Local modelling with radial basis function networks

Different types of radial basis function networks (RBFN) training algorithms are described and compared. Advantages and drawbacks of some of these algorithms are demonstrated on simulated and real data. Interpretability of the final models is emphasized.     

Using radial basis function neural networks to recognize shifts in correlated manufacturing process parameters

Traditional statistical process control (SPC) techniues of control charting are not applicable in many process industries because data from these facilities are autocorrelated. Therefore the reduction in process variability obtained through the use of SPC techniques has not been realized in process industries. Techniques are needed to serve the same function as SPC control charts, that is to identify process shifts, in correlated parameters. Radial basis function neural networks were developed to identify shifts in process parameter values from papermaking and viscosity data sets available in the literature. Time series residual control charts were also developed for the data sets. Networks were successful at separating data that were shifted 1.5 and 2 standard deviations from nonshifted data for both the papermaking and viscosity parameter values. The network developed on the basis of the papermaking data set was also able to separate shifts of 1 standard deviation from nonshifted data. The SPC control charts were not able to identify the same process shifts. The radial basis function neural networks can be used to identify shifts in process parameters, thus allowing improved process control in manufacturing processes that generate correlated process data.     

Detection of bearing defects in three-phase induction motors using Park’s transform and radial basis function neural networks

This paper investigates the application of induction motor stator current signature analysis (MCSA) using Park’s transform for the detection of rolling element bearing damages in three-phase induction motor. The paper first discusses bearing faults and Park’s transform, and then gives a brief overview of the radial basis function (RBF) neural networks algorithm. Finally, system information and the experimental results are presented. Data acquisition and Park’s transform algorithm are achieved by using LabVIEW and the neural network algorithm is achieved by using MATLAB programming language. Experimental results show that it is possible to detect bearing damage in induction motors using an ANN algorithm.     

Radial basis function neural network for pulse radar detection

A new approach using a radial basis function network (RBFN) for pulse compression is proposed. In the study, networks using 13-element Barker code, 35-element Barker code and 21-bit optimal sequences have been implemented. In training these networks, the RBFN-based learning algorithm was used. Simulation results show that RBFN approach has significant improvement in error convergence speed (very low training error), superior signal-to-sidelobe ratios, good noise rejection performance, improved misalignment performance, good range resolution ability and improved Doppler shift performance compared to other neural network approaches such as back-propagation, extended Kalman filter and autocorrelation function based learning algorithms. The proposed neural network approach provides a robust mean for pulse radar tracking     

On the optimal design of radial basis function neural networks for the analysis of nonlinear stochastic systems

In this paper, an iterative selection strategy of Gaussian neurons for radial basis function neural networks (RBFNN) is proposed when the RBFNN method is applied to obtain the steady-state solution of the Fokker–Planck–Kolmogorov (FPK) equation. A performance index is introduced to rank neurons. Top rank neurons are selected, leading to a RBFNN with optimal number and locations of Gaussian neurons for the FPK equation under consideration. The statistical properties of the performance index are studied. It is found that the index assigned to the $j$th neuron is proportional to the probability of the system falling into the small neighborhood of the mean of this neuron as well as proportional to the weight of the neuron. The RBFNN method with the optimally selected neurons is then applied to several challenging examples of nonlinear stochastic systems in 2, 3 and 4 dimensional state space. The RBFNN solutions are also compared with the results of extensive Monte Carlo simulations. It is observed that the RBFNN method with optimally selected neurons by the proposed iterative algorithm is much more efficient than the RBFNN method with uniformly distributed neurons, and is very accurate in terms of the root mean squared (RMS) errors of the FPK equation or the RMS errors of the PDF solution compared with simulation results.     

Classification of polynomial-shaped measurement signals using abackpropagation neural network

Smoothly varying signals are frequently encountered in the field of instrumentation and measurement, and they can be accurately modeled by low-order polynomials. The order identification is difficult when the measured noisy signal has frequent order variations in the underlying polynomial. In this paper, we introduce a flexible real-time order estimator, which is based on a backpropagation neural network     

Options valuation by using radial basis function approximation

This paper describes the valuation scheme of European, barrier, and Asian options of single asset by using radial basis function approximation. The option prices are governed with Black–Scholes equation. The equation is discretized with Crank–Nicolson scheme and then, the option price is approximated with the radial basis functions with unknown parameters. In the European and the barrier options, the prices are governed with Black–Scholes equation. The governing option of the Asian option, however, is different from them of the others. In that case, one has to adopt the other radial basis functions than that for the original Black–Scholes equation.Finally, numerical results are compared with theoretical and finite difference solutions in order to confirm the validity of the present formulation.     

Analysis of anisotropic plate using multiquadric radial basis function

In this paper, multiquadric radial basis function is used for dynamic and static analysis of anisotropic plates. Multiquadric radial basis function is applied for spatial discretization and Newmark implicit scheme is used for temporal discretization. The spatial discretization of the differential equations generates greater number of algebraic equations than the unknown coefficients. The multiple linear regression analysis, which is based on the least squares error norm, is employed to obtain the coefficients. Considering simple supported boundary conditions, an analogy between isotropic skew plates and rectangular anisotropic plates is used for solutions. The effect of fiber orientation is observed in clamped square and rectangular anisotropic plates. The results obtained by this method are compared with those obtained by other analytical methods.     

Self-adaptive radial basis function neural network for short-term electricity price forecasting

Effective and reliable electricity price forecast is essential for market participants in setting up appropriate risk management plans in an electricity market. A reliable price prediction model based on an advanced self-adaptive radial basis function (RBF) neural network is presented. The proposed RBF neural network model is trained by fuzzy c-means and differential evolution is used to auto-configure the structure of networks and obtain the model parameters. With these techniques, the number of neurons, cluster centres and radii of the hidden layer, and the output weights can be automatically calculated efficiently. Meanwhile, the moving window wavelet de-noising technique is introduced to improve the network performance as well. This learning approach is proven to be effective by applying the RBF neural network in predicting of Mackey-Glass chaos time series and forecasting of the electricity regional reference price from the Queensland electricity market of the Australian National Electricity Market.     

Solving high order ordinary differential equations with radial basis function networks

This paper is concerned with the application of radial basis function networks (RBFNs) for numerical solution of high order ordinary differential equations (ODEs). Two unsymmetric RBF collocation schemes, namely the usual direct approach based on a differentiation process and the proposed indirect approach based on an integration process, are developed to solve high order ODEs directly and the latter is found to be considerably superior to the former. Good accuracy and high rate of convergence are obtained with the proposed indirect method. Copyright © 2004 John Wiley & Sons, Ltd.     

Efficient image compression techniques for compressing multimodal medical images using neural network radial basis function approach

Image compression technique is used to reduce the number of bits required in representing image, which helps to reduce the storage space and transmission cost. Image compression techniques are widely used in many applications especially, medical field. Large amount of medical image sequences are available in various hospitals and medical organizations. Large images can be compressed into smaller size images, so that the memory occupation of the image is considerably reduced. Image compression techniques are used to reduce the number of pixels in the input image, which is also used to reduce the broadcast and transmission cost in efficient form. This is capable by compressing different types of medical images giving better compression ratio (CR), low mean square error (MSE), bits per pixel (BPP), high peak signal to noise ratio (PSNR), input image memory size and size of the compressed image, minimum memory requirement and computational time. The pixels and the other contents of the images are less variant during the compression process. This work outlines the different compression methods such as Huffman, fractal, neural network back propagation (NNBP) and neural network radial basis function (NNRBF) applied to medical images such as MR and CT images. Experimental results show that the NNRBF technique achieves a higher CR, BPP and PSNR, with less MSE on CT and MR images when compared with Huffman, fractal and NNBP techniques.     

Sequential approximate robust design optimization using radial basis function network

This paper proposes a sequential approximate robust design optimization (SARDO) with the radial basis function (RBF) network. In RDO, the mean and the standard deviation of objective should be minimized simultaneously. Therefore, the RDO is generally formulated as bi-objective design optimization. Our goal is to find a robust optimal solution with a small number of function evaluations, not identifying a set of Pareto-optimal solution using Multi-Objective Evolutionary Algorithms. The weighted sum is often used to find a robust optimal solution. In contrast, the weighted lp norm method is used in this paper. Through illustrative examples, some validations of the weighted lp norm method to the RDO are clarified. Next, SARDO with the RBF network is discussed. In general, the standard deviation of functions is obtained by using the finite difference method. Thus, in order to obtain the standard deviation of functions, the finite difference method is directly applied to the response surface. High accuracy of the finite difference method will leads to highly accurate robust optimal solution. In order to avoid the inaccurate numerical calculation, the standard deviation is expressed by only the Gaussian kernel. As the result, it is expected that a highly accurate robust optimal solution can be found with a small number of function evaluations. Through numerical examples, the validity of the proposed approach is examined. Finally, the variable blank holder force trajectory for reducing springback is examined.     

基于径向基函数神经网络的CFRP切削力预测

碳纤维增强树脂基复合材料(CFRP)加工中基体相极易因切削力过大而破坏,并迅速扩展至加工表面以下而形成损伤。为了准确预测其切削力并加以控制,基于实验切削力数据建立了人工神经网络切削力模型,预测了不同纤维角度、切削深度和刀具角度下加工CFRP的切削力变化规律,并完成了不同刀具角度及切削参数下典型纤维角度CFRP单向板的直角切削实验,对预测模型进行验证,其预测精度可达85%以上。结合成屑过程在线显微观测结果可知:纤维角度是影响CFRP切削力的主要因素, 0°~135°范围内,切屑形成方式为切断型和开裂后弯断型;切削力随纤维角度增大呈先减小后增大的趋势, 135°时最大,随切削深度增加,切削力总体呈增大趋势。      

Adaptive, optical, radial basis function neural network for handwritten digit recognition

An adaptive, optical, radial basis function classifier for handwritten digit recognition is experimentally demonstrated. We describe a spatially multiplexed system that incorporates an on-line adaptation of weights and basis function widths to provide robustness to optical system imperfections and system noise. The optical system computes the Euclidean distances between a 100-dimensional input vector and 198 stored reference patterns in parallel by using dual vector-matrix multipliers and a contrastreversing spatial light modulator. Software is used to emulate an electronic chip that performs the on-line learning of the weights and basis function widths. An experimental recognition rate of 92.7% correct out of 300 testing samples is achieved with the adaptive training, versus 31.0% correct for nonadaptive training. We compare the experimental results with a detailed computer model of the system in order to analyze the influence of various noise sources on the system performance.     

Classification of motor imagery electroencephalography signals using continuous small convolutional neural network

As an important part of brain-computer interface (BCI), the electroencephalography (EEG) technology of motor imagery (MI) has been gradually recognized for its great theoretical value and practical application. In this study, in view of the different MI tasks corresponding to active region of the EEG signals, we adopt a two-dimensional form including time, frequency, and electrode location information, then we design a classification method containing continuous small convolutional neural network (CSCNN). This method is mainly used for feature extraction through continuous small convolutional kernels and one rectangle convolutional kernel, and the softmax classifier for classification. In the experiment, classification accuracy and kappa value are used as evaluation criteria to verify the effectiveness of the method proposed in this study. For classification accuracy, BCI competition IV data set 2b is used to compare with the other five classification methods (CNN, CNN-SAE, stacked autoencoder [SAE], support vector machine [SVM], and one-dimensional convolution combined with gated recurrent unit [1DCGRU]). The results demonstrate that the overall accuracy of CSCNN is higher than other methods, and CSCNN obtains an average accuracy of 82.8%. For kappa value, BCI competition IV data set 2b is used to compare with the other three methods (filter bank common spatial pattern [FBCSP], Twin SVM, and CNN-SAE). The performance of CSCNN is better with an average value of 0.663. Overall, the results show that CSCNN maintains a small number of parameters and improves the classification accuracy.     

Radial basis function neural networks solution for stationary probability density function of nonlinear stochastic systems

A radial basis function neural networks (RBF-NN) solution of the reduced Fokker–Planck-Kolmogorov (FPK) equation is proposed in this paper. The activation functions consist of normalized Gaussian probability density functions (PDFs). The use of normalized Gaussian PDFs leads to a simple constraint on the coefficients for normalization of the RBF-NN solution, which as a constraint is imposed with the help of the method of Lagrange multiplier. The relationship between the proposed RBF-NN PDF solution and the generalized cell mapping with short-time Gaussian approximation is discussed, which provides a justification for Gaussian PDFs with varying means and variances in the state space. The optimal number of neurons or activation functions, which leads to the smallest error, is investigated. Four examples are presented to show the effectiveness of the proposed solution method. The results indicate that the proposed solution method is a very efficient and accurate way to compute the stationary PDF of nonlinear stochastic systems. It is also found that the distribution of the optimal coefficients as a function of the mean of Gaussian activation functions is similar to the steady-state PDF solution. Finally, we should point out that an important advantage of the RBF-NN method over methods such as finite element and finite difference is its ability to obtain solutions of the FPK equation for multi-degree-of-freedom stochastic systems.     

Solving high‐order partial differential equations with indirect radial basis function networks

This paper reports a new numerical method based on radial basis function networks (RBFNs) for solving high‐order partial differential equations (PDEs). The variables and their derivatives in the governing equations are represented by integrated RBFNs. The use of integration in constructing neural networks allows the straightforward implementation of multiple boundary conditions and the accurate approximation of high‐order derivatives. The proposed RBFN method is verified successfully through the solution of thin‐plate bending and viscous flow problems which are governed by biharmonic equations. For thermally driven cavity flows, the solutions are obtained up to a high Rayleigh number of 10⁷. Copyright © 2005 John Wiley & Sons, Ltd.     

Boundary collocation method for acoustic eigenanalysis of three-dimensional cavities using radial basis function

 In this paper, a theoretical formulation based on the collocation method is presented for the eigenanalysis of arbitrarily shaped acoustic cavities. This article can be seen as the extension of non-dimensional influence function (NDIF) method proposed by Kang et al. (1999, 2000a) extending from two-dimensional to three-dimensional case. Unlike the conventional collocation techniques in the literature, approximate functions used in this paper are two-point functions of which the argument is only the distance between the two points. Based on this radial basis expansion, the acoustic field can be represented more exactly. The field solution is obtained through the linear superposition of radial basis function, and boundary conditions can be applied at the discrete points. The influence matrix is symmetric regardless of the boundary shape of the cavity, and the calculated eigenvalues rapidly converge to the exact values by using only a few boundary nodes. Moreover, the method results in true and spurious boundary modes, which can be obtained from the right and left unitary vectors of singular value decomposition, respectively. By employing the updating term and document of singular value decomposition (SVD), the true and spurious eigensolutions can be sorted out, respectively. The validity of the proposed method are illustrated through several numerical examples. Received: 29 August 2001 / Accepted: 27 June 2002 Financial support from the National Science Council under Grant No. NSC-90-2211-E-019-006 for National Taiwan Ocean University is gratefully acknowledged.