This paper presents the architecture and learning procedure underlying ANFIS (AdaptiveNetwork -based Fuzzy Inference System), a fuzzy inference system implemented in the framework of adaptive networks. By using a hybrid learning procedure, the proposed ANFIS can construct an input-output mapping based on both human knowledge (in the form of fuzzy if-then rules) and stipulated input-output data pairs. In our simulation, we employ the ANFIS architecture to model nonlinear functions, identify nonlinear components on-linely in a control system, and predict a chaotic time series, all yielding remarkable results. Comparisons with artificail neural networks and earlier work on fuzzy modeling are listed and discussed. Other extensions of the proposed ANFIS and promising applications to automatic control and signal processing are also suggested. 1 Introduction System modeling based on conventional mathematical tools (e.g., differential equations) is not well suited for dealing with ill-define...

Fundamental and advanced developments in neuro-fuzzy synergisms for modeling and control are reviewed. The essential part of neuro-fuzzy synergisms comes from a common framework called adaptive networks, which unifies both neural networks and fuzzy models. The fuzzy models under the framework of adaptive networks is called ANFIS (Adaptive-Network-based Fuzzy Inference System), which possess certain advantages over neural networks. We introduce the design methods for ANFIS in both modeling and control applications. Current problems and future directions for neuro-fuzzy approaches are also addressed.

A new methodology of extraction, optimization, and application of sets of logical rules is described. Neural networks are used for initial rule extraction, local, or global minimization procedures for optimization, and Gaussian uncertainties of measurements are assumed during application of logical rules. Algorithms for extraction of logical rules from data with real-valued features require determination of linguistic variables or membership functions. Context-dependent membership functions for crisp and fuzzy linguistic variables are introduced and methods of their determination described. Several neural and machine learning methods of logical rule extraction generating initial rules are described, based on constrained multilayer perceptron, networks with localized transfer functions or on separability criteria for determination of linguistic variables. A tradeoff between accuracy/simplicity is explored at the rule extraction stage and between rejection/error level at the optimization stage. Gaussian uncertainties of measurements are assumed during application of crisp logical rules, leading to “soft trapezoidal” membership functions and allowing to optimize the linguistic variables using gradient procedures. Numerous applications of this methodology to benchmark and real-life problems are reported and very simple crisp logical rules for many datasets provided.

The term Soft Computing (SC) represents the combination of emerging problem-solving technologies such as Fuzzy Logic (FL), Probabilistic Reasoning (PR), Neural Networks (NNs), and Genetic Algorithms (GAs). Each of these technologies provide us with complementary reasoning and searching methods to solve complex, real-world problems. After a brief description of each of these technologies, we will analyze some of their most useful combinations, such as the use of FL to control GAs and NNs parameters; the application of GAs to evolve NNs (topologies or weights) or to tune FL controllers; and the implementation of FL controllers as NNs tuned by backpropagation-type algorithms.

This chapter provides an introduction to the field of hybrid neural systems. Hybrid neural systems are computational systems which are based mainly on artificial neural networks but also allow a symbolic interpretation, or interaction with symbolic components. In this overview, we will describe recent results of hybrid neural systems. We will give a brief overview of the main methods used, outline the work that is presented here, and provide additional references. We will also highlight some important general issues and trends.

This paper describes techniques for integrating neural networks and symbolic components into powerful hybrid systems. Neural networks have unique processing characteristics that enable tasks to be performed that would be di cult or intractable for a symbolic rule-based system. However, a stand-alone neural network requires an interpretation either by ahuman or a rulebased system. This motivates the integration of neural/symbolic techniques within a hybrid system. Anumber of integration possibilities exist: some systems consist of neural network components performing symbolic tasks while other systems are composed of several neural networks and symbolic components, each component acting as a self-contained module communicating with the others. Other hybrid systems are able to transform subsymbolic representations into symbolic ones and vice-versa. This paper providesanoverview and evaluation of the state of the artofseveral hybrid neural systems for rule-based processing. 1

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... This paper is focused on the extraction and use of logical rules for data understanding. All aspects of rule generation, optimization, and application are described, including the problem of finding good symbolic descriptors for continuous data, tradeoffs between accuracy and simplicity at the rule-extraction stage, and tradeoffs between rejection and error level at the rule optimization stage. Stability of rule-based description, calculation of probabilities from rules, and other related issues are also discussed. Major approaches to extraction of logical rules based on neural networks, decision trees, machine learning, and statistical methods are introduced. Optimization and application issues for sets of logical rules are described. Applications of such methods to benchmark and real-life problems are reported and illustrated with simple logical rules for many datasets. Challenges and new directions for research are outlined.

The shape of if-part fuzzy sets affects how well feedforward fuzzy systems approximate continuous functions. We explore a wide range of candidate if-part sets and derive supervised learning laws that tune them. Then we test how well the resulting adaptive fuzzy systems approximate a battery of test functions. No one set shape emerges as the best shape. The sinc function often does well and has a tractable learning law. But its undulating sidelobes may have no linguistic meaning. This suggests that the engineering goal of function-approximation accuracy may sometimes have to outweigh the linguistic or philosophical interpretations of fuzzy sets that have accompanied their use in expert systems. We divide the if-part sets into two large classes. The first class consists of-dimensional joint sets that factor into scalar sets as found in almost all published fuzzy systems. These sets ignore the correlations among vector components of input vectors. Fuzzy systems that use factorable if-part sets suffer in general from exponential rule explosion in high dimensions when they blindly approximate functions without knowledge of the functions. The factorable fuzzy sets themselves also suffer from what we call the second curse of dimensionality: The fuzzy sets tend to become binary spikes in high dimension. The second class of if-part sets consists of the more general but less common-dimensional joint sets that do not factor into scalar fuzzy sets. We present a method for constructing such unfactorable joint sets from scalar distance measures. Fuzzy systems that use unfactorable if-part sets need not suffer from exponential rule explosion but their increased complexity may lead to intractable learning laws and inscrutable if-then rules. We prove that some of these unfactorable join...

INTRODUCTION 1.1. WHY UNIVERSAL APPROXIMATION In some cases, fuzzy rule based model needs tuning. If we have applied some version of fuzzy rule based modeling, and the resulting model is satisfactory, great. But sometimes, the resulting model is not of very high quality: \Gamma we may have misinterpreted some of the expert's rules; \Gamma we may have missed some of the important rules. So, to improve the quality of the resulting model, we must either: \Gamma re-interpret the existing rules, or \Gamma ask the experts for extra rules. 2 V. KREINOVICH ET AL. Before we start tuning, we want to make sure that tuning will help. In order to make sure that this "tuning" will always help, we must guarantee that by appropriate tuning, we will be able to change the initial (not very successful) model into a successful one. To guarantee that the fuzzy rule based modeling me