Skip to main content
SpringerLink
Log in

Designing Interpretable Fuzzy Systems

  • Chapter
  • First Online:
Explainable Fuzzy Systems

Part of the book series: Studies in Computational Intelligence ((SCI,volume 970))

Abstract

Fuzzy sets and fuzzy logic are powerful tools widely used to represent human knowledge and mimic human reasoning capabilities, being the main constituents of fuzzy systems. Among the different approaches to fuzzy systems, fuzzy rule-based systems represent the one offering a better framework for interpretability considerations. Their applications range from classification to modeling and control. Independently on its purpose, the behavior of a fuzzy rule-based system can be evaluated in terms of its reliability and comprehensibility, two concepts usually represented by accuracy and interpretability in the context of fuzzy systems. Indeed, achieving the highest possible levels of accuracy and interpretability is one of the central aspects of designing fuzzy systems. The present chapter will go through the design process considering its different steps and analyzing the multiple options allowing us to improve interpretability or to achieve a better interpretability-accuracy balance, in the search for interpretable fuzzy systems which represent an interesting approach in the framework of explainable Artificial Intelligence. We will consider questions related to the knowledge extraction and refinement process; some examples are complexity reduction and semantic improvement. We will also analyze other questions linked to the design of the processing structure, such as the effects of applying different aggregation and implication operators.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 149.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 199.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 199.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    Some authors generalize the definition of rules by including also disjunctive consequents (Pedrycz and Gomide 1998). In such case, however, inference cannot be accomplished independently for each output variable. For such reason, disjunctive consequents are very rare in fuzzy modeling.

References

  • Alcalá, R, Alcalá-Fdez J, Gacto MJ, Herrera F (2008) On the usefulness of MOEAs for getting compact FRBSs under parameter tuning and rule selection. In: Ghosh A, Dehuri S, Ghosh S (eds) Multi-objective evolutionary algorithms for knowledge discovery from databases. Springer, Berlin, Heidelberg, pp 91–107. https://doi.org/10.1007/978-3-540-77467-9_5

  • Alcalá R, Casillas J, Cordón O, Herrera F (2001a) Building fuzzy graphs: features and taxonomy of learning for non-grid-oriented fuzzy rule-based systems. J Intell Fuzzy Syst 11:99–119

    Google Scholar 

  • Alcalá R, Casillas J, Cordón O, Herrera F (2001b) Cooperative coevolution for linguistic modeling with weighted double-consequent fuzzy rules. In: International conference in fuzzy logic and technology, pp 237–240

    Google Scholar 

  • Alcalá-Fdez J, Alcalá R, Herrera F (2011) A fuzzy association rule-based classification model for high-dimensional problems with genetic rule selection and lateral tuning. IEEE Trans Fuzzy Syst 19(5):857–872. https://doi.org/10.1109/TFUZZ.2011.2147794

    Article  Google Scholar 

  • Alonso JM, Castiello C, Magdalena L, Mencar C (2021a) An overview of fuzzy systems. In: Explainable fuzzy systems: paving the way from interpretable fuzzy systems to explainable AI systems, studies in computational intelligence, Chapter 2, vol 270. Springer, pp 25–48. https://doi.org/10.1007/978-3-030-71098-9_2

  • Alonso JM, Castiello C, Magdalena L, Mencar C (2021b) Interpretability constraints and criteria for fuzzy systems. In: Explainable fuzzy systems: paving the way from interpretable fuzzy systems to explainable AI systems, studies in computational intelligence, chapter 3, vol 270. Springer, pp 49–89. https://doi.org/10.1007/978-3-030-71098-9_3

  • Alonso JM, Castiello C, Magdalena L, Mencar C (2021c) Revisiting indexes for assessing interpretability of fuzzy systems. In: Explainable fuzzy systems: paving the way from interpretable fuzzy systems to explainable AI systems, studies in computational intelligence, Chapter 4, vol 270. Springer, pp 91–118. https://doi.org/10.1007/978-3-030-71098-9_4

  • Alonso JM, Castiello C, Mencar C (2015) Interpretability of fuzzy systems: current research trends and prospects. In: Kacprzyk J, Pedrycz W (eds) Springer handbook of computational intelligence. Springer, Berlin/Heidelberg, pp 219–237. https://doi.org/10.1007/978-3-662-43505-2_14

  • Alonso JM, Cordon O, Guillaume S, Magdalena L (2007) Highly interpretable linguistic knowledge bases optimization: genetic tuning versus solis-wetts. Looking for a good interpretability-accuracy trade-off. In: IEEE international conference on fuzzy systems (FUZZ-IEEE), pp 901–906. https://doi.org/10.1109/FUZZY.2007.4295485

  • Alonso JM, Magdalena L (2009) A conceptual framework for understanding a fuzzy system. Joint IFSA-EUSFLAT Conf 75:119–124

    Google Scholar 

  • Alonso JM, Magdalena L (2011) HILK++: an interpretability-guided fuzzy modeling methodology for learning readable and comprehensible fuzzy rule-based classifiers. Soft Comput 15(10):1959–1980. https://doi.org/10.1007/s00500-010-0628-5

  • Alonso JM, Magdalena L, Cordón O (2010) Embedding HILK in a three-objective evolutionary algorithm with the aim of modeling highly interpretable fuzzy rule-based classifiers. In: International workshop on genetic and evolutionary fuzzy systems (GEFS). IEEE, pp 15–20. https://doi.org/10.1109/GEFS.2010.5454165

  • Alonso JM, Magdalena L, Guillaume S (2008) HILK: a new methodology for designing highly interpretable linguistic knowledge bases using the fuzzy logic formalism. Int J Intell Syst 23(7):761–794. https://doi.org/10.1002/int.20288

    Article  MATH  Google Scholar 

  • Antonelli M, Bernardo D, Hagras H, Marcelloni F (2017) Multiobjective evolutionary optimization of type-2 fuzzy rule-based systems for financial data classification. IEEE Trans Fuzzy Syst 25(2):249–264

    Article  Google Scholar 

  • Antonelli M, Ducange P, Lazzerini B, Marcelloni F (2009) Multi-objective evolutionary learning of granularity, membership function parameters and rules of Mamdani fuzzy systems. Evol Intell 2(1–2):21–37. https://doi.org/10.1007/s12065-009-0022-3

    Article  Google Scholar 

  • Antonelli M, Ducange P, Lazzerini B, Marcelloni F (2011) Learning knowledge bases of multi-objective evolutionary fuzzy systems by simultaneously optimizing accuracy, complexity and partition integrity. Soft Comput 15(12):2335–2354. https://doi.org/10.1007/s00500-010-0665-0

    Article  Google Scholar 

  • Antonelli M, Ducange P, Marcelloni F, Segatori A (2016) On the influence of feature selection in fuzzy rule-based regression model generation. Inf Sci 329:649–669. https://doi.org/10.1016/j.ins.2015.09.045

    Article  Google Scholar 

  • Bardossy A, Duckstein L (1995) Fuzzy rule-based modeling with application to geophysical. CRC Press, Biological and Engineering Systems

    Google Scholar 

  • Barsacchi M, Bechini A, Ducange P, Marcelloni F (2019) Optimizing partition granularity, membership function parameters, and rule bases of fuzzy classifiers for big data by a multi-objective evolutionary approach. Cogn Comput 11(3):s12, 559–018–9613–6

    Google Scholar 

  • Bastian A (1994) How to handle the flexibility of linguistic variables with applications. Int J Uncertain Fuzziness Knowl-Based Syst 2(4):463–484

    Article  MathSciNet  MATH  Google Scholar 

  • Bikdash M (1999) A highly interpretable form of Sugeno inference systems. IEEE Trans Fuzzy Syst 7(6):686–696. https://doi.org/10.1109/91.811237

    Article  Google Scholar 

  • Bodenhofer U, Bauer P (2000) Towards an axiomatic treatment of interpretability. Int Conf Soft Comput 2:334–339

    Google Scholar 

  • Bouchon-Meunier B (1987) Linguistic variables in the knowledge base of an expert system. In: Rose J (ed) Cybernetics and systems: the way ahead. Thales Publications, Lytham St Annes, England, pp 745–752

    Google Scholar 

  • Bouchon-Meunier B, Jia Y (1992) Linguistic modifiers and imprecise categories. Int J Intell Syst 7:25–36

    Article  MATH  Google Scholar 

  • Cannone R, Alonso JM, Magdalena L (2011) Multi-objective design of highly interpretable fuzzy rule-based classifiers with semantic cointension. In: IEEE symposium series on computational intelligence (IEEE-SSCI), IV international workshop on genetic and evolutionary fuzzy systems (GEFS), Paris, pp 1–8. https://doi.org/10.1109/GEFS.2011.5949502

  • Carmona P, Castro JL, Herrero L (2012) Balancing interpretability against accuracy in fuzzy modeling by means of ACO. In: International conference on information processing and management of uncertainty in knowledge-based systems (IPMU), pp 71–80

    Google Scholar 

  • Carmona P, Castro JL, Zurita JM (2004) Learning maximal structure fuzzy rules with exceptions. Fuzzy Sets Syst 146(1):63–77. https://doi.org/10.1016/j.fss.2003.11.006

    Article  MathSciNet  MATH  Google Scholar 

  • Casillas J, Cordón O, Herrera F (2002) COR: a methodology to improve ad hoc data-driven linguistic rule learning methods by inducing cooperation among rules. IEEE Trans Syst Man Cybern-Part B: Cybern 32(4):526–537

    Article  Google Scholar 

  • Casillas J, Cordón O, Herrera F, Magdalena L (2003a) Accuracy improvements in linguistic fuzzy modeling. Studies in fuzziness and soft computing, vol 129. Springer, Heidelberg

    Google Scholar 

  • Casillas J, Cordón O, Herrera F, Magdalena L (2003b) Interpretability issues in fuzzy modeling. Studies in fuzziness and soft computing, vol 128. Springer, Heidelberg

    Google Scholar 

  • Casillas J, Cordón O, Herrera F, Merelo JJ (2002) Cooperative coevolution for learning fuzzy rule-based systems. In: International conference on artificial evolution (Evolution Artificielle), pp 311–322. Springer, Berlin, Heidelberg

    Google Scholar 

  • Casillas J, Cordón O, del Jesus MJ, Herrera F (2001) Genetic tuning of fuzzy rule-based systems integrating linguistic hedges. In: Joint IFSA-NAFIPS conference, pp 1570–1574

    Google Scholar 

  • Castiello C, Fanelli AM, Lucarelli M, Mencar C (2019) Interpretable fuzzy partitioning of classified data with variable granularity. Appl Soft Comput 74:567–582. https://doi.org/10.1016/j.asoc.2018.10.040

    Article  Google Scholar 

  • Castro JL, Castro-Schez JJ, Zurita JM (1999) Learning maximal structure rules in fuzzy logic for knowledge acquisition in expert systems. Fuzzy Sets Syst 101(3):331–342

    Article  MathSciNet  MATH  Google Scholar 

  • Chang TC, Hasegawa K, Ibbs CW (1991) The effects of membership functions in fuzzy reasoning. Fuzzy Sets Syst 44:169–186

    Article  MATH  Google Scholar 

  • Chi Z, Yan H, Pham T (1996) Fuzzy algorithms: with applications to image processing and pattern recognition. World Scientific

    Google Scholar 

  • Cococcioni M, Ducange P, Lazzerini B, Marcelloni F (2007) A Pareto-based multi-objective evolutionary approach to the identification of Mamdani fuzzy systems. Soft Comput 11(11):1013–1031. https://doi.org/10.1007/s00500-007-0150-6

    Article  Google Scholar 

  • Cordón O, Herrera F (1997) A three-stage evolutionary process for learning descriptive and approximate fuzzy logic controller knowledge bases from examples. Int J Approx Reason 17(4):369–407

    Article  MATH  Google Scholar 

  • Cordón O, Herrera F (2000) A proposal for improving the accuracy of linguistic modeling. IEEE Trans Fuzzy Syst 8(3):335–344. https://doi.org/10.1109/91.855921

    Article  Google Scholar 

  • Cordón O, Herrera F, Hoffmann F, Magdalena L (2001) Genetic fuzzy systems: evolutionary tuning and learning of fuzzy knowledge bases. World Scientific

    Google Scholar 

  • Cordón O, Herrera F, Magdalena L, Villar P (2001) A genetic learning process for the scaling factors, granularity and contexts of the fuzzy rule-based system data base. Inf Sci 136(1–4):85–107. https://doi.org/10.1016/S0020-0255(01)00143-8

    Article  MATH  Google Scholar 

  • Cordón O, Herrera F, Peregrín A (1997) Applicability of the fuzzy operators in the design of fuzzy logic controllers. Fuzzy Sets Syst 86:15–41

    Article  MATH  Google Scholar 

  • Cordón O, Herrera F, Zwir I (2002) Linguistic modeling by hierarchical systems of linguistic rules. IEEE Trans Fuzzy Syst 10(1):2–20

    Article  MATH  Google Scholar 

  • Cordón O, del Jesus MJ, Herrera F (1998) Genetic learning of fuzzy rule-based classification systems cooperating with fuzzy reasoning methods. Int J Intell Syst 13:1025–1053

    Article  Google Scholar 

  • Cpałka K (2017) Improving fuzzy systems interpretability by appropriate selection of their structure. In: Design of interpretable fuzzy systems. Springer International Publishing, Cham, pp 37–60. https://doi.org/10.1007/978-3-319-52881-6_4

  • Deb K, Pratap A, Agarwal S, Meyarivan T (2002) A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans Evol Comput 6(2):182–197. https://doi.org/10.1109/4235.996017

    Article  Google Scholar 

  • Delgado M, Vila MA, Voxman W (1998) A fuzziness measure for fuzzy numbers: applications. Fuzzy Sets Syst 94(2):205–216

    Article  MathSciNet  MATH  Google Scholar 

  • Driankov D, Hellendoorn H, Reinfrank M (1993) An introduction to fuzzy control. Springer, Heidelberg, Germany

    Google Scholar 

  • Dujet C, Vincent N (1995) Force implication: a new approach to human reasoning. Fuzzy Sets Syst 69:53–63

    Article  MathSciNet  Google Scholar 

  • Dvovrák A (1999) On linguistic approximation in the frame of fuzzy logic deduction. Soft Comput 3(2):111–116

    Article  Google Scholar 

  • Elkano M, Sanz JA, Barrenechea E, Bustince H, Galar M (2020) CFM-BD: a distributed rule induction algorithm for building compact fuzzy models in big data classification problems. IEEE Trans Fuzzy Syst 28(1):163–177. https://doi.org/10.1109/TFUZZ.2019.2900856

    Article  Google Scholar 

  • Eshragh F, Mamdani EH (1981) A general approach to linguistic approximation. In: Mamdani EH, Gaines BR (eds) Fuzzy Reasoning and its applications. Academic Press, London, UK, pp 169–187

    Google Scholar 

  • Espinosa J, Vandewalle J (2000) Constructing fuzzy models with linguistic integrity from numerical data-AFRELI algorithm. IEEE Trans Fuzzy Syst 8(5):591–600

    Article  Google Scholar 

  • Fiordaliso A (2001) A constrained Takagi-Sugeno fuzzy system that allows for better interpretation and analysis. Fuzzy Sets Syst 118(2):307–318

    Article  MathSciNet  Google Scholar 

  • Fiordaliso A (2001) Autostructuration of fuzzy systems by rules sensitivity analysis. Fuzzy Sets Syst 118(2):281–296

    Article  MathSciNet  Google Scholar 

  • Fonseca CM, Fleming PJ (1995) An overview of evolutionary algorithms in multiobjective optimization. Evol Comput 3(1):1–16. https://doi.org/10.1162/evco.1995.3.1.1

    Article  Google Scholar 

  • Gacto MJ, Alcala R, Herrera F (2009) A multi-objective evolutionary algorithm for tuning fuzzy rule-based systems with measures for preserving interpretability. In: Joint IFSA-EUSFLAT conference, Lisbon, Portugal, pp 1146–1151

    Google Scholar 

  • González A, Pérez R (1998) A fuzzy theory refinement algorithm. Int J Approx Reason 19(3–4):193–220

    Article  MathSciNet  MATH  Google Scholar 

  • González A, Pérez R (1998) Completeness and consistency conditions for learning fuzzy rules. Fuzzy Sets Syst 96(1):37–51

    Article  MathSciNet  Google Scholar 

  • González A, Pérez R (1999) A study about the inclusion of linguistic hedges in a fuzzy rule learning algorithm. Int J Uncertain Fuzziness Knowl-Based Syst 7(3):257–266

    Article  MATH  Google Scholar 

  • Gonzalez J, Rojas I, Pomares H, Herrera L, Guillen A, Palomares J, Rojas F (2007) Improving the accuracy while preserving the interpretability of fuzzy function approximators by means of multi-objective evolutionary algorithms. Int J Approx Reason 44(1):32–44. https://doi.org/10.1016/j.ijar.2006.02.006

    Article  MathSciNet  MATH  Google Scholar 

  • Gudwin RR, Gomide F, Pedrycz W (1998) Context adaptation in fuzzy processing and genetic algorithms. Int J Intell Syst 13(10–11):929–948

    Article  Google Scholar 

  • Guillaume S, Charnomordic B (2004) Generating an interpretable family of fuzzy partitions from data. IEEE Trans Fuzzy Syst 12(3):324–335. https://doi.org/10.1109/TFUZZ.2004.825979

    Article  Google Scholar 

  • Guillaume S, Magdalena L (2005) An OR and NOT implementation that improves linguistic rule interpretability. In: International fuzzy systems association world congress (IFSA), pp 88–92. Beijing, China

    Google Scholar 

  • Gupta MM, Qi J (1991) Design of fuzzy logic controllers based on generalized T-operators. Fuzzy Sets Syst 40:473–489

    Article  MathSciNet  MATH  Google Scholar 

  • Hangos KM (1995) Special issue on grey box modelling. Int J Adapt Control Signal Process 9(6)

    Google Scholar 

  • Hellendoorn H, Thomas C (1993) Defuzzification in fuzzy controllers. J Intell Fuzzy Syst 1:109–123

    Article  Google Scholar 

  • Herrera F, Lozano M, Verdegay JL (1995) Tuning fuzzy controllers by genetic algorithms. Int J Approx Reason 12:299–315

    Article  MathSciNet  MATH  Google Scholar 

  • Herrera F, Lozano M, Verdegay JL (1998) A learning process for fuzzy control rules using genetic algorithms. Fuzzy Sets Syst 100(1–3):143–158. https://doi.org/10.1016/S0165-0114(97)00043-2

    Article  Google Scholar 

  • Hong TP, Lee CY (1999) Effect of merging order on performance of fuzzy induction. Intell Data Anal 3(2):139–151

    MATH  Google Scholar 

  • Ishibuchi H, Nakashima T, Murata T (2001) Three-objective genetics-based machine learning for linguistic rule extraction. Inf Sci 136(1–4):109–133. https://doi.org/10.1016/S0020-0255(01)00144-X

    Article  MATH  Google Scholar 

  • Ishibuchi H, Nozaki K, Yamamoto N, Tanaka H (1995) Selecting fuzzy if-then rules for classification problems using genetic algorithms. IEEE Trans Fuzzy Syst 3(3):260–270. https://doi.org/10.1109/91.413232

    Article  Google Scholar 

  • Ishibuchi H, Yamamoto T (2005) Rule weight specification in fuzzy rule-based classification systems. IEEE Trans Fuzzy Syst 13(4):428–435. https://doi.org/10.1109/TFUZZ.2004.841738

    Article  Google Scholar 

  • Jain A, Duin P (2000) Statistical pattern recognition: a review. IEEE Trans Pattern Anal Mach Intell 22(1):4–37. https://doi.org/10.1109/34.824819

    Article  Google Scholar 

  • Jang JS (1993) ANFIS: adaptive-network-based fuzzy inference system. IEEE Trans Syst Man Cybern 23(3):665–685. https://doi.org/10.1109/21.256541

    Article  Google Scholar 

  • Jimenez F, Gomez-Skarmeta AF, Roubos H, Babuska R (2001) A multi-objective evolutionary algorithm for fuzzy modeling. In: Joint IFSA world congress and NAFIPS international conference. IEEE, New York, pp 1222–1228. https://doi.org/10.1109/NAFIPS.2001.944781

  • Jin Y (2000) Fuzzy modeling of high-dimensional systems: complexity reduction and interpretability improvement. IEEE Trans Fuzzy Syst 8(2):212–221. https://doi.org/10.1109/91.842154

    Article  Google Scholar 

  • Jin Y, Von Seelen W, Sendhoff B (1999) On generating FC(3) fuzzy rule systems from data using evolution strategies. IEEE Trans Syst Man Cybern Part B, Cybern 29(6):829–845. https://doi.org/10.1109/3477.809036

  • Juang CF, Chen CY (2013) Data-driven interval type-2 neural fuzzy system with high learning accuracy and improved model interpretability. IEEE Trans Cybern 43(6):1781–1795

    Article  Google Scholar 

  • Karr CL (1991) Genetic algorithms for fuzzy controllers. AI Expert 6(2):26–33

    MathSciNet  Google Scholar 

  • Kiszka J, Kochanska M, Sliwinska D (1985) The influence of some fuzzy implication operators on the accuracy of a fuzzy model—Parts I and II. Fuzzy Sets Syst 15(111–128):223–240

    Article  MATH  Google Scholar 

  • Klose A, Nurnberger A, Nauck D (1998) Some approaches to improve the interpretability of neuro-fuzzy classifiers. In: European congress on intelligent techniques and soft computing, Aachen, Germany, pp 629–633

    Google Scholar 

  • Krone A, Kiendl H (1994) Automatic generation of positive and negative rules for two-way fuzzy controllers. In: European congress on intelligent techniques and soft computing. Verlag Mainz, Aachen, Germany, pp 438–447

    Google Scholar 

  • Lee C (1990a) Fuzzy logic in control systems: fuzzy logic controller. I. IEEE Trans Syst Man Cybern 20(2):404–418. https://doi.org/10.1109/21.52551

  • Lee C (1990b) Fuzzy logic in control systems: fuzzy logic controller. II. IEEE Trans Syst Man Cybern 20(2):419–435. https://doi.org/10.1109/21.52552

  • Lindskog P (1997) Fuzzy identification from a grey box modeling point of view. In: Hellendoorn H, Driankov D (eds) Fuzzy model identification. Springer, Heidelberg, Germany, pp 3–50

    Google Scholar 

  • Linkens D, Chen M (1999) Input selection and partition validation for fuzzy modelling using neural network. Fuzzy Sets Syst 107(3):299–308. https://doi.org/10.1016/S0165-0114(97)00322-9

    Article  MathSciNet  Google Scholar 

  • Liska J, Melsheimer SS (1994) Complete design of fuzzy logic systems using genetic algorithms. In: IEEE international conference on fuzzy systems (FUZZ-IEEE), pp 1377–1382

    Google Scholar 

  • Liu BD, Chen CY, Tsao JY (2001) Design of adaptive fuzzy logic controller based on linguistic-hedge concepts and genetic algorithms. IEEE Trans Syst Man Cybern-Part B: Cybern 31(1):32–53

    Article  Google Scholar 

  • López S, Magdalena L, Velasco JR (1999) Genetic fuzzy c-means algorithm for the automatic generation of fuzzy partitions. In: Bouchon-Meunier B, Yager RR, Zadeh LA (eds) Information, uncertainty, fusion. Kluwer Scientific, Norwell, MA, USA, pp 407–418

    Google Scholar 

  • Lotfi A, Andersen H, Tsoi AC (1996) Interpretation preservation of adaptive fuzzy inference systems. Int J Approx Reason 15(4):379–394. https://doi.org/10.1016/S0888-613X(96)00070-9

    Article  MathSciNet  MATH  Google Scholar 

  • Lucas LA, Centeno TM, Delgado MR (2009) Towards interpretable general type-2 fuzzy classifiers. In: International conference on intelligent systems design and applications (ISDA), pp 584–589

    Google Scholar 

  • Magdalena L (1997) Adapting the gain of an FLC with genetic algorithms. Int J Approx Reason 17(4):327–349. https://doi.org/10.1016/S0888-613X(97)00001-7

    Article  MATH  Google Scholar 

  • Magdalena L (2002) On the role of context in hierarchical fuzzy controllers. Int J Intell Syst 17(5):471–493. https://doi.org/10.1002/int.10033

    Article  MATH  Google Scholar 

  • Magdalena L (2018) Do hierarchical fuzzy systems really improve interpretability? In: Medina J, Ojeda-Aciego M, Verdegay JL, Pelta DA, Cabrera IP, Bouchon-Meunier B, Yager RR (eds) Information processing and management of uncertainty in knowledge-based systems. Theory and foundations. Springer International Publishing, pp 16–26

    Google Scholar 

  • Magdalena L (2019) Semantic interpretability in hierarchical fuzzy systems: creating semantically decouplable hierarchies. Inf Sci 496:109–123. https://doi.org/10.1016/j.ins.2019.05.016

    Article  Google Scholar 

  • Magdalena L, Monasterio-Huelin F (1997) A fuzzy logic controller with learning through the evolution of its knowledge base. Int J Approx Reason 16(3–4):335–358. https://doi.org/10.1016/S0888-613X(97)80098-9

    Article  MATH  Google Scholar 

  • Mamdani EH (1974) Applications of fuzzy algorithm for control a simple dynamic plant. Proc Inst Electr Eng 121(12):1585–1588

    Google Scholar 

  • Márquez AA, Márquez FA, Peregrín A (2009) Rule base and adaptive fuzzy operators cooperative learning of Mamdani fuzzy systems with multi-objective genetic algorithms. Evol Intell 2(1):39. https://doi.org/10.1007/s12065-009-0026-z

    Article  Google Scholar 

  • Márquez AA, Márquez FA, Peregrín A (2012) A mechanism to improve the interpretability of linguistic fuzzy systems with adaptive defuzzification based on the use of a multi-objective evolutionary algorithms. Int J Comput Intell Syst 5(2). https://doi.org/10.1080/18756891.2012.685309

  • Martinez L, Herrera F (2000) A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans Fuzzy Syst 8(6):746–752. https://doi.org/10.1109/91.890332

  • Mencar C, Lucarelli M, Castiello C, Fanelli AM (2013) Design of strong fuzzy partitions from cuts. In: Conference of the European society for fuzzy logic and technology (EUSFLAT), advances in intelligent systems research. Atlantis Press, Paris, France, pp 424–431. https://doi.org/10.2991/eusflat.2013.65

  • Mouzouris GC, Mendel JM (1997) Nonsingleton fuzzy logic systems: theory and application. IEEE Trans Fuzzy Syst 5(1):56–71

    Article  Google Scholar 

  • Nauck D, Kruse R (1998) How the learning of rule weights affects the interpretability of fuzzy systems. In: IEEE international conference on fuzzy systems (FUZZ-IEEE), IEEE world congress on computational intelligence, pp 1235–1240 . https://doi.org/10.1109/FUZZY.1998.686295

  • Nomura H, Hayashi H, Wakami N (1992) A learning method of fuzzy inference rules by descent method. In: IEEE international conference on fuzzy systems (FUZZ-IEEE), San Diego, USA, pp 203–210

    Google Scholar 

  • Nozaki K, Ishibuchi H, Tanaka H (1997) A simple but powerful heuristic method for generating fuzzy rules from numerical data. Fuzzy Sets Syst 86(3):251–270. https://doi.org/10.1016/0165-0114(95)00413-0

    Article  Google Scholar 

  • Pal NR, Pal K (1999) Handling of inconsistent rules with an extended model of fuzzy reasoning. J Intell Fuzzy Syst 7:55–73

    Google Scholar 

  • Pedrycz W (1996) Fuzzy modelling: paradigms and practice. Kluwer Academic Press

    Google Scholar 

  • Pedrycz W (2001) Fuzzy equalization in the construction of fuzzy sets. Fuzzy Sets Syst 119(2):329–335

    Article  MathSciNet  MATH  Google Scholar 

  • Pedrycz W, Gomide F (1998) An introduction to fuzzy sets. Analysis and design. The MIT Press, Cambridge (MA)

    Google Scholar 

  • Peña-Reyes CA, Sipper M (2001) Fuzzy CoCo: a cooperative-coevolutionary approach to fuzzy modeling. IEEE Trans Fuzzy Syst 9(5):727–737. https://doi.org/10.1109/91.963759

    Article  Google Scholar 

  • Potter MA, Jong KAD (2000) Cooperative coevolution: an architecture for evolving coadapted subcomponents. Evol Comput 8(1):1–29

    Article  Google Scholar 

  • Raju GVS, Zhou J, Kisner RA (1991) Hierarchical fuzzy control. Int J Control 54(5):1201–1216. https://doi.org/10.1080/00207179108934205

    Article  MathSciNet  MATH  Google Scholar 

  • Razak TR, Garibaldi JM, Wagner C, Pourabdollah A, Soria D (2017) Interpretability indices for hierarchical fuzzy systems. In: IEEE international conference on fuzzy systems (FUZZ-IEEE))

    Google Scholar 

  • Razak TR, Garibaldi JM, Wagner C, Pourabdollah A, Soria D (2020) Towards a framework for capturing interpretability of hierarchical fuzzy systems—a participatory design approach. IEEE Trans Fuzzy Syst. https://doi.org/10.1109/TFUZZ.2020.2969901

  • Rey MI, Galende M, Fuente MJ, Sainz-Palmero GI (2017) Multi-objective based fuzzy rule based systems (FRBSs) for trade-off improvement in accuracy and interpretability: a rule relevance point of view. Knowl-Based Syst 127:67–84. https://doi.org/10.1016/j.knosys.2016.12.028

    Article  Google Scholar 

  • Riid A, Rüstern E (2000) Interpretability versus adaptability in fuzzy systems. In: Estonian Academy of sciences. Engineering, pp 76–95

    Google Scholar 

  • Riid A, Rüstern E (2011) Identification of transparent, compact, accurate and reliable linguistic fuzzy models. Inf Sci 181(20):4378–4393. https://doi.org/10.1016/j.ins.2011.01.041

    Article  MATH  Google Scholar 

  • Rojas I, Pomares H, Ortega J, Prieto A (2000) Self-organized fuzzy system generation from training examples. IEEE Trans Fuzzy Syst 8(1):23–36. https://doi.org/10.1109/91.824763

    Article  Google Scholar 

  • Roubos H, Setnes M (2001) Compact and transparent fuzzy models and classifiers through iterative complexity reduction. IEEE Trans Fuzzy Syst 9(4):516–524. https://doi.org/10.1109/91.940965

    Article  Google Scholar 

  • Rudin C (2019) Stop explaining black box machine learning models for high stakes decisions and use interpretable models instead. Nat Mach Intell 1(5):206–215. https://doi.org/10.1038/s42256-019-0048-x

    Article  Google Scholar 

  • Sánchez L, Casillas J, Cordón O, del Jesus MJ (2002) Some relationships between fuzzy and random set-based classifiers and models. Int J Approx Reason 29(2):175–213

    Article  MathSciNet  MATH  Google Scholar 

  • Setnes M, Babuska R, Kaymak U, van Nauta Lemke HR (1998a) Similarity measures in fuzzy rule base simplification. IEEE Trans Syst Man Cybern-Part B: Cybern 28(3):376–386

    Article  Google Scholar 

  • Setnes M, Babuska R, Verbruggen HB (1998b) Complexity reduction in fuzzy modeling. Math Comput Simul 46(5–6):509–518

    Google Scholar 

  • Setnes M, Hellendoorn H (2000) Orthogonal transforms for ordering and reduction of fuzzy rules. In: IEEE international conference on fuzzy systems (FUZZ-IEEE), San Antonio, TX, USA, pp 700–705

    Google Scholar 

  • Shi Y, Eberhart R, Chen Y (1999) Implementation of evolutionary fuzzy systems. IEEE Trans Fuzzy Syst 7(2):109–119

    Google Scholar 

  • Shi-Rong LIU, Jin-Shou YU (2001) Model construction optimization for a class of fuzzy models. Chin J Comput 24(2):164–172

    Google Scholar 

  • Shukla PK, Tripathi SP (2013) Interpretability issues in evolutionary multi-objective fuzzy knowledge base systems. In: Bansal JC, Singh PK, Deep K, Pant M, Nagar AK (eds) International conference on bio-inspired computing: theories and applications (BIC-TA). Springer, India, pp 473–484

    Google Scholar 

  • Sudkamp T, Knapp J, Knapp A (2001) Refine and merge: generating small bases from training data. In: Joint IFSA world congress and NAFIPS international conference, Vancouver, Canada, pp 197–202

    Google Scholar 

  • Sugeno M, Kang GT (1988) Structure identification of fuzzy model. Fuzzy Sets Syst 28:15–33

    Article  MathSciNet  MATH  Google Scholar 

  • Sugeno M, Yasukawa T (1993) A fuzzy-logic-based approach to qualitative modeling. IEEE Trans Fuzzy Syst 1(1). https://doi.org/10.1109/TFUZZ.1993.390281

  • Suzuki T, Kodama T, Furuhashi T, Tsutsui H (2001) Fuzzy modeling using genetic algorithms with fuzzy entropy as conciseness measure. Inf Sci 136(1–4):53–67. https://doi.org/10.1016/S0020-0255(01)00141-4

    Article  MATH  Google Scholar 

  • Takagi H, Hayashi I (1991) NN-driven fuzzy reasoning. Int J Approx Reason 5(3):191–212

    Article  MATH  Google Scholar 

  • Takagi T, Sugeno M (1985) Fuzzy identification of systems and its applications to modeling and control. IEEE Trans Syst Man Cybern 15(1):116–132

    Article  MATH  Google Scholar 

  • Thrift P (1991) Fuzzy logic synthesis with genetic algorithms. In: International conference on genetic algorithms (ICGA). Morgan Kaufmann, San Diego, USA, pp. 509–513

    Google Scholar 

  • Trillas E, Valverde L (1985) On implication and indistinguishability in the setting of fuzzy logic. In: Kacpryzk J, Yager RR (eds) Management decision support systems using fuzzy logic and possibility theory. Verlag TUV Rheinland, pp 198–212

    Google Scholar 

  • Valente de Oliveira J (1999) Towards neuro-linguistic modeling: constraints for optimization of membership functions. Fuzzy Sets Syst 106(3), 357–380. https://doi.org/10.1016/S0165-0114(97)00281-9

  • Wang LX (1994) Adaptive fuzzy systems and control: design and analysis. Prentice-Hall

    Google Scholar 

  • Wang LX, Mendel JM (1992) Fuzzy basis functions, universal approximation, and orthogonal least squares learning. IEEE Trans Neural Netw 3:807–814

    Article  Google Scholar 

  • Wang LX, Mendel JM (1992) Generating fuzzy rules by learning from examples. IEEE Trans Syst Man Cybern 22(6):1414–1427

    Article  MathSciNet  Google Scholar 

  • Yager RR (1998) On the construction of hierarchical fuzzy systems models. IEEE Trans Syst Man Cybern—Part C 28(1):55–66

    Article  Google Scholar 

  • Yen J, Wang L (1996) An SVD-based fuzzy model reduction strategy. In: IEEE international conference on fuzzy systems (FUZZ-IEEE). IEEE, New Orleans, LA, USA, pp 835–841

    Google Scholar 

  • Yen J, Wang L, Gillespie C (1998) Improving the interpretability of TSK fuzzy models by combining global learning and local learning. IEEE Trans Fuzzy Syst 6(4):530–537

    Article  Google Scholar 

  • Yoshinari Y, Pedrycz W, Hirota K (1993) Construction of fuzzy models through clustering techniques. Fuzzy Sets Syst 54:157–165

    Article  MathSciNet  Google Scholar 

  • Zadeh LA (1972) A fuzzy-set theoretic interpretation of linguistic hedges. J Cybern 2(2):4–34

    Article  MathSciNet  Google Scholar 

  • Zadeh LA (1973) Outline of a new approach to the analysis of complex systems and decision processes. IEEE Trans Syst Man Cybern SMC-3(1), 28–44. https://doi.org/10.1109/TSMC.1973.5408575

  • Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning—I. Inf Sci 8:199–249

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning—II. Inf Sci 8:301–357

    Article  MathSciNet  MATH  Google Scholar 

  • Zadeh LA (1975) The concept of a linguistic variable and its application to approximate reasoning—III. Inf Sci 9(1):43–80. https://doi.org/10.1016/0020-0255(75)90017-1

    Article  MathSciNet  MATH  Google Scholar 

  • Zhang Y, Ishibuchi H, Wang S (2018) Deep Takagi-Sugeno-Kang fuzzy classifier with shared linguistic fuzzy rules. IEEE Trans Fuzzy Syst 26(3):1535–1549. https://doi.org/10.1109/TFUZZ.2017.2729507

    Article  Google Scholar 

  • Zitzler E, Laumanns M, Thiele L (2001) Improving the strength pareto evolutionary algorithm for multi-objetive optimization. In: Evolutionary methods for design, optimization and control with applications to industrial problems, pp 95–100

    Google Scholar 

  • Zong-Yi X, Yong Z, Yuan-Long H, Guo-Qiang C (2008) Multi-objective fuzzy modeling using NSGA-II. In: IEEE conference on cybernetics and intelligent systems, pp 119–124. https://doi.org/10.1109/ICCIS.2008.4670812

Download references

Author information

Authors and Affiliations

  1. Centro Singular de Investigación en Tecnoloxías Intelixentes (CiTIUS), Universidade de Santiago de Compostela, Santiago de Compostela, Spain

    Jose Maria Alonso Moral

  2. Dipartimento di Informatica, Università degli Studi di Bari Aldo Moro, Bari, Italy

    Ciro Castiello & Corrado Mencar

  3. Departamento de Matemática Aplicada a las Tecnologías de la Información y las Comunicaciones, ETSI Informáticos, Universidad Politécnica de Madrid, Madrid, Spain

    Luis Magdalena

Authors
  1. Jose Maria Alonso Moral
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ciro Castiello
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Luis Magdalena
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Corrado Mencar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jose Maria Alonso Moral .

Rights and permissions

Reprints and permissions

Copyright information

© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Alonso Moral, J.M., Castiello, C., Magdalena, L., Mencar, C. (2021). Designing Interpretable Fuzzy Systems. In: Explainable Fuzzy Systems. Studies in Computational Intelligence, vol 970. Springer, Cham. https://doi.org/10.1007/978-3-030-71098-9_5

Download citation

Publish with us

Policies and ethics

Search

Navigation