Sequential Learning Methods on RBF with Novel Approach of Minimal Weight Update

Radial basis function (RBF) network can be viewed as a fuzzy rule base with specified membership functions and fuzzy inference operations. However, there is a trade-off between the approximation performance of RBF network and the number of hidden neurons. To tackle this problem, this paper proposes a dynamic RBF (DRBF) network with a constructive learning. This DRBF network not only can create the new hidden neurons, but also can prune the insignificant hidden neurons. Then, an adaptive dynamic RBF fuzzy neural control (ADRFNC) system, including a neural controller and a saturation compensator, is developed. The neural controller uses a DRBF network to on-line mimic an ideal controller and the saturation compensator is designed to dispel the approximation error introduced by the neural controller. Finally, the proposed ADRFNC system is applied to a chaotic circuit and a DC motor control system. Simulation and experimental results show the proposed ADRFNC system can achieve a favorable tracking performance when the controller’s parameters have been learned and the network structure has been constructed by the proposed learning algorithm.

Radial Basis Function (RBF) Network is popularly used for solving pattern recognition problems. The training of RBF Network is faster compared to multi layer perceptron using error backpropagation. However, the RBF Network uses the pseudo inverse matrix to calculate weights from the hidden layer to the output layer. Thus calculation cost increases when the number of data and the number of hidden units increase. In addition, in RBF Network the decision of optimum number of hidden units is difficult. It is also more prone to overtraining, needing repeated train and test cycles to ascertain a proper number of the Network hidden units, so that generalization performance is good. In this work, we propose a technique to set up RBF network parameters which is fast, as well as the number of hidden units are automatically determined. We start with training a Self-Organizing Maps (SOM), which is a unsupervised training, though our samples are labeled. SOM can find the distribution of data in multidimensional space, and map it on a two dimensional display. The results of SOM network is used to calculate the RBF parameters. It is shown by experiments that using the proposed method, RBF network parameters can be determined much faster compared to existing technique. Moreover, the recognition rate for the test data was higher, showing better generalization performance.

In the radial basis function (RBF) network, selection of width parameter of the basis functions is considered. The effect of the width parameter on multivariable discrete function approximation is analysed in the Fourier domain and based on this, an appropriate selection method for the width parameter is presented. The method basically takes the orthogonal multiresolutional decomposition by wavelet transform of the function subject to approximation and to each decomposed component, a separate RBF network having a suitable width parameter matching to the appropriate resolution level, is employed. Each outcome from these multiresolutional RBF networks is added to that from the others and a final approximation result is obtained. A marked enhancement of approximation is achieved which implies an enhancement of the generalisation capability of the network performing robust estimation at the same time.

Three learning phases for radial-basis-function networks

In order to forecast missile storage reliability better, the paper researched a forecasting method based on neural network which is with the ability of actualizing multi-nonlinear mapping from input to output, and discussed steps of forecasting based on back propagation (BP) network and radial basis function (RBF) network respectively. At last, the storage reliability of one type ship-to-ship missile is forecasted based on BP network and RBF network respectively. The results show that both of the BP and RBF are suitable for Missile storage reliability forecasting, and the precision of the train goal is better by using RBF network. RBF network is more suitable for dealing with this problem.

A new learning method of radial basis function (RBF) network based on fuzzy C-means (FCM) clustering and ant colony optimization (ACO) is presented in the paper. Generally, the crucial problem of learning RBF network is the selection of the unit number, the centers and widths of the Gaussian radial basis function in hidden layer, and the weights between hidden layer and output layer. In this work, the centers are determined by FCM clustering, and the weights are determined by least mean squares (LMS) method. In addition, ACO is introduced to optimize two important parameters of FCM including the weighting exponent which impacts its performance, and the clustering number that is equal to the hidden unit number which impacts the generalization of RBF network, by minimizing an objective function integrated the generalization error with the hidden unit number of RBF network. Simulation results of identifying a nonlinear system illustrate the effectiveness of designing the RBF network with smaller structure but stronger generalization ability, comparing with K-means and orthogonal least squares (OLS) based learning methods.

A new approach is presented to enhance fuzzy modeling using multidimensional fuzzy sets directly in the fuzzy model in place of decomposed fuzzy sets by projection. The antecedent fuzzy sets in the form of multivariable functions are represented in continuous form. This is accomplished by using a multi input and multi output neural network, which is in particular radial basis functions (RBF) network providing the required multivariable function approximation properties in a fuzzy model. The inputs of the network are fuzzy model inputs, outputs are the membership function values as to the multivariable fuzzy sets involved in the model. Thus, each output is restricted to one multivariable fuzzy set. The RBF network is trained according to the multidimensional fuzzy sets which are identified after fuzzy clustering of the data. Based on this, the linear model parameters are determined by the method of least squares in a straightforward manner. The transparency issues can be handled by means of projection process to have information about the shape and locations of the membership functions pertinent to each variable at the input. The accurate fuzzy modeling having been thus guaranteed the information on the linear model parameters are used to establish the multivariable fuzzy rules with their respective validity regions. The redundancy of the number of fuzzy sets can be circumvented by classical cluster merging algorithms available. In this approach, the fuzzy rules are determined from the local linear models, however in contrast with the conventional fuzzy modeling, the model outputs are determined by the multivariable fuzzy sets. To obtain regular sets, RBF network is trained by means of orthogonal least squares (OLS) method so that at the same time the convergence issues of general neural network training is eliminated.

Recognition of human faces is a very important task in many applications such as authentication and surveillance. An efficient face recognition system with face image representation using averaged wavelet and wavelet packet coefficients, Discriminative Common Vector (DCV) and modified Local Binary Patterns (LBP) and recognition using radial basis function (RBF) network is presented. Face images are decomposed by 2-level wavelet and wavelet packet transformation. The discriminative common vectors are obtained for averaged wavelet. The new proposed LBP operator is applied on the obtained DCV and also applied on averaged wavelet packet coefficients of all the samples of a class. The histogram values obtained from the LBP are recognized using RBF network. The proposed work is tested on three face databases such as Olivetti Oracle Research Lab (ORL), Japanese Female Facial Expression (JAFFE) and Essex face database. The proposed method results in good recognition rates along with less training time because of the extracted discriminant input from the preprocessing steps involved in the proposed work.

This article explores the use of single trial EEG signals to predict the voluntary movements of single hand and two hands. During single-hand movements, three kinds of task, grasping, releasing, and holding were considered. The tasks considered during two-hand movements are left and right grasping, left and right releasing, and holding. The subject performs the tasks spontaneously without waiting for and responding to any external cues. In addition, a neural adaptive noise canceller is developed that accomplishes eye blinks suppression. The neural adaptive filter is here implemented by means of a three-layer feed-forward neural network. The feature vectors are formed from the three channels (Fz, C3, and F3). We employ the multilayer perceptron (MLP) with back-propagation learning algorithm and Radial Basis Function (RBF) network with stochastic gradient learning rule for discriminating different patterns of the EEG signals. In the classical approach to RBF and MLP network implementation, the number of hidden units is predetermined. It, usually, results in too many hidden units. To overcome this drawback, we develop an enhanced resource-allocating network (RAN) for discriminating the EEG patterns. These networks start with no hidden units and grow by allocating new hidden units based on the novelty in the EEG signals, which arrive sequentially. The results of this analysis show that the neural networks would be able to detect the movements of a single hand and two hands with an average classification accuracy of 98.82% and 96.40%, respectively. Moreover, the RAN provides a reduction in the training epochs as compared to the MLP and RBF networks. This work represents a promising approach to control prosthesis device.

In this paper, a steady-state frequency-domain model of an uncontrolled rectifier bridge with capacitive dc smoothing, developed with radial basis function (RBF) networks, is presented. The nonlinear load model is incorporated in a balanced harmonic load-flow program in order to evaluate the harmonic distortion introduced in the networks by this device. The fundamental frequency load flow, used in the harmonic load-flow algorithm, is enhanced with the incorporation to the Jacobian matrix of active and reactive power sensitivities from nonlinear loads. These sensitivities can be computed easily from the nonlinear load model built with RBF networks.

This work introduces an artificial neural network based speed control system for AC induction motors. In motor control, Radial Basis Function (RBF) network and Volt-Hertz (V/f) method have been adopted as neural controller and scalar driving tool respectively. V/Hz method has been preferred for the sake of simplicity and well-known reliability. The proposed control scheme consists of two main parts; a RBF network and a reference model unit. RBF based main controller supplies appropriate control signals to the driving unit to compensate nonlinearity of the induction motor and track the reference model output. Reference model, is very stable linear filter which is supplying set values to be imitated by induction motor. The success of the proposed control scheme has been demonstrated by experimental results; induction motor has been able to track the prescribed speed trajectory with rather small errors and good stability under properly loading conditions.

We propose a differential radial basis function (RBF) network termed RBF-DiffNet—whose hidden layer blocks are partial differential equations (PDEs) linear in terms of the RBF—to make the baseline RBF network robust to noise in sequential data. Assuming that the sequential data derives from the discretisation of the solution to an underlying PDE, the differential RBF network learns constant linear coefficients of the PDE, consequently regularising the RBF network by following modified backward-Euler updates. We experimentally validate the differential RBF network on the logistic map chaotic timeseries as well as on 30 real-world timeseries provided by Walmart in the M5 forecasting competition. The proposed model is compared with the normalised and unnormalised RBF networks, ARIMA, and ensembles of multilayer perceptrons (MLPs) and recurrent networks with long short-term memory (LSTM) blocks. From the experimental results, RBF-DiffNet consistently shows a marked reduction in the prediction error over the baseline RBF network (e.g., 41% reduction in the root mean squared scaled error on the M5 dataset, and 53% reduction in the mean absolute error on the logistic map); RBF-DiffNet also shows a comparable performance to the LSTM ensemble but requires 99% less computational time. Our proposed network consequently enables more accurate predictions—in the presence of observational noise—in sequence modelling tasks such as timeseries forecasting that leverage the model interpretability, fast training, and function approximation properties of the RBF network.

There is significant interest in the interplay between fuzzy systems and neural networks. Jang and Sun (1993) established the functional equivalence of Gaussian radial basis function (RBF) networks and a restricted class of Takagi-Sugeno-type (1985) fuzzy systems. This result was extended to the full TS-model by Hunt et al. (1994) who employed networks with local models and ellipsoidal basis functions. The restriction to Gaussian type basis functions, and therefore to Gaussian-shaped fuzzy membership functions, was later removed by Hunt et al. through employment of spline-based networks. This covers fuzzy systems with a broad range of membership function shapes (triangular and trapezoidal shapes are common special cases). In this paper we present a generalised form of the functional equivalence theorem and discuss its relevance for the direct implementation of fuzzy control systems in the form of neural networks. >

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.

This paper proposes a generic criterion that defines the optimum number of basis functions for radial basis function (RBF) neural networks. The generalization performance of an RBF network relates to its prediction capability on independent test data. This performance gives a measure of the quality of the chosen model. An RBF network with an overly restricted basis gives poor predictions on new data, since the model has too little flexibility. By contrast, an RBF network with too many basis functions also gives poor generalization performance since it is too flexible and fits too much of the noise on the training data. Bias and variance are complementary quantities, and it is necessary to assign the number of basis function optimally in order to achieve the best compromise between them. In this paper we use Stein's unbiased risk estimator to derive an analytical criterion for assigning the appropriate number of basis functions. Two cases of known and unknown noise have been considered and the efficacy of this criterion in both situations is illustrated experimentally. The paper also shows an empirical comparison between this method and two well known classical methods, cross validation and the Bayesian information criterion, BIC.