Radial basis function

A radial basis function (RBF) is a real function , the value of which only depends on the distance to the origin, so that . The name comes from the fact that the function is radially symmetric according to this definition and that these functions are also used as basis functions of an approximation . More generally one can consider the distance to a point c, called the center , such that . ${\displaystyle \varphi }$ ${\displaystyle \varphi (\mathbf {x} )=\varphi (\|\mathbf {x} \|)}$ ${\displaystyle \varphi (\mathbf {x} ,\mathbf {c} )=\varphi (\|\mathbf {x} -\mathbf {c} \|)}$

properties

Every function that applies to is an RBF. The Euclidean norm is usually chosen as the norm ; other norms are also possible depending on the problem. For example, if one takes the norm induced by the Lukaszyk-Karmowski metric, it is possible with some RBFs to avoid problems with poorly conditioned matrices, which are necessary for calculating the coefficients that occur when approximating by RBFs. Since the radius is a scalar variable, the behavior with regard to the effort in the higher-dimensional space is benign. ${\displaystyle \varphi (x)}$ ${\displaystyle \varphi (\mathbf {x} )=\varphi (\|\mathbf {x} \|)}$ ${\displaystyle \|\cdot \|}$

Types of radial basis functions

Piecewise polynomial RBF ( ) ${\displaystyle R_{n}}$	${\displaystyle |r|^{n}}$, for odd ${\displaystyle n}$
Thin Plate Spline ( ) ${\displaystyle TPS_{n}}$	${\displaystyle |r|^{n}\ln |r|}$, for straight ${\displaystyle n}$
Multiquadric RBF (MQ)	${\displaystyle {\sqrt {1+(ar)^{2}}}}$
Inverse multiquadric RBF (IMQ)	${\displaystyle {\frac {1}{\sqrt {1+(ar)^{2}}}}}$
Inverse quadric RBF (IQ)	${\displaystyle {\frac {1}{1+(ar)^{2}}}}$
Gaussian RBF (GS)	${\displaystyle e^{-(ar)^{2}}}$

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 which have different centers and are weighted by the coefficients . It must also be ensured that the function values ​​at the centers / support points are precisely adhered to. ${\displaystyle f(\mathbf {x} )\approx \sum _{i=1}^{N}\lambda _{i}\varphi (\|\mathbf {x} -\mathbf {c} _{i}\|)}$ ${\displaystyle f(x)}$ ${\displaystyle N}$ ${\displaystyle c_{i}}$ ${\displaystyle \lambda _{i}}$

Approximation methods of this type are used to model nonlinear systems (with sufficiently simple chaotic behavior), for 3D reconstruction in computer graphics (for example hierarchical RBFs) or to create response surfaces in optimization. Other areas of application of RBFs are the solving of partial differential equations (e.g. using grid-free collocation ) or transformations in image registration .

functionality

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  Approximation with RBF (Gaussian approach)

To clarify the functionality, the function should be approximated with the aid of the Gaussian RBF. For this purpose, six evenly distributed support points are consulted in the selected area . You have to calibrate the free parameter . The figure for the approximated function curve shows two different specifications for . The black approximation shows the optimal value ( ) and the red curve shows the intuitive choice of , which shows the superposition of individual Gaussian functions. ${\displaystyle f(x)=\sin(x)+0.95+0.075x^{2}-0.001x^{4}}$ ${\displaystyle 0\leq x\leq 10}$ ${\displaystyle a}$ ${\displaystyle a}$ ${\displaystyle a=0.17}$ ${\displaystyle a=1.0}$

Determination of the coefficients

To determine the coefficients of the radial basis functions, a linear system of equations has to be solved. This is set up with the help of selected support points. The matrix consists of the values ​​of the basic function that result from the spacing of the interpolation points, the vector to be determined contains the coefficients of the radial basis functions and the right side of the linear equation system contains the function values ​​at the interpolation points. ${\displaystyle N}$ ${\displaystyle A}$

The following applies to a function value of a -th interpolation point: ${\displaystyle k}$

${\displaystyle f(\mathbf {x} _{k})=\sum _{i=1}^{N}\lambda _{i}\varphi (r_{ki})}$with the abbreviation . ${\displaystyle r_{ki}=\|\mathbf {x} _{k}-\mathbf {x} _{i}\|}$

Thus the following system of equations can be set up.

${\displaystyle {\begin{bmatrix}\varphi (r_{11})&\varphi (r_{12})&\cdots &\varphi (r_{1N})\\\varphi (r_{21})&\varphi (r_{22})&\cdots &\varphi (r_{2N})\\\vdots &\vdots &\ddots &\vdots \\\varphi (r_{N1})&\varphi (r_{N2})&\cdots &\varphi (r_{NN})\end{bmatrix}}{\begin{bmatrix}\lambda _{1}\\\lambda _{2}\\\vdots \\\lambda _{N}\end{bmatrix}}={\begin{bmatrix}f(\mathbf {x} _{1})\\f(\mathbf {x} _{2})\\\vdots \\f(\mathbf {x} _{N})\end{bmatrix}}}$

Implementation in the source code

The coefficients can be determined with the help of the specified Matlab source text.

function [lam,coreFunction] = RBFcalcCoeff(a,xk,fk)

coreFunction = @(x) exp(-(a.*x).^2);   %%% definieren der RBF (phi(x))

N = length(xk);                        %%% N ausgewählte Stützstellen

A = zeros(N,N);                        %%% Initialisierung der Matrix A

for i1 = 1:N                           %%% Schleife über alle Zeilen
  for i2 = i1:N                        %%% Schleife über alle Spalten
      r = sqrt((xk(i1)-xk(i2)).^2);    %%% Bestimmung des Radius
      A(i1,i2) = coreFunction(r);      %%% Funktionswert der Matrix
      A(i2,i1) = A(i1,i2);
  end
end

lam=A\fk;                              %%% Lösung des Gleichungssystems

Neural Networks

Insofar as artificial neural networks are a suitable type of procedure for approximation, especially with high-dimensional problems, RBF represent a special model for such networks.

literature

• Martin D. Buhmann: Radial Basis Functions: Theory and Implementations . Cambridge University Press, 2003
• E. Larsson, B. Fornberg: A Numerical Study of some Radial Basis Function based Solution Methods for Elliptic PDEs . In: Computers & Mathematics with Applications , Volume 46, 2003
• H. Wendland: Scattered Data Approximation . Cambridge University Press, 2005

Web links

• Martin Buhmann, Janin Jäger, On radial basis functions , Oberwolfach Snapshots 2019