Radial basis function

Radial basis functions. Radial interpolation uses radial basis functions in the linear combination that express the desired interpolated function, i.e.,... 

Nair, S., S. Udpa, and L. Udpa, (1993), Radial basis functions network for defect sizing . Review of Progress in QNDE, Vol. 12, 1993, pp. 819-825... 

Likewise, a basis set can be improved by uncontracting some of the outer basis function primitives (individual GTO orbitals). This will always lower the total energy slightly. It will improve the accuracy of chemical predictions if the primitives being uncontracted are those describing the wave function in the middle of a chemical bond. The distance from the nucleus at which a basis function has the most significant effect on the wave function is the distance at which there is a peak in the radial distribution function for that GTO primitive. The formula for a normalized radial GTO primitive in atomic units is... 

Girosi, F., and Anzellotti, G., Rales of convergence for radial basis functions and neural networks. Artificial Neural Networks with Applications in Speech and Vision, (R. J. Matttmone, ed.), p. 97. Chapman Hall, London, 1993. 

The partial wave basis functions with which the radial dipole matrix elements fLv constructed (see Appendix A) are S-matrix normalized continuum functions obeying incoming wave boundary conditions. 

Many different types of networks have been developed. They all consist of small units, neurons, that are interconnected. The local behaviour of these units determines the overall behaviour of the network. The most common is the multi-layer-feed-forward network (MLF). Recently, other networks such as the Kohonen, radial basis function and ART networks have raised interest in the chemical application area. In this chapter we focus on the MLF networks. The principle of some of the other networks are explained and we also discuss how these networks relate with other algorithms, described elsewhere in this book. 

Radial basis function networks (RBF) are a variant of three-layer feed forward networks (see Fig 44.18). They contain a pass-through input layer, a hidden layer and an output layer. A different approach for modelling the data is used. The transfer function in the hidden layer of RBF networks is called the kernel or basis function. For a detailed description the reader is referred to references [62,63]. Each node in the hidden unit contains thus such a kernel function. The main difference between the transfer function in MLF and the kernel function in RBF is that the latter (usually a Gaussian function) defines an ellipsoid in the input space. Whereas basically the MLF network divides the input space into regions via hyperplanes (see e.g. Figs. 44.12c and d), RBF networks divide the input space into hyperspheres by means of the kernel function with specified widths and centres. This can be compared with the density or potential methods in pattern recognition (see Section 33.2.5). 

J. Park and I.W. Sandberg, Universal approximation using radial basis function networks. Neural Computation, 3 (1991) 246-257. 

B. Carse and T.C. Fogarty, Fast evolutionary learning of minimal radial basis function neural networks using a genetic algorithm. Lecture Notes in Computer Science, 1143, (1996) 1-22. 

L. Kiernan, J.D. Mason and K. Warwick, Robust initialisation of Gaussian radial basis function networks using partitioned k-means clustering. Electron. Lett., 32 (1996) 671-672. 

W. Luo, M.N. Karim, A.J. Morris and E.B. Martin, Control relevant identification of a pH waste water neutralisation process using adaptive radial basis function networks. Computers Chem. Eng., 20(1996)S1017... 

B. Waiczack and D.L. Massart, Application of radial basis functions-partial least squares to non-linear pattern recognition problems diagnosis of process faults. Anal. Chim. Acta, 331 (1996) 187-193. 

Of the several approaches that draw upon this general description, radial basis function networks (RBFNs) (Leonard and Kramer, 1991) are probably the best-known. RBFNs are similar in architecture to back propagation networks (BPNs) in that they consist of an input layer, a single hidden layer, and an output layer. The hidden layer makes use of Gaussian basis functions that result in inputs projected on a hypersphere instead of a hyperplane. RBFNs therefore generate spherical clusters in the input data space, as illustrated in Fig. 12. These clusters are generally referred to as receptive fields. 

Among these, the most widely used is the radial basis function network (RBFN). The key distinctions among these methods are summarized in Table I and discussed in detail in Bakshi and Utojo (1998). An RBFN is an example of a method that can be used for both input analysis and input-output analysis. As discussed earlier, the basis functions in RBFNs are of the form 0( xj - tm 2), where tm denotes the center of the basis function. One of the most popular basis functions is the Gaussian,... 

Chen, S., Cowan, C. F. N., and Grant, P. M., Orthogonal least squares learning algorithm for radial basis function networks, IEEE Trans. Neur. Net. 2(2), 302-309 (1991). 

Leonard, J., and Kramer, M. A., Radial basis function networks for classifying process faults, IEEE Control Systems, April, p. 31 (1991). 

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