Radial Basis Function Algorithm Research Articles

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.

In-situ Inspection Error Compensation for Machining Accuracy Improvement of Complex Components

In order to improve in-situ inspection accuracy of complex surface components, error factors influencing the inspection accuracy of the on-machine inspection system are discussed and error compensation methods are studied. The main errors of the inspection system are analyzed which contain geometric error of machine tools, pre-travel error of the inspection system and radius error of measuring probe. The geometric error model of a three-axis numerical control machine tool is established and each individual error value is identified and obtained through laser interferometer. For the pre-travel error of the inspection system, a radial basis function algorithm based prediction method is proposed and implemented. Based on the distribution of the pre-travel probe error obtained, an real-time compensation of the error can be performed during the actual inspection process. Considering the probe radius and its compensation of the contact-type measuring system4, a new algorithm is developed to improve the calculation accuracy of the normal vector which will affect the compensation accuracy. To verify the effectiveness of the proposed method, a complex part is exampled. Through the comparison of the in-situ inspection result with that of the CMM measurement, the proposed inspection system can achieve a great improvement of the inspection accuracy.

A Method for Vibration-Cylinder Air-Pressure Sensor Temperature Compensation Based on Radial Basis Function (RBF) Neural Network

There is a nonlinear measurement error of the vibration-cylinder air-pressure sensor when the environment temperature changes; To solve the problem, the paper carried out a research on vibration-cylinder air-pressure sensor temperature compensation based on radial basis function(RBF) neural network; The temperature error characteristics of the sensor was studied; The sensor temperature compensation of RBF neural network structures and algorithms was designed; The centers, variances, weights, and the hidden layer neuron number of the radial basis function were determined. The experiments showed that the trained RBF neural network can approximate the input-output relationship of the vibration-cylinder air-pressure sensor in high accuracy.

An improved biometrics technique based on metric learning approach

A biometrics technique based on metric learning approach is proposed in this paper to achieve higher correct classification rates under the condition that the feature of the query is very different from that of the register for a given individual. Inspired by the definition of generalized distance, the criterion of this new metric learning is defined by finding an embedding that preserves local information and obtaining a subspace that best detects the essential manifold structure. Furthermore, the two transformation matrices for the query and the register are obtained by a generalized eigen-decomposition. Experiments tested on biometric applications of CASIA(B) gait database and the UMIST face database, demonstrate that our proposed method performs better than classical metric learning methods and the current radial basis function (RBF) algorithms.

Fast and efficient second-order method for training radial basis function networks.

This paper proposes an improved second order (ISO) algorithm for training radial basis function (RBF) networks. Besides the traditional parameters, including centers, widths and output weights, the input weights on the connections between input layer and hidden layer are also adjusted during the training process. More accurate results can be obtained by increasing variable dimensions. Initial centers are chosen from training patterns and other parameters are generated randomly in limited range. Taking the advantages of fast convergence and powerful search ability of second order algorithms, the proposed ISO algorithm can normally reach smaller training/testing error with much less number of RBF units. During the computation process, quasi Hessian matrix and gradient vector are accumulated as the sum of related sub matrices and vectors, respectively. Only one Jacobian row is stored and used for multiplication, instead of the entire Jacobian matrix storage and multiplication. Memory reduction benefits the computation speed and allows the training of problems with basically unlimited number of patterns. Several practical discrete and continuous classification problems are applied to test the properties of the proposed ISO training algorithm.

Comparison of divergence-free algorithms for 3D MRI with three-directional velocity encoding

Summary Two divergence-free algorithms (finite difference and radial basis functions) are evaluated for improving the flow visualization quality of PC-VIPR 4D flow data. Both methods improve streamline length, with the finite difference method performing slightly better. Background The velocity-to-noise ratio in 4D MR Flow acquisitions can suffer, particularly for low velocity segments, because the velocity encoding settings are adjusted to the highest velocities in the volume. Blood is essentially incompressible so vascular velocity fields should be divergence free. Therefore, any remaining divergence in an acquired 3D or 4D MR flow dataset can be assumed to originate from image noise, which can be reduced by imposing divergence free conditions. Several methods have been proposed to reconstruct a divergence free velocity vector field, yet to our knowledge no comparison studies exist to explore their properties in flow MRI data. This study compares two methods in their effectiveness at improving flow visualizations, the finite difference method (FDM)[1] and the radial basis function (RBF)[2] based approach, which differ in their handling of noise and boundary conditions. Methods MR flow data were acquired in a straight tube flow phantom and the internal carotid arteries of a volunteer on a clinical 3T scanner with a 3D radially undersampled acquistion (PC-VIPR)[3]. Time resolved PCVIPR images were acquired as follows: scan time: 6 min acquired isotropic spatial resolution: 0.68 mm, TR/TE = 8.3/2.9 ms, temporal resolution: 39.5 ms. The FDM algorithm is solved with a fast Fourier solver for the Poisson equation. The RBF algorithm includes a normalized convolution, and is solved using iterative least squares. The quality of the resulting flow fields was quantified by average streamline lengths and streamline counts that extended from the emitter plane to a distal portion in the flow phantom or vessel. Data fidelity was measured by comparing flow values pre and post processing in the emitter and analysis planes. Results

Design of an aperture-coupled microstrip antenna using a hybrid neural network

In this study, an artificial neural network (ANN) model using hybrid neural network is proposed for the design of aperture-coupled microstrip antennas (ACMSAs). The new hybrid model is developed by combining radial basis function (RBF) and back-propagation algorithm (BPA). The performances evaluation of the hybrid model reveals superiority over the conventional BPA and RBF models in terms of error and time. The results obtained by the proposed model are compared with the simulation results obtained from the IE3D software package and also with the experimental results obtained from the fabricated ACMSA. The results show good agreement.

Modeling of Suspended Sediment Concentration at Kasol in India Using ANN, Fuzzy Logic, and Decision Tree Algorithms

The prediction of the sediment loading generated within a watershed is an important input in the design and management of water resources projects. High variability of hydro-climatic factors with sediment generation makes the modelling of the sediment process cum- bersome and tedious. The methods for the estimation of sediment concentration based on the properties of flow and sediment have limitations attributed to the simplification of important parameters and boundary conditions. Under such circumstances, soft computing approaches have proven to be an efficient tool in modelling the sediment concentration. The focus of this paper is to present the development of models using Artificial Neural Network (ANN) with back propagation and Levenberg-Maquardt algorithms, radial basis function (RBF), Fuzzy Logic, and decision tree algorithms such as M5 and REPTree for predicting the suspended sediment concentration at Kasol, upstream of the Bhakra reservoir, located in the Sutlej basin in northern India. The input vector to the various models using different algorithms was derived con- sidering the statistical properties such as auto-correlation function, partial auto-correlation, and cross-correlation function of the time series. It was found that the M5 model performed well compared to other soft computing techniques such as ANN, fuzzy logic, radial basis function, and REPTree investigated in this study, and results of the M5 model indicate that all ranges of sediment concentration values were simulated fairly well. This study also suggests that M5 model trees, which are analogous to piecewise linear functions, have certain advantages over other soft computing techniques because they offer more insight into the generated model, are acceptable to decision makers, and always converge. Further, the M5 model tree offers explicit expressions for use by field engineers. DOI: 10.1061/(ASCE)HE.1943-5584.0000445. © 2012 American Society of Civil Engineers. CE Database subject headings: Suspended sediment; Neural networks; Fuzzy sets; Reservoirs. Author keywords: Suspended sediment concentration; Neural networks; Fuzzy Logic; M5; REPTree; Bhakra reservoir.

Stochastic radial basis function algorithms for large-scale optimization involving expensive black-box objective and constraint functions

This paper presents a new algorithm for derivative-free optimization of expensive black-box objective functions subject to expensive black-box inequality constraints. The proposed algorithm, called ConstrLMSRBF, uses radial basis function (RBF) surrogate models and is an extension of the Local Metric Stochastic RBF (LMSRBF) algorithm by Regis and Shoemaker (2007a) [1] that can handle black-box inequality constraints. Previous algorithms for the optimization of expensive functions using surrogate models have mostly dealt with bound constrained problems where only the objective function is expensive, and so, the surrogate models are used to approximate the objective function only. In contrast, ConstrLMSRBF builds RBF surrogate models for the objective function and also for all the constraint functions in each iteration, and uses these RBF models to guide the selection of the next point where the objective and constraint functions will be evaluated. Computational results indicate that ConstrLMSRBF is better than alternative methods on 9 out of 14 test problems and on the MOPTA08 problem from the automotive industry (Jones, 2008 [2]). The MOPTA08 problem has 124 decision variables and 68 inequality constraints and is considered a large-scale problem in the area of expensive black-box optimization. The alternative methods include a Mesh Adaptive Direct Search (MADS) algorithm (Abramson and Audet, 2006 [3]; Audet and Dennis, 2006 [4]) that uses a kriging-based surrogate model, the Multistart LMSRBF algorithm by Regis and Shoemaker (2007a) [1] modified to handle black-box constraints via a penalty approach, a genetic algorithm, a pattern search algorithm, a sequential quadratic programming algorithm, and COBYLA (Powell, 1994 [5]), which is a derivative-free trust-region algorithm. Based on the results of this study, the results in Jones (2008) [2] and other approaches presented at the ISMP 2009 conference, ConstrLMSRBF appears to be among the best, if not the best, known algorithm for the MOPTA08 problem in the sense of providing the most improvement from an initial feasible solution within a very limited number of objective and constraint function evaluations.

Constructive Approximation to Multivariate Function by Decay RBF Neural Network

It is well known that single hidden layer feedforward networks with radial basis function (RBF) kernels are universal approximators when all the parameters of the networks are obtained through all kinds of algorithms. However, as observed in most neural network implementations, tuning all the parameters of the network may cause learning complicated, poor generalization, overtraining and unstable. Unlike conventional neural network theories, this brief gives a constructive proof for the fact that a decay RBF neural network with n+1 hidden neurons can interpolate n+1 multivariate samples with zero error. Then we prove that the given decay RBFs can uniformly approximate any continuous multivariate functions with arbitrary precision without training. The faster convergence and better generalization performance than conventional RBF algorithm, BP algorithm, extreme learning machine and support vector machines are shown by means of two numerical experiments.

PetRBF — A parallel O( N) algorithm for radial basis function interpolation with Gaussians

We have developed a parallel algorithm for radial basis function ( rbf) interpolation that exhibits O( N) complexity, requires O( N) storage, and scales excellently up to a thousand processes. The algorithm uses a gmres iterative solver with a restricted additive Schwarz method ( rasm) as a preconditioner and a fast matrix-vector algorithm. Previous fast rbf methods — achieving at most O( Nlog N) complexity — were developed using multiquadric and polyharmonic basis functions. In contrast, the present method uses Gaussians with a small variance with respect to the domain, but with sufficient overlap. This is a common choice in particle methods for fluid simulation, our main target application. The fast decay of the Gaussian basis function allows rapid convergence of the iterative solver even when the subdomains in the rasm are very small. At the same time we show that the accuracy of the interpolation can achieve machine precision. The present method was implemented in parallel using the petsc library (developer version). Numerical experiments demonstrate its capability in problems of rbf interpolation with more than 50 million data points, timing at 106 s (19 iterations for an error tolerance of 10 − 15 ) on 1024 processors of a Blue Gene/L (700 MHz PowerPC processors). The parallel code is freely available in the open-source model.

A REPAIR ALGORITHM FOR RADIAL BASIS FUNCTION NEURAL NETWORK AND ITS APPLICATION TO CHEMICAL OXYGEN DEMAND MODELING

This paper presents a repair algorithm for the design of a Radial Basis Function (RBF) neural network. The proposed repair RBF (RRBF) algorithm starts from a single prototype randomly initialized in the feature space. The algorithm has two main phases: an architecture learning phase and a parameter adjustment phase. The architecture learning phase uses a repair strategy based on a sensitivity analysis (SA) of the network's output to judge when and where hidden nodes should be added to the network. New nodes are added to repair the architecture when the prototype does not meet the requirements. The parameter adjustment phase uses an adjustment strategy where the capabilities of the network are improved by modifying all the weights. The algorithm is applied to two application areas: approximating a non-linear function, and modeling the key parameter, chemical oxygen demand (COD) used in the waste water treatment process. The results of simulation show that the algorithm provides an efficient solution to both problems.

Rotational transport on a sphere: Local node refinement with radial basis functions

This paper develops an algorithm for radial basis function (RBF) local node refinement and implements it for vortex roll-up and transport on a sphere. A heuristic based on an electrostatic repulsion type principle is used to re-distribute the nodes, clustering in areas where higher resolution is needed. It is then important to have a scheme that varies the shape of the RBFs over the domain so as to counteract the effects of Runge phenomena where the nodes are sparse. The roll-up of two diametrically opposed moving vortices are studied. The performance differences between near-uniform and refined nodes are addressed in terms of convergence, time stability, and computational cost. RBF results are put into context by comparison with published results for methods such as finite volume and discontinuous Galerkin.

An optimised radial basis function algorithm for fast non-rigid registration of medical images

The registration of multi-modal medical image data is important in the fields of image guided surgery and computer aided medical diagnosis. Registration accuracy is of utmost importance in both fields, however in the former, the speed of registration is equally important. In this paper, we present a point-based ‘fast’ non-rigid registration algorithm which exhibits significant speedups as compared to the non-optimised equivalent algorithm. Additionally, we make use of the parallel nature of the graphics processing unit (GPU) of the video adapter card of a standard PC to gain further speedups. The algorithm achieved sub-second performance when tested on the registration of MR with CT image data of size 256 3 .

Parallel Stochastic Global Optimization Using Radial Basis Functions

We develop a parallel implementation of a stochastic radial basis function (RBF) algorithm for global optimization by Regis and Shoemaker [Regis, R. G., C. A. Shoemaker. 2007a. A stochastic radial basis function method for the global optimization of expensive functions. INFORMS J. Comput. 19(4) 497–509]. The proposed parallel algorithm is suitable for the global optimization of computationally expensive objective functions and does not require derivatives. Each iteration of the algorithm consists of building an RBF model to approximate the expensive function and using this model to select multiple points for simultaneous function evaluation on multiple processors. The function evaluation points are selected from a set of random candidate points according to two criteria: estimated function value based on the RBF model, and minimum distance from previously evaluated points and previously selected points within each iteration. We compare the performance of our parallel stochastic RBF algorithm against alternative parallel global optimization methods, including two multistart parallel finite-difference quasi-Newton methods, a multistart implementation of Asynchronous Parallel Pattern Search [Hough, P., T. G. Kolda, V. J. Torczon. 2001. Asynchronous parallel pattern search for nonlinear optimization. SIAM J. Sci. Comput. 23(1) 134–156], a parallel implementation of Probabilistic Global Search Lausanne [Raphael, B., I. F. C. Smith. 2003. A direct stochastic algorithm for global search. Appl. Math. Comput. 146 729–758], a parallel evolutionary algorithm, and a deterministic parallel RBF algorithm by Regis and Shoemaker [Regis, R. G., C. A. Shoemaker. 2007c. Parallel radial basis function methods for the global optimization of expensive functions. Eur. J. Oper. Res. 182(2) 514–535]. We report good results for our parallel stochastic RBF method when using one, four, or eight processors in comparison with the alternatives on 20 test problems and on 3 optimization problems involving groundwater bioremediation.

A Growing and Pruning Method for Radial Basis Function Networks

A recently published generalized growing and pruning (GGAP) training algorithm for radial basis function (RBF) neural networks is studied and modified. GGAP is a resource-allocating network (RAN) algorithm, which means that a created network unit that consistently makes little contribution to the network's performance can be removed during the training. GGAP states a formula for computing the significance of the network units, which requires a d-fold numerical integration for arbitrary probability density function p(x) of the input data x (x in R(d)) . In this work, the GGAP formula is approximated using a Gaussian mixture model (GMM) for p(x) and an analytical solution of the approximated unit significance is derived. This makes it possible to employ the modified GGAP for input data having complex and high-dimensional p(x), which was not possible in the original GGAP. The results of an extensive experimental study show that the modified algorithm outperforms the original GGAP achieving both a lower prediction error and reduced complexity of the trained network.

Field Programmable Gate Array based floating point hardware design of recursive k-means clustering algorithm for Radial Basis Function Neural Network

In this paper, the hardware design of Radial Basis Function Neural Network (RBFNN), which is capable of dealing with floating point arithmetic operations and hardware architecture to calculate the centres of hidden layer using the k-means algorithm are proposed. The RBFNN are very much useful in adaptive control applications. Hardware implementation of neural network will give much faster training than traditional processors and also relatively inexpensive. The architecture of RBFNN is based on a computational model whose main features are: the capability to exploit the inherent parallelism of neural networks and to increase or decrease the number of neurons, aiming flexibility of the network. The design has been done with Very High Speed Integrated Circuit Hardware Description Language (VHDL). The results are verified and analysed in the MATLAB environment. In this work, the floating point hardware gives best precision and also very much useful for wide dynamic range requirements. The design was tested and synthesised with the help of Virtex-II pro device. The simulation and synthesis results show the effectiveness and speed of training.

Node adaptive domain decomposition method by radial basis functions

AbstractDuring the last years, there has been increased interest in developing efficient radial basis function (RBF) algorithms to solve partial differential problems of great scale. In this article, we are interested in solving large PDEs problems, whose solution presents rapid variations. Our main objective is to introduce a RBF dynamical domain decomposition algorithm which simultaneously performs a node adaptive strategy. This algorithm is based on the RBFs unsymmetric collocation setting. Numerical experiments performed with the multiquadric kernel function, for two stationary problems in two dimensions are presented. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009

Radial basis function neural network for hydrologic inversion: an appraisal with classical and spatio-temporal geostatistical techniques in the context of site characterization

This paper investigates three techniques for spatial mapping and the consequential hydrologic inversion, using hydraulic conductivity (or transmissivity) and hydraulic head as the geophysical parameters of concern. The data for the study were obtained from the Waste Isolation and Pilot Plant (WIPP) site and surrounding area in the remote Chihuahuan Desert of southeastern New Mexico. The central technique was the Radial Basis Function algorithm for an Artificial Neural Network (RBF-ANN). An appraisal of its performance in light of classical and temporal geostatistical techniques is presented. Our classical geostatistical technique of concern was Ordinary Kriging (OK), while the method of Bayesian Maximum Entropy (BME) constituted an advanced, spatio-temporal mapping technique. A fusion technique for soft or inter-dependent data was developed in this study for use with the neural network. It was observed that the RBF-ANN is capable of hydrologic inversion for transmissivity estimation with features remaining essentially similar to that obtained from kriging. The BME technique, on the other hand, was found to reveal an ability to map localized lows and highs that were otherwise not as apparent in OK or RBF-ANN techniques.

Affine modeling of nonlinear multivariable processes using a new adaptive neural network-based approach

This paper presents a new method for on-line identification of exact affine model for multivariable processes with nonlinear and time-varying behaviors. A self-generating radial basis function (RBF) neural network trained by growing and pruning algorithm for RBF (GAP–RBF) is utilized for deriving the affine model. The extended Kalman filter (EKF) is used for parameter adaptation in the GAP–RBF neural network. The growing and pruning criteria of the original GAP–RBF have been modified with the objective to enhance its performance in on-line identification. Simulation results on two nonlinear multivariable CSTR benchmark problems show an excellent performance of the proposed approach, incorporated with the modified GAP–RBF (MGAP–RBF) neural network, for affine modeling.