The concept of evolving Fuzzy Systems (eFSs) was introduced in the 1990s with neural networks with the works of Fritzke [1], Williamson [2], and Kwok and Yeung [3], and then, extended to rule-based models with Angelov and Buswell [4]. There are also a few works with evolving Fuzzy Trees [5-7]. One of the main advantages of eFSs is their ability to create and update their structure autonomously from data streams without retraining the model to new data. On the other hand, eFSs are continuously updated as new data are available to the system, so an eFS can capture shifts and drifts in the data by self-updating its structure to include the pattern's change, making this class suitable to deal with highly complex non-linear data. One of the main advantages of rule-based eFSs is that they combine explainability and accuracy. Rule-based eFSs can transform the learned knowledge into human-interpretable fuzzy rules efficiently [8-9].
In this paper, six remarkable rule-based eFSs are implemented: evolving Takagi-Sugeno (eTS) [10-11], simplified eTS (Simpl_eTS) [12], evolving extended Takagi-Sugeno (exTS) [13], evolving Multivariable Gaussian (eMG) [14], evolving Participatory Learning (ePL+) [15], and evolving Participatory Learning with Kernel Recursive Least Square and Distance Correlation (ePL-KRLS-DISCO) [8]. All these models have two main parts: consequent and antecedent. The antecedent is responsible for forming the rules, and the consequent part comprises the parameters used to compute the output. The first model, eTS, uses a scatter concept to create the rules and the recursive least square (RLS) to calculate the consequent parameters. The model has a few hyperparameters, and the user cannot control the number of rules. Simpl_eTS and exTS introduce some improvements in the quality of the clusters compared to eTS. The eMG includes the concept of a covariance matrix to create clusters with a more flexible shape. The ePL+ model discusses the participatory learning (PL) concept, a technique used to avoid the effect of outliers and noise data in the model. And finally, ePL-KRLS-DISCO proposes the implementation of coefficient correlation in the antecedent and the kernel recursive least square (KRLS) in the consequent. Consequently, the model not only can capture the tendency of the data but also can deal with highly complex data.
REFERENCES
[1] Fritzke, B.: Growing cell structures—A self-organizing network for unsupervised and supervised learning. Neural Networks 7(9), 1441–1460 (1994).
https://doi.org/10.1016/0893-6080(94)90091-4
[2] Williamson, J.R.: Gaussian ARTMAP: A neural network for fast incremental learning of noisy multidimensional maps. Neural Networks 9(5), 881–897 (1996).
https://doi.org/10.1016/0893-6080(95)00115-8
[3] Kwok, T.-Y., Yeung, D.-Y.: Constructive algorithms for structure learning in feedforward neural networks for regression problems. IEEE Transactions on Neural Networks 8(3), 630–645 (1997).
https://doi.org/10.1109/72.572102
[4] Angelov, P., Buswell, R.: Evolving rule-based models: A tool for intelligent adaptation. In: Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Cat. No. 01TH8569), vol. 2, pp. 1062–1067 (2001). https://doi.org/10.1109/nafips.2001.944752. IEEE
[5] Eggermont, J.: Evolving fuzzy decision trees with genetic programming and clustering. In: Genetic Programming: 5th European Conference, EuroGP 2002 Kinsale, Ireland, April 3–5, 2002 Proceedings, pp. 71–82 (2002).
https://doi.org/10.1007/3-540-45984-7 7. Springer
[6] Lai, R.K., Fan, C.-Y., Huang, W.-H., Chang, P.-C.: Evolving and clustering fuzzy decision tree for financial time series data forecasting. Expert Systems with Applications 36(2), 3761–3773 (2009).
https://doi.org/10.1016/j.eswa.2008.02.025
[7] Lemos, A., Caminhas, W., Gomide, F.: Fuzzy evolving linear regression trees. Evolving Systems 2, 1–14 (2011).
https://doi.org/10.1007/s12530-011-9028-z
[8] Alves, K.S.T.R., de Aguiar, E.P.: A novel rule-based evolving fuzzy system applied to the thermal modeling of power transformers. Applied Soft Computing 112, 107764 (2021).
https://doi.org/10.1016/j.asoc.2021.107764
[9] Gu, X., Han, J., Shen, Q., Angelov, P.P.: Autonomous learning for fuzzy systems: a review. Artificial Intelligence Review, 1–47 (2022). https://doi.org/10.1007/s10462-022-10355-6
[10] Angelov, P., Filev, D.: On-line design of takagi-sugeno models. In: International Fuzzy Systems Association World Congress, pp. 576–584 (2003).
https://doi.org/10.1007/3-540-44967-1 69. Springer
[11] Angelov, P.P., Filev, D.P.: An approach to online identification of takagi-sugeno fuzzy models. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 34(1), 484–498 (2004).
https://doi.org/10.1109/TSMCB.2003.817053
[12] Angelov, P., Filev, D.: Simpl ets: A simplified method for learning evolving takagi-sugeno fuzzy models. In: The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ’05., pp. 1068–1073 (2005).
https://doi.org/10.1109/FUZZY.2005.1452543. IEEE
[13] Angelov, P., Zhou, X.: Evolving fuzzy systems from data streams in real-time. In: 2006 International Symposium on Evolving Fuzzy Systems, pp. 29–35 (2006).
https://doi.org/10.1109/ISEFS.2006.251157. IEEE
[14] Lemos, A., Caminhas, W., Gomide, F.: Multivariable gaussian evolving fuzzy modeling system. IEEE Transactions on Fuzzy Systems 19(1), 91–104 (2010).
https://doi.org/10.1109/TFUZZ.2010.2087381
[15] Maciel, L., Gomide, F., Ballini, R.: An enhanced approach for evolving participatory learning fuzzy modeling. In: 2012 IEEE Conference on Evolving and Adaptive Intelligent Systems, pp. 23–28 (2012).
https://doi.org/10.1109/EAIS.2012.6232799. IEEE
In this paper, six remarkable rule-based eFSs are implemented: evolving Takagi-Sugeno (eTS) [10-11], simplified eTS (Simpl_eTS) [12], evolving extended Takagi-Sugeno (exTS) [13], evolving Multivariable Gaussian (eMG) [14], evolving Participatory Learning (ePL+) [15], and evolving Participatory Learning with Kernel Recursive Least Square and Distance Correlation (ePL-KRLS-DISCO) [8]. All these models have two main parts: consequent and antecedent. The antecedent is responsible for forming the rules, and the consequent part comprises the parameters used to compute the output. The first model, eTS, uses a scatter concept to create the rules and the recursive least square (RLS) to calculate the consequent parameters. The model has a few hyperparameters, and the user cannot control the number of rules. Simpl_eTS and exTS introduce some improvements in the quality of the clusters compared to eTS. The eMG includes the concept of a covariance matrix to create clusters with a more flexible shape. The ePL+ model discusses the participatory learning (PL) concept, a technique used to avoid the effect of outliers and noise data in the model. And finally, ePL-KRLS-DISCO proposes the implementation of coefficient correlation in the antecedent and the kernel recursive least square (KRLS) in the consequent. Consequently, the model not only can capture the tendency of the data but also can deal with highly complex data.
REFERENCES
[1] Fritzke, B.: Growing cell structures—A self-organizing network for unsupervised and supervised learning. Neural Networks 7(9), 1441–1460 (1994).
https://doi.org/10.1016/0893-6080(94)90091-4
[2] Williamson, J.R.: Gaussian ARTMAP: A neural network for fast incremental learning of noisy multidimensional maps. Neural Networks 9(5), 881–897 (1996).
https://doi.org/10.1016/0893-6080(95)00115-8
[3] Kwok, T.-Y., Yeung, D.-Y.: Constructive algorithms for structure learning in feedforward neural networks for regression problems. IEEE Transactions on Neural Networks 8(3), 630–645 (1997).
https://doi.org/10.1109/72.572102
[4] Angelov, P., Buswell, R.: Evolving rule-based models: A tool for intelligent adaptation. In: Proceedings Joint 9th IFSA World Congress and 20th NAFIPS International Conference (Cat. No. 01TH8569), vol. 2, pp. 1062–1067 (2001). https://doi.org/10.1109/nafips.2001.944752. IEEE
[5] Eggermont, J.: Evolving fuzzy decision trees with genetic programming and clustering. In: Genetic Programming: 5th European Conference, EuroGP 2002 Kinsale, Ireland, April 3–5, 2002 Proceedings, pp. 71–82 (2002).
https://doi.org/10.1007/3-540-45984-7 7. Springer
[6] Lai, R.K., Fan, C.-Y., Huang, W.-H., Chang, P.-C.: Evolving and clustering fuzzy decision tree for financial time series data forecasting. Expert Systems with Applications 36(2), 3761–3773 (2009).
https://doi.org/10.1016/j.eswa.2008.02.025
[7] Lemos, A., Caminhas, W., Gomide, F.: Fuzzy evolving linear regression trees. Evolving Systems 2, 1–14 (2011).
https://doi.org/10.1007/s12530-011-9028-z
[8] Alves, K.S.T.R., de Aguiar, E.P.: A novel rule-based evolving fuzzy system applied to the thermal modeling of power transformers. Applied Soft Computing 112, 107764 (2021).
https://doi.org/10.1016/j.asoc.2021.107764
[9] Gu, X., Han, J., Shen, Q., Angelov, P.P.: Autonomous learning for fuzzy systems: a review. Artificial Intelligence Review, 1–47 (2022). https://doi.org/10.1007/s10462-022-10355-6
[10] Angelov, P., Filev, D.: On-line design of takagi-sugeno models. In: International Fuzzy Systems Association World Congress, pp. 576–584 (2003).
https://doi.org/10.1007/3-540-44967-1 69. Springer
[11] Angelov, P.P., Filev, D.P.: An approach to online identification of takagi-sugeno fuzzy models. IEEE Transactions on Systems, Man, and Cybernetics, Part B (Cybernetics) 34(1), 484–498 (2004).
https://doi.org/10.1109/TSMCB.2003.817053
[12] Angelov, P., Filev, D.: Simpl ets: A simplified method for learning evolving takagi-sugeno fuzzy models. In: The 14th IEEE International Conference on Fuzzy Systems, 2005. FUZZ’05., pp. 1068–1073 (2005).
https://doi.org/10.1109/FUZZY.2005.1452543. IEEE
[13] Angelov, P., Zhou, X.: Evolving fuzzy systems from data streams in real-time. In: 2006 International Symposium on Evolving Fuzzy Systems, pp. 29–35 (2006).
https://doi.org/10.1109/ISEFS.2006.251157. IEEE
[14] Lemos, A., Caminhas, W., Gomide, F.: Multivariable gaussian evolving fuzzy modeling system. IEEE Transactions on Fuzzy Systems 19(1), 91–104 (2010).
https://doi.org/10.1109/TFUZZ.2010.2087381
[15] Maciel, L., Gomide, F., Ballini, R.: An enhanced approach for evolving participatory learning fuzzy modeling. In: 2012 IEEE Conference on Evolving and Adaptive Intelligent Systems, pp. 23–28 (2012).
https://doi.org/10.1109/EAIS.2012.6232799. IEEE