smile.interpolation

Class RBFInterpolation1D

java.lang.Object

smile.interpolation.RBFInterpolation1D

All Implemented Interfaces:

Interpolation

public class RBFInterpolation1D
extends java.lang.Object
implements Interpolation

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 scalings. Gaussians, multi-quadrics and inverse multi-quadrics are infinitely smooth and 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 and Description
RBFInterpolation1D(double[] x, double[] y, RadialBasisFunction rbf) Constructor.
RBFInterpolation1D(double[] x, double[] y, RadialBasisFunction rbf, boolean normalized) Constructor.

Method Summary

Modifier and Type	Method and Description
double	interpolate(double x) Given a value x, return an interploated value.

Methods inherited from class java.lang.Object

equals, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

RBFInterpolation1D

public RBFInterpolation1D(double[] x,
                          double[] y,
                          RadialBasisFunction rbf)

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

Parameters:

x - the point set.

y - the function values at given points.

rbf - the radial basis function used in the interpolation

RBFInterpolation1D

public RBFInterpolation1D(double[] x,
                          double[] y,
                          RadialBasisFunction rbf,
                          boolean normalized)

Constructor.

Parameters:

x - the point set.

y - the function values at given points.

rbf - the radial basis function used in the interpolation

normalized - true for the normalized RBF interpolation.

Method Detail

interpolate

public double interpolate(double x)

Description copied from interface: Interpolation

Given a value x, return an interploated value.

Specified by:

interpolate in interface Interpolation