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203
ANFIS: AdaptiveNetworkBased Fuzzy Inference System”,
 IEEE Trans. on System, Man and Cybernetics,
, 1993
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Functional Equivalence between Radial Basis Function Networks and Fuzzy Inference Systems
, 1993
"... This short article shows that under some minor restrictions, the functional behavior of radial basis function networks and fuzzy inference systems are actually equivalent. This functional equivalence implies that advances in each literature, such as new learning rules or analysis on representational ..."
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Cited by 169 (4 self)
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This short article shows that under some minor restrictions, the functional behavior of radial basis function networks and fuzzy inference systems are actually equivalent. This functional equivalence implies that advances in each literature, such as new learning rules or analysis on representational power, etc., can be applied to both models directly. It is of interest to observe that twomodels stemming from different origins turn out to be functional equivalent.
An online selfconstructing neural fuzzy inference network and its applications
 IEEE. TRANS. FUZZY. SYS
, 1998
"... A selfconstructing neural fuzzy inference network (SONFIN) with online learning ability is proposed in this paper. The SONFIN is inherently a modified Takagi–Sugeno–Kang (TSK)type fuzzy rulebased model possessing neural network’s learning ability. There are no rules initially in the SONFIN. The ..."
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Cited by 92 (22 self)
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A selfconstructing neural fuzzy inference network (SONFIN) with online learning ability is proposed in this paper. The SONFIN is inherently a modified Takagi–Sugeno–Kang (TSK)type fuzzy rulebased model possessing neural network’s learning ability. There are no rules initially in the SONFIN. They are created and adapted as online learning proceeds via simultaneous structure and parameter identification. In the structure identification of the precondition part, the input space is partitioned in a flexible way according to a aligned clusteringbased algorithm. As to the structure identification of the consequent part, only a singleton value selected by a clustering method is assigned to each rule initially. Afterwards, some additional significant terms (input variables) selected via a projectionbased correlation measure for each rule will be added to the consequent part (forming a linear equation of input variables) incrementally as learning proceeds. The combined precondition and consequent structure identification scheme can set up an economic and dynamically growing network, a main feature of the SONFIN. In the parameter identification, the consequent parameters are tuned optimally by either least mean squares (LMS) or recursive least squares (RLS) algorithms and the precondition parameters are tuned by backpropagation algorithm. Both the structure and parameter identification are done simultaneously to form a fast learning scheme, which is another feature of the SONFIN. Furthermore, to enhance the knowledge representation ability of the SONFIN, a linear transformation for each input variable can be incorporated into the network so that much fewer rules are needed or higher accuracy can be achieved. Proper linear transformations are also learned dynamically in the parameter identification phase of the SONFIN. To demonstrate the capability of the proposed SONFIN, simulations in different areas including control, communication, and signal processing are done. Effectiveness of the SONFIN is verified from these simulations.
Designing fuzzy inference systems from data: an interpretabilityoriented review
 IEEE TRANS. FUZZY SYSTEMS
, 2001
"... Fuzzy inference systems (FIS) are widely used for process simulation or control. They can be designed either from expert knowledge or from data. For complex systems, FIS based on expert knowledge only may suffer from a loss of accuracy. This is the main incentive for using fuzzy rules inferred from ..."
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Cited by 88 (16 self)
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Fuzzy inference systems (FIS) are widely used for process simulation or control. They can be designed either from expert knowledge or from data. For complex systems, FIS based on expert knowledge only may suffer from a loss of accuracy. This is the main incentive for using fuzzy rules inferred from data. Designing a FIS from data can be decomposed into two main phases: automatic rule generation and system optimization. Rule generation leads to a basic system with a given space partitioning and the corresponding set of rules. System optimization can be done at various levels. Variable selection can be an overall selection or it can be managed rule by rule. Rule base optimization aims to select the most useful rules and to optimize rule conclusions. Space partitioning can be improved by adding or removing fuzzy sets and by tuning membership function parameters. Structure optimization is of a major importance: selecting variables, reducing the rule base and optimizing the number of fuzzy sets. Over the years, many methods have become available for designing FIS from data. Their efficiency is usually characterized by a numerical performance index. However, for humancomputer cooperation another criterion is needed: the rule interpretability. An implicit assumption states that fuzzy rules are by nature easy to be interpreted. This could be wrong when dealing with complex multivariable systems or when the generated partitioning is meaningless for experts. This paper analyzes the main methods for automatic rule generation and structure optimization. They are grouped into several families and compared according to the rule interpretability criterion. For this purpose, three conditions for a set of rules to be interpretable are defined.
Neural Networks in Designing Fuzzy Systems for Real World Applications”, Fuzz y
 University of Sydney
, 1995
"... AbstractA special multilayer perceptron architecture known as FuNe I is successfully used for generating fuzzy systems for a number of real world applications. The FuNe I trained with supervised learning can be used to extract fuzzy rules from a given representative input/output data set. Furthermo ..."
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Cited by 59 (8 self)
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AbstractA special multilayer perceptron architecture known as FuNe I is successfully used for generating fuzzy systems for a number of real world applications. The FuNe I trained with supervised learning can be used to extract fuzzy rules from a given representative input/output data set. Furthermore, optimization of the knowledge base is possible including the tuning of membership functions. The new method employed to identify the rule relevant nodes before the rules are extracted makes FuNe I suitable for applications with large number of inputs. Some of the real world applicationsinareas of state identi cation and image classi cation show encouraging results in a shorter development time. Expert knowledge is not compulsory but can be included in the automatically extracted knowledge base. The generated fuzzy system can be implemented in hardware very easily. A exible prototype board is developed with a FPGA chip in order to run applications with up to 128 inputs and 4 outputs in realtime (1.25 million rules per second).
Improving the interpretability of TSK fuzzy models by combining global and local learning
 IEEE TRANS. FUZZY SYST
, 1998
"... The fuzzy inference system proposed by Takagi, Sugeno, and Kang, known as the TSK model in fuzzy system literature, provides a powerful tool for modeling complex nonlinear systems. Unlike conventional modeling where a single model is used to describe the global behavior of a system, TSK modeling is ..."
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Cited by 47 (1 self)
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The fuzzy inference system proposed by Takagi, Sugeno, and Kang, known as the TSK model in fuzzy system literature, provides a powerful tool for modeling complex nonlinear systems. Unlike conventional modeling where a single model is used to describe the global behavior of a system, TSK modeling is essentially a multimodel approach in which simple submodels (typically linear models) are combined to describe the global behavior of the system. Most existing learning algorithms for identifying the TSK model are based on minimizing the square of the residual between the overall outputs of the real system and the identified model. Although these algorithms can generate a TSK model with good global performance (i.e., the model is capable of approximating the given system with arbitrary accuracy, provided that sufficient rules are used and sufficient training data are available), they cannot guarantee the resulting model to have a good local performance. Often, the submodels in the TSK model may exhibit an erratic local behavior, which is difficult to interpret. Since one of the important motivations of using the TSK model (also other fuzzy models) is to gain insights into the model, it is important to investigate the interpretability issue of the TSK model. In this paper, we propose a new learning algorithm that integrates global learning and local learning in a single algorithmic framework. This algorithm uses the idea of local weighed regression and local approximation in nonparametric statistics, but remains the component of global fitting in the existing learning algorithms. The algorithm is capable of adjusting its parameters based on the user’s preference, generating models with good tradeoff in terms of global fitting and local interpretation. We illustrate the performance of the proposed algorithm using a motorcycle crash modeling example.
Combined Genetic Algorithm Optimization and Regularized Orthogonal Least Squares Learning for Radial Basis Function Networks
, 1999
"... The paper presents a twolevel learning method for radial basis function (RBF) networks. A regularized orthogonal least squares (ROLS) algorithm is employed at the lower level to construct RBF networks while the two key learning parameters, the regularization parameter and the RBF width, are optimiz ..."
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Cited by 40 (14 self)
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The paper presents a twolevel learning method for radial basis function (RBF) networks. A regularized orthogonal least squares (ROLS) algorithm is employed at the lower level to construct RBF networks while the two key learning parameters, the regularization parameter and the RBF width, are optimized using a genetic algorithm (GA) at the upper level. Nonlinear time series modeling and prediction is used as an example to demonstrate the effectiveness of this hierarchical learning approach.
Support Vector Learning for Fuzzy RuleBased Classification Systems
, 2003
"... To design a fuzzy rulebased classi cation system (fuzzy classi er) with good generalization abilityina high dimensional feature space has been an active research topic for a long time. As a powerful machine learning approach for pattern recognition problems, support vector machine (SVM) is known to ..."
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Cited by 29 (1 self)
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To design a fuzzy rulebased classi cation system (fuzzy classi er) with good generalization abilityina high dimensional feature space has been an active research topic for a long time. As a powerful machine learning approach for pattern recognition problems, support vector machine (SVM) is known to have good generalization ability. More importantly, an SVM can work very well on a high (or even in nite) dimensional feature space. This paper investigates the connection between fuzzy classi ers and kernel machines, establishes a link between fuzzy rules and kernels, and proposes a learning algorithm for fuzzy classi ers. We rst show that a fuzzy classi er implicitly de nes a translation invariant kernel under the assumption that all membership functions associated with the same input variable are generated from location transformation of a reference function. Fuzzy inference on the IFpart of a fuzzy rule can be viewed as evaluating the kernel function. The kernel function is then proven to be a Mercer kernel if the reference functions meet certain spectral requirement. The corresponding fuzzy classi er is named positive de  nite fuzzy classi er (PDFC). A PDFC can be built from the given training samples based on a support vector learning approach with the IFpart fuzzy rules given by the support vectors. Since the learning process minimizes an upper bound on the expected risk (expected prediction error) instead of the empirical risk (training error), the resulting PDFC usually has good generalization. Moreover, because of the sparsity properties of the SVMs, the number of fuzzy rules is irrelevant to the dimension of input space. In this sense, we avoid the "curse of dimensionality." Finally, PDFCs with dierent reference functions are constructed using the su...
The Shape of Fuzzy Sets in Adaptive Function Approximation
, 2001
"... The shape of ifpart fuzzy sets affects how well feedforward fuzzy systems approximate continuous functions. We explore a wide range of candidate ifpart sets and derive supervised learning laws that tune them. Then we test how well the resulting adaptive fuzzy systems approximate a battery of test ..."
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Cited by 27 (3 self)
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The shape of ifpart fuzzy sets affects how well feedforward fuzzy systems approximate continuous functions. We explore a wide range of candidate ifpart 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 functionapproximation 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 ifpart sets into two large classes. The first class consists ofdimensional 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 ifpart 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 ifpart sets consists of the more general but less commondimensional 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 ifpart sets need not suffer from exponential rule explosion but their increased complexity may lead to intractable learning laws and inscrutable ifthen rules. We prove that some of these unfactorable join...