Abstract
Fuzzy systems are well suited for nonlinear modeling. They can be effectively used if their structure and structure parameters are properly chosen. Moreover, it should be ensured that system rules are clear and interpretable. In this paper we propose a new algorithm for automatic learning and new interpretability criteria of fuzzy systems. Interpretability criteria are related to all aspects of those systems, not only their fuzzy sets and rules. Therefore, proposed criteria also concern parameterized triangular norms, discretization points and weights of importance from the rule base. As of the present time similar solutions have not been discussed in the literature. The proposed criteria are taken into account in the learning process, which is carried out with the use of a new learning algorithm. It was created by combining the genetic and the firework algorithms (this particular combination makes it possible to automatically choose not only system parameters but also its structure). It is an important advantage as most of the learning algorithms can only select system parameters when their structure has been specified by the designer. Proposed solutions were tested using typical simulation problems of nonlinear modeling.
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Alcalá, R., Ducange, P., Herrera, F., Lazzerini, B., Marcelloni, F.: A multi-objective evolutionary approach to concurrently learn rule and data base soft linguistic fuzzy rule-based systems. IEEE Trans. Fuzzy Syst. 17, 1106–1122 (2009)
Alonso, J.M., Magdalena, L., Cordón, O.: Embedding HILK in a three-objective evolutionary algorithm with the aim of modeling highly interpretable fuzzy rule-based classifiers. In: 4th International Workshop on Genetic and Evolving Fuzzy Systems (GEFS2010), pp. 15–20 (2010)
Alonso, J.M.: Modeling highly interpretable fuzzy systems. Eur. Centre Soft Comput. (2010)
Alonso, J.M., Magdalena, L.: HILK++: an interpretability-guided fuzzy modeling methodology for learning readable and comprehensible fuzzy rule-based classifiers. Soft Comput. 15(10), 1959–1980 (2011)
Althoff, M., Stursberg, O., Buss, M.: Reachability analysis of nonlinear systems with uncertain parameters using conservative linearization. In: Proceedings of the 47th IEEE Conference on Decision and Control, pp. 4042–4048 (2008)
Amor, N.B., Salem, B., Zied, E.: Naive Bayes vs decision trees in intrusion detection systems. In: Proceedings of the 2004 ACM Symposium on Applied Computing (2004)
Andrieu, C., Doucet, A.: Particle filtering for partially observed Gaussian state space models. JR Stat. Soc. B 64(4), 827–836 (2002)
Bartczuk, Ł., Przybył, A., Koprinkova-Hristova, P.: New method for non-linear correction modelling of dynamic objects with genetic programming. In: Artificial Intelligence and Soft Computing. Lecture Notes in Computer Science, vol. 9120, pp. 318–329 (2015)
Botta, A., Lazzerini, B., Marcelloni, F., Stefanescu, D.C.: Context adaptation of fuzzy systems through a multi-objective evolutionary approach based on a novel interpretability index. Soft Comput. 13, 437–449 (2009)
Box, G., Jenkins, G.: Time Series Analysis: Forecasting and Control. Holden-Day, San Francisco (1970)
Brasileiro, Í., Santos, I., Soares, A., Rablo, R., Mazullo, F.: Ant colony optimization applied to the problem of choosing the best combination among M combinations of shortest paths in transparent optical networks. J. Artif. Intell. Soft Comput. Res. 6(4), 231–242 (2016)
Brooks, T.F., Pope, D.S., Marcolini, A.M.: Airfoil self-noise and prediction. Technical report, NASA RP-1218 (1989)
Chen, K.: Global modeling of different vehicles. IEEE Veh. Technol. Mag. 4(2), 80–89 (2009)
Chen, X., Abraham, E., Sankaranarayanan, S.: Flow*: an analyzer for non-linear hybrid systems. In: Proceedings of the 25th International Conference on Computer Aided Verification, vol. 8044, pp. 258–263 (2013)
Cpałka, K.: A new method for design and reduction of neuro-fuzzy classification systems. IEEE Trans. Neural Netw. 20, 701–714 (2009)
Cpałka, K.: On evolutionary designing and learning of flexible neuro-fuzzy structures for nonlinear classification. In: Nonlinear Analysis Series A: Theory, Methods and Applications, vol. 71, pp. 1659–1672. Elsevier (2009)
Cpałka, K.: Design of Interpretable Fuzzy Systems. Springer (2017)
Cpałka, K., Rebrova, O., Nowicki, R., Rutkowski, L.: On design of flexible neuro-fuzzy systems for nonlinear modelling. Int. J. Gen. Syst. 42(6), 706–720 (2013)
Cpałka, K., Rutkowski, L.: Flexible Takagi-Sugeno fuzzy systems. In: Proceedings of the 2005 IEEE International Joint Conference on Neural Networks IJCNN ’05, vol. 3, pp. 1764–1769 (2005)
Cyran, A.K., Kozielski, S., Peters, F.P., Stanczyk, U., Wakulicz-Deja, A.: Adaptable graphical user interfaces for player-based applications. Adv. Intell. Soft Comput. 59, 69–76 (2009)
Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6, 182–197 (2002)
Duch, W., Korbicz, J., Rutkowski, L., Tadeusiewicz, R.: Biocybernetics and biomedical engineering EXIT, Warszawa (2013)
Duda, P., Hayashi, Y., Jaworski, M.: On the strong convergence of the orthogonal series-type kernel regression neural networks in a non-stationary environment. In: Artificial Intelligence and Soft Computing, vol. 7267, pp. 47–54. Springer (2012)
El-Samak, A.F., Ashour, W.: Optimization of traveling salesman problem using affinity propagation clustering and genetic algorithm. J. Artif. Intell. Soft Comput. Res. 5, 239–246 (2015)
Er, M.J., Duda, P.: On the weak convergence of the orthogonal series-type kernel regresion neural networks in a non-stationary environment. In: International Conference on Parallel Processing and Applied Mathematics. Lecture Notes in Computer Science, vol. 7203, pp. 90–98. Springer (2012)
Espinosa, J., Vandewalle, J.: Constructing fuzzy models with linguistic integrity from numerical data-AFRELI algorithm. IEEE Trans. Fuzzy Syst. 8, 591–600 (2000)
Farahbod, F., Eftekhari, M.: Comparsion of different T-norm operators in classification problems. Int. J. Fuzzy Logic Syst. 2(3), 33–41 (2012)
Fazendeiro, P., de Oliveira, J.V., Pedrycz, W.: A multiobjective design of a patient and anaesthetist-friendly neuromuscular blockade controller. IEEE Trans. Biomed. Eng. 54, 1667–1678 (2007)
Fraser, A., Burnell, D.: Computer Models in Genetics. McGraw-Hill, New York (1970)
Gabryel, M., Cpałka, K., Rutkowski, L.: Evolutionary strategies for learning of neuro-fuzzy systems. In: Proceedings of the I Workshop on Genetic Fuzzy Systems, Granada, vol. 119, p. 123 (2005)
Gacto, M.J., Alcalá, R., Herrera, F.: Integration of an index to preserve the semantic interpretability in the multi-objective evolutionary rule selection and tuning of linguistic fuzzy systems. IEEE Trans. Fuzzy Syst. 18(3), 515–531 (2010)
Gacto, M.J., Alcalá, R., Herrera, F.: Interpretability of linguistic fuzzy rule-based systems: an overview of interpretability measures. Inf. Sci. 181(20), 4340–4360 (2011)
Gorzalczany, M.B., Rudzinski, F.: Accuracy vs. interpretability of fuzzy rule-based classifiers: an evolutionary approach. In: Proceedings of the 2012 International Conference on Swarm and Evolutionary Computation SIDE’12, pp. 222–230 (2012)
Guillaume, S., Charnomordic, B.: Generating an interpretable family of fuzzy partitions from data. IEEE Trans. Fuzzy Syst. 12(3), 324–335 (2004)
Ibrahim, S.S., Bamatraf, M.A.: Interpretation trained neural networks based on genetic algorithms. Int. J. Artif. Intell. Appl. (IJAIA) 4(1), 13–22 (2013)
Icke, I., Rosenberg, A.: Multi-objective genetic programming for visual analytics. In: Silva, S., et al. (eds.) EuroGP 2011. LNCS, vol. 6621, pp. 322–334 (2011)
Ishibuchi, H., Nakashima, T., Murata, T.: Comparsion of the Michigan and Pittsburgh approaches to the design of fuzzy classification systems. Electron. Commun. Jpn. Part 3 80(12), 379–387 (1997)
Ishibuchi, H., Nakashima, T., Murata, T.: Performance evaluation of fuzzy classifier systems for multidimensional pattern classification problems. IEEE Trans. SMC B Cybern. 29, 601–618 (1999)
Ishibuchi, H.: Rule weight specification in fuzzy rule-based classification systems. IEEE Trans. Fuzzy Syst. 13(4), 428–436 (2005)
Ishibuchi, H., Nojima, Y.: Analysis of interpretability-accuracy tradeoff of fuzzy systems by multiobjective fuzzy genetics-based machine learning. Int. J. Approximate Reasoning 44, 4–31 (2007)
Jaworski, M., Er, M.J., Pietruczuk, L.: On the application of the Parzen-type kernel regression neural network and order statistics for learning in a non-stationary environment. In: International Conference on Artificial Intelligence and Soft Computing. Lecture Notes in Artificial Intelligence, vol. 7267, pp. 90–98. Springer (2012)
Kacprzyk, J.: Studies in Computational Intelligence, vol. 143 (2008)
Kaczorek, T.: A modified state variable diagram method for determination of positive realizations of linear continous-time systems with delays. Int. J. Appl. Math. Comput. Sci. 22(4), 897–905 (2012)
Kar, S., Das, S., Ghosh, P.K.: Applications of neuro-fuzzy systems: a brief review and future outline. Appl. Soft Comput. 15, 243–259 (2014)
Kamyar, M.: Takagi-Sugeno fuzzy modeling for process control industrial automation. In: Robotics and Artificial Intelligence (EEE8005), School of Electrical, Electronic and Computer Engineering (2008)
Kenesei, T., Abonyi, J.: Interpretable support vector machines in regression and classification-application in process engineering. Hung. J. Ind. Chem. 35, 101–108 (2007)
Klement, E.P., Mesiar, R., Pap, E.: Triangular Norms. Kluwer Academic Publishers (2000)
Leekwijck, W.V., Kerre, E.E.: Defuzzification: criteria and classification. Fuzzy Sets Syst. 108(2), 159–178 (1999)
Leon, M., Xiong, N.: Adapting differential evolution algorithms for continuous optimization via greedy adjustment of control parameters. J. Artif. Intell. Soft Comput. Res. 6(2), 103–118 (2016)
Liu, F., Quek, C., Ng, G.S.: A novel generic hebbian ordering-based fuzzy rule base reduction approach to Mamdani neuro-fuzzy system. Neural Comput. 19, 1656–1680 (2007)
Loh, W.-Y.: Classification and regression trees. Wiley Interdisc. Rev.: Data Min. Knowl. Discovery 1(1), 14–23 (2011)
Łapa, K., Cpałka, K., Wang, L.: New method for design of fuzzy systems for nonlinear modelling using different criteria of interpretability. Lect. Notes Comput. Sci. 8467, 217–232 (2014)
Łapa, K., Szczypta, J., Venkatesan, R.: Aspects of structure and parameters selection of control systems using selected multi-population algorithms. Lect. Notes Comput. Sci. 9120, 247–260 (2015)
Marquez, A.A, Marquez, F.A., Peregrin, A.: A multi-objective evolutionary algorithm with an interpretability improvement mechanism for linguistic fuzzy systems with adaptive defuzzification. IEEE Int. Conf. Fuzzy Syst. 1–7 (2010)
Mehran, K.: Takagi-Sugeno fuzzy modeling for process control. In: Industrial Automation, Robotics and Artificial Intelligence (EEE8005) (2008)
Mencar, C., Castellano, G., Fanelli, A.M.: Some fundamental interpretability issues in fuzzy modeling. In: Proceedings of the Joint 4th Conference of the European Society for Fuzzy Logic and Technology, pp. 100–105 (2005)
Mencar, C., Castellano, G., Fanelli, A.M.: On the role of interpretability in fuzzy data mining. Int. J. Uncertainty Fuzziness Knowl. Based Syst. 521–537 (2007)
Mencar, C., Castiello, C., Cannone, R., Fanelli, A.M.: Interpretability assessment of fuzzy knowledge bases: a cointension based approach. Int. J. Approximate Reasoning 52(4), 501–518 (2011)
Miller, G.A.: The magical number seven, plus or minus two: some limits on our capacity for processing information. Psychol. Rev. 63, 81–97 (1956)
Miyajima, H., Shigei, N., Miyajima, H.: Performance comparison of hybrid electromagnetism-like mechanism algorithms with descent method. J. Artif. Intell. Soft Comput. Res. 5(4), 271–282 (2015)
Musa, A.A.H., Muawia, M.A.: Analysis of the DC motor speed control using state variable transition matrix. Int. J. Sci. Res. (IJSR) 2758–2763 (2012)
Nguyen, K.P., Fujita, G., Dieu, V.N.: Cuckoo search algorithm for optimal placement and sizing of static VAR compensator in large-scale power systems. J. Artif. Intell. Soft Comput. Res. 6(2), 59–68 (2016)
Patan, K., Korbicz, J.: Nonlinear model predictive control of a boiler unit: a fault tolerant control study. Int. J. Appl. Math. Comput. Sci. 22(1), 225–237 (2012)
Pietruczuk, L., Duda, P., Jaworski, M.: Adaptation of decision trees for handling concept drift. In: International Conference on Artificial Intelligence and Soft Computing. Lecture Notes in Artificial Intelligence, vol. 7894, pp. 459–473. Springer (2013)
Przybył, A., Cpałka, K.: A new method to construct of interpretable models of dynamic systems. Lect. Notes Artif. Intell. 697–705 (2012)
Pulkkinen, P., Koivisto, H.: A dynamically constrained multiobjective genetic fuzzy system for regression problems. IEEE Trans. Fuzzy Syst. 18(1), 161–177 (2010)
Riid, A., Rustern, E.: Interpretability improvement of fuzzy systems: reducing the number of unique singletons in zeroth order Takagi-Sugeno systems. IEEE Int. Conf. Fuzzy Syst. 1–6 (2010)
Riid, A., Rustern, E.: Interpretability, interpolation and rule weights in linguistic fuzzy modeling. In: Petrosino, A., et al. (eds.) WILF 2011. LNAI, vol. 6857, pp. 91–98 (2011)
Riid, A., Rustern, E.: Adaptability, interpretability and rule weights in fuzzy rule-based systems. Inf. Sci. 257(1), 301–312 (2014)
Rosfariedzah, R., Nagarajan, R., Rahim, M.: Fuzzy variable structure control with reduced-order observer for micro satellite stabilization in space. In: Proceedings of the International Conference on Man-Machine Systems (ICoMMS), pp. 11–13 (2009)
Rutkowski, L.: Flexible Neuro-Fuzzy Systems. Kluwer Academic Publishers (2004)
Rutkowski, L.: Computational Intelligence. Springer (2008)
Rutkowski, L., Cpałka, K.: A general approach to neuro-fuzzy systems. In: The 10th IEEE International Conference on Fuzzy Systems, 2001, Melbourne, pp. 1428–1431 (2001)
Rutkowski, L., Cpałka, K.: Compromise approach to neuro-fuzzy systems. In: 2nd Euro-International Symposium on Computation Intelligence, vol. 76, pp. 85–90, Kosice, Slovakia, 16–19 June 2002
Rutkowski, L., Cpałka, K.: A neuro-fuzzy controller with a compromise fuzzy reasoning. Control Cybern. 31(2), 297–308 (2002)
Rutkowski, L., Cpałka, K.: Neuro-fuzzy systems derived from quasi-triangular norms. In: Proceedings of the IEEE International Conference on Fuzzy Systems, vol. 2, pp. 1031–1036, Budapest, 26–29 July 2004
Rutkowski, L., Cpałka, K.: Designing and learning of adjustable quasi-triangular norms with applications to neuro-fuzzy systems. IEEE Trans. Fuzzy Syst. 13, 140–151 (2005)
Rutkowski, L., Cpałka, K.: Flexible neuro fuzzy systems. IEEE Trans. Neural Netw. 14(2003), 554–574 (2013)
Rutkowski, L., Przybył, A., Cpałka, K.: Novel online speed profile generation for industrial machine tool based on flexible neuro-fuzzy approximation. IEEE Trans. Ind. Electron. 59(2), 1238–1247 (2012)
Rutkowski, L., Przybył, A., Cpałka, K., Er, M.J.: Online speed profile generation for industrial machine tool based on neuro-fuzzy approach. Lect. Notes Artif. Intell. 114, 645–650 (2010)
Sánchez, G., Jiménez, F., Sánchez, J.M., Alcaraz, J.M.: A multi-objective neuro-evolutionary algorithm to obtain interpretable fuzzy models. In: Current Topics in Artificial Intelligence. Lecture Notes in Computer Science, vol. 5988, pp. 51–60 (2010)
Scherer, R.: Neuro-fuzzy systems with relation matrix. Artif. Intell. Soft Comput. 6113, 210–215 (2010)
Shukla, P.K., Tripathi, S.P.: A review on the interpretability-accuracy trade-off in evolutionary multi-objective fuzzy systems (EMOFS). Information 3, 256–277 (2012)
Shukla, P.K., Tripathi, S.P.: Handling high dimensionality and interpretability-accuracy trade-off issues in evolutionary multiobjective fuzzy classifiers. Int. J. Sci. Eng. Res. 5(6), 665–671 (2014)
Shukla, P.K., Tripathi, S.P.: A new approach for tuning interval type-2 fuzzy knowledge bases using genetic algorithms. J. Uncertainty Anal. Appl. 2, 4 (2014)
Siminski, K.: Rule weights in a neuro-fuzzy system with a hierarchical domain partition. Int. J. Appl. Math. Comput. Sci. 20(2), 337–347 (2010)
Singh, L., Kumar, S., Paul, S.: Automatic simultaneous architecture and parameter search in fuzzy neural network learning using novel variable length crossover differential evolution. In: IEEE International Conference on Fuzzy Systems, pp. 1795–1802 (2008)
Tadeusiewicz, R.: Place and role of intelligent systems in computer science. Comput. Methods Mater. Sci. 10(4), 193–206 (2010)
Tan, Y., Shi, Y., Tan, K.C.: Fireworks algorithm for optimization. In: ICSI 2010, Part I. LNCS, vol. 6145, pp. 355–364 (2010)
Tan, C.: More than Accuracy: Interpretability. @MLDG 08/15/2013. https://chenhaot.com/pubs/mldg-interpretability.pdf (2013)
Tikk, D., Gedeon, T., Wong, K.: A feature ranking algorithm for fuzzy modeling problems. In: Interpretability Issues in Fuzzy Modeling, pp. 176–192. Springer (2003)
Tsanas, A., Xifara, A.: Accurate quantitative estimation of energy performance of residential buildings using statistical machine learning tools. Energy Build. 49, 560–567 (2012)
Vanhoucke, V., Silipo, R.: Interpretability in multidimensional classification. In: Interpretability Issues in Fuzzy Modeling, pp. 193–217. Springer (2003)
Viharos, Z.J., Kis, K.B.: Survey on neuro-fuzzy systems and their applications in technical diagnostics. In: 13th IMEKO TC10 Workshop on Technical Diagnostics Advanced Measurement Tools in Technical Diagnostics for Systems’ Reliability and Safety (2014)
Wang, H., Kwong, S., Jin, Y., Wei, W., Man, K.F.: Multi-objective hierarchical genetic algorithm for interpretable fuzzy rule-based knowledge extraction. Fuzzy Sets Syst. 149(1), 149–186 (2005)
Yang, C.H., Moi, S.H., Lin, Y.D., Chuang, L.Y.: Genetic algorithm combined with a local search method for identifying susceptibility genes. J. Artif. Intell. Soft Comput. Res. 6, 203–212 (2016)
Yeh, I.C.: Modeling slump flow of concrete using second-order regressions and artificial neural networks. Cement Concr. Compos. 29(6), 474–480 (2007)
Yin, Z., O’Sullivan C, Brabazon A.: An analysis of the performance of genetic programming for realised volatility forecas. J. Artif. Intell. Soft Computing Res. 6, 155–172 (2016)
Zalasiński, M.: New algorithm for on-line signature verification using characteristic global features. Adv. Intell. Syst. Comput. 432, 137–146 (2016)
Zalasiński, M., Cpałka, K.: New algorithm for on-line signature verification using characteristic hybrid partitions. Adv. Intell. Syst. Comput. 432, 147–157 (2016)
Zalasiński, M., Cpałka, K., Hayashi, Y.: A new approach to the dynamic signature verification aimed at minimizing the number of global features. Lect. Notes Comput. Sci. 9693, 218–231 (2016)
Zalasiński, M., Cpałka, K., Rakus-Andersson, E.: An idea of the dynamic signature verification based on a hybrid approach. Lect. Notes Comput. Sci. 9693, 232–246 (2016)
Żurada, J.M.: Introduction to Artificial Neural Systems. Jaico Publishing House (2005)
Acknowledgements
The project was financed by the National Science Centre (Poland) on the basis of the decision number DEC-2012/05/B/ST7/02138.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG
About this chapter
Cite this chapter
Łapa, K., Cpałka, K., Rutkowski, L. (2018). New Aspects of Interpretability of Fuzzy Systems for Nonlinear Modeling. In: Gawęda, A., Kacprzyk, J., Rutkowski, L., Yen, G. (eds) Advances in Data Analysis with Computational Intelligence Methods. Studies in Computational Intelligence, vol 738. Springer, Cham. https://doi.org/10.1007/978-3-319-67946-4_9
Download citation
DOI: https://doi.org/10.1007/978-3-319-67946-4_9
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-67945-7
Online ISBN: 978-3-319-67946-4
eBook Packages: EngineeringEngineering (R0)