Radial basis function
A radial basis function (RBF ) is a real function whose value depends only on the distance to the origin, so that. The name comes from the fact that the function according to this definition is radially symmetrical and further these functions are used as basis functions of approximation. General can determine the distance to a point c look, the center is named, so that.
Properties
Any function for which it holds is a RBF. As standard, the Euclidean norm is usually selected, depending on the problem, other standards are possible. For example, assuming that induced by the Lukaszyk - Karmowski metric standard, so it is possible for some RBFs to avoid problems with ill-conditioned matrices that are necessary to calculate the coefficients which occur in approximation by RBFs. Since it is in the radius of a scalar quantity is the behavior with respect to the expenditure in the higher-dimensional space benign.
Types of radial basis functions
Approximation using RBFs
Typically, linear combinations are used by radial basis functions for the approximation of functions. Here, the function to be approximated is approximated by a sum of radial basis functions have different centers, and are weighted by the coefficients. Furthermore, it should be noted that the function values are maintained exactly at the centers / nodes.
Approximation methods of this type are used to model non-linear systems ( with sufficiently simple chaotic behavior ), for 3D reconstruction in computer graphics ( for example, hierarchical RBFs ) or for creating response surfaces used in the optimization. Other applications of RBFs are solving partial differential equations (eg using lattice -free collocation) or transformations in image registration.
Operation
To illustrate the operation the function using the Gaussian RBF to be approximated. These are drawn in the selected range of five evenly spaced grid points to rate. The free parameters have to calibrate it. The figure for the approximated function curve shows two different specifications for. This is illustrated with the black approximation of the optimum value () and the red curve intuitive choice of which shows the superposition of single Gaussian functions.
Determination of the coefficients
For determining the coefficients of the radial basis function is a linear equation system to be solved. This is set up with the help of selected interpolation points. In this case, the matrix of the values of the basis function, resulting from the spacing of the support points to be determined is vector containing the coefficients of the radial basis function and the right side of the linear equation system contains the function values at the interpolation points.
For a function value of a -th grid point is valid:
With the abbreviation.
Thus, the following system of equations can be set up.
Implementation in the source code
The determination of the coefficients may be performed using the specified MATLAB source code.
Function [b, core function] = RBFcalcCoeff ( rbfConstant, coords, values) [ m, n ] = size ( coords ); core function = @ (x ) exp ( - ( rbfConstant. * x ) ^ 2. ); % % % Define rbf core function A = zeros (m, m); %%% Init for i1 = 1: m %%% loops to fill matrix A for i2 = i1: m cummSum = 0; %%% Calc radius in n- dimensional space for k = 1: n cummSum = cummSum ( coords (I1, k) - coords (i2, k)) ^ 2. ; end radius = sqrt ( cummSum ); A ( I1, I2) = core function ( radius); A ( I2, I1 ) = A ( I1, I2); %%% Save in matrix end end b = A \ values ; %%% Solve linear equation system Neural Networks
Inasmuch as Artificial neural networks are suitable for the approximation procedure, especially in high-dimensional problems, RBF provide a specific model for such networks dar.