Abstract: The radial basis function (RBF) network has its foundation in the conventional approximation theory. It has the capability of universal approximation. The RBF network is a popular alternative to the well-known multilayer perceptron (MLP), since it has a simpler structure and a much faster training process. In this paper, we give a comprehensive survey on the RBF network and its learning. Many aspects associated with the RBF network, such as network structure, universal approimation capability, radial basis functions, RBF network learning, structure optimization, normalized RBF networks, application to dynamic system modeling, and nonlinear complex-valued signal processing, are described. We also compare the features and capability of the two models.

Abstract: Deep Brain Stimulation has been used in the study of and for treating Parkinson's Disease (PD) tremor symptoms since the 1980s. In the research reported here we have carried out a comparative analysis to classify tremor onset based on intraoperative microelectrode recordings of a PD patient's brain Local Field Potential (LFP) signals. In particular, we compared the performance of a Support Vector Machine (SVM) with two well known artificial neural network classifiers, namely a Multiple Layer Perceptron (MLP) and a Radial Basis Function Network (RBN). The results show that in this study, using specifically PD data, the SVM provided an overall better classification rate achieving an accuracy of 81% recognition.

Abstract: This paper presents a novel model reference adaptive control of five-phase interior-permanent-magnet (IPM) motor drives. The primary controller is designed based on an artificial neural network (ANN) to simulate the nonlinear characteristics of the system without knowledge of accurate motor models or parameters. The proposed motor drive decouples the torque and flux components of five-phase IPM motors by applying multiple-reference-frame transformation. Therefore, the motor can be easily driven below the rated speed with the maximum-torque-per-ampere operation or above the rated speed with the flux weakening operation. The ANN-based primary controller consists of a radial basis function network which is trained online to adapt system uncertainties. The complete IPM motor drive is simulated in Matlab/Simulink environment and implemented experimentally utilizing a dSPACE DS1104 controller board on a five-phase prototype IPM motor.

Abstract: In this article, two robust adaptive control schemes are investigated for a class of completely non-affine pure-feedback non-linear systems with input non-linearity and perturbed uncertainties using radial basis function neural networks (RBFNNs). Based on the dynamic surface control (DSC) technique and using the quadratic Lyapunov function, the explosion of complexity in the traditional backstepping design is avoided when the gain signs are known. In addition, the unknown virtual gain signs are dealt with using the Nussbaum functions. Using the mean value theorem and Young's inequality, only one learning parameter needs to be tuned online at each step of recursion. It is proved that the proposed design method is able to guarantee semi-global uniform ultimate boundedness (SGUUB) of all signals in the closed-loop system. Simulation results verify the effectiveness of the proposed approach.

Abstract: In this paper, a bio-inspired parallel manipulator with one translation along z-axis and two rotations along x- and y- axes is developed as the hybrid head mechanism of a groundhog robotic system. Several important issues including forward kinematic modeling, performance mapping, and multi-objective improvement are investigated with specific methods or technologies. Accordingly, the forward kinematics is addressed based on the integration of radial basis function network and inverse kinematics. A novel performance index called dexterous stiffness is defined, derived and mapped. The multi-objective optimization with particle swarm algorithm is conducted to search for the optimal dexterous stiffness and reachable workspace.

Abstract: This paper presents new multi-objective evolutionary hybrid algorithms for the design of Radial Basis Function Networks (RBFNs) for classification problems. The algorithms are memetic Pareto particle swarm optimization based RBFN (MPPSON), Memetic Elitist Pareto non dominated sorting genetic algorithm based RBFN (MEPGAN) and Memetic Elitist Pareto non dominated sorting differential evolution based RBFN (MEPDEN). The proposed methods integrate accuracy and structure of RBFN simultaneously. These algorithms are implemented on two-class and multiclass pattern classification problems with one complex real problem. The results reveal that the proposed methods are viable, and provide an effective means to solve multi-objective RBFNs with good generalization ability and simple network structure. The accuracy and complexity of the network obtained by the proposed algorithms are compared through statistical tests. This study shows that the proposed methods obtain RBFNs with an appropriate balance between accuracy and simplicity.

Abstract: Radial Basis Function Neural Networks (RBFNNs) have been successfully employed in several function approximation and pattern recognition problems. The use of different RBFs in RBFNN has been reported in the literature and here the study centres on the use of the Generalized Radial Basis Function Neural Networks (GRBFNNs). An interesting property of the GRBF is that it can continuously and smoothly reproduce different RBFs by changing a real parameter @t. In addition, the mixed use of different RBF shapes in only one RBFNN is allowed. Generalized Radial Basis Function (GRBF) is based on Generalized Gaussian Distribution (GGD), which adds a shape parameter, @t, to standard Gaussian Distribution. Moreover, this paper describes a hybrid approach, Hybrid Algorithm (HA), which combines evolutionary and gradient-based learning methods to estimate the architecture, weights and node topology of GRBFNN classifiers. The feasibility and benefits of the approach are demonstrated by means of six gene microarray classification problems taken from bioinformatic and biomedical domains. Three filters were applied: Fast Correlation-Based Filter (FCBF), Best Incremental Ranked Subset (BIRS), and Best Agglomerative Ranked Subset (BARS); this was done in order to identify salient expression genes from among the thousands of genes in microarray data that can directly contribute to determining the class membership of each pattern. After different gene subsets were obtained, the proposed methodology was performed using the selected gene subsets as new input variables. The results confirm that the GRBFNN classifier leads to a promising improvement in accuracy.

Abstract: This paper presents a wearable module and neural-network-based activity classification algorithm for energy expenditure estimation. The purpose of our design is first to categorize physical activities with similar intensity levels, and then to construct energy expenditure regression (EER) models using neural networks in order to optimize the estimation performance. The classification of physical activities for EER model construction is based on the acceleration and ECG signal data collected by wearable sensor modules developed by our research lab. The proposed algorithm consists of procedures for data collection, data preprocessing, activity classification, feature selection, and construction of EER models using neural networks. In order to reduce the computational load and achieve satisfactory estimation performance, we employed sequential forward and backward search strategies for feature selection. Two representative neural networks, a radial basis function network (RBFN) and a generalized regression neural network (GRNN), were employed as EER models for performance comparisons. Our experimental results have successfully validated the effectiveness of our wearable sensor module and its neural-network-based activity classification algorithm for energy expenditure estimation. In addition, our results demonstrate the superior performance of GRNN as compared to RBFN.

Abstract: Handwriting recognition has been one of the active and challenging research areas in the field of image processing and pattern recognition. It has numerous applications that includes, reading aid for blind, bank cheques and conversion of any hand written document into structural text form. Neural Network (NN) with its inherent learning ability offers promising solutions for handwritten character recognition. This paper identifies the most suitable NN for the design of hand written English character recognition system. Different Neural Network (NN) topologies namely, back propagation neural network, nearest neighbour network and radial basis function network are built to classify the characters. All the NN based Recognition systems use the same training data set and are trained for the same target mean square error. Two hundred different character data sets for each of the 26 English characters are used to train the networks. The performance of the recognition systems is compared extensively using test data to draw the major conclusions of this paper

Abstract: This brief presents a contouring control scheme for robotic manipulators. The geometric properties of the desired contour are incorporated in the controller design phase, and the resulting controller has been structured as a two-layered hierarchical control scheme that consists of an outer loop and an inner loop. The outer loop is formed by kinematic control system in operational space, which can be designed to assign different dynamics to the tangential, normal, and binormal direction of the desired contour. It is shown that the outer loop can provide a joint velocity reference signal to the inner one. The inner loop is used to implement a velocity servo control system at the robot joint level. Meanwhile, a radial basis function network is adopted to compensate for the nonlinear dynamics of the robotic manipulator, where a robust control strategy is used to suppress the modeling error of neural networks. Experimental results on the Zebra-Zero robotic manipulator have demonstrated the effectiveness of the proposed control scheme in comparison with other control strategies.

Abstract: The radial basis function (RBF) network is a type of neural network that uses a radial basis function as its activation function (Ou, Oyang & Chen, 2005). Because of the better approximation capabilities, simpler network structure and faster learning speed, the RBF networks have attracted considerable attention in many science and engineering field. Horng (2010) used the RBF for multiple classifications of supraspinatus ultrasonic images. Korurek & Dogan (2010) used the RBF networks for ECG beat classifications. Wu, Warwick, Jonathan, Burgess, Pan & Aziz (2010) applied the RBF networks for prediction of Parkinson’s disease tremor onset. Feng & Chou (2011) use the RBF network for prediction of the financial time series data. In spite of the fact that the RBF network can effectively be applied, however, the number of neurons in the hidden layer of RBF network always affects the network complexity and the generalizing capabilities of the network. If the number of neurons of the hidden layer is insufficient, the learning of RBF network fails to correct convergence, however, the neuron number is too high, the resulting over-learning situation may occur. Furthermore, the position of center of the each neuron of hidden layer and the spread parameter of its activation function also affect the network performance considerably. The determination of three parameters that are the number of neuron, the center position of each neuron and its spread parameter of activation function in the hidden layer is very important.

Abstract: Varying-coefficient models have attracted great attention in nonlinear time series analysis recently. In this paper, we consider a semi-parametric functional-coefficient autoregressive model, called the radial basis function network-based state-dependent autoregressive (RBF-AR) model. The stability conditions and existing conditions of limit cycle of the RBF-AR model are discussed. An efficient structured parameter estimation method and the modified multi-fold cross-validation criterion are applied to identify the RBF-AR model. Application of the RBF-AR model to the famous Canadian lynx data is presented. The forecasting capability of the RBF-AR model is compared to those of other competing time series models, which shows that the RBF-AR model is as good as or better than other models for the postsample forecasts.

Abstract: Sequential approximate optimization (SAO) is an attractive approach for design optimization. In this paper, the radial basis function (RBF) network is employed for the SAO. First, we examine the width of the Gaussian kernel, which affects the response surface. By examining the simple estimate proposed by Nakayama, four sufficient conditions are introduced. Then, a new simple estimate of the width in the Gaussian kernel is proposed. Second, a new sampling strategy with the RBF network is also proposed. In order to find the sparse region, the density function with the RBF network is developed. The proposed width and sampling strategy are examined through benchmark problems. Finally, the proposed SAO is applied to the optimal variable blank holder force (VBHF) trajectory for square cup deep drawing. The objective is taken as the minimization of the deviation of whole thickness. The constraints are quantitatively defined with the forming limit diagram in which no wrinkling and tearing can be observed. The design variables are the blank holder force. In particular, the risk of both tearing and wrinkling can be handled as the constraints separately. Numerical simulation is carried out by the optimal VBHF trajectory with SAO. It is clear from the numerical simulation that no tearing and wrinkling can be observed.

Abstract: We consider the problem of computing the survival (first-passage) probability density function of jump-diffusion models with two stochastic factors. In particular the Fokker–Planck partial integro-differential equation associated to these models is solved using a meshless collocation approach based on radial basis functions (RBF). To enhance the computational efficiency of the method, the calculation of the jump integrals is performed using a suitable Chebyshev interpolation procedure. In addition, the RBF discretization is carried out in conjunction with an ad hoc change of variables, which allows to use radial basis functions with equally spaced centers and at the same time yields an accurate resolution of the gradients of the survival probability density function near the barrier. Numerical experiments are presented showing that the RBF approach is extremely accurate and fast, and performs significantly better than the conventional finite difference method.

Abstract: Accurate weather forecasts are essential for various human activities. Weather forecasting is a complex process that can exhaust the resources of many computational devices. Out of numerous weather forecasting techniques Artificial Neural Networks (ANN) methodology is one of the most widely used techniques. In this study the application of Neural Network Ensembles in Rainfall Forecasting is investigated by using an Ensemble Neural Network (ENN) to forecast the rainfall in Colombo, Sri Lanka. The ensemble consist of a combination of Multi Layer Feed Forward Network with Back Propagation Algorithm (BPN), Radial Basis Function Network (RBFN) and General Regression Neural Network (GRNN). The performance of ensemble is compared with the performance of BPN, RBFN and GRNN. The ANNs are trained, validated and tested using daily observed weather data for 41 years. The results of our experiment show that the performance of the ensemble model is better than the performance of the other models for this application.

Abstract: The paper presents hybrid artificial intelligence system in constraint based scheduling of integrated manufacturing ERP systems. The system includes neural networks. The models were created by use simple neural networks (linear network - L, multi-layer network with error backpropagation - MLP and Radial Basis Function network - RBF) and hybrid neural networks in the form of: L - MLP network, L - RBF network, MLP-RBF network and L - MLP - RBF network. Neural networks as classification models were used to selection of tool for manufacturing operation. Next models as forecasting models were used to forecasting of tool use in different time intervals for manufacturing operation. These models were used at the stage of constraint bases scheduling and preventing standstill due to lack of tools, and special tools in particular. The created models were tested on real data from an enterprise.

Abstract: In this paper, a robust radial basis function (RBF) network based classifier is proposed for polarimetric synthetic aperture radar (SAR) images. The proposed feature extraction process utilizes the covariance matrix elements, the H/@a/A decomposition based features combined with the backscattering power (span), and the gray level co-occurrence matrix (GLCM) based texture features, which are projected onto a lower dimensional feature space using principal components analysis. For the classifier training, both conventional backpropagation (BP) and multidimensional particle swarm optimization (MD-PSO) based dynamic clustering are explored. By combining complete polarimetric covariance matrix and eigenvalue decomposition based pixel values with textural information (contrast, correlation, energy, and homogeneity) in the feature set, and employing automated evolutionary RBF classifier for the pattern recognition unit, the overall classification performance is shown to be significantly improved. An experimental study is performed using the fully polarimetric San Francisco Bay and Flevoland data sets acquired by the NASA/Jet Propulsion Laboratory Airborne SAR (AIRSAR) at L-band to evaluate the performance of the proposed classifier. Classification results (in terms of confusion matrix, overall accuracy and classification map) compared with the major state of the art algorithms demonstrate the effectiveness of the proposed RBF network classifier.

Abstract: — A major challenge in biomedical studies in recent years has been the classification of gene expression profiles into categories, such as cases and controls. This is done by first training a classifier by using a labeled training set containing labeled samples from the two populations, and then using that classifier to predict the labels of new samples. Such predictions have recently been shown to improve the diagnosis and treatment selection practices for several diseases. This procedure is complicated, however, by the high dimensionality of the data. While microarrays can measure the levels of thousands of genes per sample, case-control microarray studies usually involve no more than several dozen samples. Standard classifiers do not work well in these situations where the number of features (gene expression levels measured in these microarrays) far exceeds the number of samples. Selecting only the features that are most relevant for discriminating between the two categories can help construct better classifiers, in terms of both accuracy and efficiency. This paper provides a comparison between dimension reduction technique, namely Partial Least Squares (PLS)method and a hybrid feature selection scheme, and evaluates the relative performance of four different supervised classification procedures such as Radial Basis Function Network (RBFN), Multilayer Perceptron Network (MLP), Support Vector Machine using Polynomial kernel function(Polynomial- SVM) and Support Vector Machine using RBF kernel function (RBF-SVM) incorporating those methods. Experimental results show that the Partial Least-Squares(PLS) regression method is an appropriate feature selection method and a combined use of different classification and feature selection approaches makes it possible to construct high performance classification models for microarray data.

Abstract: This paper presents an on-line learning adaptive neural control scheme for helicopters performing highly nonlinear maneuvers. The online learning adaptive neural controller compensates the nonlinearities in the system and uncertainties in the modeling of the dynamics to provide the desired performance. The control strategy uses a neural controller aiding an existing conventional controller. The neural controller is based on a online learning dynamic radial basis function network, which uses a Lyapunov based on-line parameter update rule integrated with a neuron growth and pruning criteria. The online learning dynamic radial basis function network does not require a priori training and also it develops a compact network for implementation. The proposed adaptive law provides necessary global stability and better tracking performance. Simulation studies have been carried-out using a nonlinear (desktop) simulation model similar to that of a BO105 helicopter. The performances of the proposed adaptive controller clearly shows that it is very effective when the helicopter is performing highly nonlinear maneuvers. Finally, the robustness of the controller has been evaluated using the attitude quickness parameters (handling quality index) at different speed and flight conditions. The results indicate that the proposed online learning neural controller adapts faster and provides the necessary tracking performance for the helicopter executing highly nonlinear maneuvers.

Abstract: The paper describes an on-line prediction algorithm to estimate, over a determined time horizon, the solar irradiation of a specific site The learning algorithm is based on a radial basis function network and combines the growing criterion and the pruning strategy of the minimal resource allocating network technique An Extended Kalman Filter (EKF) is used to update all the parameters of the network The on-line algorithm is able to avoid the initial training of the neural network A comparison of the performance obtained by the MRAN EKF RBF Neural Network with respect to the standard RBF Neural Network is presented

Abstract: A novel neural architecture for prediction in industrial control: the 'Double Recurrent Radial Basis Function network' (R2RBF) is introduced for dynamic monitoring and prognosis of industrial processes. Three applications of the R2RBF network on the prediction values confirmed that the proposed architecture minimizes the prediction error. The proposed R2RBF is excited by the recurrence of the output looped neurons on the input layer which produces a dynamic memory on both the input and output layers. Given the learning complexity of neural networks with the use of the back-propagation training method, a simple architecture is proposed consisting of two simple Recurrent Radial Basis Function networks (RRBF). Each RRBF only has the input layer with looped neurons using the sigmoid activation function. The output of the first RRBF also presents an additional input for the second RRBF. An unsupervised learning algorithm is proposed to determine the parameters of the Radial Basis Function (RBF) nodes. The K-means unsupervised learning algorithm used for the hidden layer is enhanced by the initialization of these input parameters by the output parameters of the RCE algorithm.

Abstract: The use of radial basis functions have attracted increasing attention in recent years as an elegant scheme for high-dimensional scattered data approximation, an accepted method for machine learning, one of the foundations of mesh-free methods, an alternative way to construct higher order methods for solving partial differential equations (PDEs), an emerging method for solving PDEs on surfaces, a novel method for mesh repair and so on. All these applications share one mathematical foundation: high dimensional approximation/interpolation. This paper explains why radial basis functions are preferred to multi-variate polynomials for scattered data approximation in high-dimensional space; and gives a brief description on how to construct the most commonly used compactly supported radial basis functions. Without sophisticated mathematics, one can construct a compactly supported (radial) basis function with required smoothness according to procedures described here. Short programs and tables for compactly supported radial basis functions are supplied.