Radial Basis Function Centers Research Articles

A Fast and Efficient Method for Training Categorical Radial Basis Function Networks

This brief presents a novel learning scheme for categorical data based on radial basis function (RBF) networks. The proposed approach replaces the numerical vectors known as RBF centers with categorical tuple centers, and employs specially designed measures for calculating the distance between the center and the input tuples. Furthermore, a fast noniterative categorical clustering algorithm is proposed to accomplish the first stage of RBF training involving categorical center selection, whereas the weights are calculated through linear regression. The method is applied on 22 categorical data sets and compared with several different learning schemes, including neural networks, support vector machines, naïve Bayes classifier, and decision trees. Results show that the proposed method is very competitive, outperforming its rivals in terms of predictive capabilities in the majority of the tested cases.

LCA based RBF training algorithm for the concurrent fault situation

In the construction of a radial basis function (RBF) network, one of the most important issues is the selection of RBF centers. However, many selection methods are designed for the fault free situation only. This paper first assumes that all the training samples are used for constructing a fault tolerant RBF network. We then add an l1 norm regularizer into the fault tolerant objective function. According to the nature of the l1 norm regularizer, some unnecessary RBF nodes are removed automatically during training. Based on the local competition algorithm (LCA) concept, we propose an analog method, namely fault tolerant LCA (FTLCA), to minimize the fault tolerant objective function. We prove that the proposed fault tolerant objective function has a unique optimal solution, and that the FTLCA converges to the global optimal solution. Simulation results show that the FTLCA is better than the orthogonal least square approach and the support vector regression approach.

One‐day‐ahead hourly forecasting for photovoltaic power generation using an intelligent method with weather‐based forecasting models

An intelligent method is proposed in this study to predict one-day-ahead hourly photovoltaic (PV) power generation. The proposed method comprises data classification, training, forecasting and forecasting updating stages. In the first stage, a fuzzy k-means algorithm is used to classify the historical data of daily PV power generation into various weather types. In the second stage, five training models are established, according to the verbal weather forecast of the Taiwan Central Weather Bureau (TCWB), in terms such as the sunny, sunny and cloudy, cloudy, cloudy and rainy and rainy. Each training model is constructed using a radial basis function neural network (RBFNN), for which the parameters of each RBFNN, including the position of the radial basis function (RBF) centres, the width of the RBFs and the weights between the hidden and the output layers, are optimised using a harmony search algorithm (HSA). In the forecasting stage, fuzzy inference is used to select an adequate forecasting model from the trained models. To cope with the possible fluctuation of PV power generation, the forecasts are updated every 3 h, according to the updated weather forecasts of the TCWB. The proposed approach is tested on a practical PV power generation system. The results show that the proposed method provides better forecasting results than the existing methods over 1-year testing data.

A Reduced Radial Basis Function Method for Partial Differential Equations on Irregular Domains

We propose and test the first Reduced Radial Basis Function Method for solving parametric partial differential equations on irregular domains. The two major ingredients are a stable Radial Basis Function (RBF) solver that has an optimized set of centers chosen through a reduced-basis-type greedy algorithm, and a collocation-based model reduction approach that systematically generates a reduced-order approximation whose dimension is orders of magnitude smaller than the total number of RBF centers. The resulting algorithm is efficient and accurate as demonstrated through two- and three-dimensional test problems.

快速Hermite径向基函数曲面重构

In this paper, a fast method with compactly supported radial basis functions (CSRBFs) is presented for Hermite surface interpolation or approximation from scattered points. By constructing a hierarchy of the given points, a global surface reconstruction is achieved in a coarse-to- ne way which overcome the problems resulted from using CSRBFs. Moreover, we design a radial basis function center selection approach based on approximat- ing errors, to reduce the interpolating points on each level, which make the Hermite radial basis function implicits to be able to deal with point clouds more than one million of points. The experiments demonstrate our method is also suitable for handling extremely nonuniform or noisy point clouds.

Improved Radio Frequency Identification Indoor Localization Method via Radial Basis Function Neural Network

Indoor localization technique has received much attention in recent years. Many techniques have been developed to solve the problem. Among the recent proposed methods, radio frequency identification (RFID) indoor localization technology has the advantages of low-cost, noncontact, non-line-of-sight, and high precision. This paper proposed two radial basis function (RBF) neural network based indoor localization methods. The RBF neural networks are trained to learn the mapping relationship between received signal strength indication values and position of objects. Traditional method used the received signal strength directly as the input of neural network; we added another input channel by taking the difference of the received signal strength, thus improving the reliability and precision of positioning. Fuzzy clustering is used to determine the center of radial basis function. In order to reduce the impact of signal fading due to non-line-of-sight and multipath transmission in indoor environment, we improved the Gaussian filter to process received signal strength values. The experimental results show that the proposed method outperforms the existing methods as well as improves the reliability and precision of the RFID indoor positioning system.

EVOLVING RBF NEURAL NETWORKS FOR ADAPTIVE SOFT-SENSOR DESIGN

This work presents an adaptive framework for building soft-sensors based on radial basis function (RBF) neural network models. The adaptive fuzzy means algorithm is utilized in order to evolve an RBF network, which approximates the unknown system based on input-output data from it. The methodology gradually builds the RBF network model, based on two separate levels of adaptation: On the first level, the structure of the hidden layer is modified by adding or deleting RBF centers, while on the second level, the synaptic weights are adjusted with the recursive least squares with exponential forgetting algorithm. The proposed approach is tested on two different systems, namely a simulated nonlinear DC Motor and a real industrial reactor. The results show that the produced soft-sensors can be successfully applied to model the two nonlinear systems. A comparison with two different adaptive modeling techniques, namely a dynamic evolving neural-fuzzy inference system (DENFIS) and neural networks trained with online backpropagation, highlights the advantages of the proposed methodology.

Optimizing widths with PSO for center selection of Gaussian radial basis function networks

The radial basis function (RBF) centers play different roles in determining the classification capability of a Gaussian radial basis function neural network (GRBFNN) and should hold different width values. However, it is very hard and time-consuming to optimize the centers and widths at the same time. In this paper, we introduce a new insight into this problem. We explore the impact of the definition of widths on the selection of the centers, propose an optimization algorithm of the RBF widths in order to select proper centers from the center candidate pool, and improve the classification performance of the GRBFNN. The design of the objective function of the optimization algorithm is based on the local mapping capability of each Gaussian RBF. Further, in the design of the objective function, we also handle the imbalanced problem which may occur even when different local regions have the same number of examples. Finally, the recursive orthogonal least square (ROLS) and genetic algorithm (GA), which are usually adopted to optimize the RBF centers, are separately used to select the centers from the center candidates with the initialized widths, in order to testify the validity of our proposed width initialization strategy on the selection of centers. Our experimental results show that, compared with the heuristic width setting method, the width optimization strategy makes the selected centers more appropriate, and improves the classification performance of the GRBFNN. Moreover, the GRBFNN constructed by our method can attain better classification performance than the RBF LS-SVM, which is a state-of-the-art classifier.

A new a priori net length estimation technique for integrated circuits using radial basis functions

Placement is a stage in the design of digital circuits where the locations of the circuit components are determined, while minimizing the total length of wires connecting them. A priori individual length estimates can be used to improve the quality of a placement solution. However, finding such estimates is a daunting task. A technique based on Radial Basis Functions (RBFs) is developed in this paper. Unlike polynomials, the RBFs provide flexible basis elements with only local support, which greatly enhances both their robustness and their ability to fit highly non-linear data sets. Today’s placement problems deal with a very large number of components making it impossible to apply traditional RBF modeling techniques. Thus specialized methods for determining the RBF centers and shape parameters are developed. The proposed technique is tested on popular benchmark circuits, and shows improvements of up to 24% over the best existing model for mixed-size circuits.

An online self-adaptive modular neural network for time-varying systems

We propose an online self-adaptive modular neural network (OSAMNN) for time-varying systems. Starting with zero subnetworks, OSAMNN uses a single-pass subtractive cluster algorithm to update the centers of radial-basis function (RBF) neurons for learning. Then the input space can be partitioned. The OSAMNN structure is capable of growing or merging subnetworks to maintain suitable model complexity, and the centers of RBF neurons can also be dynamically adjusted according to changes in the data environment. A fuzzy strategy is applied to select suitable subnetworks to learn the current sample. This method yields improved learning efficiency and accuracy. OSAMNN can adapt its architecture to realize online modeling of time-varying nonlinear input–output maps. Results for experiments on benchmark and real-world time-varying systems support the proposed techniques.

Radial basis function network training using a nonsymmetric partition of the input space and particle swarm optimization.

This paper presents a novel algorithm for training radial basis function (RBF) networks, in order to produce models with increased accuracy and parsimony. The proposed methodology is based on a nonsymmetric variant of the fuzzy means (FM) algorithm, which has the ability to determine the number and locations of the hidden-node RBF centers, whereas the synaptic weights are calculated using linear regression. Taking advantage of the short computational times required by the FM algorithm, we wrap a particle swarm optimization (PSO) based engine around it, designed to optimize the fuzzy partition. The result is an integrated framework for fully determining all the parameters of an RBF network. The proposed approach is evaluated through its application on 12 real-world and synthetic benchmark datasets and is also compared with other neural network training techniques. The results show that the RBF network models produced by the PSO-based nonsymmetric FM algorithm outperform the models produced by the other techniques, exhibiting higher prediction accuracies in shorter computational times, accompanied by simpler network structures.

Reproducing kernel hilbert space approach for the online update of radial bases in neuro-adaptive control.

Classical work in model reference adaptive control for uncertain nonlinear dynamical systems with a radial basis function (RBF) neural network adaptive element does not guarantee that the network weights stay bounded in a compact neighborhood of the ideal weights when the system signals are not persistently exciting (PE). Recent work has shown, however, that an adaptive controller using specifically recorded data concurrently with instantaneous data guarantees boundedness without PE signals. However, the work assumes fixed RBF network centers, which requires domain knowledge of the uncertainty. Motivated by reproducing kernel Hilbert space theory, we propose an online algorithm for updating the RBF centers to remove the assumption. In addition to proving boundedness of the resulting neuro-adaptive controller, a connection is made between PE signals and kernel methods. Simulation results show improved performance.

Locally regularised two-stage learning algorithm for RBF network centre selection

Nonlinear system models constructed from radial basis function (RBF) networks can easily be over-fitted due to the noise on the data. While information criteria, such as the final prediction error (FPE), can provide a trade-off between training error and network complexity, the tunable parameters that penalise a large size of network model are hard to determine and are usually application dependent. This article introduces a new locally regularised, two-stage stepwise construction algorithm for RBF networks. The main objective is to produce a parsimonious network that generalises well over unseen data. This is achieved by utilising Bayesian learning within a two-stage stepwise construction procedure to penalise centres that are mainly interpreted by the noise. Specifically, each output layer weight is assigned a hyperparameter, a large value of such a parameter forcing the associated output layer weight to be near to zero. Sparsity is achieved by removing irrelevant RBF centres from the network. The efficacy of proposed algorithm from the original two-stage construction method is retained. Numerical analysis shows that this new method only needs about half of the computation involved in the locally regularised orthogonal least squares (LROLS) alternative. Results from two simulation examples are presented to show that the nonlinear system models resulting from this new approach are superior in terms of both sparsity and generalisation capability.

Detection of Fraudulent Emails by Authorship Extraction

Fraudulent emails can be detected by extraction of authorship information from the contents of emails. This paper presents information extraction based on unique words from the emails. These unique words will be used as representative features to train Radial Basis function (RBF). Final weights are obtained and subsequently used for testing. The percentage of identification of email authorship depends upon number of RBF centers and the type of functional words used for training RBF. One hundred and fifty authors with over one hundred files from the sent folder of Enron email dataset are considered. A total of 300 unique words of number of characters in each word ranging from three to seven are considered. Training and testing of RBF are done by taking different lengths of words. Our simulation shows the effectiveness of the proposed RBF network for email authorship identification. The accuracy of authorship identification ranges from 95% to 97%.

Wind Power Generation Prediction by Particle Swarm Optimization Algorithm and RBF Neural Network

Wind power generation trend prediction is the important to make the plan on the development of wind power generation. Wind power generation prediction by particle swarm optimization algorithm and RBF neural network in the paper. As the connection weights between the hidden layer and output layer, the centers of radial basis function in hidden layer and the widths of radial basis function in hidden layer have a great influence on the prediction results of RBF neural network,particle swarm optimization which has a great global optimization ability is used to optimize the three parameters including the connection weights between the hidden layer and output layer, the centers of radial basis function in hidden layer and the widths of radial basis function in hidden layer. It is indicated that the hybrid model of particle swarm optimization algorithm and RBF neural network has better prediction ability than BP neural network.

A Fast Treecode for Multiquadric Interpolation with Varying Shape Parameters

A treecode algorithm is presented for the fast evaluation of multiquadric radial basis function (RBF) approximations. The method is a dual approach to one presented by Krasny and Wang, which applies far-field expansions to clusters of RBF centers (source points). The new approach clusters evaluation points instead and is therefore easily able to cope with basis functions that have different multiquadric shape parameters. The new treecode is able to evaluate an approximation on N centers at M points in $O((N+M) \log M)$ time in the ideal case when evaluation points are uniformly distributed. When coupled with a two-level restricted additive Schwarz preconditioner for GMRES iterations, the treecode is well suited for use within an adaptive RBF iteration, previously described by Driscoll and Heryudono, as is demonstrated by experiments on test functions.

Research for RBF Neural Networks Modeling Accuracy of Determining the Basis Function Center Based on Clustering Methods

The radial basis function (RBF) neural network is superior to other neural network on the aspects of approximation ability, classification ability, learning speed and global optimization etc., it has been widely applied as feedforward networks, its performance critically rely on the choice of RBF centers of network hidden layer node. K-means clustering, as a commonly method used on determining RBF center, has low neural network generalization ability, due to its clustering results are not sensitive to initial conditions and ignoring the influence of dependent variable. In view of this problem, fuzzy clustering and grey relational clustering methods are proposed to substitute K-means clustering, RBF center is determined by the results of fuzzy clustering or grey relational clustering, and some researches of RBF neural networks modeling accuracy are done. Practical modeling cases demonstrate that the modeling accuracy of fuzzy clustering RBF neural networks and grey relational clustering RBF neural networks are significantly better than K-means clustering RBF neural networks, applying of fuzzy clustering or grey relational clustering to determine the basis function center of RBF neural networks hidden layer node is feasible and effective.

Two-stage RBF network construction based on particle swarm optimization

The conventional radial basis function (RBF) network optimization methods, such as orthogonal least squares or the two-stage selection, can produce a sparse network with satisfactory generalization capability. However, the RBF width, as a nonlinear parameter in the network, is not easy to determine. In the aforementioned methods, the width is always pre-determined, either by trial-and-error, or generated randomly. Furthermore, all hidden nodes share the same RBF width. This will inevitably reduce the network performance, and more RBF centres may then be needed to meet a desired modelling specification. In this paper we investigate a new two-stage construction algorithm for RBF networks. It utilizes the particle swarm optimization method to search for the optimal RBF centres and their associated widths. Although the new method needs more computation than conventional approaches, it can greatly reduce the model size and improve model generalization performance. The effectiveness of the proposed technique is confirmed by two numerical simulation examples.

Real Time Economic and Emission Dispatch using RBF Network with OLS and MPSO Algorithms

paper proposes a new approach to real time economic and emission dispatch by using orthogonal least-squares (OLS) and modified particle swarm optimization (MPSO) algorithms to construct the radial basis function (RBF) network. The objectives considered are fuel cost and NOx/CO2 emissions. The RBF network is composed of input, hidden, and output layers. The OLS algorithm provides a simple and efficient means for fitting radial basis function networks. The MPSO algorithm is implemented to tune the parameters in the network, including the dilation and translation of RBF centers and the weights between the hidden and output layer. The proposed approach has been tested on the IEEE 30-bus six-generator system. Testing results indicate that the proposed approach can make a quick response and yield accurate Real time economic and emission solutions.

A Fast Incremental Learning for Radial Basis Function Networks Using Local Linear Regression

To avoid the catastrophic interference in incremental learning, we have proposed Resource Allocating Network with Long Term Memory (RAN-LTM). In RAN-LTM, not only new training data but also some memory items stored in long-term memory are trained either by a gradient descent algorithm or by solving a linear regression problem. In the latter approach, radial basis function (RBF) centers are not trained but selected based on output errors when connection weights are updated. The proposed incremental learning algorithm belongs to the latter approach where the errors not only for a training data but also for several retrieved memory items and pseudo training data are minimized to suppress the catastrophic interference. The novelty of the proposed algorithm is that connection weights to be learned are restricted based on RBF activation in order to improve the efficiency in learning time and memory size. We evaluate the performance of the proposed algorithm in one-dimensional and multi-dimensional function approximation problems in terms of approximation accuracy, learning time, and average memory size. The experimental results demonstrate that the proposed algorithm can learn fast and have good performance with less memory size compared to memory-based learning methods.