com.imsl.math

Class CsSmoothC2

java.lang.Object

com.imsl.math.Spline

com.imsl.math.CsSmoothC2

All Implemented Interfaces:

Serializable, Cloneable

public class CsSmoothC2
extends Spline

Extension of the Spline class used to construct a spline for noisy data points using an alternate method.

Class CsSmoothC2 is designed to produce a \(C^2\) cubic spline approximation to a data set in which the function values are noisy. This spline is called a smoothing spline. It is a natural cubic spline with knots at all the data abscissas x, but it does not interpolate the data \((x_i, f_i)\). The smoothing spline \(S_\sigma\) is the unique \(C^2\) function that minimizes $$\int\limits_a^b {s''_\sigma \left( x \right)^2 dx}$$

subject to the constraint

$$\sum\limits_{i=0}^{n-1} {\left| {s_\sigma \left( {x_i } \right) - f_i } \right|} ^2 \le \sigma$$.

Recommended values for \(\sigma\) depend on the weights, w. If an estimate for the standard deviation of the error in the y-values is availiable, then \(w_i\) should be set to this value and the smoothing parameter should be choosen in the confidence interval corresponding to the left side of the above inequality. That is, $$n-\sqrt{2n} \le \sigma \le n+\sqrt{2n}\) CsSmoothC2 is based on an algorithm of Reinsch (1967). This algorithm is also discussed in de Boor (1978, pages 235-243).

See Also:

Example, Serialized Form

Field Summary

Fields inherited from class com.imsl.math.Spline

breakPoint, coef, EPSILON_LARGE

Constructor Summary

Constructors

Constructor and Description
CsSmoothC2(double[] xData, double[] yData, double sigma) Constructs a smooth cubic spline from noisy data using an algorithm based on Reinsch (1967).
CsSmoothC2(double[] xData, double[] yData, double[] weight, double sigma) Constructs a smooth cubic spline from noisy data using an algorithm based on Reinsch (1967) with weights supplied by the user.

Method Summary

Methods inherited from class com.imsl.math.Spline

copyAndSortData, copyAndSortData, derivative, derivative, derivative, getBreakpoints, integral, value, value

Methods inherited from class java.lang.Object

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

Constructor Detail

CsSmoothC2

public CsSmoothC2(double[] xData,
                  double[] yData,
                  double sigma)

Constructs a smooth cubic spline from noisy data using an algorithm based on Reinsch (1967). All of the points have equal weights.

Parameters:

xData - A double array containing the x-coordinates of the data. Values must be distinct.

yData - A double array containing the y-coordinates of the data. The arrays xData and yData must have the same length.

sigma - A double value specifying the smoothing parameter. Sigma must not be negative.

CsSmoothC2

public CsSmoothC2(double[] xData,
                  double[] yData,
                  double[] weight,
                  double sigma)

Constructs a smooth cubic spline from noisy data using an algorithm based on Reinsch (1967) with weights supplied by the user.

Parameters:

xData - A double array containing the x-coordinates of the data. Values must be distinct.

yData - A double array containing the y-coordinates of the data. The arrays xData and yData must have the same length.

weight - A double array containing the weights. The arrays xData and weight must have the same length.

sigma - A double value specifying the smoothing parameter. Sigma must not be negative.