com.imsl.math
Class RadialBasis

java.lang.Object
  com.imsl.math.RadialBasis

All Implemented Interfaces:

Cloneable, Serializable

public class RadialBasis

extends Object

implements Serializable, Cloneable

RadialBasis computes a least-squares fit to scattered data in , where d is the dimension. More precisely, we are given data points

and function values

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

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:

The are the centers.

A radial function, , maps into . The default radial function is the Hardy multiquadric,

with . An alternate radial function is the Gaussian, .

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

See Also:

Example, Serialized Form

Nested Class Summary

static interface	RadialBasis.Function Public interface for the user supplied function to the RadialBasis object.
static class	RadialBasis.Gaussian The Gaussian basis function, .
static class	RadialBasis.HardyMultiquadric The Hardy multiquadric basis function, .

Field Summary

static long

serialVersionUID

Constructor Summary

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

Method Summary

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

Field Detail

serialVersionUID

public static final long serialVersionUID

See Also:

Constant Field Values

Constructor Detail

RadialBasis

public RadialBasis(int nDim,
                   int nCenters)

Creates a new instance of RadialBasis.

Parameters:

nDim - is the number of dimensions.

nCenters - is the number of centers.

Method Detail

getANOVA

public ANOVA getANOVA()

Returns the ANOVA statistics from the linear regression.

Returns:

an ANOVA table and related statistics

See Also:

LinearRegression, ANOVA

getRadialFunction

public RadialBasis.Function getRadialFunction()

Returns the radial function.

Returns:

the current radial function.

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.

setRadialFunction

public void setRadialFunction(RadialBasis.Function radialFunction)

Sets the radial function.

Parameters:

radialFunction - is the radial function.

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.

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.

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:

the value 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[], the point at which the radial basis is to be evaluated

Returns:

a double giving the value of the radial basis at the point x