Decision Aid , 2005 .

[16]  James M. Keller,et al.  Learning Fuzzy-Valued Fuzzy Measures for the Fuzzy-Valued Sugeno Fuzzy Integral , 2010, IPMU.

[17]  Marek Gagolewski,et al.  Fitting Symmetric Fuzzy Measures for Discrete Sugeno Integration , 2017, EUSFLAT/IWIFSGN.

[18]  Eyke Hüllermeier,et al.  Learning monotone nonlinear models using the Choquet integral , 2011, Machine Learning.

[19]  Huy Quan Vu,et al.  A Choquet Integral Toolbox and Its Application in Customer Preference Analysis , 2014 .

[20]  Salvatore Greco,et al.  Robust ordinal regression for value functions handling interacting criteria , 2014, Eur. J. Oper. Res..

[21]  Jean-Luc Marichal,et al.  k-intolerant capacities and Choquet integrals , 2007, Eur. J. Oper. Res..

[22]  Michel Grabisch,et al.  A review of methods for capacity identification in Choquet integral based multi-attribute utility theory: Applications of the Kappalab R package , 2008, Eur. J. Oper. Res..

[23]  Michel Grabisch,et al.  A new algorithm for identifying fuzzy measures and its application to pattern recognition , 1995, Proceedings of 1995 IEEE International Conference on Fuzzy Systems..

[24]  Gleb Beliakov,et al.  Nonadditivity index and capacity identification method in the context of multicriteria decision making , 2018, Inf. Sci..

[25]  Gleb Beliakov,et al.  Learning fuzzy measures from data: Simplifications and optimisation strategies , 2019, Inf. Sci..

[26]  Jian-Zhang Wu,et al.  Compromise principle based methods of identifying capacities in the framework of multicriteria decision analysis , 2014, Fuzzy Sets Syst..

[27]  Endre Pap,et al.  Two kinds of explicit preference information oriented capacity identification methods in the context of multicriteria decision analysis , 2018, Int. Trans. Oper. Res..

[28]  Patrick Meyer,et al.  Package R kappalab : " Non-additive measure and integral manipulation functions ", version 0.4-5 , 2010 .

[29]  M. Grabisch,et al.  A representation of preferences by the Choquet integral with respect to a 2-additive capacity , 2011 .

[30]  Humberto Bustince,et al.  A Practical Guide to Averaging Functions , 2015, Studies in Fuzziness and Soft Computing.

[31]  Jean-Luc Marichal,et al.  Determination of weights of interacting criteria from a reference set , 2000, Eur. J. Oper. Res..

[32]  Jean-Luc Marichal,et al.  An axiomatic approach to the definition of the entropy of a discrete Choquet capacity , 2005, Inf. Sci..

[33]  Jean-Luc Marichal,et al.  Tolerant or intolerant character of interacting criteria in aggregation by the Choquet integral , 2004, Eur. J. Oper. Res..

[34]  Salvatore Greco,et al.  Non Additive Robust Ordinal Regression for urban and territorial planning: an application for siting an urban waste landfill , 2016, Ann. Oper. Res..

[35]  George J. Klir,et al.  Constructing fuzzy measures: a new method and its application to cluster analysis , 1996, Proceedings of North American Fuzzy Information Processing.

[36]  Jean-Luc Marichal,et al.  Entropy of discrete Choquet capacities , 2002, Eur. J. Oper. Res..

[37]  Gleb Beliakov,et al.  Marginal contribution representation of capacity‐based multicriteria decision making , 2020, Int. J. Intell. Syst..

[38]  Serge Guillaume,et al.  k-maxitive fuzzy measures: A scalable approach to model interactions , 2017, Fuzzy Sets Syst..

[39]  Jean-Luc Marichal,et al.  Behavioral analysis of aggregation in multicriteria decision aid , 2000 .

[40]  R. Fletcher,et al.  Practical Methods of Optimization: Fletcher/Practical Methods of Optimization , 2000 .

[41]  Shanlin Yang,et al.  2-Additive Capacity Identification Methods From Multicriteria Correlation Preference Information , 2015, IEEE Transactions on Fuzzy Systems.

[42]  Marek Gagolewski,et al.  Supervised Learning to Aggregate Data With the Sugeno Integral , 2019, IEEE Transactions on Fuzzy Systems.

[43]  Eyke Hüllermeier,et al.  Efficient Learning of Classifiers Based on the 2-Additive Choquet Integral , 2013 .

[44]  Jean-Luc Marichal,et al.  Entropy of bi-capacities , 2007, EUSFLAT Conf..