CsSmooth Class

CsSmooth Class

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

Inheritance Hierarchy

System.Object
Imsl.Math.Spline
Imsl.Math.CsSmooth

Namespace: Imsl.Math
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class CsSmooth : Spline

<SerializableAttribute>
Public Class CsSmooth
	Inherits Spline

[SerializableAttribute]
public ref class CsSmooth : public Spline

[<SerializableAttribute>]
type CsSmooth =  
    class
        inherit Spline
    end

The CsSmooth type exposes the following members.

Constructors

	Name	Description
	CsSmooth(Double[],Double[])	Constructs a smooth cubic spline from noisy data using cross-validation to estimate the smoothing parameter. All of the points have equal weights.
	CsSmooth(Double[],Double[],Double[])	Constructs a smooth cubic spline from noisy data using cross-validation to estimate the smoothing parameter. Weights are supplied by the user.

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Methods

	Name	Description
	Derivative(Double)	Returns the value of the first derivative of the spline at a point. (Inherited from Spline.)
	Derivative(Double, Int32)	Returns the value of the derivative of the spline at a point. (Inherited from Spline.)
	Derivative(Double[], Int32)	Returns the value of the derivative of the spline at each point of an array. (Inherited from Spline.)
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Eval(Double)	Returns the value of the spline at a point. (Inherited from Spline.)
	Eval(Double[])	Returns the value of the spline at each point of an array. (Inherited from Spline.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetBreakpoints	Returns a copy of the breakpoints. (Inherited from Spline.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	Integral	Returns the value of an integral of the spline. (Inherited from Spline.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Remarks

Class CsSmooth is designed to produce a 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 . The smoothing spline S is the unique 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 , 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).

Reference

Imsl.Math Namespace

Other Resources

Example