Abstract: It is generally recognized that digital channel equalization can be interpreted as a problem of nonlinear classification. Networks capable of approximating nonlinear mappings can be quite useful in such applications. The radial basis function network (RBFN) is one such network. We consider an extension of the RBFN for complex-valued signals (the complex RBFN or CRBFN). We also propose a stochastic-gradient (SG) training algorithm that adapts all free parameters of the network. We then consider the problem of equalization of complex nonlinear channels using the CRBFN as part of an equalizer. Results of simulations we have carried out show that the CRBFN with the SG algorithm can be quite effective in channel equalization. >

Abstract: A description and original application of a type of neural network, called the radial basis function network (RBFN), to the short-term system load forecasting (SLF) problem is presented. The predictive capability of the RBFN models and their ability to produce accurate measures that can be used to estimate confidence intervals for the short-term forecasts are illustrated, and an indication of the reliability of the calculations is given. Performance results are given for daily peak and total load forecasts for one year using data from a large-scale power system. A comparison between results from the RBFN model and the back-propagation neural network are also presented.

Abstract: Adaptive neural network processing of phased-array antenna received signals promises to decrease antenna manufacturing and maintenance costs while increasing mission uptime and performance between repair actions. We introduce one such neural network which performs aspects of digital beamforming with imperfectly manufactured, degraded, or failed antenna components. This paper presents measured results achieved with an adaptive radial basis function (ARBF) artificial neural network architecture which learned the single source direction finding (DF) function of an eight-element X-band array having multiple, unknown failures and degradations. We compare the single source DF performance of this ARBF neural network, whose internal weights are computed using a modified gradient descent algorithm, with another radial basis function network, Linnet, whose weights are calculated using linear algebra. Both networks are compared to a traditional DF approach using monopulse.

Abstract: Approximation by radial basis functions with “quasi-uniformly” distributed centres inR d is discussed. A construction of new polynomially decaying functions that span the approximation space is presented and the properties of the quasi-interpolation operator with these functions are investigated. It is shown that the quasi-interpolant reproduces polynomials and gives approximation orders identical to those in the uniform square-grid case.

Abstract: This paper presents a radial basis function neural network which is trained to learn the dynamics of nonlinear autonomous systems. Contrary to conventional approaches, not only the output layer weights, but also the other parameters of the RBF network are trained using the extended Kalman filter algorithm. The advantages over conventional methods are that centers and variances of the hidden layer nodes need not be calculated before the optimum output weight matrix is determined, and that on-line training is possible. Due to a suitable factorization of the Riccati divergence equation as contained in the Kalman filter, the algorithm can be implemented local to the nodes in the network, and a matrix inversion replaced by simple divisions, thereby significantly reducing the computational complexity. Finally, the network is applied to the task of learning the dynamics of speech signals obtained from sustained vowels, and subsequently used to re-synthesize these vowels autonomously.

Abstract: The two-layer radial basis function network, with fixed centers of the basis functions, is analyzed within a stochastic training paradigm. Various definitions of generalization error are considered, and two such definitions are employed in deriving generic learning curves and generalization properties, both with and without a weight decay term. The generalization error is shown analytically to be related to the evidence and, via the evidence, to the prediction error and free energy. The generalization behavior is explored; the generic learning curve is found to be inversely proportional to the number of training pairs presented. Optimization of training is considered by minimizing the generalization error with respect to the free parameters of the training algorithms. Finally, the effect of the joint activations between hidden-layer units is examined and shown to speed training.

Abstract: The paper describes an improved version of the Radial Basis Function algorithm, which integrates the advantages of Multi-Layer Perceptrons and Radial Basis Functions alone. The proposed paradigm is more general in nature, since it has the other two as particular subcases. It finds applications in several pattern recognition and classification tasks. Furthermore it can also be used as a method to map Fuzzy Inference Systems on Artificial Neural Networks.

Abstract: This paper compares radial basis function networks for identification of nonlinear dynamic systems with classical methods derived from the Volterra series. The performance of these different approaches, such as Hammerstein, Wiener and NDE models, is analysed. Since the centres and variances of the Gaussian radial basis functions will be fixed before learning and only the weights are learned, a linear optimization problem arises. Therefore training the network and parameter estimation becomes comparable in computational effort. It is shown that the classical methods can compete or even perform better than the neural network, if the assumptions for the structure are valid. However, in practical applications when the structure is not known the radial basis function network performs much better than the classical methods.

Abstract: Recognizing the sex of conspecifics is important. Humans rely primarily on visual pattern recognition for this task. A wide variety of linear and nonlinear models have been developed to understand this task of sex recognition from human faces.' These models have used both pixelbased and feature-based representations of the face for input. Fleming and Cottrell (1990) and Golomb et a!. (1991) utilized first an autoencoder compression network on a pixel-based representation, and then a classification network. Brunelli and Poggio (1993) used a type of radial basis function network with geometrical face measurements as input. O'Toole and colleagues (1991, 1993) represented faces as principal components. When the hidden units of an autoencoder have a linear output function, then the N hidden units in the network span the first N principal components of the input (Baldi and Hornik 1989). Bruce et al. (1993) constructed a discriminant function for sex with 2-D and 3-D facial measures. In this note we compare the performance of a simple perceptron and a standard multilayer perceptron (MLP) on the sex classification task. We used a range of spatial resolutions of the face to determine how the reliability of sex discrimination is related to resolution. A normalized pixel-based representation was used for the faces because it explicitly retained texture and shape information while also maintaining geometric relationships. We found that the linear perceptron model can classify sex from facial images with 81% accuracy, compared to 92% accuracy with compression coding on the same data set (Golomb et al. 1991). The advantage of using a simple linear perceptron with normalized pixelbased inputs is that it allows us to see explicitly those regions of the face

Abstract: This paper describes an experimental implementation of a novel paradigm for a model-free design of the trajectory tracking controller. The design is based on a nonlinear approximation of the feedforward dependence on control task parameters. These task parameters comprise initial and final set points of the system and define the trajectory to be tracked. As an approximation method, we use a radial basis function network. The initial feedforward data for the approximation are obtained by performing learning control iterations for a number of selected task parameter values. In our experiments with a direct-drive industrial robot AdeptOne, high performance of the designed approximation-based controller is achieved despite strongly nonlinear system dynamics and large Coulomb-friction. The obtained results open an avenue for industrial applications of the developed approach in robotics and elsewhere.

Abstract: This paper is concerned with the types of invariance exhibited by Radial Basis Function (RBF) neural networks when used for human face classification, and the generalisation abilities arising from this behaviour. Experiments using face images in ranges from face-on to profile show the RBF network's invariance to 2-D shift, scale and y-axis rotation. Finally, the suitability of RBF techniques for future, more automated face classification purposes is discussed.

Abstract: The paper investigates the application of inversion of a radial basis function network (RBFN) to nonlinear control problems for which the structure of the nonlinearity is unknown. Initially, the RBF network is trained to learn the forward dynamics of the plant. Two different controller structures are then proposed based on this identified RBFN model. In one scheme, a feedback control law is derived based on the input prediction by inversion of the RBFN model so that the system is Lyapunov stable. The second kind of controller structure predicts the feedforward control action, while the fixed controller actuates the feedback stabilising signal. An extended Kalman filtering based algorithm is employed to carry out the network inversion during each sampling interval. Two examples are presented to verify the proposed scheme. Simulation results show that the performance of the controller based on the proposed network inversion scheme is efficient.

Abstract: : The paper studies L-infinity (IR d)-norm approximations from a space spanned by a discrete set of translates of a basis function theta. Attention here is restricted to functions theta whose Fourier transform is smooth on IRd/ 0, and has a singularity at the origin. Examples of such basis functions are the thin-plate splines and the multiquadrics, as well as other types of radial basis functions that are employed in Approximation Theory. The above approximation problem is well-understood in case the set of points used for translating theta forms a lattice in IR d, and many optimal and quasi-optimal approximation schemes can already be found in the literature. In contrast, only few, mostly specific, results are known for a set of scattered points. The main objective of this paper is to provide a general tool for extending approximation schemes that use integer translates of a basis function to the non-uniform case. We introduce a single, relatively simple, conversion method that preserves the approximation orders provided by a large number of schemes presently in the literature (more precisely, to almost all stationary schemes ). In anticipation of future introduction of new schemes for uniform grids, an effort is made to impose only a few mild conditions on the function theta, which still allow for a unified error analysis to hold. In the course of the discussion here, the recent results of BuDL on scattered center approximation are reproduced and improved upon.

Abstract: This paper compares the approximation accuracy of two basis functions that share a common radial basis function (RBF) neural network architecture used for approximating a known function on the unit sphere. The basis function types considered are that of a new spherical basis function, the von Mises function, and the now well-known Gaussian basis function. Gradient descent learning rules were applied to optimize (learn) the solution for both approximating basis functions. A benchmark approximation problem was used to compare the performance of the two types of basis functions, in this case the mathematical expression for the scattering of an acoustic wave striking a rigid sphere. >

Abstract: This paper evaluates the classification accuracy of three neural network classifiers on a satellite image-based pattern classification problem. The neural network classifiers used include two types of the Multi-Layer-Perceptron (MLP) and the Radial Basis Function Network. A normal (conventional) classifier is used as a benchmark to evaluate the performance of neural network classifiers. The satellite image consists of 2,460 pixels selected from a section (270 x 360) of a Landsat-5 TM scene from the city of Vienna and its northern surroundings. In addition to evaluation of classification accuracy, the neural classifiers are analysed for generalization capability and stability of results. Best overall results (in terms of accuracy and convergence time) are provided by the MLP-1 classifier with weight elimination. It has a small number of parameters and requires no problem-specific system of initial weight values. Its in-sample classification error is 7.87% and its out-of-sample classification error is 10.24% for the problem at hand. Four classes of simulations serve to illustrate the properties of the classifier in general and the stability of the result with respect to control parameters, and on the training time, the gradient descent control term, initial parameter conditions, and different training and testing sets.

Abstract: During a flight test program, certain instrumented parameters are subject to degradation due to wear and tear. Over the course of a flight test program, data from several flights will be lost due to undetected instrumentation faults. The application of artificial neural networks as flight test data estimators has been proposed with the intention of reducing the aforementioned cost and wasted test flights. Several network topologies have been studied. A simulation program has been used to identify the appropriate network topology for this task. It is shown that a single network is not capable of predicting all of the correct values of the suspect parameters based solely on the information received from the reliable instruments. However, it is shown that a collection of smaller networks can succeed in this task, with each network predicting one suspect parameter. Limited training and test results based on simulation-generated data are presented. Actual flight test data from a typical business jet have been used to verify this concept and these results are presented as well. It is demonstrated that in most cases these networks are capable of predicting the measured parameter outputs with sufficient accuracy to enable identification of instrumentation system degradation.

Abstract: This paper describes a new approach to the analysis of weather radar data for short-range rainfall forecasting based on a neural network model. This approach consists in extracting synthetic information from radar images using the approximation capabilities of multilayer neural networks. Each image in a sequence is approximated using a modified radial basis function network trained by a competitive mechanism. Prediction of the rain field evolution is performed by analysing and extrapolating the time series of weight values. This method has been compared to the conventional cross-correlation technique and the persistence method for three different rainfall events, showing significant improvement in 30 and 60 min ahead forecast accuracy.

Abstract: This paper evaluates the classification accuracy of three neural network classifiers on a satellite image-based pattern classification problem. The neural network classifiers used include two types of the Multi-Layer-Perceptron (MLP) and the Radial Basis Function Network. A normal (conventional) classifier is used as a benchmark to evaluate the performance of neural network classifiers. The satellite image consists of 2,460 pixels selected from a section (270 x 360) of a Landsat-5 TM scene from the city of Vienna and its northern surroundings. In addition to evaluation of classification accuracy, the neural classifiers are analysed for generalization capability and stability of results. Best overall results (in terms of accuracy and convergence time) are provided by the MLP-1 classifier with weight elimination. It has a small number of parameters and requires no problem-specific system of initial weight values. Its in-sample classification error is 7.87% and its out-of-sample classification error is 10.24% for the problem at hand. Four classes of simulations serve to illustrate the properties of the classifier in general and the stability of the result with respect to control parameters, and on the training time, the gradient descent control term, initial parameter conditions, and different training and testing sets. (authors' abstract)

Abstract: In this paper a functional equivalence property of feedforward networks is introduced and studied for the case of radial basis function networks with Gaussian activation function and metrics induced by an inner product. The description of functional equivalent parameterizations is used for proposition of new genetic learning rules that operate only on a small part of the whole weight space.

Abstract: It is invariably the case that when an application is developed using the radial basis function network in the neural network domain, it is constructed using Gaussian basis functions. This paper discusses the rationale for employing alternative basis functions to the prevalent Gaussian. In particular, we argue the case in support of unbounded basis functions and nonpositive definite basis functions. The use of unbounded and nonpositive basis functions, though counterintuitive in application domains such as classification and time series forecasting, have a good theoretical motivation from the domains of functional interpolation and, somewhat surprisingly, kernel-based density estimation. In addition to collating the theoretical arguments, we present a performance comparison between Gaussian and unbounded, nonpositive definite basis functions in a radial basis function network applied to a financial derivatives regression problem: estimating the price of $/DM options contracts.

Abstract: Most research to date using genetic algorithms to evolve neural networks has focused on the multi-layer perceptron. Altemative neural network approaches such as the radial basis function network, and their representations appear to have received relatively little attention as grist for the GA mill. This is perhaps surprising since, for example, the radial basis function network has also been proved to be universal function approximator. Here we focus on evolution of radial basis function networks. While the multilayer perceptron network approximates functions through global interaction between network nodes, the radial basis function network uses local interactions between network nodes. It is suggested, that this difference may be of significance in tem of epistatic interactions in encoded genomes for the two types of network, which affects the ability of the genetic algorithm to evolve successful networks. A representation and attendant genetic operators for evolving radial basis function networks are proposed, drawing on recent work on evolutionary fuzzy logic systems. Experimental results in applying a hybrid leaming technique, using a genetic algorithm for evolving the radial basis function hidden layer (number of hidden nodes and hidden node centres and widths) and supervised leaming for tuning of network connection weights, are presented.

Abstract: Considers the problem of how to incorporate prior knowledge in supervised learning techniques. The authors set the problem in the framework of regularization theory, and consider the case in which one knows that the approximated function has radial symmetry. The problem can be solved in two alternative ways: 1) use the invariance as a constraint in the regularization theory framework to derive a rotation invariant version of radial basis functions; 2) use the radial symmetry to create new, "virtual" examples from a given data set. The authors show that these two apparently different methods of learning from "hints" (Abu-Mostafa, 1993) lead to exactly the same analytical solution.

Abstract: Image interpolation using radial basis function (RBF) neural networks is accomplished. In this work the RBF network is first trained with the given image, satisfying the constraint of the gray value at each pixel. With the desired magnification ratio, each pixel is then divided into subpixels. The subpixel gray values are calculated using the trained network. Two dimensional Gaussian basis functions are used as the neurons in the hidden layer.

Abstract: Many schemes for the employment of neural networks in control systems have been proposed [9] and some practical applications have also been made [2]. It is possible to apply a neural network to just about every conceivable control problem, however in many cases, although of interest, the network might not be the best or even a good solution, due to its relatively complex nonlinear operation. A neural network is in essence a nonlinear mapping device and in this respect, at the present time, most of the reported work describing the use of neural networks in a control environment is concerned solely with the problem of process modelling or system identification.

Abstract: Abstract The network models discussed in Chapters 3 and 4 are based on units which compute a non-linear function of the scalar product of the input vector and a weight vector. Here we consider the other major class of neural network model, in which the activation of a hidden unit is determined by the distance between the input vector and a prototype vector. An interesting and important property of these radial basis function networks is that they form a unifying link between a number of disparate concepts as we shall demonstrate in this chapter. In particular, we shall motivate the use of radial basis functions from the point of view of function approximation, regularization, noisy interpolation, density estimation, optimal classification theory, and potential functions.