Radial Basis Function Approach Research Articles

A response surface method-based hybrid optimizer

In this article, we describe a hybrid optimizer based on a highly accurate response surface method, which uses several radial basis functions and polynomials as interpolants. The response surface is capable to interpolate linear as well as highly non-linear functions in multi-dimensional spaces having up to 500 dimensions. The accuracy, robustness, efficiency, transparency and conceptual simplicity are discussed. Based on the extensive testing performed on 296 test functions, the radial basis functions (RBFs) approach seems computationally easy to implement and results are superior, requiring small computing time. The performance of the RBF approximation is compared with wavelets neural networks for several selected test cases and the optimizer is compared with other hybrid optimizers, as well as with the IOSO commercial code.

A Radial Basis Function Approach to Credit Barrier Model

Albanese and et. al. (2003) and Avellaneda and Zhu (2001) develop the framework of Credit Barrier Model. They provide special solutions to the model in case of simple stochastic structure. The technical aspect of the model remains wide open for general stochastic structure that is crucial when the model is required to calibrate with aggregate amount of empirical data. This paper provides a technical solution to this problem with the use of radial basis function (RBF). This paper discusses the numerical implementation of the credit barrier model using the RBF method. It also demonstrates that the RBF method is numerically tractable in this problem and allows in the model richer stochastic structure capable of capturing relevant market information.

A Stable Algorithm for Flat Radial Basis Functions on a Sphere

When radial basis functions (RBFs) are made increasingly flat, the interpolation error typically decreases steadily until some point when Runge-type oscillations either halt or reverse this trend. Because the most obvious method to calculate an RBF interpolant becomes a numerically unstable algorithm for a stable problem in the case of near-flat basis functions, there will typically also be a separate point at which disastrous ill-conditioning enters. We introduce here a new method, RBF-QR, which entirely eliminates such ill-conditioning, and we apply it in the special case when the data points are distributed over the surface of a sphere. This algorithm works even for thousands of node points, and it allows the RBF shape parameter to be optimized without the limitations imposed by stability concerns. Since interpolation in the flat RBF limit on a sphere is found to coincide with spherical harmonics interpolation, new insights are gained as to why the RBF approach (with nonflat basis functions) often is the more accurate of the two methods.

Nondestructive determination of compound amoxicillin powder by NIR spectroscopy with the aid of chemometrics

Near-infrared (NIR) spectroscopy, in combination with chemometrics, enables nondestructive analysis of solid samples without time-consuming sample preparation methods. A new method for the nondestructive determination of compound amoxicillin powder drug via NIR spectroscopy combined with an improved neural network model based on principal component analysis (PCA) and radial basis function (RBF) neural networks is investigated. The PCA technique is applied to extraction relevant features from lots of spectra data in order to reduce the input variables of the RBF neural networks. Various optimum principal component analysis–radial basis function (PCA–RBF) network models based on conventional spectra and preprocessing spectra (standard normal variate (SNV) and multiplicative scatter correction (MSC)) have been established and compared. Principal component regression (PCR) and partial least squares (PLS) multivariate calibrations are also used, which are compared with PCA–RBF neural networks. Experiment results show that the proposed PCA–RBF method is more efficient than PCR and PLS multivariate calibrations. And the PCA–RBF approach with SNV preprocessing spectra is found to provide the best performance.

A parallel time stepping approach using meshfree approximations for pricing options with non-smooth payoffs

In this paper we consider a meshfree radial basis function approach for the valuation of pricing options with non-smooth payoffs. By taking advantage of parallel architecture, a strongly stable and highly accurate time stepping method is developed with computational complexity comparable to the implicit Euler method implemented concurrently on each processor. This, in collusion with the radial basis function approach, provides an efficient and reliable valuation of exotic options, such as American digital options.

A radial basis function approach for the free vibration analysis of functionally graded plates using a refined theory

The free vibration analysis of functionally graded plates is performed by the multiquadric radial basis function method and a higher-order shear deformation theory. The multiquadric method is a truly meshless method, allowing a fast and simple domain and boundary discretization. We use an homogenization technique for material properties based on the Mori–Tanaka scheme. Numerical tests show that the method is reliable and produces good results.

Indirect T‐Trefftz and F‐Trefftz methods for solving boundary value problem of Poisson equation

The Poisson equation can be solved by first finding a particular solution and then solving the resulting Laplace equation. In this paper, a computational procedure based on the Trefftz method is developed to solve the Poisson equation for two‐dimensional domains. The radial basis function approach is used to find an approximate particular solution for the Poisson equation. Then, two kinds of Trefftz methods, the T‐Trefftz method and F‐Trefftz method, are adopted to solve the resulting Laplace equation. In order to deal with the possible ill‐posed behaviors existing in the Trefftz methods, the truncated singular value decomposition method and L‐curve concept are both employed. The Poisson equation of the type, ?2 u = f(x, u), in which x is the position and u is the dependent variable, is solved by the iterative procedure. Numerical examples are provided to show the validity of the proposed numerical methods and some interesting phenomena are carefully discussed while solving the Helmholtz equation as a Poisson equation. It is concluded that the F‐Trefftz method can deal with a multiply connected domain with genus p(p > 1) while the T‐Trefftz method can only deal with a multiply connected domain with genus 1 if the domain partition technique is not adopted.

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.

Kernal Width Selection for SVM Classification

The most critical component of kernel-based learning algorithms is the choice of an appropriate kernel and its optimal parameters. In this paper, we propose a rule-based meta-learning approach for automatic radial basis function (RBF) kernel and its parameter selection for Support Vector Machine (SVM) classification. First, the best parameter selection is considered on the basis of prior information of the data with the help of Maximum Likelihood (ML) method and Nelder-Mead (N-M) simplex method. Then, the new rule-based meta-learning approach is constructed and tested on different sizes of 112 datasets with binary class as well as multi-class classification problems. We observe that our rule-based methodology provides significant improvement of computational time as well as accuracy in some specific cases.

Toward the optimization of operating conditions for hydrogen polymer electrolyte fuel cells

A study is performed to find the optimal operating conditions of hydrogen polymer electrolyte fuel cells using an efficient optimization approach based on validated multi-resolution fuel cell simulation tool developed in house. Through the design of experiment method, a set of designed simulation runs were carried out using the fuel cell simulation tool. Based on the simulation results, an analytic metamodel was then constructed using the radial basis function approach. A feasible sequential quadratic programming scheme was then employed to optimize the metamodel to achieve the global optimal solutions. To illustrate the optimization approach, four control parameters including cell temperature, cathode stoichiometry, cathode pressure, and cathode relative humidity were considered. The optimization objective is defined as the maximization of the overall efficiency of the fuel cell system under ideal or realistic system assumptions. The study shows that different optimal solutions exist for different system assumptions, as well as different current loading levels, classified into small, medium, and large current densities. The approach adopted in this study is generic and can be readily applied to a larger number of control parameters and further to the fuel cell design optimizations.

Extended Radial Basis Functions: More Flexible and Effective Metamodeling

Economic competitiveness is driving industry to new frontiers of engineering design. Robustness and reliability based design, multidisciplinary simulation based design, increased complexity and sophistication of our design, and optimization aided design are four such areas, which are seriously challenging our ability to keep pace with the need to adequately model the systems we seek to design – in spite of the exponential growth of computing power. An emerging consensus within the community is that the effective development of computationally benign models (metamodels) will help us navigate the challenging road ahead. The process of constructing metamodels for computationally expensive models, such as finite element, aerodynamic, and heat transfer models, is representative of the tasks we must address. Among the available metamodeling techniques, Radial Basis Functions (RBFs) have recently generated much interest for their effectiveness and versatility. Radial basis functions offer numerous advantages over traditional the Response Surface Methodology (RSM), including their ability to effectively generate multidimensional interpolative approximations. This paper, however, shows how the typical RBF approach lacks the critical flexibility required to handle the wide variety of complex models arising from the use of advanced techniques, such as uncertainty handling and multiobjective optimization, often encountered in modern design. Further, in this paper, we propose a powerful new approach – the Extended Radial Basis Function (E-RBF) approach – that provides the designer with significant flexibility and freedom in the metamodeling process, compared to conventional RBFs. Examples are provided that demonstrate the effectiveness of the new approach, and explore its potential superiority over traditional RBF and response surface methodologies. Initial investigation indicates that the E-RBF possesses unique and novel properties not available in any other single method.

Adaptive online correction and interpolation of quadrature encoder signals using radial basis functions

This paper considers the development of an adaptive online approach for the correction and interpolation of quadrature encoder signals, suitable for application to precision motion control systems. It is based on the use of a two-stage double-layered radial basis function (RBF) neural network. The first RBF stage is used to adaptively correct for the imperfections in the encoder signals such as mean, phase offsets, amplitude deviation and waveform distortion. The second RBF stage serves as the inferencing machine to adaptively map the quadrature encoder signals to higher order sinusoids, thus, enabling intermediate positions to be derived. Experimental and simulation results are provided to verify the effectiveness of the RBF approach.

Radial basis function approach to nonlinear Granger causality of time series

We consider an extension of Granger causality to nonlinear bivariate time series. In this frame, if the prediction error of the first time series is reduced by including measurements from the second time series, then the second time series is said to have a causal influence on the first one. Not all the nonlinear prediction schemes are suitable to evaluate causality; indeed, not all of them allow one to quantify how much knowledge of the other time series counts to improve prediction error. We present an approach with bivariate time series modeled by a generalization of radial basis functions and show its application to a pair of unidirectionally coupled chaotic maps and to physiological examples.

Local Function Approximation in Evolutionary Algorithms for the Optimization of Costly Functions

We develop an approach for the optimization of continuous costly functions that uses a space-filling experimental design and local function approximation to reduce the number of function evaluations in an evolutionary algorithm. Our approach is to estimate the objective function value of an offspring by fitting a function approximation model over the k nearest previously evaluated points, where k=(d+1)(d+2)/2 and d is the dimension of the problem. The estimated function values are used to screen offspring to identify the most promising ones for function evaluation. To fit function approximation models, a symmetric Latin hypercube design (SLHD) is used to determine initial points for function evaluation. We compared the performance of an evolution strategy (ES) with local quadratic approximation, an ES with local cubic radial basis function (RBF) interpolation, an ES whose initial parent population comes from an SLHD, and a conventional ES. These algorithms were applied to a twelve-dimensional (12-D) groundwater bioremediation problem involving a complex nonlinear finite-element simulation model. The performances of these algorithms were also compared on the Dixon-Szego test functions and on the ten-dimensional (10-D) Rastrigin and Ackley test functions. All comparisons involve analysis of variance (ANOVA) and the computation of simultaneous confidence intervals. The results indicate that ES algorithms with local approximation were significantly better than conventional ES algorithms and ES algorithms initialized by SLHDs on all Dixon-Szego test functions except for Goldstein-Price. However, for the more difficult 10-D and 12-D functions, only the cubic RBF approach was successful in improving the performance of an ES. Moreover, the results also suggest that the cubic RBF approach is superior to the quadratic approximation approach on all test functions and the difference in performance is statistically significant for all test functions with dimension d/spl ges/4.

A radial basis function approach in the meshless local Petrov-Galerkin method for Euler-Bernoulli beam problems

A meshless local Petrov-Galerkin (MLPG) method that uses radial basis functions rather than generalized moving least squares (GMLS) interpolations to develop the trial functions in the study of Euler-Bernoulli beam problems is presented. The use of radial basis functions (RBF) in meshless methods is demonstrated for C1 problems for the first time. This interpolation choice yields a computationally simpler method as fewer matrix inversions and multiplications are required than when GMLS interpolations are used. Test functions are chosen as simple weight functions as in the conventional MLPG method. Patch tests, mixed boundary value problems, and problems with complex loading conditions are considered. The radial basis MLPG method yields accurate results for deflections, slopes, moments, and shear forces, and the accuracy of these results is better than that obtained using the conventional MLPG method.

A deformable digital brain atlas system according to Talairach and Tournoux

Brain atlases are valuable tools which assist neurosurgeons during the planning of an intervention. Since a printed atlas book has several disadvantages—among them the difficulty to map the information onto a patient’s individual anatomy—we have developed a digital version of the well-established stereotaxic brain atlas of Talairach and Tournoux. Our atlas system is mainly dedicated to assist neurosurgical planning, and its benefits are: (i) a three-dimensional (3D) representation of most brain structures contained in the Talairach atlas; (ii) a nonrigid matching capability which warps the standard atlas anatomy to an individual brain magnetic resonance imaging (MRI) dataset in a few minutes and which is able to take deformations due to tumors into account; (iii) the integration of several sources of neuroanatomical knowledge; (iv) an interface to a navigation system which allows utilization of atlas information intraoperatively. In this paper we outline the algorithm we have developed to achieve 3D surface models of the brain structures. Moreover, we describe the nonrigid matching method which consists of two tasks: firstly, point correspondences between the atlas and the patient are established in an automatic fashion, and secondly these displacement vectors are interpolated using a radial basis function approach to form a continuous transformation function. To generate appropriate target structures for the first of these tasks, we implemented a quick segmentation tool which is capable to segment the cortex and ventricles in less than 5 min. An evaluation shows that our nonrigid approach is more precise than the conventional piecewise linear matching, though it should be further improved for the region around the deep grey nuclei. Summarizing, we developed a Win32 program which permits the convenient and fast application of standardized anatomy to individual brains which potentially contain tumors.

A Fault Classification Method by RBF Neural Network with OLS Learning Procedure

This paper presents a new approach to identify fault types and phases. A transmission line fault classification method based on a radial basis function (RBF) neural network with orthogonal-least-square (OLS) learning procedure was used to identify various patterns of associated voltages and currents. The RBF neural network was also compared with the back-propagation (BP) neural network in this paper. It is shown that the RBF approach can provide a fast and precise operation for various faults. The simulation results also show that the proposed approach can be used as an effective tool for high speed relaying.

A Radial Basis Function Approach to Pattern Recognition and Its Applications

Pattern recognition is one of the most common problems encountered in engineering and scientific disciplines, which involves developing prediction or classification models from historic data or training samples. This paper introduces a new approach, called the Representational Capability (RC) algorithm, to handle pattern recognition problems using radial basis function (RBF) models. The RC algorithm has been developed based on the mathematical properties of the interpolation and design matrices of RBF models. The model development process based on this algorithm not only yields the best model in the sense of balancing its parsimony and generalization ability, but also provides insights into the design process by employing a design parameter (δ). We discuss the RC algorithm and its use at length via an illustrative example. In addition, RBF classification models are developed for heart disease diagnosis.

A combined method for the global optimization using radial basis function and deterministic approach

The present paper proposes a combined method for the global optimization of realistic engineering problems using radial basis functions and a deterministic approach. A die mold model for orientation of magnetic powder is optimized to show the effectiveness of the proposed method.

Wavelet transforms and neural networks for compression and recognition

Robust recognition for image and speech processing needs data compression that preserves features. To accomplish this, we have utilized the discrete wavelet transforms and the continuous wavelet transforms (CWT) together with artificial neural networks (ANN) to achieve automatic pattern recognition. Our approach is motivated by the mathematical analog of the CWT to the human hearing and visual systems, e.g., the so-called Mexican hat and Gabor functions, Gaussian window, respectively. We develop an ANN method to construct an optimum mother wavelet that can organize sensor input data in the multiresolution format that seems to become essential for brainstyle computing. In one realization, the architecture of our ANN is similar to that of a radial basis function approach, except that each node is a wavelet having three learnable parameters: weight W ij, scale a, and shift b. The node is not a McCullouch-Pitts neuron but a “wave-on”. We still use a supervised learning conjugate gradient descent algorithm in these parameters to construct a “super-mother” wavelet from a superposition of a set of waveons-mother wavelets. Using these techniques, we can accomplish the signal-enhanced and feature-preserving compression, e.g., on the infrared images, that avoids the overtraining and overfitting that have plagued ANN's ability to generalize and abstract information.