Abstract: The purpose of this paper is to explore the representation capability of radial basis function (RBF) neural networks. The main results are: 1) the necessary and sufficient condition for a function of one variable to be qualified as an activation function in RBF network is that the function is not an even polynomial, and 2) the capability of approximation to nonlinear functionals and operators by RBF networks is revealed, using sample data either in frequency domain or in time domain, which can be used in system identification by neural networks. >

Abstract: A Bayesian solution is derived for digital communication channel equalization with decision feedback. This is an extension of the maximum a posteriori probability symbol-decision equalizer to include decision feedback. A novel scheme utilizing decision feedback that not only improves equalization performance but also reduces computational complexity greatly is proposed. It is shown that the Bayesian equalizer has a structure equivalent to that of the radial basis function network, the latter being a one-hidden-layer artificial neural network widely used in pattern classification and many other areas of signal processing. Two adaptive approaches are developed to realize the Bayesian solution. The maximum-likelihood Viterbi algorithm and the conventional decision feedback equalizer are used as two benchmarks to asses the performance of the Bayesian decision feedback equalizer. >

Abstract: We prove that an artificial neural network with multiple hidden layers and akth-order sigmoidal response function can be used to approximate any continuous function on any compact subset of a Euclidean space so as to achieve the Jackson rate of approximation. Moreover, if the function to be approximated has an analytic extension, then a nearly geometric rate of approximation can be achieved. We also discuss the problem of approximation of a compact subset of a Euclidean space with such networks with a classical sigmoidal response function.

Abstract: The use of radial basis function networks (RBFNs) for diagnosis and classification is discussed. Even though RBFNs can be trained quickly compared to backpropagation networks, the training effort is still significant for large-scale diagnosis problems. Rho-Net, an architecture that decomposes the dynamic classification problem in two ways, making such training tractable, is presented. The first decomposition reduces the amount of training data needed for any stage of the training process by constructing separate networks for each fault class. The second decomposition reduces the dimensionality of the input space by incorporating temporal information at the output of the network, instead of as a temporal window at the input of the net. Application of Rho-Nets to chemical process simulation is discussed. >

Abstract: The authors propose a self-generating algorithm for radial basis functions to automatically determine the minimal number of basis functions to achieve the specified model error. This model is also regarded as a multilayered neural network or fuzzy model of class C/sup infinity /. The self-generating algorithm consists of two processes: model parameter tuning by the gradient method for a fixed number of rules, and a basis function generation procedure by which a new basis function is generated in such a way that the center is located at the point where maximal inference error takes place in the input space, when the effect of parameter tuning is diminished. A numerical example shows that the algorithm can achieve the specified model error with fewer basis functions than other methods by which only coefficients of the basis functions are tuned. The method is applied to the nonlinear prediction of optical chaotic time series. >

Abstract: The complex radial basis function (RBF) network proposed has complex centres and weights but the response of its hidden nodes remains real. Several learning algorithms for the existing real RBF network are extended to this complex network. The proposed network is capable of generating complicated nonlinear decision surface or approximating an arbitrary nonlinear function in multidimensional complex space and it provides a powerful tool for nonlinear signal processing involving complex signals. This is demonstrated using two practical applications to communication systems. The first case considers the equalisation of time-dispersive communication channels, and the authors show that the underlying Bayesian solution has an identical structure to the complex RBF network. In the second case, they use the complex RBF network to model nonlinear channels, and this application is typically found in channel estimation and echo cancellation involving nonlinear distortion. >

Abstract: Many nonlinear deterministic models of time series have large numbers of parameters and tend to overfit in the presence of noise. This paper shows how to generate radial basis function models with small numbers of parameters for a given quality of fit. It also addresses questions of how to select subsets from candidate sets of centers for radial basis function models, and what kinds of basis functions to use, as well as how large a model is appropriate.

Abstract: Radial basis function (RBF) networks are finding increasing use in applications involving multidimensional function interpolation and pattern classification. The major obstacle to the wider use of RBFs is the complex nature of their calculations, in particular the requirement to evaluate Euclidean distance repeatedly. Almost no hardware implementations have been reported. The Letter presents analogue VLSI circuits which can calculate the Euclidean norm, and compute programmable width basis functions of this norm. These novel circuits can be combined with existing ‘conventional’ neural circuits, to implement complete RBF networks.

Abstract: Randomly initialized radial basis function neural networks are compared to networks whose centers are obtained by using vector quantization. It is shown that the error rate for small networks can be decreased by about 28%. To achieve the same performance with a trained network as with a randomly initialized network, only half of the number of hidden neurons is needed. This may be important for time critical applications. The time used for the training and initialization of a smaller network is comparable to the time used for the initialization of a larger network. >

Abstract: This paper presents a local neural network structure called spatiotemporally local network, by combining the radial basis function network (RBFN) and the local recurrent networks. Three local structures are proposed and the algorithms are compared for nonlinear dynamic system identification. System dynamics can be fully modeled with the fast learning of the proposed neural network structure.

Abstract: Several empirical studies have demonstrated the feasibility of employing neural networks as models of nonlinear dynamical systems. This paper presents a stability theory approach to synthesizing and analyzing neural network based identification schemes. First static network architectures are combined with dynamical elements in the form of stable filters to construct a type of recurrent network configuration which is shown to be capable of approximating a large class of dynamical systems. Identification schemes, based on neural network models, are then developed using the Lyapunov synthesis approach with the projection modification method. These identification schemes are shown to guarantee stability of the overall system, even in the presence of modeling errors.

Abstract: Considerable progress has been made in recent years in the analysis of time series arising from chaotic systems. In particular, a variety of schemes for the short-term prediction of such time series has been developed. However, hitherto all such algorithms have used batch processing and have not been able to continuously update their estimate of the dynamics using new observations as they are made. This severely limits their usefulness in real time signal processing applications. In this paper we present a continuous update prediction scheme for chaotic time series that overcomes this difficulty. It is based on radial basis function approximation combined with a recursive least squares estimation algorithm. We test this scheme using simulated data and comment on its relationship to adaptive transversal filters, which are widely used in conventional signal processing.

Abstract: The fuzzy min-max function approximation neural network is introduced, and results of its performance on a sample problem are presented. The function approximation network is realized by modifying the previously developed fuzzy min-max clustering network to include an output layer that sums and thresholds the hidden layer membership functions. The approximation of a test function to a small tolerance and robustness when trained on sparse data is demonstrated. >

Abstract: In this paper, the computational and convergence properties of Gaussian radial basis function approximations employed in the Dual Reciprocity Boundary Element Method are studied. The Gaussian function has some desirable features for practical use; however, its performance is strongly influenced by its decay parameter since too small a value produces isolated peaks and too large values will make the approximation procedure ill-conditioned. It is shown herein that the approximating series of Gaussians is convergent, and that error bounds of the approximation can be estimated.

Abstract: Radial Basis Function (RBF) network have been studied intensively ([1], [9], [8], [3], [2], [12], [13]). Besides its applications several theoretical results have been obtained. E.g., (1) RBF net can be naturally derived from regularization theory ([9]), (2) RBF net has universal approximation ability [4], (3) RBF net has also best approximation ability ([3],[5]).

Abstract: The authors apply the radial basis function (RBF) neural network to low-angle radar tracking. Computer simulations show that the RBF network is capable of tracking both stationary and moving targets with high accuracy. As well, the tracking performance of the RBF network is evaluated under different signal-to-noise ratio situations. Furthermore, real-life data are used to test the RBF network. The results demonstrate the robustness and effectiveness of the network in terms of its independence of array errors and of the nature of the noise background.

Abstract: A new paradigm for learning control of a generic nonlinear system is presented. The method is based on a special radial basis function network architecture and is adaptive as the input/output properties of the system are estimated and the learning control gain is computed online. The controlled system is described with a mapping between the feedforward control and an array of sampled output values over a given time interval. The network approximates the system input/output mapping as a function of the task parameter vector comprising the initial and a desired final system state. The approximation is affine and linear in the control input. For given network weights, this allows inverting the network approximation and an optimal control program. The network weights are updated depending on the results of the learning trials. The application of the algorithm to the control of an arbitrary planar motion of a two link arm demonstrates fast learning and the high accuracy of the method. >

Abstract: In this paper we examine the representational and expressive power of two types of linearly weighted neural network: the polynomial discriminators (PDFs) and the radial basis function networks (RBFNs). A {O, 1}-valued function on Rn is a polyne mial discriminator of degree at most k if there is a surface in Rn which separates the positive examples of ! from the negative examples and which can be described by a polynomial equation of degree at mat k. The set of such functions is denoted P(n, k). In a similar way, ~B (n, k) denotes the set of boolean functions on {O, 1}n whose positive and negative examples can be separated by a surface whose equation is a polynomial of degree at most k. The threshold order of a boolean function f is the least value of k such that / c PB (n, k). By bounding the number of boole~ functions having at most a given threshold order, we prove that for large n, all but a negligible pr~ portion of boolean functions on n variables have threshold order at least [n/2J and that if n is odd, at most 1/2 of all such functions have threshold order at most [n/2J. We then determine precisely the VC dimensions of ‘PB (n, k) and P(n, k). These dimensions coincide for k = 1 but thereafter they diverge. In passing, we provide a new proof of a result of Dudley. We then examine radial basis function networks. Using results of Micchelli and others on the interpolation properties of such networks, we obtain bounds on the VC dimensions of RBFNs with certain standard basis functions. Permission to copy without fee all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice IS given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission. ACM COLT ’93 171931 CA, USA @ 1993 ACM 0-89791-61 1-5/93 /0007 /0158 . ..$l .50 . Sean B. Holden Engineering Department University of Cambridge Trumpington Street Cambridge CB2 lPZ United Kingdom sbh@eng. cam. ac. uk

Abstract: In this paper, an alternative connectionist architecture, called a Radial Basis Function (RBF) network, is considered for visual road- following. A controller based on each network architecture is built and the performances of the two network controllers are evaluated using a driving simulator. The RBF network was able to overcome many of the problems encountered by ALVINN (Autonomous Land Vehicle In a Neural Network).

Abstract: This paper investigates the modelling capabilities of neural nets for a dynamic nonlinear process. Different neural structures are compared: multilayer perceptron (MLP) and radial basis function network (RBF) with an external tapped delay line, and modifications of both network types using internal delays, called time-delay MLP (TDMLP) and time-delay RBF (TDRBF). The nonlinear process to be modelled is a drive system including some nonlinearities, e.g. saturation effects. A special clustering procedure is introduced in order to increase the modelling accuracy, reduce computation and provide better generalisation.

Abstract: We present a new constructive algorithm for building Radial-Basis-Function (RBF) network classifiers and a tree based associated algorithm for fast processing of the network. This method, named Constructive Tree Radial-Basis-Function (CTRBF), allows to build and train a RBF network in one pass over the training data set. The training can be in supervised or unsupervised mode. Furthermore, the algorithm is not restricted to fixed input size problems. Several construction and pruning strategies are discussed. We tested and compared this algorithm with classical RBF and multilayer perceptrons on a real world problem: on-line handwritten character recognition. While instantaneous incremental learning is the major property of the architecture, the tree associated to the RBF network gives impressive speed improvement with minimal performance losses. Speed-up factors of 20 over classical RBF have been obtained.

Abstract: Feedforward neural networks are divided into two classes according to characteristics of hidden units: one based on units with non-local activation functions, such as multilayer perceptrons (MLPs), the other based on units with local activation functions, such as radial basis function networks (RBFNs). Though both MLPs and RBFNs theoretically have ability to represent arbitrary functions, they cannot practically acquire sufficient approximation accuracy in most cases. In order to obtain more precise approximation, a hybrid model composed of an MLP and an RBFN is proposed. The performance of a single MLP, a single RBFN, and the proposed model is compared by function approximation problems. The results show that the proposed model is superior to the others in most cases, with regard to not only approximation accuracy but also learning speed.

Abstract: An important aspect of non-destructive testing is the interpretation and classification of signal obtained by NDT methods such as eddy current and ultrasound. These signals are typically complex, non-stationary waveforms, with signals corresponding to a particular class of defect in a specimen having similar form and shape. However, distortions and noise introduced by the measurement system make the manual classification of these signals a time-consuming and unreliable process, with the results affected by operator fatigue and measurement quality. The design of traditional classifiers for this task also poses many difficulties, due to a number of parameters that influence measurement, and the limited understanding of the effect of these parameters on the signal. Recently, artificial neural networks have been applied to a variety of NDT problems, including signal classification, with encouraging results. Artificial neural networks consist of a dense interconnection of simple computational elements, whose interconnection strengths are determined using a predefined learning algorithm, specific to the network. These networks do not require an explicit mathematical modeling of the data they have to process, and are robust even in the presence of noisy data and data generated by strongly non-linear processes [1]. An example of a neural network that has been extensively used in NDT applications is the multilayer perception. However, the error backpropagation algorithm used for training the multilayer perceptron has several disadvantages, such as long training times and susceptibility to local minima. This paper presents a novel approach to defect sizing that involves the use of a radial basis functions network. The network has the advantages of having shorter training times and a parametric nature that allows network optimization on an analytic basis. The application of such a network in the inversion of ultrasonic data to obtain flaw sizing is described. Results from the sizing of defects in aluminium blocks are presented.

Abstract: Neuro controllers have recently been applied to practical systems. The commonest network in these applications has been the multilayer perceptron trained by backpropagation. The objective of this paper is to present a new neuro control scheme for servomotors. An important feature of the proposed control scheme is that the radial basis function network, instead of normal backpropagation neural net, is used to tune a conventional controller. Another goal is to introduce a two layer radial basis network structure to be trained with the novel algorithm. Simulations are performed with both radial basis function networks showing that the proposed neuro controller can be trained in a short period of time and is robust. >

Abstract: A variety of standard basis functions have long been used for characterizing signals including Fourier, Walsh, Haar, and Slant. Optimum Karhunen-Loeve basis functions derived from correlation data can also be used. Radial basis functions are a new type. They are based on actual data rather than predefined or generated sets. This paper relates radial basis functions and standard basis functions. It reviews how they are derived and illustrates their broad use with EKG and stock market examples. They have great utility and may become as common as their standard predecessors. >

Abstract: An application of the radial basis function network to seismic waveform classification is presented. The network performs generalization and discrimination of input patterns using an external teacher. Modifications to this scheme are described. They include: (1) changing the size of the spheres; (2) using a random walk scheme during testing; (3) gradually decreasing the initial radii to avoid overlap of two distinct regions; (4) a conflict resolution mechanism; and (5) a simple means of decreasing the sphere radius. The applications to seismic signals include using the moments over a sliding window and the first several points of a wavelet. The speed of training of this network exceeds that of backpropagation with the same error rate.

Abstract: In the paper, a multilayer radial basis function network is proposed for nonlinear time series modeling and prediction. It uses successive approximations, first obtaining a number of coarse approximations, and then optimally linearly combining the coarsely defined functions to achieve an accurate end result. Compared with the conventional approaches using radial basis functions, the present method considerably reduces computation time, and can improve the predictive ability of radial basis function networks while retaining good training accuracy. The method is particularly useful for modeling and prediction of nearly cyclical nonlinear time series in the presence of observational noise. Numerical examples for chaotic time series and some classical real world time series are presented. It is shown that the successive approximation radial basis function network presented in the paper is a very useful tool for nonlinear modeling and prediction.

Abstract: A method of radar target recognition by range profiles is developed, based on the radial basis function network (RBFN). The problem of producing suitable patterns for recognition is discussed. Then a heuristic clustering algorithm for training RBFN is proposed. It is shown, from theoretical analysis and experimental results of rotating platform imaging based on experimental data acquired in a microwave anechoic chamber, that target recognition based on range profile is promising in application of ATR system.