SIDE EFFECTS OF NORMALISING RADIAL BASIS FUNCTION NETWORKS

1996 ◽  
Vol 07 (02) ◽  
pp. 167-179 ◽  
Author(s):  
ROBERT SHORTEN ◽  
RODERICK MURRAY-SMITH
Keyword(s):  
Side Effects ◽  
Radial Basis Function ◽  
Basis Function ◽  
Function Approximation ◽  
Partition Of Unity ◽  
Basis Functions ◽  
Radial Basis Function Networks ◽  
Rbf Network ◽  
Linear Function Approximation ◽  
Radial Basis

Normalisation of the basis function activations in a Radial Basis Function (RBF) network is a common way of achieving the partition of unity often desired for modelling applications. It results in the basis functions covering the whole of the input space to the same degree. However, normalisation of the basis functions can lead to other effects which are sometimes less desirable for modelling applications. This paper describes some side effects of normalisation which fundamentally alter properties of the basis functions, e.g. the shape is no longer uniform, maxima of basis functions can be shifted from their centres, and the basis functions are no longer guaranteed to decrease monotonically as distance from their centre increases—in many cases basis functions can ‘reactivate’, i.e. re-appear far from the basis function centre. This paper examines how these phenomena occur, discusses their relevance for non-linear function approximation and examines the effect of normalisation on the network condition number and weights.

scholarly journals Adaptive Linear and Normalized Combination of Radial Basis Function Networks for Function Approximation and Regression

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Yunfeng Wu ◽  
Xin Luo ◽  
Fang Zheng ◽  
Shanshan Yang ◽  
Suxian Cai ◽  
...  
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Function Approximation ◽  
Mean Squared Error ◽  
Weighted Average ◽  
Radial Basis Function Networks ◽  
Data Sets ◽  
Radial Basis ◽  
Synthetic Function ◽  
Adaptively Adjusting

This paper presents a novel adaptive linear and normalized combination (ALNC) method that can be used to combine the component radial basis function networks (RBFNs) to implement better function approximation and regression tasks. The optimization of the fusion weights is obtained by solving a constrained quadratic programming problem. According to the instantaneous errors generated by the component RBFNs, the ALNC is able to perform the selective ensemble of multiple leaners by adaptively adjusting the fusion weights from one instance to another. The results of the experiments on eight synthetic function approximation and six benchmark regression data sets show that the ALNC method can effectively help the ensemble system achieve a higher accuracy (measured in terms of mean-squared error) and the better fidelity (characterized by normalized correlation coefficient) of approximation, in relation to the popular simple average, weighted average, and the Bagging methods.

Ray Tracing Using Radial Basis Function Networks

2016 ◽  
Vol 138 (2) ◽  
Author(s):  
Travis Wiens
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Refractive Index ◽  
Radial Basis Function ◽  
Ray Tracing ◽  
Basis Function ◽  
Radial Basis Function Networks ◽  
Rbf Network ◽  
Radial Basis ◽  
Ray Path

This paper presents a numerical method of tracing of sound or other refracted rays through a medium with arbitrarily varying refractive index. The method uses a radial basis function (RBF) network to define the refractive index of the medium, allowing continuous gradients to be determined analytically and the ray path to be solved using standard numerical ordinary differential equation (ODE) solution techniques.

scholarly journals Using Radial Basis Function Networks for Function Approximation and Classification

2012 ◽  
Vol 2012 ◽  
pp. 1-34 ◽  
Author(s):  
Yue Wu ◽  
Hui Wang ◽  
Biaobiao Zhang ◽  
K.-L. Du
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Function Approximation ◽  
System Modeling ◽  
Rbf Network ◽  
Network Learning ◽  
Radial Basis ◽  
Comprehensive Survey ◽  
Dynamic System Modeling ◽  
Complex Valued

The radial basis function (RBF) network has its foundation in the conventional approximation theory. It has the capability of universal approximation. The RBF network is a popular alternative to the well-known multilayer perceptron (MLP), since it has a simpler structure and a much faster training process. In this paper, we give a comprehensive survey on the RBF network and its learning. Many aspects associated with the RBF network, such as network structure, universal approimation capability, radial basis functions, RBF network learning, structure optimization, normalized RBF networks, application to dynamic system modeling, and nonlinear complex-valued signal processing, are described. We also compare the features and capability of the two models.

Recursive Particle Swarm Optimization Application for the Radial Basis Function Networks Modeling System

2015 ◽  
Vol 713-715 ◽  
pp. 1817-1820
Author(s):  
Ling Liu ◽  
Min Chen ◽  
Hong Yi Guo
Keyword(s):  
Particle Swarm Optimization ◽  
Radial Basis Function ◽  
Basis Function ◽  
Optimization Problems ◽  
Particle Swarm ◽  
Basis Functions ◽  
Radial Basis Function Networks ◽  
Historical Knowledge ◽  
Swarm Optimization ◽  
Radial Basis

A Recursive Particle Swarm Optimization (RPSO) is proposed to solve dynamic optimization problems where the data is obtained not once but one by one. The position of each particle swarm is updated recursively based on the continuous data and the historical knowledge. The experiment results indicate that RPSO-based radial basis function networks needs fewer radial basis functions and gives more accurate results than traditional PSO in solving dynamic problems.

Indexed Families of Functionals and Gaussian Radial Basis Functions

2003 ◽  
Vol 15 (2) ◽  
pp. 455-468 ◽  
Author(s):  
Irwin W. Sandberg
Keyword(s):  
Radial Basis Function ◽  
Basis Function ◽  
Basis Functions ◽  
Inner Product ◽  
Radial Basis Function Networks ◽  
Product Spaces ◽  
Inner Product Spaces ◽  
Finite Dimensional ◽  
Radial Basis

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.

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