Robustness of radial basis functions

Neural networks are intended to be used in future nanoelectronic technology since these architectures seem to be robust to malfunctioning elements and noise in its inputs and parameters. In this work, the robustness of radial basis function networks is analyzed in order to operate in noisy and unreliable environment. Furthermore, upper bounds on the mean square error under noise contaminated parameters and inputs are determined if the network parameters are constrained. To achieve robuster neural network architectures fundamental methods are introduced to identify sensitive parameters and neurons.     

Metamodeling using extended radial basis functions: a comparative approach

The process of constructing computationally benign approximations of expensive computer simulation codes, or metamodeling, is a critical component of several large-scale multidisciplinary design optimization (MDO) approaches. Such applications typically involve complex models, such as finite elements, computational fluid dynamics, or chemical processes. The decision regarding the most appropriate metamodeling approach usually depends on the type of application. However, several newly proposed kernel-based metamodeling approaches can provide consistently accurate performance for a wide variety of applications. The authors recently proposed one such novel and effective metamodeling approach—the extended radial basis function (E-RBF) approach—and reported highly promising results. To further understand the advantages and limitations of this new approach, we compare its performance to that of the typical RBF approach, and another closely related method—kriging. Several test functions with varying problem dimensions and degrees of nonlinearity are used to compare the accuracies of the metamodels using these metamodeling approaches. We consider several performance criteria such as metamodel accuracy, effect of sampling technique, effect of sample size, effect of problem dimension, and computational complexity. The results suggest that the E-RBF approach is a potentially powerful metamodeling approach for MDO-based applications, as well as other classes of computationally intensive applications.     

Fast evaluation of radial basis functions: I

This paper describes some new techniques for the rapid evaluation and fitting of radial basic functions. The techniques are based on the hierarchical and multipole expansions recently introduced by several authors for the calculation of many-body potentials. Consider in particular the N term thin-plate spline, s(x) = Σ_j=1^N d_jφ(x−x_j), where φ(u) = |u|²log|u|, in 2-dimensions. The direct evaluation of s at a single extra point requires an extra O(N) operations. This paper shows that, with judicious use of series expansions, the incremental cost of evaluating s(x) to within precision ε, can be cut to O(1+|log ε|) operations. In particular, if A is the interpolation matrix, a_i,j = φ(x_i−x_j, the technique allows computation of the matrix-vector product Ad in O(N), rather than the previously required O(N²) operations, and using only O(N) storage. Fast, storage-efficient, computation of this matrix-vector product makes pre-conditioned conjugate-gradient methods very attractive as solvers of the interpolation equations, Ad = y, when N is large.     

Gaussian radial basis function interpolant for the different data sites and basis centers

The present study introduces results about unique solvability of Gaussian RBF interpolation with the different data sites and basis centers. For \( N=2 \), we show that the interpolation matrix is singular only when the vector of difference between basis centers and the vector of difference between data sites are perpendicular to each other. For \(N>2\), we show certain states that the interpolation matrix is singular, then we provide several mild conditions which guarantee the interpolation matrix to be non-singular. We propose an algorithm to describe how to choose the basis centers and data sites. The results show that if the basis centers are chosen different from the data sites, the interpolation is uniquely solvable under mild conditions.     

Using radial basis functions to approximate a function and itserror bounds

A novel network called the validity index network (VI net) is presented. The VI net, derived from radial basis function networks, fits functions and calculates confidence intervals for its predictions, indicating local regions of poor fit and extrapolation     

Indexed families of functionals and gaussian radial basis functions

We report on results concerning the capabilities of gaussian radial basis function networks in the setting of inner product spaces that need not be finite dimensional. Specifically, we show that important indexed families of functionals can be uniformly approximated, with the approximation uniform also with respect to the index. Applications are described concerning the classification of signals and the synthesis of reconfigurable classifiers.     

A new class of oscillatory radial basis functions

Radial basis functions (RBFs) form a primary tool for multivariate interpolation, and they are also receiving increased attention for solving PDEs on irregular domains. Traditionally, only nonoscillatory radial functions have been considered. We find here that a certain class of oscillatory radial functions (including Gaussians as a special case) leads to nonsingular interpolants with intriguing features especially as they are scaled to become increasingly flat. This flat limit is important in that it generalizes traditional spectral methods to completely general node layouts. Interpolants based on the new radial functions appear immune to many or possibly all cases of divergence that in this limit can arise with other standard types of radial functions (such as multiquadrics and inverse multiquadratics).     

An interactive approach for CBIR using a network of radial basis functions

An important requirement for constructing effective content-based image retrieval (CBIR) systems is accurate characterization of visual information. Conventional nonadaptive models, which are usually adopted for this task in simple CBIR systems, do not adequately capture all aspects of the characteristics of the human visual system. An effective way of addressing this problem is to adopt a "human-computer" interactive approach, where the users directly teach the system about what they regard as being significant image features and their own notions of image similarity. We propose a machine learning approach for this task, which allows users to directly modify query characteristics by specifying their attributes in the form of training examples. Specifically, we apply a radial-basis function (RBF) network for implementing an adaptive metric which progressively models the notion of image similarity through continual relevance feedback from users. Experimental results show that the proposed methods not only outperform conventional CBIR systems in terms of both accuracy and robustness, but also previously proposed interactive systems.     

Approximation of insurance liability contracts using radial basis functions

ABSTRACT

We introduce the Option Interpolation Model (OIM) for accurate approximation of embedded option values in insurance liabilities. Accurate approximation is required for ex-ante risk management applications. The OIM is based on interpolation with radial basis functions, which can interpolate scattered data, and does not suffer from the curse of dimensionality. To reduce computation time we present an inversion method to determine the interpolation function weights. The robustness, accuracy and efficiency of the OIM are investigated in several numerical experiments. We show that the OIM results in highly accurate approximations.     

The relationship between the Laplace transform and Fourier transform

In this note, a relationship between the one-sided Laplace transform and the one-sided Fourier transform is discussed.     

A meshless multilayer model for a coastal system by radial basis functions

A multilayer computational model for simulating three-dimensional tidal flows in coastal waters is presented in this paper. A truly meshless numerical scheme based on radial basis functions (RBFs) is employed to obtain an accurate approximation to the solution of the model. For computational efficiency in solving large-scale problems, a noniterative domain decomposition method is combined with the use of the RBFs scheme. To smooth the numerical simulation of the advection across each subdomain, the commonly used upstream technique is also incorporated. Finally, for numerical verification, the proposed multilayer model is successfully used to obtain a stable and convergent simulation of the water flows in the Pearl River Estuary of the South China Sea. The numerical approximations exhibit good agreement with the observed tidal and current data.     

高斯径向基函数重构特征对表面肌电信号识别

针对在不同动作模式下对表面肌电信号提取的特征信息总是有较大差异,而相同动作模式下提取的特征信息较为接近这一特点,提出了高斯径向基函数重构算法对肌电信号进行识别。该算法在对表面肌电信号提取特征信息后,用高斯径向基函数对特征矢量进行重构,使得重构的特征矢量的空间分布存在很大差异而直接进行识别。用该重构算法对提取的AR系数重构,然后进行识别,平均识别率为97.2%;对小波系数重构,平均识别率为99%。     

Transient analysis of composite plates by radial basis functions in a pseudospectral framework

This paper presents a study of the linear transient response of composite plates using radial basis functions and collocation method in a pseudospectral framework. The first-order shear deformation plate theory is used to define a set of algebraic equations from the equations of motion and boundary conditions. The transient analysis is performed by a Newmark algorithm. In order to assess the quality of the present numerical method, an analytical solution was also developed. Numerical tests on square and rectangular cross-ply laminated plates demonstrate that the present method produces highly accurate displacements and stresses when compared with the available results.     

Shakespeare vs. fletcher: A stylometric analysis by radial basis functions

In this paper we show, for the first time, how Radial Basis Function (RBF) network techniques can be used to explore questions surrounding authorship of historic documents. The paper illustrates the technical and practical aspects of RBF's, using data extracted from works written in the early 17th century by William Shakespeare and his contemporary John Fletcher. We also present benchmark comparisons with other standard techniques for contrast and comparison.David Lowe is Professor of Neural Computing at Aston University, UK. His research interests span from the theoretical aspects of dynamical systems theory and statistical pattern processing, to a wide range of application domains, from financial market analysis (Novel Exploitation of Neural Network Methods in Financial Markets, invited paper,World Conference on Computational Intelligence, vol. VI, pp. 3623–28, 1994) to the artificial nose (Novel Topographic Nonlinear Feature Extraction using Radial Basis Functions for Concentration Coding in the Artificial Nose,3 ^rd IEE International Conference on Artificial Neural Networks, pp. 95–99, Conference Publication number 372, The Institute of Electrical Engineers, 1993).Robert Matthews is a visiting research fellow at Aston University. His research interests include probability, number theory and astronomy. His recent paper inNature (vol. 374, pp. 681–82, 1995) somehow managed to combine all three.     

Application of radial basis functions in solving fuzzy integral equations

Neural Computing and Applications - In the present paper, a numerical method based on radial basis functions (RBFs) is proposed to approximate the solution of fuzzy integral equations. By applying...     

Preconditioned conjugate gradients,radial basis functions,and Toeplitz matrices

Radial basis functions provide highly useful and flexible interpolants to multivariate functions. Further, they are beginning to be used in the numerical solution of partial differential equations. Unfortunately, their construction requires the solution of a dense linear system. Therefore, much attention has been given to iterative methods. In this paper, we present a highly efficient preconditioner for the conjugate gradient solution of the interpolation equations generated by gridded data. Thus, our method applies to the corresponding Toeplitz matrices. The number of iterations required to achieve a given tolerance is independent of the number of variables.     

Online modeling and adaptive control of robotic manipulators using Gaussian radial basis function networks

Neural Computing and Applications - Radial basis function network (RBFN) is used in this paper for predefined trajectory control of both one-link and two-link robotic manipulators. The updating...     

Moving least squares interpolation with thin-plate splines and radial basis functions

Moving least-squares methods for interpolation or approximation of scattered data are well known, and can suffer from defects, such as flat spots in the Shepard method, and edge effects inherited from a polynomial basis in the higher degree cases. We investigate methods based on thin-plate splines and on other radial basis functions. It turns out that a small support of the weight function leads to a small support for the “spline basis” and associated efficiency in the evaluation of the approximant. The edge effects seem minimal and good interpolants of scattered data can be obtained.     

Gaussian radial basis function neural network controller of a synchronous reluctance motor in electric motorcycle applications

In this article, a sliding mode control (SMC) design based on a Gaussian radial basis function neural network (GRBFNN) is proposed for the synchronous reluctance motor (SynRM) system in electrical motorcycle applications. The conventional SMC assumes that the upper lumped boundaries of parameter variations and external disturbances are known, and the sign function is used. This causes high-frequency chattering and the high-gain phenomenon. In order to avoid these drawbacks, the proposed method utilizes the Lyapunov stability method and the steep descent rule to guarantee the convergence asymptotically, and reduce the magnitude of the chattering or avoid it completely. Finally, numerical simulations are shown to illustrate the good performance of our controller design.