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

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
[SerializableAttribute]
public class CsSmooth : Spline
Visual Basic (Declaration)
<SerializableAttribute> _
Public Class CsSmooth _
	Inherits Spline
Visual C++
[SerializableAttribute]
public ref class CsSmooth : public Spline

Remarks

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).

Inheritance Hierarchy

See Also