smile.math.rbf.RadialBasisFunction Maven / Gradle / Ivy

Go to download
Show more of this group Show more artifacts with this name
Show all versions of smile-base Show documentation
smile-base
There is a newer version: 4.2.0
/*
 * Copyright (c) 2010-2021 Haifeng Li. All rights reserved.
 *
 * Smile is free software: you can redistribute it and/or modify
 * it under the terms of the GNU General Public License as published by
 * the Free Software Foundation, either version 3 of the License, or
 * (at your option) any later version.
 *
 * Smile is distributed in the hope that it will be useful,
 * but WITHOUT ANY WARRANTY; without even the implied warranty of
 * MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
 * GNU General Public License for more details.
 *
 * You should have received a copy of the GNU General Public License
 * along with Smile.  If not, see .
 */

package smile.math.rbf;

import smile.math.Function;

/**
 * A radial basis function (RBF) is a real-valued function whose value depends
 * only on the distance from the origin, so that φ(x)=φ(||x||); or
 * alternatively on the distance from some other point c, called a center, so
 * that φ(x,c)=φ(||x-c||). Any function φ that satisfies the
 * property  is a radial function. The norm is usually Euclidean distance,
 * although other distance functions are also possible. For example by
 * using probability metric it is for some radial functions possible
 * to avoid problems with ill conditioning of the matrix solved to
 * determine coefficients w_i (see below), since the ||x|| is always
 * greater than zero.
 * 
 * Sums of radial basis functions are typically used to approximate given
 * functions:
 * 

 * y(x) = Σ w_i φ(||x-c_i||)
 * 

 * where the approximating function y(x) is represented as a sum of N radial
 * basis functions, each associated with a different center c_i, and weighted
 * by an appropriate coefficient w_i. The weights w_i can
 * be estimated using the matrix methods of linear least squares, because
 * the approximating function is linear in the weights.
 * 
 * This approximation process can also be interpreted as a simple kind of neural
 * network and has been particularly used in time series prediction and control
 * of nonlinear systems exhibiting sufficiently simple chaotic behavior,
 * 3D reconstruction in computer graphics (for example, hierarchical RBF).
 *
 * @author Haifeng Li
 */
public interface RadialBasisFunction extends Function {

}