Abstract: A technique for approximating a continuous function of n variables with a radial basis function (RBF) neural network is presented. The method uses an n-dimensional raised-cosine type of RBF that is smooth, yet has compact support. The RBF network coefficients are low-order polynomial functions of the input. A simple computational procedure is presented which significantly reduces the network training and evaluation time. Storage space is also reduced by allowing for a nonuniform grid of points about which the RBFs are centered. The network output is shown to be continuous and have a continuous first derivative. When the network is used to approximate a nonlinear dynamic system, the resulting system is bounded-input bounded-output stable. For the special case of a linear system, the RBF network representation is exact on the domain over which it is defined, and it is optimal in terms of the number of distinct storage parameters required. Several examples are presented which illustrate the effectiveness of this technique.

Abstract: River flow forecasting is required to provide basic information on a wide range of problems related to the design and operation of river systems. The availability of extended records of rainfall and other climatic data, which could be used to obtain stream flow data, initiated the practice of rainfall-runoff modelling. While conceptual or physically-based models are of importance in the understanding of hydrological processes, there are many practical situations where the main concern is with making accurate predictions at specific locations. In such situation it is preferred to implement a simple “black box” (data-driven, or machine learning) model to identify a direct mapping between the inputs and outputs without detailed consideration of the internal structure of the physical process. Artificial neural networks (ANNs) is probably the most successful machine learning technique with flexible mathematical structure which is capable of identifying complex non-linear relationships between input and output data without attempting to reach understanding as to the nature of the phenomena. In this study the applicability of ANNs for downstream flow forecasting in the Apure river basin (Venezuela) was investigated. Two types of ANN architectures, namely multi-layer perceptron network (MLP) and a radial basis function network (RBF) were implemented. The performances of these networks were compared with a conceptual rainfall-runoff model and they were found to be slightly better for this river flow-forecasting problem.

Abstract: Radial basis functions are well-known and successful tools for the interpolation of data in many dimensions. Several radial basis functions of compact support that give rise to nonsingular interpolation problems have been proposed, and in this paper we study a new, larger class of smooth radial functions of compact support which contains other compactly supported ones that were proposed earlier in the literature.

Abstract: The paper presents a new approach for the protection of power transmission lines using a minimal radial basis function neural network (MRBFNN). This type of RBF neural network uses a sequential learning procedure to determine the optimum number of neurons in the hidden layer without resorting to trial and error. The input data to this network comprises fundamental peak values of relaying point voltage and current signals, the zero-sequence component of current and system operating frequency. These input variables are obtained by a Kalman filtering approach. Further, the parameters of the network are adjusted using a variant of extended Kalman filter known as locally iterated Kalman filter to produce better accuracy in the output for harmonics, DC offset and noise in the input data. The number of training patterns and the training time are drastically reduced and significant accuracy is achieved in different types of fault classification and location in transmission lines using computer simulated tests.

Abstract: This chapter presents a broad overview of Radial Basis Function Networks (RBFNs), and facilitates an understanding of their properties by using concepts from approximation theory, catastrophy theory and statistical pattern recognition. While this chapter is aimed to provide an adequate theoretical background for the subsequent application oriented chapters in this book, it also covers several aspects with immediate practical implications: alternative ways of training RBFNs, how to obtain an appropriate network size for a given problem, and the impact of the resolution (width) of the radial basis functions on the solution obtained. Some prominent applications of RBFNs are also outlined.

Abstract: Fully Tuned Radial Basis Function Neural Networks for Flight Control presents the use of the Radial Basis Function (RBF) neural networks for adaptive control of nonlinear systems with emphasis on flight control applications A Lyapunov synthesis approach is used to derive the tuning rules for the RBF controller parameters in order to guarantee the stability of the closed loop system Unlike previous methods that tune only the weights of the RBF network, this book presents the derivation of the tuning law for tuning the centers, widths, and weights of the RBF network, and compares the results with existing algorithms It also includes a detailed review of system identification, including indirect and direct adaptive control of nonlinear systems using neural networks

Abstract: In this article we discuss artificial neural networks-based fault detection and isolation (FDI) applications for robotic manipulators. The artificial neural networks (ANNs) are used for both residual generation and residual analysis. A multilayer perceptron (MLP) is employed to reproduce the dynamics of the robotic manipulator. Its outputs are compared with actual position and velocity measurements, generating the so-called residual vector. The residuals, when properly analyzed, provides an indication of the status of the robot (normal or faulty operation). Three ANNs architectures are employed in the residual analysis. The first is a radial basis function network (RBFN) which uses the residuals of position and velocity to perform fault identification. The second is again an RBFN, except that it uses only the velocity residuals. The third is an MLP which also performs fault identification utilizing only the velocity residuals. The MLP is trained with the classical back-propagation algorithm and the RBFN is trained with a Kohonen self-organizing map (KSOM). We validate the concepts discussed in a thorough simulation study of a Puma 560 and with experimental results with a 3-joint planar manipulator. © 2001 John Wiley & Sons, Inc.

Abstract: A review of interpolation to values of a function f(x), x 2 R d , by radial basis function methods is given. It addresses the nonsingularity of the interpolation equations, the inclusion of polynomial terms, and the accuracy of the approximation sf, where s is the interpolant. Then some numerical experiments investigate the situation when the data points are on a low dimensional nonlinear manifold in R d. They suggest that the number of data points that are necessary for good accuracy on the manifold is independent of d, even if d is very large. The experiments employ linear and multiquadric radial functions, because an iterative algorithm for these cases was developed at Cambridge recently. The algorithm is described brieey. Fortunately, the number of iterations is small when the data points are on a low dimensional manifold. We expect these ndings to be useful, because the manifold setting for large d is similar to typical distributions of interpolation points in data mining applications.

Abstract: In this paper, a classification method combined with a simple regression model is presented to predict the discrete PDF of power market clearing prices (MCP), which is critical for much decision-making such as optimizing bidding strategies, and is very difficult to predict because of high market uncertainties. Our basic idea is to form a number of classes by discretizing the variable to be predicted, and thus convert the time series prediction problem into a pattern classification problem. The classes are clustered based on historical data, and the input statistics (e.g., input mean and covariance matrix, etc.) corresponding to each class are obtained via clustering. Given a new input, these classes are sorted according to the posterior probabilities calculated by using the Bayes' formula. The prediction is then generated by using an auto-regression method (AR) for the classes with high posterior probabilities. The classes are then updated after the actual output is available. The method developed can be seen as an RBF with variable structure. The posterior probabilities can be viewed as an approximation of the discrete PDF, providing valuable information for decision-makers and for what if analysis. The developed method has been used to predict the on-peak and off-peak average MCPs in the New England market. The testing results show that it outperforms the RBF network in terms of prediction accuracy and computational time.

Abstract: A e ight control scheme in which a radial basis function network (RBFN)aids a conventional controller has been developed. The RBFN controller, consisting of variable Gaussian functions, uses only online learning to represent the local inverse dynamics of the aircraft system. With a Lyapunov synthesis approach, a tuning rule for updating all of the parameters of the RBFN (including centers, widths, as well as the weights of the output layer ) is derived, which extends Gomi and Kawato’ s strategy, where only the weights were adaptable. (Gomi, H., and Kawato, M., “ Neural Network Control for a Closed-Loop System Using Feedback-Error Learning,” Neural Networks , Vol. 6, No. 7, 1993, pp. 933 ‐946). The proposed tuning rule guarantees the convergence of the overall system and greatly improves the tracking accuracy. Simulation studies using an F8 aircraft longitudinal model illustrate the superior performance of the proposed scheme. The simulation studies further indicate that the results can be extended to a dynamic RBFN in which the hidden neurons can be added /pruned, thus producing a more compact network structure.

Abstract: This paper describes how artificial neural networks can be used to classify multivariate data. Two types of neural networks were applied: a counter propagation neural network (CP-ANN) and a radial basis function network (RBFN). These strategies were used to classify soil samples from different geographical regions in Brazil by means of their near-infrared (diffuse reflectance) spectra. The results were better with CP-ANN (classification error 8.6%) than with RBFN (classification error 11.0%).

Abstract: In this paper, the problem of time series prediction is studied. A Bayesian procedure based on Gaussian process models is proposed and compared to the radial basis function networks. In our experiments, Gaussian process models show excellent prediction. The conceptual simplicity, and good performance of Gaussian process models should make them very attractive for a wide range of problems.

Abstract: This paper proposes an automated facial expression recognition system using neural network classifiers. First, we use the rough contour estimation routine, mathematical morphology, and point contour detection method to extract the precise contours of the eyebrows, eyes, and mouth of a face image. Then we define 30 facial characteristic points to describe the position and shape of these three facial features. Facial expressions can be described by combining different action units, which are specified by the basic muscle movements of a human face. We choose six main action units, composed of facial characteristic point movements, as the input vectors of two different neural network‐based expression classifiers including a radial basis function network and a multilayer perceptron network. Using these two networks, we have obtained recognition rates as high as 92.1% in categorizing the facial expressions neutral, anger, or happiness. Simulation results by the computer demonstrate that computers are c...

Abstract: Neural models for calculating the resonant frequency of electrically thin and thick rectangular microstrip antennas, based on the multilayered perceptrons and the radial basis function networks, are presented. Six learning algorithms, backpropagation, delta-bar-delta, extended-delta-bar-delta, quick-propagation, directed random search and genetic algorithms, are used to train the multilayered perceptrons. The radial basis function network is trained according to its learning strategy. The reason for using six different learning algorithms and two different structures is to speed up the training time and to compare the performance of neural models for this specific application. The resonant frequency results obtained by using neural models are in very good agreement with the experimental results available in the literature. When the performances of neural models are compared with each other, the best results for training and test were obtained from the radial basis function network

Abstract: The appropriate operation of a radial basis function (RBF) neural network depends mainly upon an adequate choice of the parameters of its basis functions. The simplest approach to train an RBF network is to assume fixed radial basis functions defining the activation of the hidden units. Once the RBF parameters are fixed, the optimal set of output weights can be determined straightforwardly by using a linear least squares algorithm, which generally means reduction in the learning time as compared to the determination of all RBF network parameters using supervised learning. The main drawback of this strategy is the requirement of an efficient algorithm to determine the number, position, and dispersion of the RBFs. The approach proposed here is inspired by models derived from the vertebrate immune system, that will be shown to perform unsupervised cluster analysis. The algorithm is introduced and its performance is compared to that of the random, k-means center selection procedures and other results from the literature. By automatically defining the number of RBF centers, their positions and dispersions, the proposed method leads to parsimonious solutions. Simulation results are reported concerning regression and classification problems.

Abstract: In content-based image retrieval (CBIR), content of an image can be expressed in terms of different features such as color, texture, shape, or text annotations. Retrieval based on these features can be various by the way how to combine the feature values. Most of the existing approaches assume a linear relationship between different features, and the usefulness of such systems was limited due to the difficulty in representing high-level concepts using low-level features. In this paper, we introduce Neural Network-based Flexible Image Retrieval (NNFIR) system, a human-computer interaction approach to CBIR using Radial Basis Function (RBF) network to combine the values of the heterogeneous features. By using the RBF network, this approach determines nonlinear relationship between features so that more accurate similarity comparison between images can be supported. The experimental results show that the proposed approach has the superior retrieval performance than the existing linear combining method, the rank-based method and the BackPropagation-based method. Although the proposed retrieval model is for CBIR, it can be easily expanded to handle other media types, such as video and audio.

Abstract: When two or more robotic manipulators are working cooperatively, faults can put at risk the task, the robots, or the manipulated load. In this work, two artificial neural networks are employed in a fault detection and isolation system for cooperative robotic manipulators. A multilayer perceptron is utilized to reproduce the dynamics of the cooperative system The difference between its outputs and the actual velocity measurements generates the residual vector. This vector is classified by a radial basis function network that produces the fault information. Simulations with two robotic manipulators performing a cooperative task are presented, indicating that free-swinging joint faults are correctly detected and isolated. The main contribution of this work is the first application of fault detection and isolation to cooperative manipulators with faults at the robots' joints.

Abstract: In many practical engineering design problems, the form of objective function is not given explicitly in terms of design variables. Given the value of design variables, under this circumstance, the value of objective function is obtained by some analysis such as structural analysis, fluidmechanic analysis, thermodynamic analysis, and so on. Usually, these analyses are considerably time consuming to obtain a value of objective function. In order to make the number of analyses as few as possible, we suggest a method by which optimization is performed in parallel with predicting the form of objective function. In this paper, radial basis function networks (RBFN) are employed in predicting the form of objective function, and genetic algorithms (GA) in searching the optimal value of the predicted objective function. The effectiveness of the suggested method will be shown through some numerical examples.

Abstract: This paper presents the application of the recently developed Minimal Resource Allocating Network (MRAN) for aircraft flight control, with special emphasis on its robustness and fault tolerance properties. MRAN is a dynamic Radial Basis Function network (RBFN) incorporating a growing and pruning strategy resulting in a compact network structure. For the aircraft control application presented here, a simple scheme in which MRAN aids a conventional controller using a feedback error learning mechanism is presented. The robustness and the fault tolerant nature of the neuro controller is illustrated using a F8 fighter aircraft model in an autopilot mode. The objective of the controller is to follow the velocity and pitch rate pilot commands under large parameter variations and sudden changes in actuator time constants. Simulation results demonstrate the satisfactory performance of the MRAN neuro-flight controller even under these faulty conditions.

Abstract: A method for driving the dynamics of a nonlinear system to a sliding mode is discussed. The approach is based on a sliding mode control methodology, i.e., the system under control is driven towards a sliding mode by tuning the parameters of the controller. In this loop, the parameters of the controller are adjusted such that a zero learning error level is reached in one dimensional phase space defined on the output of the controller. A Gaussian radial basis function neural network is used as the controller.

Abstract: An original application of a special type of neural network called radial basis function network toward systematization of neural network forecaster design is made. The network itself is used to select its input variables and parameters. The network has a characteristic of convergence to the lowest possible training error for a given set of network parameters and input variables. The advantage of this network lies in selection of input variables on the basis of network performance, and this selection includes the load time series and weather variables. The training of the network is considerably fast and does not need monitoring of training process for nonconvergence and parameter tuning during design and testing. The design method is tested for short-term load forecast on hourly basis for one of the systems, and excellent results were obtained. The comparison with existing feed forward neural network forecaster shows that the proposed forecaster outperforms the former.

Abstract: Proposes a radial basis function network (RBFN) approach to the solution of the tracking problem for mobile robots. RBFN-based controllers are investigated in order to introduce some degree of robustness in the control system and to avoid the main disadvantage of multilayer neural networks (MNN) to be highly nonlinear in the parameters. The training of the nets and the control performances analysis have been done in a real experimental setup. The proposed solutions are implemented on a PC-based control architecture for the real-time control of the LabMate mobile base and are compared with MNN-based control schemes. The experimental results are satisfactory in terms of tracking errors and computational efforts.