Package com.imsl.math

Class RadialBasis

java.lang.Object
com.imsl.math.RadialBasis
All Implemented Interfaces:
Serializable, Cloneable
public class RadialBasis extends Object implements Serializable, Cloneable

RadialBasis computes a least-squares fit to scattered data in \( {\bf R}^d\), where d is the dimension. More precisely, we are given data points $$x_0,\ldots,x_{n-1}\in{\bf R}^d $$ and function values $$f_0,\ldots,f_{n-1}\in{\bf R}^1$$

The radial basis fit to the data is a function sF which approximates the above data in the sense that it minimizes the sum-of-squares error

$$\sum_{i=0}^{n-1}w_i\left(F(x_i)-f_i\right)^2 $$

where w are the weights. Of course, we must restrict the functional form of F. Here we assume it is a linear combination of radial functions:

$$F(x)\equiv\sum_{j=0}^{m-1}\alpha_j\phi(\|x-c_j\|) $$ The \(c_j\) are the centers.

A radial function, \(\phi(r)\), maps \([0,\infty) \) into \({\bf R}^1\). The default radial function is the Hardy multiquadric,

$$\phi(r)\equiv\sqrt{r^2+\delta^2}$$

with \(\delta=1\). An alternate radial function is the Gaussian, \(e^{-ax^2}\).

By default, the centers are points in a Faure sequence, scaled to cover the box containing the data.

Two update methods allow the user to specify weights for each data point in the approximation scheme. In this way the user can influence the fit of the radial basis function. For example, if weights are in the range [0,1] then 0-weighted points are effectively removed from computations and 1-weighted points will have more influence than any others. When the number of centers equals the number of data points, the RBF fit will be "exact", otherwise it will be an approximation (useful for large or noisy data sets). Provided the ratios of the weights are not too extreme, weights will not appreciably change the accuracy of the fit to the data, but they will affect the shape of the approximating function away from the data: Greater weights result in greater influence at greater distances.

See Also:

  • Nested Class Summary

    Nested Classes
    Modifier and Type
    Class
    Description
    static interface 
    RadialBasis.Function
    Public interface for the user supplied function to the RadialBasis object.
    static class 
    RadialBasis.Gaussian
    The Gaussian basis function, \(e^{-ax^2}\).
    static class 
    RadialBasis.HardyMultiquadric
    The Hardy multiquadric basis function, \(\sqrt{r^2+\delta^2} \).

  • Constructor Summary

    Constructors
    Constructor
    Description
    RadialBasis(int nDim, int nCenters)
    Creates a new instance of RadialBasis.

  • Method Summary

    Modifier and Type
    Method
    Description
    ANOVA
    getANOVA()
    Returns the ANOVA statistics from the linear regression.
    RadialBasis.Function
    getRadialFunction()
    Returns the radial function.
    double[]
    gradient(double[] x)
    Returns the gradient of the radial basis approximation at a point.
    void
    setRadialFunction(RadialBasis.Function radialFunction)
    Sets the radial function.
    void
    update(double[][] x, double[] f)
    Adds a set of data points, all with weight = 1.
    void
    update(double[][] x, double[] f, double[] w)
    Adds a set of data points with user-specified weights.
    void
    update(double[] x, double f)
    Adds a data point with weight = 1.
    void
    update(double[] x, double f, double w)
    Adds a data point with a specified weight.
    double
    value(double[] x)
    Returns the value of the radial basis approximation at a point.
    double[]
    value(double[][] x)
    Returns the value of the radial basis at a point.

    Methods inherited from class java.lang.Object

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

  • Constructor Details

    • RadialBasis

      public RadialBasis(int nDim, int nCenters)
      Creates a new instance of RadialBasis.
      Parameters:
      nDim - an int specifying the number of dimensions.
      nCenters - an int specifying the number of centers.

  • Method Details

    • setRadialFunction

      public void setRadialFunction(RadialBasis.Function radialFunction)
      Sets the radial function.
      Parameters:
      radialFunction - a RadialBasis.Function to be used in the approximation. The default is Hardy Multiquadric with \(\delta=1\).

    • getRadialFunction

      public RadialBasis.Function getRadialFunction()
      Returns the radial function.
      Returns:
      a RadialBasis.Function which is the current radial function.

    • update

      public void update(double[] x, double f)
      Adds a data point with weight = 1.
      Parameters:
      x - is a double array containing the locations of the data point.
      f - is a double containing the function value at the data point.

    • update

      public void update(double[] x, double f, double w)
      Adds a data point with a specified weight.
      Parameters:
      x - is a double array containing the locations of the data point.
      f - is a double containing the function value at the data point.
      w - is a double containing the weight of this data point.

    • update

      public void update(double[][] x, double[] f)
      Adds a set of data points, all with weight = 1.
      Parameters:
      x - is a double matrix of size n by nDim containing the locations of the data points for each dimension.
      f - is a double array containing the function values at the data points.

    • update

      public void update(double[][] x, double[] f, double[] w)
      Adds a set of data points with user-specified weights.
      Parameters:
      x - is a double matrix of size n by nDim containing the locations of the data points for each dimension.
      f - is a double array containing the function values at the data points.
      w - is a double array containing the weights associated with the data points.

    • value

      public double value(double[] x)
      Returns the value of the radial basis approximation at a point.
      Parameters:
      x - is a double array containing the locations of the data point at which the approximation is to be computed.
      Returns:
      a double containing the value of the radial basis approximation at x.

    • gradient

      public double[] gradient(double[] x)
      Returns the gradient of the radial basis approximation at a point.
      Parameters:
      x - is a double array containing the locations of the data point at which the approximation's gradient is to be computed.
      Returns:
      a double array, of length nDim containing the value of the gradient of the radial basis approximation at x.

    • value

      public double[] value(double[][] x)
      Returns the value of the radial basis at a point.
      Parameters:
      x - a double matrix of size n by nDim containing the points at which the radial basis is to be evaluated.
      Returns:
      a double array giving the value of the radial basis at the point x

    • getANOVA

      public ANOVA getANOVA()
      Returns the ANOVA statistics from the linear regression.
      Returns:
      an ANOVA table and related statistics
      See Also: