JMSLTM Numerical Library 6.1

com.imsl.math
Class Spline2DLeastSquares

java.lang.Object
  extended by com.imsl.math.Spline2D
      extended by com.imsl.math.Spline2DLeastSquares
All Implemented Interfaces:
Serializable, Cloneable

public class Spline2DLeastSquares
extends Spline2D

Computes a two-dimensional, tensor-product spline approximant using least squares.

The Spline2DLeastSquares class computes a tensor-product spline least-squares approximation to weighted tensor-product data. The input consists of data vectors to specify the tensor-product grid for the data, two vectors with the weights, the values of the surface on the grid, and the specification for the tensor-product spline. The grid is specified by the two vectors x = xData and y = yData of length
n = xData.length and m = yData.length, respectively. A two-dimensional array f = fData contains the data values which are to be fit. The two vectors w_x = xWeights and w_y = yWeights contain the weights for the weighted least-squares problem. The information for the approximating tensor-product spline can be provided using the setXOrder, setYOrder, setXKnots and setYKnots methods. This information is contained in k_x = xOrder, t_x = xKnots, and N = xSplineSpaceDim for the spline in the first variable, and in k_y = yOrder, t_y = yKnots and M = ySplineSpaceDim for the spline in the second variable. This class computes coefficients for the tensor-product spline by solving the normal equations in tensor-product form as discussed in de Boor (1978, Chapter 17). The interested reader might also want to study the paper by Grosse (1980).

As the computation proceeds, we obtain coefficients c minimizing

sumlimits_{i = 0}^{n - 1} {sumlimits_{j = 0}^{m - 1} {w_x left( i right)w_y left( j right)} left[ {sumlimits_{k = 0}^{N - 1} {sumlimits_{l = 0}^{M - 1} {c_{kl} } B_{kl} left( {x_i ,y_i } right) - f_{ij} } } right]} ^2

where the function B_{kl} is the tensor-product of two B-splines of order k_x and k_y. Specifically, we have

B_{kl} left( {x,y} right) = B_{k,k_x ,t_x } left( x right)B_{l,k_y ,t_y } left( y right)

The spline

sumlimits_{k = 0}^{N - 1} {sumlimits_{l = 0}^{M - 1} {c_{kl} } } B_{kl}

and its partial derivatives can be evaluated using the value method.

See Also:
Example 1, Serialized Form

Constructor Summary
Spline2DLeastSquares(double[] xData, double[] yData, double[][] fData, int xSplineSpaceDim, int ySplineSpaceDim)
          Constructor for Spline2DLeastSquares.
 
Method Summary
 void compute()
          Computes a two-dimensional, tensor-product spline approximant using least squares.
 double getErrorSumOfSquares()
          Returns the weighted error sum of squares.
 int getXOrder()
          Returns the order of the spline in the x-direction.
 double[] getXWeights()
          Returns the weights for the least-squares fit in the x-direction.
 int getYOrder()
          Returns the order of the spline in the y-direction.
 double[] getYWeights()
          Returns the weights for the least-squares fit in the y-direction.
 void setXKnots(double[] xKnots)
          Sets the knot sequences of the spline in the x-direction.
 void setXOrder(int xOrder)
          Sets the order of the spline in the x-direction.
 void setXWeights(double[] xWeights)
          Sets the weights for the least-squares fit in the x-direction.
 void setYKnots(double[] yKnots)
          Sets the knot sequences of the spline in the y-direction.
 void setYOrder(int yOrder)
          Sets the order of the spline in the y-direction.
 void setYWeights(double[] yWeights)
          Sets the weights for the least-squares fit in the y-direction.
 
Methods inherited from class com.imsl.math.Spline2D
derivative, derivative, getCoefficients, getXKnots, getYKnots, integral, value, value
 
Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
 

Constructor Detail

Spline2DLeastSquares

public Spline2DLeastSquares(double[] xData,
                            double[] yData,
                            double[][] fData,
                            int xSplineSpaceDim,
                            int ySplineSpaceDim)
Constructor for Spline2DLeastSquares.

Parameters:
xData - a double array containing the data points in the x-direction.
yData - a double array containing the data points in the y-direction.
fData - a double matrix of size xData.length by yData.length containing the values to be approximated.
xSplineSpaceDim - an int scalar value specifying the linear dimension of the spline subspace for the x variable. It should be smaller than xData.length and greater than or equal to xOrder (whose default value is 4).
ySplineSpaceDim - an int scalar value specifying the linear dimension of the spline subspace for the y variable. It should be smaller than yData.length and greater than or equal to yOrder (whose default value is 4).
Method Detail

compute

public void compute()
Computes a two-dimensional, tensor-product spline approximant using least squares.


getErrorSumOfSquares

public double getErrorSumOfSquares()
Returns the weighted error sum of squares.

Returns:
a double scalar containing the weighted error sum of squares.

getXOrder

public int getXOrder()
Returns the order of the spline in the x-direction.

Returns:
an int scalar containing the order of the spline in the x-direction.

getXWeights

public double[] getXWeights()
Returns the weights for the least-squares fit in the x-direction.

Returns:
a double array containing the weights for the least-squares fit in the x-direction.

getYOrder

public int getYOrder()
Returns the order of the spline in the y-direction.

Returns:
an int scalar containing the order of the spline in the y-direction.

getYWeights

public double[] getYWeights()
Returns the weights for the least-squares fit in the y-direction.

Returns:
a double array containing the weights for the least-squares fit in the y-direction.

setXKnots

public void setXKnots(double[] xKnots)
Sets the knot sequences of the spline in the x-direction.

Parameters:
xKnots - a double array of size xSplineSpaceDim + xOrder specifying the knot sequences of the spline in the x-direction. Default knot sequences are selected by the class.

setXOrder

public void setXOrder(int xOrder)
Sets the order of the spline in the x-direction.

Parameters:
xOrder - an int scalar value specifying the order of the spline in the x-direction. xOrder must be at least 1. Default: xOrder = 4, implying a tensor-product cubic spline.

setXWeights

public void setXWeights(double[] xWeights)
Sets the weights for the least-squares fit in the x-direction.

Parameters:
xWeights - a double array of size xData.length specifying the weights for the least-squares fit in the x-direction. Default: all weights are equal to 1.

setYKnots

public void setYKnots(double[] yKnots)
Sets the knot sequences of the spline in the y-direction.

Parameters:
yKnots - a double array of size ySplineSpaceDim + yOrder specifying the knot sequences of the spline in the y-direction. Default knot sequences are selected by the class.

setYOrder

public void setYOrder(int yOrder)
Sets the order of the spline in the y-direction.

Parameters:
yOrder - an int scalar value specifying the order of the spline in the y-direction. yOrder must be at least 1. Default: yOrder = 4, implying a tensor-product cubic spline.

setYWeights

public void setYWeights(double[] yWeights)
Sets the weights for the least-squares fit in the y-direction.

Parameters:
yWeights - a double array of size yData.length specifying the weights for the least-squares fit in the y-direction. Default: all weights are equal to 1.

JMSLTM Numerical Library 6.1

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Built July 30 2010.