Neurofuzzy approach for nonlinear dynamical systems modeling

Orientadores: Fernando Antonio Campos Gomide, Pyramo Pires Costa JuniorTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica e de ComputaçãoResumo: Este trabalho propõe um procedimento sistemático para obtenção de modelos de sistemas dinâmicos não-lineares complexos utilizando redes neurais nebulosas. As redes neurais nebulosas aplicadas em modelagem são capazes de extrair conhecimento de dados entrada/saída e representar este conhecimento na forma de regras nebulosas do tipo se-então, gerando modelos lingüísticos convenientes para compreensão humana. Duas novas classes de redes neurais nebulosas são propostas a partir de generalizações dos neurônios lógicos AND e OR. Estas generalizações, denominadas unineurons e nullneurons, implementam, além da plasticidade sináptica, outra importante característica dos neurônios biológicos, a plasticidade neuronal. Desta forma, os neurônios propostos neste trabalho são capazes de modificar parâmetros internos em resposta à alterações, permitindo que unineurons e nullneurons variem individualmente de um neurônio AND para um neurônio OR (e vice-e-versa), dependendo da necessidade do problema. Conseqüentemente, uma rede neural nebulosa composta por unineurons e nullneurons é mais geral do que as redes neurais nebulosas similares sugeridas na literatura. Além da introdução de redes neurais com unineurons e nullneurons, um novo algoritmo de treinamento para obtenção de modelos nebulosos de sistemas dinâmicos é proposto utilizando aprendizado participativo. Neste algoritmo, uma nova informação fornecida à rede por meio de um dado entrada/saída é comparada com o conhecimento que já se tem a respeito do sistema. A nova informação só tem influência na atualização do conhecimento se não entrar em conflito com o conhecimento adquirido anteriormente. Como conseqüência, redes neurais nebulosas que utilizam este novo algoritmo de treinamento são mais robustas a dados de treinamento com valores que correspondem a comportamentos anômalos ou mesmo a erros durante a obtenção destes dados. As abordagens propostas foram utilizadas para desenvolver modelos para previsão de séries temporais e modelagem térmica de transformadores de potência. Os resultados experimentais mostram que os modelos aqui propostos são mais robustos e apresentam os melhores desempenhos, tanto em termos de precisão quanto em termos de custos computacionais, quando comparados com abordagens alternativas sugeridas na literaturaAbstract: This work suggests a systematic procedure to develop models of complex nonlinear dynamical systems using neural fuzzy networks. The neural fuzzy networks are able to extract knowledge from input/output data and to encode it explicitly in the form of if-then rules. Therefore, linguistic models are obtained in a form suitable for human understanding. Two new classes of fuzzy neurons are introduced to generalize AND and OR logic neurons. These generalized login neurons, called unineurons and nullneurons, provide a mechanism to implement synaptic plasticity and an important characteristic of biological neurons, the neuronal plasticity. Unineurons and nullneurons achieve synaptic and neuronal plasticity modifying their internal parameters in response to external changes. Thus, unineurons and nullneurons may individually vary from a AND neuron to a OR neuron (and vice-versa), depending upon the necessity of the modeling task. Neural fuzzy networks constructed with unineurons and nullneurons are more general than similar fuzzy neural approaches suggested in literature. Training algorithms for neural fuzzy networks with unineurons and nullneurons are also studied. In particular, a new training algorithm based on the participatory learning is introduced to develop fuzzy models of dynamical systems. In the participatory learning algorithm, a new information brought to the network through an input/output data is first compared with the knowledge that the network already has about the model. The new information influences the update of the knowledge only if it does not conflict with the current knowledge. As a result, neural fuzzy networks trained with participatory learning show greater robustness to training data with anomalous values than their counterparts. The neural fuzzy network and training algorithms suggested herein are used to develop time series forecast models and thermal models of power transformers. Experimental results show that the models proposed here are more robust and perform best in terms of accuracy and computational costs when compared against alternative approaches suggested in the literatureDoutoradoAutomaçãoDoutor em Engenharia Elétric

Fuzzy Mathematics

This book provides a timely overview of topics in fuzzy mathematics. It lays the foundation for further research and applications in a broad range of areas. It contains break-through analysis on how results from the many variations and extensions of fuzzy set theory can be obtained from known results of traditional fuzzy set theory. The book contains not only theoretical results, but a wide range of applications in areas such as decision analysis, optimal allocation in possibilistics and mixed models, pattern classification, credibility measures, algorithms for modeling uncertain data, and numerical methods for solving fuzzy linear systems. The book offers an excellent reference for advanced undergraduate and graduate students in applied and theoretical fuzzy mathematics. Researchers and referees in fuzzy set theory will find the book to be of extreme value

Mathematical Fuzzy Logic in the Emerging Fields of Engineering, Finance, and Computer Sciences

Mathematical fuzzy logic (MFL) specifically targets many-valued logic and has significantly contributed to the logical foundations of fuzzy set theory (FST). It explores the computational and philosophical rationale behind the uncertainty due to imprecision in the backdrop of traditional mathematical logic. Since uncertainty is present in almost every real-world application, it is essential to develop novel approaches and tools for efficient processing. This book is the collection of the publications in the Special Issue “Mathematical Fuzzy Logic in the Emerging Fields of Engineering, Finance, and Computer Sciences”, which aims to cover theoretical and practical aspects of MFL and FST. Specifically, this book addresses several problems, such as:- Industrial optimization problems- Multi-criteria decision-making- Financial forecasting problems- Image processing- Educational data mining- Explainable artificial intelligence, etc

Neurons And Neural Fuzzy Networks Based On Nullnorms

This paper suggests a new type of elementary unit for neural fuzzy networks based on the concept of nullnorm. A nullnorm is a category of fuzzy set-oriented operators that generalizes triangular norms and conorms. The new unit, called nullneuron, is a generalization of and/or logic-based neurons parametrized by an element u, called the absorbing element. If the absorbing element u = 0, then the nullneuron becomes an and neuron and if u = 1, then the nullneuron becomes a dual or neuron. The paper also addresses two learning schemes for a class of hybrid neural fuzzy networks with nullneurons. The first scheme uses the gradient descent technique and the second reinforcement learning. Both learning schemes adjust not only the weights associated with the inputs of the nullneurons, but also the role of the nullneuron in the network (and or or) by individually adjusting the parameter u of each nullneuron. The neurofuzzy network presented here is more general than alternative approaches discussed in the literature because they allow different triangular norms in the same network structure. Experimental results show that nullneuron-based networks provide accurate results with low computational effort. © 2008 IEEE.123128Box, G.E.P., Jenkins, G.M., (1976) Time Series Analysis- Forecasting and Control, , Holden Day, CA, USA, second editionDelgado, M., Gmez-Skarmeta, A.F., Martin, F., Fuzzy clustering based rapid prototyping for fuzzy rule-based modeling (1997) IEEE Trans. on Fuzzy Systems, 5, pp. 223-233. , MayFigueiredo, M., Ballini, R., Soares, S., Andrade, M., Gomide, F., Learning algorithms for a class of neurofuzzy network and application (2004) IEEE Trans. on System, Man and Cybernetics - Part C, 34 (3), pp. 293-301Gomide, F., Pedrycz, W., (2007) Fuzzy Systems Engineering: Toward Human-Centric Computing, , Wiley Interscience, NJ, USAM. Hell, P. Costa-Jr., and F. Gomide. Nullneurons-based hybrid neurofuzzy network. In Proceedings of Annual Meeting of the North American Fuzzy Information Processing Society, NAFIPS 07, pages 331-336, San Diego-CA, USA, June 2007Hell, M., Costa-Jr, P., Gomide, F., Hybrid neurofuzzy computing with nullneurons (2008) Proceedings of IEEE World Congress on Computational Intelligence - WCCI 2008, , Hong Kong, China, JuneJang, R., Anfis: Adaptive network based fuzzy inference system (1993) IEEE Trans. on Systems Man and Cybernetics, 23 (3), pp. 665-685Lin, C.T., Lee, C.S., (1996) Neural Fuzzy Systems, , Prentice-Hall, NJ, USAPedrycz, W., A identification algorithm in fuzzy relational systems (1984) Fuzzy Sets and Systems, 13, pp. 153-167Pedrycz, W., Neurocomputations in relational systems (1991) IEEE Trans. on Pattern Analysis and Machinne Intelig, 13 (3). , 289-297Pedrycz, W., Logic-based neurocomputing with unineurons (2006) IEEE Trans. on Fuzzy Systems, 14 (6), pp. 860-873Rutkowski, L., Cpalka, K., Design and learning of adjustable quasi-triangular norms with applications to neurofuzzy systems (2005) IEEE Trans. on Neural Networks, 13 (1), pp. 140-151Calvo, B.D.B.T., Fodor, J., The functional equations of frank ans alsina for uninorms and nullnorms (2001) Fuzzy Sets and Systems, 120, pp. 385-394Caminhas, F.G.W., Tavares, H., Pedrycz, W., Fuzzy set based neural networks: Structure, learning and application (1999) Journal of Advanced Computational Intelligence, 3 (3), pp. 151-157Xu, W., Lu, Y.Z., Fuzzy model identification and selflearning for dynamic systems (1987) IEEE Trans. on System, Man and Cybernetics, SMC-17, pp. 683-689. , JulyYager, R., Rybalov, A., Uninorm aggregation operators (1996) Fuzzy Sets and Systems, 80 (1), pp. 111-120Yoshinari, Y., Pedrycz, W., Hirota, K., Construction of fuzzy models through clustering techniques (1993) Fuzzy Sets and Systems, 54, pp. 157-16

Hybrid Neurofuzzy Computing With Nullneurons

In this paper we address a new type of elementary neurofuzzy unit called nullneuron. A nullneuron is a generalization of and/or neurons based on the concept of nullnorm, a category of fuzzy sets operators that generalizes triangular norms and conorms. The nullneuron model is parametrized by an element u, called the absorbing element. If the absorbing element u = 0, then the nullneuron becomes a and neuron and if u = 1, then the nullneuron becomes a dual or neuron. Also, we introduce a new learning scheme for hybrid neurofuzzy networks based on nullneurons using reinforcement learning. This learning scheme adjusts the weights associated with the individual inputs of the nullneurons, and learns the role of the nullneuron in the network (and or or) by individually adjusting the parameter u of each nullneuron. Nullneuron-based neural networks and the associated learning scheme is more general than similar neurofuzzy networks because they allow different triangular norms in the same network structure. Experimental results show that nullneuron-based networks provide accurate results with low computational effort. © 2008 IEEE.36533659Lin, C.C., Lee, C.S.J., Neural network based fuzzy logic control and decision system (1991) IEEE Trans. on Systems Man and Cybernetics, 40 (12), pp. 1320-1336Jang, R., ANFIS: Adaptive network based fuzzy inference system (1993) IEEE Trans. on Systems Man and Cybernetics, 23 (3), pp. 665-685Lin, C.T., Lee, C.S., (1996) Neural Fuzzy Systems, , Prentice Hall, NJ, USAKosko, B., (1991) Neural Networks and fuzzy Systems: A Dynamical System Approach to Machine Intelligence, , Prentice HallFigueiredo, M., Ballini, R., Soares, S., Andrade, M., Gomide, F., Learning Algorithms for a class of Neurofuzzy Network and Application (2004) IEEE Trans. on System, Man and Cybernetics - Part C, 34 (3), pp. 293-301Pedrycz, W., Fuzzy neural networks and neurocomputations (1993) Fuzzy Sets and Systems, 56 (1), pp. 1-28Yager, R., Rybalov, A., Uninorm aggregation operators (1996) Fuzzy Sets and Systems, 80 (1), pp. 111-120Calvo, T., De Baets, B., Fodor, J., The functional equations of Frank ans Alsina for uninorms and nullnorms (2001) Fuzzy Sets Syst, 120, pp. 385-394Rutkowski, L., Cpalka, K., Flexible neuro-fuzzy systems (2003) IEEE Trans. on Neural Networks, 14 (3), pp. 554-574Rutkowski, L., Cpalka, K., Design and learning of adjustable quasi-triangular norms with applications to neuro-fuzzy systems (2005) IEEE Trans. on Neural Networks, 13 (1), pp. 140-151Pedrycz, W., Logic-based neurocomputing with unineurons (2006) IEEE Trans. on Fuzzy Syst, 14 (6), pp. 860-873Hell, M., Costa Jr., P., Gomide, F., Nullneurons-Based Hybrid Neurofuzzy Network (2007) Proceedings of Annual Meeting of the North American Fuzzy Information Processing Society, 2007. NAFIPS '07, pp. 331-336Gomide, F., Pedrycz, W., Fuzzy Systems Engineering: Toward Human-Centric Computing (2007) Wiley Interscience, , Hoboken, NJ, USAHetch-Nielsen, R., (1989) Neurocomputing, , San Diego, CA: Addison WesleyBarto, A.G., Jordan, M.I., Gradient following without backpropagation in layered networks (1987) Proc. of the IEEE First International Conference on Neural Networks, 2, pp. 629-636. , San DiegoCaminhas, W., Tavares, H., Gomide, F., Pedrycz, W., Fuzzy set based neural networks: Structure, Learning and Application (1999) Journal of Advanced Computational Intelligence, 3 (3), pp. 151-157Box, G.E.P., Jenkins, G.M., (1976) Time Series Analysis- Forecasting and Control, , 2.ed, Holden Day, CA, USATong, R.M., The Evaluation of Fuzzy Models Derived from Experimental Data (1980) Fuzzy Sets and Systems, 4, pp. 1-12Farag, W.A., Quintana, V.H., Lambert-Torres, G., A genetic-based neuro-fuzzy approach for modeling and control of dynamical systems (1998) IEEE Trans. Neural Networks, 9 (5), pp. 756-767. , SepXiao-Zhi, G., Ovaska, S.J., Linguistic information feedforward-based dynamical fuzzy systems (2006) IEEE Transactions on Systems, Man and Cybernetics, Part C, 36 (4), pp. 453-463. , JulyPedrycz, W., A Identification Algorithm in Fuzzy Relational Systems (1984) Fuzzy Sets and Systems, 13, pp. 153-167Xu, W., Lu, Y.Z., Fuzzy Model Identification and Self-learning for Dynamic Systems (1987) IEEE Trans. on System, Man and Cybernetics, SMC-17, pp. 683-689. , JulyDelgado, M., Gmez-Skarmeta, A.F., Martin, F., Fuzzy Clustering based Rapid Prototyping for Fuzzy Rule-based Modeling (1997) IEEE Trans. on Fuzzy Systems, 5, pp. 223-233. , MayYoshinari, Y., Pedrycz, W., Hirota, K., Construction of Fuzzy Models Through Clustering Techniques (1993) Fuzzy Sets and Systems, 54, pp. 157-16

A Fast Learning Algorithm For Uninorm-based Fuzzy Neural Networks

This paper suggests a fast learning algorithm for weighted uninorm-based neural networks. Fuzzy neural networks are models capable to approximate functions with high accuracy and to generate transparent models through extraction of linguistic information from the resulting topology. A fuzzy neural network model based on weighted uninorms has been developed recently. It was shown that this model approximates any continuous real function on a compact subset. In this paper we introduce a fast learning algorithm for this class of fuzzy neural networks based on ideas from extreme learning machine. The algorithm is detailed and computational experiments reported to illustrate the accuracy and time efficiency of the learning approach. The results show that neural fuzzy model is accurate and learning speed is as good as or faster than alternative neural network models. © 2012 IEEE.Minist. Commun. Inf. Technol. Republic AzerbaijanPedrycz, W., Fuzzy Neural Networks and Neurocomputations (1993) Fuzzy Sets and Systems, 56 (1), pp. 1-28. , MAY 25Caminhas, W., Tavares, H., Gomide, F., Pedrycz, W., Fuzzy sets based neural networks: Structure, learning and applications (1999) Journal of Advanced Computational Intelligence, 3 (3), pp. 151-157Gomide, F., Pedrycz, W., (2007) Fuzzy Systems Engineering: Toward Human-Centric Computing, , NJ, USA: Wiley InterscienceBallini, R., Gomide, F., Learning in recurrent, hybrid neurofuzzy networks (2002) IEEE International Conference on Fuzzy Systems, pp. 785-791Hell, M., Costa, P., Gomide, F., Participatory learning in power transformers thermal modeling (2008) IEEE Transactions on Power Delivery, 23 (4), pp. 2058-2067. , OctGobi, A.E., Pedrycz, D., Logic minimization as an efficient means of fuzzy structure discovery (2008) IEEE Transactions on Fuzzy Systems, 16 (3), pp. 553-566. , JUNPedrycz, W., Logic-based fuzzy neurocomputing with unineurons (2006) IEEE Transactions on Fuzzy Systems, 14 (6), pp. 860-873. , DECPedrycz, W., Hirota, K., Uninorm-based logic neurons as adaptive and interpretable processing constructs (2007) Soft Computing, 11 (1), pp. 41-52. , JANHell, M., Gomide, F., Ballini, R., Costa, P., Uninetworks in time series forecasting (2009) NAFIPS 2009. Annual Meeting of the North American Fuzzy Information Processing Society, 2009, pp. 1-6. , juneLemos, A., Caminhas, W., Gomide, F., New uninorm-based neuron model and fuzzy neural networks (2010) 2010 Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), pp. 1-6Lemos, A., Kreinovich, V., Caminhas, W., Gomide, F., Universal approximation with uninorm-based fuzzy neural networks (2011) 2011 Annual Meeting of the North American Fuzzy Information Processing Society (NAFIPS), pp. 1-6. , marchYager, R., Rybalov, A., Uninorm aggregation operators (1996) Fuzzy Sets and Systems, 80 (1), pp. 111-120. , MAY 27Calvo, T., Baets, B.D., Fodor, J., The functional equations of frank and alsina for uninorms and nullnorms (2001) Fuzzy Sets and Systems, 120 (3), pp. 385-394Herrera, F., Lozano, M., Verdegay, J.L., Tackling real-coded genetic algorithms: Operators and tools for behavioural analysis (1998) Artif. Intell. Rev., 12 (4), pp. 265-319Huang, G.-B., Zhu, Q.-Y., Siew, C.-K., Extreme learning machine: A new learning scheme of feedforward neural networks (2004) 2004 IEEE International Joint Conference on Neural Networks (IJCNN), 2, pp. 985-990. , july vol.2Huang, G.-B., Chen, L., Siew, C.-K., Universal approximation using incremental constructive feedforward networks with random hidden nodes (2006) Neural Networks, IEEE Transactions on, 17 (4), pp. 879-892. , julyYager, R., Uninorms in fuzzy systems modeling (2001) Fuzzy Sets and Systems, 122 (1), pp. 167-175. , AUG 16Huang, G.-B., Siew, C.-K., Extreme learning machine with randomly assigned rbf kernels (2005) International Journal of Information Technology, 11 (1), pp. 16-24Montesino-Pouzols, F., Lendasse, A., Evolving fuzzy optimally pruned extreme learning machine for regression problems (2010) Evolving Systems, 1 (1), pp. 43-58. , AugustBartlett, P., The sample complexity of pattern classification with neural networks: The size of the weights is more important than the size of the network (1998) IEEE Transactions on Information Theory, 44 (2), pp. 525-536. , marSerre, D., (2002) Matrices: Theory and Applications, , New York, US: Springer- VerlagBox, G.E.P., Jenkins, G., (1990) Time Series Analysis, Forecasting and Control, , Holden-Day, IncorporatedRiedmiller, M., Braun, H., A direct adaptive method for faster backpropagation learning: The rprop algorithm (1993) IEEE International Conference on Neural Networks, 1993, 1, pp. 586-591Jang, J., ANFIS - Adaptive-Network-Based Fuzzy Inference System (1993) IEEE Transactions on Systems Man and Cybernetics, 23 (3

Uninetworks In Time Series Forecasting

This paper presents an approach for time series forecasting using a new class of fuzzy neural networks called uninetworks. Uninetworks are constructed using a recent generalization of the classic and and or logic neurons. These generalized logic neurons, called unineurons, provide a mechanism to implement general nonlinear processing and introduce important characteristics of biological neurons such as neuronal AND synaptic plasticity. Unineurons achieve synaptic and neuronal plasticity modifying their internal parameters in response to external changes. Thus, unineurons may individually vary from an and neuron to an or neuron (and vice-versa), depending upon the necessity of the modeling task. Besides, the proposed neural fuzzy networks are able to extract knowledge from input/output data and to encode it explicitly in the form of if-then rules. Therefore, linguistic models are obtained in a form suitable for human understanding. Experimental results show that the models proposed here are more general and perform best in terms of accuracy and computational costs when compared against alternative approaches suggested in the literature. ©2009 IEEE.Minist. Commun. Inf. Technol. AzerbaijanInfield, D.G., Hill, D.C., Optimal smoothing for trend removal in short term electricity demand forecasting (1998) IEEE Trans. on Power Systems, 13 (3), pp. 1115-1120. , AugPapalexopoulos, A.D., Hesterberg, T.C., A regression-based approach to short-term load forecasting (1990) IEEE Trans. on Power Systems, 5 (4), pp. 1535-1550. , NovRahman, S., Hazim, O., A generalized knowledge-based short term load-forecasting technique (1993) IEEE Trans. on Power Systems, 8 (2), pp. 508-514. , MayHuang, S.J., Shih, K.R., Short-term load forecasting via ARMA model identification including non-Gaussian process considerations (2003) IEEE Trans. Power Systems, 18 (2), pp. 673-679. , MayBox, G.E.P., Jenkins, G.M., Time Series Analysis-Forecasting and Control (1976) Holden-Day, , San Francisco, CA-USAIrisarri, G.D., Widergren, S.E., Yehsakul, P.D., On-line load forecasting for energy control center application (1982) IEEE Trans. on Power Apparatus and Systems, 101 (1), pp. 71-78. , JanHaykin, S., Neural Networks: A Comprehensive Foundation (1998) Prentice Hall, , NJ-USA, ed. 2Pedrycz, W., Rocha, A., Fuzzy set based models of neuron and knowledge-based networks (1993) IEEE Trans. on Fuzzy Systems, 1 (4), pp. 254-266Rutkowski, L., Cpalka, K., Design and learning of adjustable quasitriangular norms with applications to neuro-fuzzy systems (2005) IEEE Trans. on Neural Networks, 13 (1), pp. 140-151M. Hell, P. Costa Jr., F. Gomide, Nullneurons-based hybrid neurofuzzy network, In Proceedings of Annual Meeting of the North American Fuzzy Information Processing Society, NAFIPS 07, pp. 331-336, San Diego-CA, USA, June 2007Hell, M., Costa Jr., P., Gomide, F., Hybrid neurofuzzy computing with nullneurons (2008) Proceedings of IEEE World Congress on Computational Intelligence - WCCI, , Hong Kong, China, JuneYager, R., Rybalov, A., Uninorm aggregation operators (1996) Fuzzy Sets and Systems, 80 (1), pp. 111-120Calvo, T., De Baets, B., Fodor, J., The functional equations of Frank and Alsina for uninorms and nullnorms (2001) Fuzzy Sets Systems, 120, pp. 385-394Pedrycz, W., Logic-based neurocomputing with unineurons (2006) IEEE Trans. on Fuzzy Systems, 14 (6), pp. 860-873Kandel, E.R., Schwartz, J.H., Jessel, T.M., Principles of Neural Science (2000) McGraw-Hill, , NY-USA, 4th. edFodor, J.C., On fuzzy implication operators (1991) Fuzzy Sets and Systems, 42, pp. 293-300J. C. Fodor, R. R. Yager, and R. Rybalov, Structure of uninorms, Int. Journal Uncertainty Fuzziness Knowledge-Based Systems, 45, no. 4, pp. 411-427, 1997Yager, R.R., Uninorms in fuzzy systems modeling (2001) Fuzzy Sets and Systems, 122, p. 167175Pedrycz, W., Neurocomputations in relational systems (1991) IEEE Trans. on Pattern Analysis and Machinne Intellig, 13, pp. 289-296Caminhas, W., Tavares, H., Gomide, F., Pedrycz, W., Fuzzy set based neural networks: Structure, Learning and Application (1999) Journal of Advanced Computational Intelligence, 3 (3), pp. 151-157Figueiredo, M., Ballini, R., Soares, S., Andrade, M., Gomide, F., Learning Algorithms for a class of Neurofuzzy Network and Application (2004) IEEE Trans. on System, Man and Cybernetics - Part C, 34 (3), pp. 293-301Pedrycz, W., Gomide, F., Fuzzy Systems Engineering: Toward Human-Centric Computing (2007) Wiley Interscience, , Hoboken, NJ, USARutkowski, L., Cpalka, K., Flexible neuro-fuzzy systems (2003) IEEE Trans. on Neural Networks, 14 (3), pp. 554-574Hetch-Nielsen, R., Neurocomputing (1989) Addison Wesley, , San Diego, CABarto, A.G., Jordan, M.I., Gradient following without backpropagation in layered networks (1987) Proc. of the IEEE First International Conference on Neural Networks, 2, pp. 629-636. , San DiegoGross, G., Galiana, F.D., Short-Term load forecasting (1987) Proceedings of the IEEE, 75 (12), pp. 1558-1573. , DecJang, R., ANFIS: Adaptive network based fuzzy inference system (1993) IEEE Trans. on Systems Man and Cybernetics, 23 (3), pp. 665-68