com.imsl.math

Class CsSmooth

java.lang.Object

com.imsl.math.Spline

com.imsl.math.CsSmooth

All Implemented Interfaces:

Serializable, Cloneable

public class CsSmooth
extends Spline

Extension of the Spline class to construct a smooth cubic spline from noisy data points.

Class CsSmooth 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 = xData, but it does not interpolate the data \((x_i, f_i)\). The smoothing spline S is the unique \(C^2\) function that minimizes

$$\int\limits_a^b {S''\left( x \right)^2 dx}$$

subject to the constraint

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

where \(\sigma\) is the smoothing parameter. The reader should consult Reinsch (1967) for more information concerning smoothing splines. CsSmooth solves the above problem when the user provides the smoothing parameter \(\sigma\). CsSmoothC2 attempts to find the "optimal" smoothing parameter using the statistical technique known as cross-validation. This means that (in a very rough sense) one chooses the value of \(\sigma\) so that the smoothing spline \((S_\sigma)\) best approximates the value of the data at \(x_I\), if it is computed using all the data except the i-th; this is true for all \(i = 0, \ldots, n-1\). For more information on this topic, we refer the reader to Craven and Wahba (1979).

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
CsSmooth(double[] xData, double[] yData) Constructs a smooth cubic spline from noisy data using cross-validation to estimate the smoothing parameter.
CsSmooth(double[] xData, double[] yData, double[] weight) Constructs a smooth cubic spline from noisy data using cross-validation to estimate the smoothing parameter.

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

CsSmooth

public CsSmooth(double[] xData,
                double[] yData)

Constructs a smooth cubic spline from noisy data using cross-validation to estimate the smoothing parameter. 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.

CsSmooth

public CsSmooth(double[] xData,
                double[] yData,
                double[] weight)

Constructs a smooth cubic spline from noisy data using cross-validation to estimate the smoothing parameter. Weights are 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 relative weights. This array must have the same length as xData.