## Neuro-Fuzzy Architectures and Hybrid LearningThe advent of the computer age has set in motion a profound shift in our perception of science -its structure, its aims and its evolution. Traditionally, the principal domains of science were, and are, considered to be mathe matics, physics, chemistry, biology, astronomy and related disciplines. But today, and to an increasing extent, scientific progress is being driven by a quest for machine intelligence - for systems which possess a high MIQ (Machine IQ) and can perform a wide variety of physical and mental tasks with minimal human intervention. The role model for intelligent systems is the human mind. The influ ence of the human mind as a role model is clearly visible in the methodolo gies which have emerged, mainly during the past two decades, for the con ception, design and utilization of intelligent systems. At the center of these methodologies are fuzzy logic (FL); neurocomputing (NC); evolutionary computing (EC); probabilistic computing (PC); chaotic computing (CC); and machine learning (ML). Collectively, these methodologies constitute what is called soft computing (SC). In this perspective, soft computing is basically a coalition of methodologies which collectively provide a body of concepts and techniques for automation of reasoning and decision-making in an environment of imprecision, uncertainty and partial truth. |

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### Contents

Introduction | 1 |

Description of Fuzzy Inference Systems | 5 |

212 Operations on Fuzzy Sets | 12 |

213 Fuzzy Relations | 19 |

214 Operations on Fuzzy Relations | 22 |

22 Approximate Reasoning | 25 |

222 Implications | 27 |

223 Linguistic Variables | 29 |

46 Architectures of Systems with NonSingleton Fuzzifier | 124 |

NeuroFuzzy Architectures Based on the Logical Approach | 127 |

52 NOCFS Architectures | 133 |

53 OCFS Architectures | 136 |

54 Performance Analysis | 145 |

55 Computer Simulations | 157 |

552 Control Examples | 158 |

553 Classification Problems | 160 |

224 Calculus of Fuzzy Rules | 34 |

225 Granulation and Fuzzy Graphs | 37 |

226 Computing with Words | 41 |

23 Fuzzy Systems | 43 |

231 RuleBased Fuzzy Logic Systems | 44 |

232 The Mamdani and Logical Approaches to Fuzzy Inference | 49 |

233 Fuzzy Systems Based on the Mamdani Approach | 51 |

234 Fuzzy Systems Based on the Logical Approach | 60 |

Neural Networks and NeuroFuzzy Systems | 69 |

311 Model of an Artificial Neuron | 70 |

312 MultiLayer Perceptron | 73 |

313 BackPropagation Learning Method | 76 |

314 RBF Networks | 80 |

315 Supervised and Unsupervised Learning | 84 |

316 Competitive Learning | 85 |

317 Hebbian Learning Rule | 88 |

318 Kohonens SelfOrganizing Neural Network | 89 |

319 Learning Vector Quantization | 94 |

3110 Other Types of Neural Networks | 97 |

32 Fuzzy Neural Networks | 98 |

33 Fuzzy Inference Neural Networks | 101 |

NeuroFuzzy Architectures Based on the Mamdani Approach | 105 |

42 General Form of the Architectures | 109 |

43 Systems with Inference Based on Bounded Product | 114 |

44 Simplified Architectures | 116 |

45 Architectures Based on Other Defuzzification Methods | 119 |

452 Neural Networks as Defuzzifiers | 122 |

Hybrid Learning Methods | 165 |

611 Learning of Fuzzy Systems | 166 |

612 Learning of NeuroFuzzy Systems | 171 |

613 FLiNN Architecture Based Learning | 174 |

62 Genetic Algorithms | 175 |

622 Evolutionary Algorithms | 181 |

63 Clustering Algorithms | 185 |

632 Fuzzy Clustering | 189 |

64 Hybrid Learning | 191 |

641 Combinations of Gradient Methods GAs and Clustering Algorithms | 192 |

642 Hybrid Algorithms for Parameter Tuning | 194 |

643 Rule Generation | 195 |

65 Hybrid Learning Algorithms for NeuroFuzzy Systems | 198 |

651 Examples of Hybrid Learning NeuroFuzzy Systems | 199 |

652 Description of Two Hybrid Learning Algorithms for Rule Generation | 201 |

653 Medical Diagnosis Applications | 204 |

Intelligent Systems | 209 |

72 Expert Systems | 212 |

722 Fuzzy and Neural Expert Systems | 214 |

73 Intelligent Computational Systems | 217 |

74 PerceptionBased Intelligent Systems | 220 |

Summary | 229 |

List of Figures | 233 |

List of Tables | 239 |

References | 241 |

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### Common terms and phrases

applied approximation back-propagation basic Cartesian product chromosomes classification connectionist consequent fuzzy sets constraint corresponding crisp crossover data vector defined Definition defuzzification denoted depicted in Fig elements employed example expert systems FIGURE formula fuzzifier fuzzy controllers fuzzy graph fuzzy IF-THEN rules fuzzy implication fuzzy inference fuzzy logic fuzzy logic systems fuzzy neural networks fuzzy relation fuzzy rules fuzzy systems Gaussian functions Gaussian membership functions genetic algorithms given by Equation gradient granulation group of implications Hebbian learning hybrid learning IEEE illustrated in Fig input vector intelligent systems learning algorithm learning methods linguistic variables logical approach Mamdani approach multi-layer neuro-fuzzy architectures neuro-fuzzy system based NOCFS OCFS neuro-fuzzy system OCFS system based optimization output fuzzy set output neurons output values perceptron performance problem product operation rule base rule of inference shown in Fig singleton Soft Computing supervised learning T-norm tion triangular membership functions universe of discourse Zadeh