smile.math.rbf.InverseMultiquadricRadialBasis Maven / Gradle / Ivy
/*******************************************************************************
* Copyright (c) 2010 Haifeng Li
*
* Licensed under the Apache License, Version 2.0 (the "License");
* you may not use this file except in compliance with the License.
* You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*******************************************************************************/
package smile.math.rbf;
/**
* Inverse multiquadric RBF. φ(r) = (r2 + r20)-1/2
* where r0 is a scale factor. Although it sounds odd, the inverse
* multiquadric gives results that are comparable to the multiquadric,
* sometimes better. The reason is what really matters is smoothness, and
* certain properties of the function's Fourier transform that are not very
* different between the multiquadric and its reciprocal. The fact that one
* increases monotonically and the other decreases turns out to be almost
* irrelevant. Besides, inverse multiquadric will extrapolate any function to
* zero far from the data.
*
* 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.
*
* @author Haifeng Li
*/
public class InverseMultiquadricRadialBasis implements RadialBasisFunction {
private double r02;
public InverseMultiquadricRadialBasis() {
this(1.0);
}
public InverseMultiquadricRadialBasis(double scale) {
r02 = scale * scale;
}
@Override
public double f(double r) {
return 1 / Math.sqrt(r*r + r02);
}
}