Class RBFInterpolation
y(x) = Σ w_{i} φ(xc_{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
socalled 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, multiquadrics
and inverse multiquadrics are infinitely smooth and involve a scale
or shape parameter, r_{0} > 0
.
Decreasing r_{0} tends to
flatten the basis function. For a given function, the quality of
approximation may strongly depend on this parameter. In particular,
increasing r_{0} 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
ConstructorDescriptionRBFInterpolation
(double[][] x, double[] y, RadialBasisFunction normalized) Constructor.RBFInterpolation
(double[][] x, double[] y, RadialBasisFunction rbf, boolean normalized) Constructor. 
Method Summary
Modifier and TypeMethodDescriptiondouble
interpolate
(double... x) Interpolate the function at given point.toString()

Constructor Details

RBFInterpolation
Constructor. By default, it is a regular rbf interpolation without normalization. Parameters:
x
 the data points.y
 the function values atx
.normalized
 the radial basis function used in the interpolation

RBFInterpolation
Constructor. Parameters:
x
 the data points.y
 the function values atx
.rbf
 the radial basis function used in the interpolationnormalized
 true for the normalized RBF interpolation.


Method Details