Package smile.interpolation

Class RBFInterpolation

java.lang.Object

smile.interpolation.RBFInterpolation

public class RBFInterpolation
extends Object

Radial basis function interpolation is a popular method for the data points are irregularly distributed in space. In its basic form, radial basis function interpolation is in the form

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 φ, each associated with a different center c_i, and weighted by an appropriate coefficient w_i. For distance, one usually chooses Euclidean distance. The weights w_i can be estimated using the matrix methods of linear least squares, because the approximating function is linear in the weights.

The points c_i often called the centers or collocation points of the RBF interpolant. Note also that the centers c_i can be located at arbitrary points in the domain, and do not require a grid. For certain RBF exponential convergence has been shown. Radial basis functions were successfully applied to problems as diverse as computer graphics, neural networks, for the solution of differential equations via collocation methods and many other problems.

Other popular choices for φ comprise the Gaussian function and the so-called thin plate splines. Thin plate splines result from the solution of a variational problem. The advantage of the thin plate splines is that their conditioning is invariant under scaling. Gaussians, multi-quadrics and inverse multi-quadrics are infinitely smooth and involve a scale or shape parameter, r₀ > 0. Decreasing r₀ tends to flatten the basis function. For a given function, the quality of approximation may strongly depend on this parameter. In particular, increasing r₀ has the effect of better conditioning (the separation distance of the scaled points increases).

A variant on RBF interpolation is normalized radial basis function (NRBF) interpolation, in which we require the sum of the basis functions to be unity. NRBF arises more naturally from a Bayesian statistical perspective. However, there is no evidence that either the NRBF method is consistently superior to the RBF method, or vice versa.

Constructor Summary

Constructors

Constructor	Description
RBFInterpolation(double[][] x, double[] y, RadialBasisFunction normalized)	Constructor.
RBFInterpolation(double[][] x, double[] y, RadialBasisFunction rbf, boolean normalized)	Constructor.

Method Summary

Modifier and Type	Method	Description
double	interpolate(double... x)	Interpolate the function at given point.
String	toString()

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, wait, wait, wait

Constructor Details

RBFInterpolation

public RBFInterpolation(double[][] x,
 double[] y,
 RadialBasisFunction normalized)

Constructor. By default, it is a regular rbf interpolation without normalization.

Parameters:

x - the data points.

y - the function values at x.

normalized - the radial basis function used in the interpolation

RBFInterpolation

public RBFInterpolation(double[][] x,
 double[] y,
 RadialBasisFunction rbf,
 boolean normalized)

Constructor.

Parameters:

x - the data points.

y - the function values at x.

rbf - the radial basis function used in the interpolation

normalized - true for the normalized RBF interpolation.

Method Details

interpolate

public double interpolate(double... x)

Interpolate the function at given point.

Parameters:

x - a point.

Returns:

the interpolated function value.

toString

public String toString()

Overrides:

toString in class Object