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In this paper, a generalized nonlinear multi-regression model based on Choquet integral (NMRCI) is proposed and applied for estimation of non-additive systems that include outliers under inherent interaction among inputs. The parameters estimation for the proposed model is also performed via a modified algorithm based on particle swarm optimization with quantum-behavior (QPSO), named MQPSO. From the proposed model, the high breakdown regression estimator, a least trimmed squares (LTS) is applied to eliminate the influence caused by these observations that contain outliers. Besides, elitist crossover of genetic algorithm (GA) and adaptive decay of simulated annealing (SA) are used for conquering premature and controlling search policy, respectively. Hence, the proposed MQPSO algorithm which combines the mechanisms of GA, SA and LTS within the QPSO algorithm can deal with the proposed model with outliers. From simulation results, the proposed MQPSO readily corrects the deviation caused by outliers without losing precisions and swiftly achieves convergences on estimating the parameters of the proposed generalized NMRCI model for the non-additive systems with outliers.

Principal Component Analysis (PCA) has long been used as a tool in exploratory data analysis, making predictive models and reducing the size of data. However, many situations in which the use of single-valued variable may cause loss of information. In this paper, an interval PCA is proposed to find out the feature with interval weather data under the season. Besides, interval PCA reveal underlying structure, as well as expose the inside variation of the observations. That is, using weather datasets with different length of time (e.g., month, season) are presented to show each patterns. Finally, the compared results between single-valued and interval-valued weather data illustrates that interval-valued data analysis can display more information.

The paper presents an improved redial basis function network to degrade the influence of the heteroscedasticity noises in the training data. A general purpose learning algorithm is regarded as the statistical nonlinear regression model which is assumed the constant noise level. However, the heteroscedasticity noises always exist in the real data. The transformation based least trimmed squares-support vector regression radial basis function network (LTS-SVR RBFN) employs the Box-Cox transformation and the LTS-SVR to address this problem. From the experiment results, it is evident that our proposed method can be used in a more realistic data.

In general, the inherent interaction among attributes must be considered circumspectly in the study of data mining and information fusion. A nonlinear model with a nonlinear multi-regression model based on the Choquet integral (NMRCI) is suitable for dealing with these problems. However, this NMRCI is an over-determined system and it is difficult to find the analytic solution. Hence, many researchers have proposed many algorithms: namely, the genetic algorithm, the neural network, particle swarm optimization, quantum-behaved particle swarm optimization (QPSO), etc., to estimate the parameters of NMRCI. In this study, a modified QPSO (MQPSO) algorithm, which is used to estimate the parameters of NMRCI, is proposed. That is, the proposed MQPSO algorithm applies the concept of the GA to the QPSO algorithm so that it can improve the convergent speed and conquer the phenomenon of premature. From the simulation results, the proposed MQPSO gives a more precise estimation and faster convergent speed for the estimated parameters of NMRCI.

In this study, an interval competitive agglomeration (ICA) clustering algorithm is proposed to overcome the problems of the unknown clusters number and the initialization of prototypes in the clustering algorithm for the symbolic interval-values data. In the proposed ICA clustering algorithm, both the Euclidean distance measure and the Hausdorff distance measure for the symbolic interval-values data are independently considered. Besides, the advantages of both hierarchical clustering algorithm and partitional clustering algorithm are also incorporated into the ICA clustering algorithm. Hence, the ICA clustering algorithm can be fast converges in a few iterations regardless of the initial number of clusters. Moreover, it is also converges to the same optimal partition regardless of its initialization. Experiments with simply data sets and real data sets show the merits and usefulness of the ICA clustering algorithm for the symbolic interval-values data.

In this article, a Box-Cox transformation-based annealing robust radial basis function networks (ARRBFNs) is proposed for an identification algorithm with outliers. Firstly, a fixed Box-Cox transformation-based ARRBFN model with support vector regression (SVR) is derived to determine the initial structure. Secondly, the results of the SVR are used as the initial structure in the fixed Box-Cox transformation-based ARRBFNs for the identification algorithm with outliers. At the same time, an annealing robust learning algorithm (ARLA) is used as the learning algorithm for the fixed Box-Cox transformation-based ARRBFNs, and applied to adjust the parameters and weights. Hence, the fixed Box-Cox transformation-based ARRBFNs with an ARLA have a fast convergence speed for an identification algorithm with outliers. Finally, the proposed algorithm and its efficacy are demonstrated with an illustrative example in comparison with Box-Cox transformation-based radial basis function networks.

This study proposes a hybrid robust approach for constructing Takagi-Sugeno-Kang (TSK) fuzzy models with outliers. The approach consists of a robust fuzzy C-regression model (RFCRM) clustering algorithm in the coarse-tuning phase and an annealing robust back-propagation (ARBP) learning algorithm in the fine-tuning phase. The RFCRM clustering algorithm is modified from the fuzzy C-regression models (FCRM) clustering algorithm by incorporating a robust mechanism and considering input data distribution and robust similarity measure into the FCRM clustering algorithm. Due to the use of robust mechanisms and the consideration of input data distribution, the fuzzy subspaces and the parameters of functions in the consequent parts are simultaneously identified by the proposed RFCRM clustering algorithm and the obtained model will not be significantly affected by outliers. Furthermore, the robust similarity measure is used in the clustering process to reduce the redundant clusters. Consequently, the RFCRM clustering algorithm can generate a better initialization for the TSK fuzzy models in the coarse-tuning phase. Then, an ARBP algorithm is employed to obtain a more precise model in the fine-tuning phase. From our simulation results, it is clearly evident that the proposed robust TSK fuzzy model approach is superior to existing approaches in learning speed and in approximation accuracy.

When a radial basis function network (RBFN) is used for identification of a nonlinear multi-input multi-output (MIMO) system, the number of hidden layer nodes, the initial parameters of the kernel, and the initial weights of the network must be determined first. For this purpose, a systematic way that integrates the support vector regression (SVR) and the least squares regression (LSR) is proposed to construct the initial structure of the RBFN. The first step of the proposed method is to determine the number of hidden layer nodes and the initial parameters of the kernel by the SVR method. Then the weights of the RBFN are determined by solving a simple minimization problem based on the concept of LSR. After initialization, an annealing robust learning algorithm (ARLA) is then applied to train the RBFN. With the proposed initialization approach, one can find that the designed RBFN has few hidden layer nodes while maintaining good performance. To show the feasibility and superiority of the annealing robust radial basis function networks (ARRBFNs) for identification of MIMO systems, several illustrative examples are included.

The back propagation (BP) algorithm for function approximation is multilayer feed-forward perceptions to learn parameters from sampling data. The BP algorithm uses the least squares method to obtain a set of weights minimizing the object function. One of main issues on the BP algorithm is to deal with data sets having variety of data distributions and bound with noises and outliers. In this paper, in order to overcome the problems of function approximation for a nonlinear system with noise and outliers, a robust fuzzy clustering method is proposed to greatly mitigate the influence of noise and outliers and then a fuzzy-based data sifter is used to partition the nonlinear system's domain into several piecewise linear subspaces to be represented by neural networks. There are two core ideas in the proposed method: (1) The robust fuzzy c-means algorithm is proposed to greatly mitigate the influence of data noise and outliers. (2) A fuzzy-based data sifter is proposed to locate good turning-points to partition a given nonlinear data domain into piecewise clusters so that a neural network can be constructed with fewer rules. Two experiments are illustrated and these results have shown that the proposed approach has good performance in various kinds of data domains with data noise and outliers.

In this paper, a new method, applying the fuzzy logic system, is proposed to discretize the continuous-time controller in computer-controlled systems. All the continuous-time controllers can be reconstructed by the proposed method under the Sampling Theorem. That is, the fuzzy logic systems are used to add nonlinearity and to approximate smooth functions. Hence, the proposed controller is a new smooth controller that can replace the original controller, independent of the sampling time under the Sampling Theorem. Consequently, the proposed controller not only can discretize the continuous-time controllers, but also can tolerate a wider range of sampling time uncertainty. Besides, the input-output stability is proposed for discretizing the continuous-time controller of the fuzzy logic systems. Finally, computer simulation shows that the proposed method can easily reconstruct the continuous-time controller and has very good robustness for different sampling times.

Microarray analysis had been became a widely used tool for disease detection. It used tens of thousands of genes that would be a huge computational problem. The proposed approach applied feature genes selection and classification with support vector machine (SVM) for the microarray data of lung tissue. Based on the proposed approach, feature genes could be finding out according to the epsilon-support vector regression (epsilon-SVR) and selection ranked genes from each class. Moreover, applied multi-class support vector classification (multi-class SVC), cross-validation and parameter search methods to acquire great prediction classification accuracy and less computing time. That is, the effective dimension reduction for finding out feature genes is an important process in the proposed approach. The results shows the feature genes which our purposed approach finding out could acquire great prediction classification accuracy.

Recently, the interval-valued data analysis is popular research topic in symbolic data analysis (SDA). For some of applications, it is natural to use interval-valued data because of uncertainty existing in the measurements, variability for defining a term (the minimum and the maximum temperature during a day), extremely behavior description (maximum wind speed in a given country), etc. However, the obtained data are always subject to outliers in some of applications. Moreover, the outliers may occur due to various reasons, such as erroneous measurements or noisy data from the tail of noise distribution functions. In order to handle the interval-valued data with outliers, a novel approach, called the robust interval support vector interval regression networks (RISVIRNs), is proposed. The RISVIRNs is extended from our previous work (e.g. the support vector interval regression networks; SVIRNs). It is easy to find that SVIRNs can have interval-valued data for outputs, but only can deal with the crisp input. Moreover, the outlier's effects are not discussed in SVIRNs. Hence, the support vector regression with interval-valued for input data (SVRI2) approach is proposed to determine the initial structure of RISVIRNs and to remove outliers form the interval-valued data set. Due to such approach can provide a better initial structure of RISVIRNs, the proposed approach can have fast convergent speed and robust against outliers. The experimental results with real data sets show the validity of the proposed RISVIRNs.

In this paper, the annealing robust radial basis function networks (ARRBFNs) are proposed to improve the problems of the robust radial basis function networks (RBFNs) for function approximation with outliers. Firstly, a support vector regression (SVR) approach is proposed to determine an initial structure of ARRBFNs in this paper. Because an SVR approach is equivalent to solving a linear constrained quadratic programming problem under a fixed structure of SVR, the number of hidden nodes, initial parameters and initial weights of the ARRBFNs are easily obtained. Secondly, the results of SVR are used as the initial structure in ARRBFNs. At the same time, an annealing robust learning algorithm (ARLA) is used as the learning algorithm for ARRBFNs, and applied to adjust the parameters as well as weights of ARRBFNs. That is, an ARLA is proposed to overcome the problems of initialization and the cut-off points in the robust learning algorithm. Hence, when an initial structure of ARRBFNs is determined by an SVR approach, the ARRBFNs with ARLA have fast convergence speed and are robust against outliers. Simulation results are provided to show the validity and applicability of the proposed ARRBFNs.

In this paper, the support vector interval regression networks (SVIRNs) are proposed for the interval regression analysis. The SVIRNs consist of two radial basis function networks. One network identifies the upper side of data interval, and the other network identifies the lower side of data intervals. Because the support vector regression (SVR) approach is equivalent to solving a linear constrained quadratic programming problem, the number of hidden nodes and the initial values of adjustable parameters can be easily obtained. Since the selection of a parameter e in the SVR approach may seriously affect the modeling performance, a two-step approach is proposed to properly select the e value. After the SVR approach with the selected e, an initial structure of SVIRNs can be obtained. Besides, outliers will not significantly affect the upper and lower bound interval obtained through the proposed two-step approach. Consequently, a traditional back-propagation (BP) learning algorithm can be used to adjust the initial structure networks of SVIRNs under training data sets without or with outliers. Due to the better initial structure of SVIRNs are obtained by the SVR approach, the convergence rate of SVIRNs is faster than the conventional networks with BP learning algorithms or with robust BP learning algorithms for interval regression analysis. Four examples are provided to show the validity and applicability of the proposed SVIRNs.

Clustering algorithms have been widely used in many different applications such as pattern recognition, data mining. It is unsupervised learning algorithm. At the same, the data sets of similarity partition belong to the same group; otherwise data sets divide other groups in the clustering algorithms. The interval fuzzy c-means (IFCM) clustering method was proposed to deal with symbolic interval data. However, it still has noisy and outliers problems. Hence, in this paper we propose interval fuzzy possibilistic c-means (IFPCM) clustering algorithm to overcome the IFCM clustering algorithm for the symbolic interval data clustering in noisy and outlier environments under smart phone. From the results of simulation shows that the proposed IFPCM clustering algorithm is implemented on windows mobile (smart) phone and demonstrated nice performance as expected.

We propose a neural gain scheduling network controller (NGSNC) to improve the gain scheduling controller for nonholonomic systems. We derive the neural networks that can approximate the gain scheduling controller arbitrarily well when the sampling frequency satisfies the sampling theorem. We also show that the NGSNC is independent of the sampling time. The proposed NGSNC has the following important properties: 1) same performance as the continuous-parameter gain scheduling controller; 2) less computing time than the continuous-parameter gain scheduling controller; 3) good robustness against the sampling intervals; and 4) straightforward stability analysis. We then show that some of nonholonomic systems can be converted to equivalent linear parameter-varying systems. As a result, the NGSNC can stabilize nonholonomic systems.

The traditional robust learning algorithms are based on the estimated errors, which is not correct in the early stage of the training process. Therefore, the use of those approaches still cannot provide very decent learning performance in face of outliers unless a set of good initial weights is used. In this paper, a novel approach, termed as NRBP (Near set based Robust Back Propagation learning algorithm) is proposed. In this learning algorithm, the training (\estimated) data sets are separated into overlapping (or nonoverlapping) subsets of those data. It uses the set error measure instead of one-step error in robust back propagation based on near set. The set error measure is an estimated error measure between a subset of training data set and corresponding subset of estimated data set. Its benefit is it includes error messages and also reduces the outlier effect.

This paper introduces a new structure of radial basis function networks (RBFNs) that can successfully model symbolic interval-valued data. In the proposed structure, to handle symbolic interval data, the Gaussian functions required in the RBFNs are modified to consider interval distance measure, and the synaptic weights of the RBFNs are replaced by linear interval regression weights. In the linear interval regression weights, the lower and upper bounds of the interval-valued data as well as the center and range of the interval-valued data are considered. In addition, in the proposed approach, two stages of learning mechanisms are proposed. In stage 1, an initial structure (i.e., the number of hidden nodes and the adjustable parameters of radial basis functions) of the proposed structure is obtained by the interval competitive agglomeration clustering algorithm. In stage 2, a gradient-descent kind of learning algorithm is applied to fine-tune the parameters of the radial basis function and the coefficients of the linear interval regression weights. Various experiments are conducted, and the average behavior of the root mean square error and the square of the correlation coefficient in the framework of a Monte Carlo experiment are considered as the performance index. The results clearly show the effectiveness of the proposed structure.

In the literature, some of methods are proposed for the symbolic interval-values data. They have the Centre method (CM), the MinMax method and the Centre and Range (CR) method. For the above methods, they needs solve the inverse of matrix. However, the dimension of this matrix is increased as increasing the variables. Moreover, the condition number of this matrix is large, it may cause a large error of solution (i.e. ill-conditioned). The above methods do not guarantee that the predicted values of the lower bounds will be lower than the predicted values of the upper bounds. To overcome the above problems, the particle swarm optimization (PSO) algorithm is applied to estimate the parameters of linear regression models. From the simulation results, the proposed method can provide satisfactory results.

In this paper, the hybrid support vector machines (SVM) and Gaussian process (GPs) are proposed to modeling of mitogen-activated protein kinases systems with noise. In the proposed approach, there are two-stage strategies. In stage 1, the support vector machine regression (SVMR) approach is used to filter out the some larger data set in the mitogen-activated protein kinases systems data set with noise. Because of the larger noise data in the training data set are almost removed, the large noise data's effects are reduce, so the concepts of robust statistic theory are not used to reduce the large noise data's effects. The rest of the training data set after stage 1 is directly used to training the Gaussian process for regression (GPR) in stage 2. According to the simulation results, the performance of the proposed approach is superior to the least squares support vector machines for regression, and GPR when the noise is existed in the mitogen-activated protein kinases systems.

In this study, a hybrid robust LS-SVMR approach is proposed to deal with training data sets with outliers dor MIMO system. The proposed approach consists of two stages of strategies. The first stage is for data preprocessing and a support vector regression is used to filter out outliers. Then, the training data set except for outliers, called as the reduced training data set, is directly used in training the non-robust least squares support vector machines for regression (LS-SVMR) for MIMO system in the second stage. Consequently, the learning mechanism of the proposed approach is much easier than the weighted LS-SVMR approach. Based on the simulation results, the performance of the proposed approach with non-robust LS-SVMR is superior to the weighted LS-SVMR approach for MIMO system when the outliers exist.

In this paper, the annealing robust neural fuzzy networks (ARNFNs) are proposed to improve the problems of neural fuzzy networks for the modeling of mitogen-activated protein kinases (MAPK) systems with outliers. Firstly, the support vector regression (SVR) approach is proposed to determine the initial structure of ARNFNs for the modeling of the MAPK systems with outliers.Because of a SVR approach is equivalent to solving a linear constrained quadratic programming problem under a fixed structure of SVR, the number of hidden nodes, the initial parameters and the initial weights of ARNFNs are easy obtained via the SVR approach. Secondly, the results of SVR are used as initial structure in ARNFNs for the modeling of the MAPK systems with outliers. At the same time, an annealing robust learning algorithm (ARLA) is used as the learning algorithm for ARNFNs, and applied to adjust the parameters in the membership function as well as weights of ARNFNs. Hence, when an initial structure of ARNFNs are determined by a SVR approach, the ARNFNs with ARLA have fast convergence speed for the modeling of the MAPK systems with outliers.

In this study, the concepts of competitive agglomeration clustering algorithm is incorporated into fuzzy c-means (FCM) clustering algorithm for symbolic interval-values data. In the proposed approach, called as IFCMwUNC clustering algorithm, the problems of the unknown clusters number and the initialization of prototypes in the FCM clustering algorithm for symbolic interval-values data are overcome and discussed. Due to the competitive agglomeration clustering algorithm possess the advantages of the hierarchical clustering algorithm and the partitional clustering algorithm, IFCMwUNC clustering algorithm can be fast converges in a few iterations regardless of the initial number of clusters. Moreover, it is also converges to the same optimal partition regardless of its initialization. Experiments results show the merits and usefulness of IFCMwUNC clustering algorithm for the symbolic interval-values data.

In this paper, we propose the approximate transformable technique, which includes the direct transformation and indirect transformation, to obtain a Chebyshev-Polynomials-Based (CPB) unified model neural networks for feedforward/recurrent neural networks via Chebyshev polynomials approximation. Based on this approximate transformable technique, we have derived the relationship between the single-layer neural networks and multilayer perceptron neural networks. It is shown that the CPB unified model neural networks can be represented as a functional link networks that are based on Chebyshev polynomials, and those networks use the recursive least square method with forgetting factor as learning algorithm. It turns out that the CPB unified model neural networks not only has the same capability of universal approximator, but also has faster learning speed than conventional feedforward/recurrent neural networks. Furthermore, we have also derived the condition such that the unified model generating by Chebyshev polynomials is optimal in the sense of error least square approximation in the single variable ease. Computer simulations show that the proposed method does have the capability of universal approximator in some functional approximation with considerable reduction in learning time.

In general, the support vector regression (SVR) is very suitable to approximate a high dimensionality space and ill-posed problem in modeling. That is, the SVR consists of a quadratic programming problem that can be solved efficiently and guaranteed to find a global extremism. Therefore, for the complex data, the SVR is easy to reconstruct an approximated model based on the linear programming technique. On the other hand, a typical microarray data consists of expression levels for a large number of genes on a relatively small number of samples. In order to avoid higher computational complexity and larger prediction errors on high-dimensional problem, we proposed the dimension reduction with SVR for the ovarian cancer microarray data. The SVR can reduce dimension on each sample from 9600 genes to about three hundreds genes. Besides, we can choose the epsiv value in the loss-function of SVR to obtain the variable number of gene and the proposed method can also overcome the block effect of microarray data. Finally, these results can provide for gene class discovery and gene class prediction

Recently, the support vector machine (SVM) that is a new learning methodology based on Vapnik Chervonenkis (VC) theory is proposed. With introducing Vapnik's epsiv-insensitive loss function, the SVM has been extended to solve a nonlinear regression estimation problem, called the support vector regression (SVR), which has been shown to exhibit excellent performance. Due to its good properties, the SVR are successfully applied to the various applications. In this paper, the SVR applies to approximate controller in the inverted pendulum system. That is, the controller can be reconstructed by the proposed method. Hence, the proposed controller is a new controller that can replace original controller. Additionally, the selection of parameters of SVR are also discussed and analysis for this application. Simulation results are provided to show the validity and applicability of the proposed approach.

In this paper, a new method, applied the annealing robust Walsh function networks, is proposed to discretize the continuous-time controller in computer-controlled systems. That is, the annealing robust Walsh function networks are used to add nonlinearly and to approximate smooth controller with digital neural networks. Hence, the proposed controller is a new smooth controller that can replace original controller and independent of the sampling time under the Sample Theorem. Besides, the input-output stability is proposed for this discretizating continuous-time controller with the annealing robust Walsh function networks. Consequently, the proposed annealing robust Walsh function networks controller cannot only discretize the continuous-time controllers, but can also tolerate a wider range of sampling time uncertainty.

The design of robust adaptive tracking control for MIMO unknown nonlinear systems, which is the combination of the approximation method of Chebyshev polynomial based unified model (CPBUM) neural network, adaptive control algorithm and H/sup /spl infin// control technique, have been successfully proposed. The proposed robust adaptive tracking control structure uses the good approximation ability, fast learning speed, and systematic design method of CPBUM neural network in controller with rough tuning and the H/sup /spl infin// disturbance attenuation technique in fine tuning. Hence, this proposed structure has good adaptability and robustness for the close-loop system. Finally, a two-degree of freedom robotic manipulator is given to demonstrate the validity of proposed method.

The TSK type of fuzzy models has attracted a great attention of the fuzzy modeling community due to their good performance in various applications. In the literature, some approaches for modeling TSK fuzzy rules have been proposed. Most of them define their fuzzy subspaces based on the idea of training data being close enough instead of having similar functions. Recently, a new approach, fuzzy c-regression model (FCRM) clustering algorithm, is proposed to construct TSK fuzzy models. However, this approach does not take into account the training data with outliers. In order to overcome the effects of outliers, a robust TSK fuzzy modeling with outliers has been proposed. It is worth noting that this approach may need more computation time due to complicated formulas. Hence, a novel TSK fuzzy modeling approach with outliers is presented in this paper. In this approach, robust fuzzy regression (RFR) clustering algorithm is proposed to simultaneously define fuzzy subspaces and find the parameters in the consequent parts of TSK rules. In the clustering process, the similarity measure is used to reduce the redundant rules. To obtain a more precision model that is not affected by outliers, an annealing robust BP learning algorithm is employed. From the simulation results, the proposed TSK fuzzy model approach indeed showed superior performance.

In this paper, the annealing robust fuzzy basis function (ARFBF) is proposed to improve the problems of the fuzzy basis function for modeling with outliers. Firstly, the repeated support vector regression (RSVR) approach is proposed to determine the initial structure of ARFBF in this paper. Because of a RSVR approach is equivalent to solving twice linear constrained quadratic programming problem under a fixed structure of SVR, the number of hidden nodes, initial parameters and initial weights of the ARFBF are easily obtained in the second SVR. Secondly, the results of RSVR are used as initial structure in ARFBF. At the same time, an annealing robust learning algorithm (ARLA) is used as the learning algorithm for ARFBF. That is, an ARLA is proposed to overcome the problems of initialization and the cut-off points in the robust learning algorithm. Hence, when an initial structure of ARFBF is determined by a RSVR approach, the ARFBF with ARLA have fast convergence speed and robust against outliers. Simulation results are provided to show the validity and applicability of the proposed ARFBF.

A new support vector regression (SVR) algorithm based on the statistical method is proposed to select an initial structure for the certain neural networks and fuzzy neural networks. There are repeated SVR methods to use in the proposed method. Because the SVR approach is equivalent to solve a linear constrained quadratic programming problem the number of hidden nodes and adjustable parameters in the proposed networks are easily obtained. Based on the better initial structure for the proposed networks, the convergence epoch of the proposed networks are faster than the conventional neural networks and fuzzy neural networks with BP learning algorithm and robust BP learning algorithm. Simulation results are provided to show the validity and applicability of the proposed method.

Various robust learning algorithms have been proposed in the literature to overcome the effects of outliers. In those learning algorithms, the robust learning algorithms are divided into two categories. One is directly adopts the theory of the M-estimator. The other is the least mean log square learning algorithm in which errors are assumed to be the Cauchy distribution. However, outliers not only affect the fatness, but also may slightly affect the skewness of the distribution. In the paper, the errors distribution is assumed to belong to the generalized exponential family and an adaptive robust learning algorithm is then proposed. The idea of our approach is to adaptively construct the influence function based on the current observed error distribution. Since the kurtosis can be used to measure the fatness of a distribution, it is employed for identifying a coefficient in the generalized exponential distribution family through the moment recursive relations. The proposed algorithm has been employed under various noise distributions and the results have demonstrated superiority over other robust learning algorithms.

In this paper, we propose an approximate equivalence neural network model with a fast learning speed as well as a good function approximation capability, and a new objective function, which satisfies the H/sup /spl infin// induced norm to solve the worst-case identification and control of nonlinear problems. The approximate equivalence neural network not only has the same capability of universal approximator, but also has a faster learning speed than the conventional feedforward/recurrent neural networks. Based on this approximate transformable technique, the relationship between the single-layered neural network and multilayered perceptrons neural network is derived. It is shown that a approximate equivalence neural network can be represented as a functional link network that is based on Chebyshev polynomials. We also derive a new learning algorithm such that the infinity norm of the transfer function from the input to the output is under a prescribed level. It turns out that the approximate equivalence neural network can be extended to do the worst-case problem, in the identification and control of nonlinear problems.

We propose support vector machines (SVM) to improve the simplified fuzzy inference system for the fuzzy neural network. Firstly, we apply SVM to determine the number of simplified fuzzy inference system rules. Because training a SVM is equivalent to solving a linear constrained quadratic programming problem under a fixed structure of SVM, we can easily determine the number of simplified fuzzy inference system rules. Secondly, we use the solution of the SVM as initial weights in fuzzy neural networks. Based on these initial weights, the fuzzy neural networks have a fast convergent speed. Finally, we derive a learning algorithm for the proposed structure and apply the proposed method to approximate a nonlinear function. Simulation results are provided to show the validity and applicability of the developed method.

We apply the Walsh function to approximate the linear and nonlinear consequent parts of a fuzzy model. For the digital implementation of the fuzzy model, we use the Walsh function to approximate each fuzzy rule under the proposed structure. That is, the consequent part of the fuzzy model can be approximated via the Walsh function. It turns out that the proposed approximated fuzzy model could be realized via some digital logical devices. We use the recursive least squares method with a forgetting factor to tune the coefficient of the proposed method. Hence, the proposed method not only can implement the fuzzy model via digital logical devices, but also has fast computation speed.

A fuzzy controller is proposed to improve the drawbacks of fuzzy modeling via universal regulators for the incomplete information on the parameters of the considered nonlinear systems. In general, in a fuzzy controller, for fuzzy modeling it is assumed that the information of nonlinear systems is complete in the consequent part of each rule. When the information is incomplete, it is difficult to design the fuzzy controller in fuzzy modeling. We propose a fuzzy modeling structure with unknown parameter. That is, the proposed structure can deal with incomplete information systems. Next, the universal regulator is used to design the fuzzy controller and stabilize the fuzzy modeling. It turns out that in such problems the optimal fuzzy regulator is unique. As a result, a special optimal fuzzy regulator can be found that does not depend on unknown parameters.

We apply input-output pseudo-linearization and Walsh function to improve and implement the proposed fuzzy modeling. For the improvement of fuzzy modeling, we employ the input-output pseudo-linearization to extend its applications. Hence, the proposed method, independent of the operating point, can reduce the number of fuzzy rule in the approximation of a nonlinear system. It turns out that the proposed method is more effective and simpler in the design of the number of rule than conventional linearization that is based on operating point. For the implement of fuzzy modeling, we use Walsh function to approximate each fuzzy rule under the proposed fuzzy system. Therefore, this proposed fuzzy modeling could be realized via some digital logical devices.

In this paper, an annealing robust fuzzy basis function (ARFBF) is proposed to improve the problems of the fuzzy basis function (FBF) for modelling with noise and outliers. Firstly, the repeated support vector regression (RSVR) approach is proposed to determine the initial structure of ARFBF. Because a RSVR approach is equivalent to solving twice linear constrained quadratic programming problem under a fixed structure of SVR, the number of hidden nodes, the initial parameters and the initial weights of ARFBF are easily obtained in the RSVR. Secondly, the results of RSVR are used as initial structure in the ARFBF. At the same time, an annealing robust learning algorithm (ARLA) is used as the learning algorithm for ARFBF, and applied to adjust the parameters as well as weights of ARFBF. That is, an ARLA is proposed to overcome the problems of initialisation and the cutoff points in the robust learning algorithm. Hence, when an initial structure of ARFBF is determined by a RSVR approach, the ARFBF with ARLA has faster convergence speed and is robust against outliers. Simulation results are provided to show the validity and applicability of the proposed ARFBF.