smile.math.rbf.package-info Maven / Gradle / Ivy
/*******************************************************************************
* Copyright (c) 2010 Haifeng Li
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*******************************************************************************/
/**
* Radial basis functions. A radial basis function 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 wi (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 wi φ(||x-ci||)
*
* where the approximating function y(x) is represented as a sum of N radial
* basis functions, each associated with a different center ci, and weighted
* by an appropriate coefficient wi. The weights wi 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
*/
package smile.math.rbf;