Abstract
In this paper, an immune inspired multi-objective fuzzy modeling (IMOFM) mechanism is proposed specifically for high-dimensional regression problems. For such problems, prediction accuracy is often the paramount requirement. With such a requirement in mind, however, one should also put considerable efforts in eliciting models which are as transparent as possible, a ‘tricky’ exercise in itself. The proposed mechanism adopts a multistage modeling procedure and a variable length coding scheme to account for the enlarged search space due to simultaneous optimisation of the rule-base structure and its associated parameters. We claim here that IMOFM can account for both Singleton and Mamdani Fuzzy Rule-Based Systems (FRBS) due to the carefully chosen output membership functions, the inference scheme and the defuzzification method. The proposed modeling approach has been compared to other representatives using a benchmark problem, and was further applied to a high-dimensional problem, taken from the steel industry, which concerns the prediction of mechanical properties of hot rolled steels. Results confirm that IMOFM is capable of eliciting not only accurate but also transparent FRBSs from quantitative data.
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References
B. Kosko, Fuzzy Systems as Universal Approximators, IEEE Transactions on Computers, vol. 43(11), pp. 1329–1333, 1994.
K. M. Passino, S. Yurkovich, Fuzzy Control, MA: Addison-Wesley, 1998, pp. 246–252.
J. H. Holland, Adaptation in Natural and Artificial Systems, MI: The University of Michigan Press, 1975.
J. D. Farmer, N. H. Packard, The Immune System, Adaptation, and Machine Learning, Physica, vol. 22D, pp. 187–204, 1986.
S. Guillaume, Designing Fuzzy Inference Systems from Data: An Interpretability-Oriented Review, IEEE Transactions on Fuzzy Systems, vol. 9(3), pp. 426–443, 2001.
M. Delgado, F. Gómez-Skarmeta Antonio, F. Martin, A Fuzzy Clustering-Based Rapid Prototyping for Fuzzy Rule-Based Modelling, IEEE Transactions on Fuzzy Systems, vol. 5(2), pp. 223–233, 1997.
C. L. Karr, Genetic Algorithms for Fuzzy Controllers, AI Expert, vol. 6 (2), pp. 26–33, 1991.
J. Chen, M. Mahfouf, Interpretable Fuzzy Modeling using Multi-Objective Immune Inspired Optimisation Algorithms, FUZZ-IEEE 2010, 2010.
J. Chen, Biologically Inspired Optimisation Algorithms for Transparent Knowledge Extraction Allied to Engineering Materials Processing, The University of Sheffield, Ph.D. Thesis, 2009.
J. Chen, M. Mahfouf, A Population Adaptive Based Immune Algorithm for Solving Multi-objective Optimisation Problems, in H. Bersini & J. Carneiro (Eds.): ICARIS 2006, LNCS 4163, pp. 280–293, 2006.
J. Chen, M. Mahfouf, Artificial Immune Systems as a Bio-inspired Optimisation Technique and Its Engineering Applications, in H. W. Mo (Eds.): Artificial Immune Systems and Natural Computing: Applying Complex Adaptive Technologies, pp. 22–48, 2008.
E. H. Mamdani, Applications of Fuzzy Algorithm for Control a Simple Dynamic Plant, Proc. Inst. Electr. Eng., vol. 121 (12), pp. 1585–1588, 1974.
T. Takagi, M. Sugeno, Fuzzy Identification of Systems and Its Applications to Modelling and Control, IEEE Transactions on Systems, Man, and Cybernetics, vol. 15, pp. 116–132, 1985.
J. Casillas, O. Cordon, J. Del Jesus Mara, F. Herrera, Genetic Tuning of Fuzzy Rule Deep Structures for Linguistic Modelling, IEEE Transactions on Fuzzy Systems, vol. 13, pp. 13–29, 2001.
L. Zadeh, Outline of A New Approach to the Analysis of Complex Systems and Decision Processes, IEEE Transactions on Systems, Man, and Cybernetics, vol. 3, pp. 28–44, 1973.
J. V. de Oliveira, Semantic Constraints for Membership Function Optimisation, IEEE Trans. Syst. Man. Cybern. Part A, vol. 29 (1), pp. 128–138, 1999.
S. M. Zhou, J. Q. Gan, Low-level Interpretability and High-level Interpretability: A Unified View of Data-Driven Interpretable Fuzzy System Modeling, Fuzzy Sets and Systems, vol. 159, pp. 2091–3131, 2008.
J. M. Alonso, L. Magdalena, G. Gonzalez-Rodriguez, Looking for a Good Fuzzy System Interpretability Index: An Experimental Approach, Int. J. Approx. Reasoning, vol. 51, pp. 115–134, 2009.
G. A. Miller, The Magical Number Seven, Plus or Minus Two: Some Limits on Our Capacity for Processing Information, The Psychological Review, vol. 63(2), pp. 81–97, 1956.
F. Herrera, Genetic Fuzzy Systems: Taxonomy, Current Research Treads and Prospects, Evol. Intel., vol. 1 (1), pp. 27–46, 2008.
H. Ishibuchi, K. Nozaki, N. Yamamoto, H. Tanaka, Selecting Fuzzy If-Then Rules for Classification Problems Using Genetic Algorithms, IEEE Transactions on Fuzzy Systems, vol. 3 (3), pp.260–270, 1995.
H. Ishibuchi, T. Nakashima, T. Murata, Three-objective Genetics-based Machine Learning for Linguistic Rule Extraction, Information Sciences, vol. 136, pp. 109–133, 2001.
H. Ishibuchi, T. Yamamoto, Fuzzy Rule Selection by Multi-Objective Genetic Local Search Algorithms and Rule Evaluation Measures in Data Mining, Fuzzy Sets and Systems, vol. 141, pp. 59–88, 2004.
M. Antonelli, P. Ducange, B. Zazzerini, F. Marcelloni, Learning Concurrently Partition Granularities and Rule Bases of Mamdani Fuzzy Systems in a Multi-objectie Evolutionary Framework, International Journal of Approximate Reasoning, vol. 50 (7), pp. 1066–1080, 2009.
M. J. Gacto, R. Alcalá, F. Herrera, Adaptation and Application of Multi-objective Evolutionary Algorithms for Rule Reduction and Parameter Tuning of Fuzzy Rule-based Systems, Soft Computing, vol. 13 (5), pp. 419–436, 2009.
M. J. Gacto, R. Alcalá, F. Herrera, Integration of an Index to Preserve the Semantic Interpretability in the Multi-Objective Evolutionary Rule Selection and Tuning of Linguistic Fuzzy Systems. IEEE Transactions on Fuzzy Systems, vol. 18(3), pp. 515–531, 2010.
M. Setnes, R. Babuška, U. Kaymak, H. Lemke, Similarity Measures in Fuzzy Rule Base Simplification, IEEE Transactions on Systems, Man, and Cybernetics-Part B, vol. 28 (3), pp. 376–386, 1998.
F. Jiménez, G. Sánchez, A. F. Gómez-Skarmeta, H. Roubos, R. Babuška, Fuzzy Modeling with Multi-Objective Neuro-Evolutionary Algorithms, IEEE International Conference on Systems, Man, and Cybernetics, vol. 3, 2002.
Y. Jin, W. Von Seelen, B. Sendhoff, On Generating FC3 Fuzzy Rule Systems From Data Using Evolution Strategies, IEEE Transactions on Systems, Man, and Cybernetics, vol. 29 (6), pp. 829–845, 1999.
H. L. Wang, S. Kwong, Y. C. Jin, W. Wei, K. F. Man, Multi-objective Hierarchical Genetic Algorithm for Interpretable Fuzzy Rule-based Knowledge Extraction, Fuzzy Sets and Systems, Vol. 149 (1), pp. 149–186, 2005.
J. González, I. Rojas, H. Pomares, L. J. Herrera, A. Guill é n, J. M. Palomares, F. Rojas, Improving the Accuracy While Preserving the Interpretability of Fuzzy Function Approximators by means of Multi-objective Evolutionary Algorithms, International Journal of Approximate Reasoning, vol. 44(1), pp. 32–44, 2007.
R. Alcalá, P. Ducange, F. Herrera, B. Lazzerini, A Multiobjective Evolutionary Approach to Concurrently Learn Rule and Data Bases of Linguistic Fuzzy-Rule-Based Systems, IEEE Transactions on Fuzzy Systems, vol. 17(5), pp. 1106–1122, 2009.
M. Setnes, H. Roubos, GA-Fuzzy Modeling and Classification: Complexity and Performance, IEEE Transactions on Fuzzy Systems, vol. 8 (5), pp. 509–522, 2000.
H. Roubos, M. Setnes, Compact and Transparent Fuzzy Models and Classifiers Through Iterative Complexity Reduction, IEEE Transactions on Fuzzy Systems, vol. 9 (4), pp. 516–524, 2001.
M. Y. Chen, D. A. Linkens, A Systematic Neuro-Fuzzy Modeling Framework With Application to Material Property Prediction, IEEE Transactions on Systems, Man, and Cybernetics, vol. 31 (5), pp.781–790, 2001.
F. Jiménez, A. F. Gómez-Skarmeta, H. Roubos, R. Babuška, Accurate, Transparent, and Compact Fuzzy Models for Function Approximation and Dynamic Modeling through Multi-Objective Evolutionary Optimisation, in E. Zitzler et al. (Eds.): EMO 2001, LNCS 1993, pp. 653–667, 2001.
M. Cococcioni, P. Ducange, B. Lazzerini, F. Marcelloni, A Pareto-based Multi-objective Evolutionary Approach to the Identification of Mamdani Fuzzy Systems, Soft Computing, vol. 11, pp. 1013–1031, 2007.
R. Acalά, M. J. Gacto, F. Herrera, A Multi-Objective Genetic Algorithm for Tuning and Rule Selection to Obtain Accurate and Compact Linguistic Fuzzy Rule-Based Systems, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 15 (5), pp. 539–557, 2007.
Q. Zhang, Nature-Inspired Multi-Objective Optimisation and Transparent Knowledge Discovery via Hierarchical Fuzzy Modelling, Ph.D. Thesis, Department of Automatic Control and Systems Engineering, The University of Sheffield, U.K, 2009.
L. Magdalena, Crossing Unordered Sets of Rules in Evolutionary Fuzzy Controllers, International Journal of Intelligent Systems, vol. 13 (10/11), pp. 993–1010, 1998.
M. G. Cooper, J. J. Vidal, Genetic Design of Fuzzy Controllers: The Cart and Jointed-Pole Problem, in Proceedings of the Third IEEE Conference on Fuzzy Systems, vol. 2, pp. 1332–1337, 1994.
Y. Jin, Fuzzy Modeling of High-Dimensional Systems: Complexity Reduction and Interpretability Improvement, IEEE Transactions on Fuzzy Systems, vol. 8 (2), pp. 212–221, 2000.
M. Sugeno, T. Yasukawa, A Fuzzy-Logic-Based Approach to Qualitative Modeling, IEEE Transactions on Fuzzy Systems, vol. 1 (1), pp. 7–31, 1993.
Y. H. Lin, III G. A. Cunningham, S. V. Coggeshall, Using Fuzzy Partitions to Create Fuzzy Systems from Input-Output Data and Set the Initial Weights in a Fuzzy Neural Network, IEEE Transactions on Fuzzy Systems, vol. 5 (4), pp. 614–621, 1997.
M. Y. Chen, D. A. Linkens, Rule-base Self-generation and Simplification for Data-driven Fuzzy Models, Fuzzy Sets and Systems, vol. 142, pp. 243–265, 2004.
J. Tenner, Optimisation of the Heat Treatment of Steel using Neural Networks, Ph.D. Thesis, Department of Automatic Control and Systems Engineering, University of Sheffield, U.K, 1999.
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This work was supported in part by the EPSRC under Grant EP/F023464/1.
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Chen, J., Mahfouf, M. Improving Transparency in Approximate Fuzzy Modeling Using Multi-objective Immune-Inspired Optimisation. Int J Comput Intell Syst 5, 322–342 (2012). https://doi.org/10.1080/18756891.2012.685311
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DOI: https://doi.org/10.1080/18756891.2012.685311