smile.math.rbf.GaussianRadialBasis Maven / Gradle / Ivy
/******************************************************************************
* Confidential Proprietary *
* (c) Copyright Haifeng Li 2011, All Rights Reserved *
******************************************************************************/
package smile.math.rbf;
import smile.math.Math;
/**
* Gaussian RBF. φ(r) = e-0.5 * r2 / r20
* where r0 is a scale factor. The interpolation accuracy using
* Gaussian basis functions can be very sensitive to r0, and they
* are often avoided for this reason. However, for smooth functions and with
* an optimal r0, very high accuracy can be achieved. The Gaussian
* also will extrapolate any function to zero far from the data, and it gets
* to zero quickly.
*
* In general, r0 should be larger than the typical separation of
* points but smaller than the "outer scale" or feature size of the function
* to interplate. There can be several orders of magnitude difference between
* the interpolation accuracy with a good choice for r0, versus a
* poor choice, so it is definitely worth some experimentation. One way to
* experiment is to construct an RBF interpolator omitting one data point
* at a time and measuring the interpolation error at the omitted point.
*
*
References
*
* - Nabil Benoudjit and Michel Verleysen. On the kernel widths in radial-basis function networks. Neural Process, 2003.
*
*
* @author Haifeng Li
*/
public class GaussianRadialBasis implements RadialBasisFunction {
/**
* The scale factor.
*/
private double r0;
/**
* Constructor. The default bandwidth is 1.0.
*/
public GaussianRadialBasis() {
this(1.0);
}
/**
* Constructor.
*
* @param scale the scale (bandwidth/sigma) parameter.
*/
public GaussianRadialBasis(double scale) {
r0 = scale;
}
@Override
public double f(double r) {
r /= r0;
return Math.exp(-0.5 * r * r);
}
}