BsLeastSquares Class

Extension of the BSpline class to compute a least squares spline approximation to data points.

Inheritance Hierarchy

System.Object
Imsl.Math.BSpline
Imsl.Math.BsLeastSquares

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

Syntax

C++

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[SerializableAttribute]
public class BsLeastSquares : BSpline

<SerializableAttribute>
Public Class BsLeastSquares
	Inherits BSpline

[SerializableAttribute]
public ref class BsLeastSquares : public BSpline

[<SerializableAttribute>]
type BsLeastSquares =  
    class
        inherit BSpline
    end

The BsLeastSquares type exposes the following members.

Constructors

	Name	Description
	BsLeastSquares(Double[],Double[], Int32)	Constructs a least squares B-spline approximation to the given data points.
	BsLeastSquares(Double[],Double[], Int32, Int32)	Constructs a least squares B-spline approximation to the given data points.
	BsLeastSquares(Double[],Double[], Int32, Int32,Double[],Double[])	Constructs a least squares B-spline approximation to the given data points.

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Methods

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

Let's make the identifications

n = xData.Length

x = xData

f = yData

m = nCoef

k = order

For convenience, we assume that the sequence x is increasing, although the class does not require this.

By default, k = 4, and the knot sequence we select equally distributes the knots through the distinct ${x_i}'s$ . In particular, the m + k knots will be generated in with k knots stacked at each of the extreme values. The interior knots will be equally spaced in the interval.

Once knots ${\bf t}$ and weights w are determined, then the spline least-squares fit to the data is computed by minimizing over the linear coefficients

$\sum_{i=0}^{n-1} {w_i\biggl[f_i-\sum_{j=1}^{m}{a_jB_j(x_i)}\biggr]^2}$

where the are a (B-spline) basis for the spline subspace.

This algorithm is based on the routine L2APPR by deBoor (1978, p. 255).

Reference

Imsl.Math Namespace

Other Resources

Example