Exploiting the separability of linear and nonlinear parameters in radial basis function networks

Three learning phases for radial-basis-function networks

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.

This work presents the solution based on the augmenting sequence of linear programming problems (LPP) as a tool for intellectualizing home environment. The proposed solution empowers the intelligent decision making procedure which can be applied to various intelligent control applications. The augmenting self-training procedure based on LPP approach is presented as well, which allows making reasonable decisions having only limited data about the controlled environment. The method permits retraining the decision making system when new data is available. As a proof of concept, this solution is applied to intelligent light control application. The obtained simulation results show the method’s capability in making reasonable decisions according to users preferences. DOI: http://dx.doi.org/10.5755/j01.itc.44.2.10235

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.

In fuzzy C-means (FCM) clustering, each data point belongs to a cluster to a degree specified by a membership grade. FCM partitions a collection of vectors in c fuzzy groups and finds a cluster center in each group such that the dissimilarity measure is minimized. This paper presents a training algorithm for the radial basis function (RBF) network using symmetry-based Fuzzy C-means (SFCM) clustering method which is the modified version of FCM clustering method based on point symmetry distance measure. The training algorithm which uses SFCM clustering method to train the network has a number of advantages such as faster training time, more accurate predictions and reduced network architecture compared to the standard RBF networks. The proposed training algorithm has been implemented in the RBF networks created by the newrb function of MATLAB which uses gradient based iterative method as learning strategy, therefore the new network will undergo a hybrid learning process. The networks called Symmetry-based Fuzzy C-means Clustering–Radial Basis Function Network (SFCM/RBF) has been tested against the standard RBF network and the networks called standard Fuzzy C-means Clustering (FCM)-RBF network (FCM/RBF) in forecasting. The experimental models has been tested on three real world application problems, particularly in Air pollutant problem, Biochemical Oxygen Demand (BOD) problem, and Phytoplankton problem. Keywords: Fuzzy c-means clustering; SFCM; Radial basis function network; point symmetry distance; forecasting.

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 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.

Bare-bones languages serve small systems with ease: Cushmann, R HEDN Vol 25 No 16 (5 September 1980) pp 171–177

This research work focuses on the development of meta-cognitive sequential learning algorithms in Radial Basis Function (RBF) network classiers, and their application to the early diagnosis of neurodegenerative diseases. The important issues in existing sequential learning algorithms are proper selection of training samples, nding minimal network structure and selection of an appropriate learning strategy. In addition, the random sequence of sample arrival inuences the performance signicantly. It has been reported in human learning that best learning strategies employ meta-cognition (metacognition means cognition about cognition) to address fundamental problems of what-tolearn, when-to-learn and how-to-learn. This thesis develops such meta-cognitive sequential learning algorithms in RBF network for classication problems. We call a RBF network employing meta-cognitive algorithm as `meta-cognitive RBF network' (McRBFN). McRBFN is developed based on Nelson and Narens model of meta-cognition for human learning. Accordingly, McRBFN has two components, namely cognitive and meta-cognitive components. A RBF network with evolving structure is the cognitive component and a self-regulatory learning mechanism is its meta-cognitive component. List of Figures 1.1 Nelson and Narens model of meta-cognition . . . . . . . . . . . . . . . .

The present paper proposes a novel non-intrusive model reduction approach based on Proper Orthogonal Decomposition (POD), the Discrete Empirical Interpolation Method (DEIM) and Radial Basis Function (RBF) networks to efficiently predict production of oil and gas reservoirs. Provided a representative set of training reservoir scenarios, either POD or DEIM allows for effectively projecting input parameters (e.g., permeability, porosity), states (e.g., pressure, saturations) and outputs (e.g., well production curves) into a much lower dimension that retains the main features contained in the simulation system. In this work, these projections are applied across multiple levels to be able to collapse a large number of spatio-temporal correlations. It is observed that these projections can be effectively performed at a large extent regardless of the underlying geological complexity and operational constraints associated with the reservoir model. The RBF network provides a powerful means for developing learning functions from input-output relationships described by the reservoir dynamics entailed by multiple combinations of inputs and controls. In order to achieve a high degree of predictability from the resulting reduced model, the RBF network exploits locality by a means of Gaussian basis functions that are maximal at the sampled point and decrease monotonically with distance. Compared to multilayer perceptron networks (i.e., traditional artificial neural networks) RBF networks require less training and are less sensitive to the presence of noise in the data. In this regard, POD or DEIM acts as a data filter that additionally aids at designing a more compact RBF network representation suitable for targeting fast reservoir predictions. Numerical results show significant accelerations with respect to running the original simulation model on a set of field-motivated cases.

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.

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.

Generalized [formula omitted]-sparse algorithms for constructing fault tolerant RBF networks

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.

This paper presents a Sequential Approximate Optimization (SAO) procedure that uses the Radial Basis Function (RBF) network. If the objective and constraints are not known explicitly but can be evaluated through a computationally intensive numerical simulation, the response surface, which is often called meta-modeling, is an attractive method for finding an approximate global minimum with a small number of function evaluations. An RBF network is used to construct the response surface. The Gaussian function is employed as the basis function in this paper. In order to obtain the response surface with good approximation, the width of this Gaussian function should be adjusted. Therefore, we first examine the width. Through this examination, some sufficient conditions are introduced. Then, a simple method to determine the width of the Gaussian function is proposed. In addition, a new technique called the adaptive scaling technique is also proposed. The sufficient conditions for the width are satisfied by introducing this scaling technique. Second, the SAO algorithm is developed. The optimum of the response surface is taken as a new sampling point for local approximation. In addition, it is necessary to add new sampling points in the sparse region for global approximation. Thus, an important issue for SAO is to determine the sparse region among the sampling points. To achieve this, a new function called the density function is constructed using the RBF network. The global minimum of the density function is taken as the new sampling point. Through the sampling strategy proposed in this paper, the approximate global minimum can be found with a small number of function evaluations. Through numerical examples, the validities of the width and sampling strategy are examined in this paper.