smile.interpolation.RBFInterpolation1D Maven / Gradle / Ivy
/*
* Copyright (c) 2010-2021 Haifeng Li. All rights reserved.
*
* Smile is free software: you can redistribute it and/or modify
* it under the terms of the GNU General Public License as published by
* the Free Software Foundation, either version 3 of the License, or
* (at your option) any later version.
*
* Smile is distributed in the hope that it will be useful,
* but WITHOUT ANY WARRANTY; without even the implied warranty of
* MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
* GNU General Public License for more details.
*
* You should have received a copy of the GNU General Public License
* along with Smile. If not, see
* y(x) = Σ wi φ(||x-ci||)
*
* where the approximating function y(x) is represented as a sum of N radial
* basis functions φ, each associated with a different center ci,
* and weighted by an appropriate coefficient wi. For distance,
* one usually chooses Euclidean distance. The weights wi can
* be estimated using the matrix methods of linear least squares, because
* the approximating function is linear in the weights.
*
* The points ci often called the centers or collocation points
* of the RBF interpolant. Note also that the centers ci 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 {@code > 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.
*
* @author Haifeng Li
*/
public class RBFInterpolation1D implements Interpolation {
/**
* The control points.
*/
private final double[] x;
/**
* The linear weights.
*/
private final double[] w;
/**
* The radial basis function.
*/
private final RadialBasisFunction rbf;
/**
* True to fit a normalized rbf interpolation.
*/
private final boolean normalized;
/**
* Constructor. By default, it is a regular rbf interpolation without
* normalization.
*
* @param x the tabulated points.
* @param y the function values at x.
* @param rbf the radial basis function used in the interpolation
*/
public RBFInterpolation1D(double[] x, double[] y, RadialBasisFunction rbf) {
this(x, y, rbf, false);
}
/**
* Constructor.
*
* @param x the tabulated points.
* @param y the function values at x.
* @param rbf the radial basis function used in the interpolation
* @param normalized true for the normalized RBF interpolation.
*/
public RBFInterpolation1D(double[] x, double[] y, RadialBasisFunction rbf, boolean normalized) {
if (x.length != y.length) {
throw new IllegalArgumentException("x.length != y.length");
}
this.x = x;
this.rbf = rbf;
this.normalized = normalized;
int n = x.length;
Matrix G = new Matrix(n, n);
double[] rhs = new double[n];
for (int i = 0; i < n; i++) {
double sum = 0.0;
for (int j = 0; j <= i; j++) {
double r = rbf.f(Math.abs(x[i] - x[j]));
G.set(i, j, r);
G.set(j, i, r);
sum += 2 * r;
}
if (normalized) {
rhs[i] = sum * y[i];
} else {
rhs[i] = y[i];
}
}
if (rbf instanceof GaussianRadialBasis) {
G.uplo(UPLO.LOWER);
Matrix.Cholesky cholesky = G.cholesky(true);
w = cholesky.solve(rhs);
} else {
Matrix.LU lu = G.lu(true);
w = lu.solve(rhs);
}
}
@Override
public double interpolate(double x) {
double sum = 0.0, sumw = 0.0;
for (int i = 0; i < this.x.length; i++) {
double f = rbf.f(Math.abs(x - this.x[i]));
sumw += w[i] * f;
sum += f;
}
return normalized ? sumw / sum : sumw;
}
@Override
public String toString() {
return String.format("RBF Interpolation(%s)", rbf);
}
}