FaureSequence Class

FaureSequence Class

Generates the low-discrepancy Faure sequence.

Inheritance Hierarchy

System.Object
Imsl.Stat.FaureSequence

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

Syntax

C++

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[SerializableAttribute]
public class FaureSequence : IRandomSequence

<SerializableAttribute>
Public Class FaureSequence
	Implements IRandomSequence

[SerializableAttribute]
public ref class FaureSequence : IRandomSequence

[<SerializableAttribute>]
type FaureSequence =  
    class
        interface IRandomSequence
    end

The FaureSequence type exposes the following members.

Constructors

	Name	Description
	FaureSequence(Int32)	Creates a Faure sequence with the default base.
	FaureSequence(Int32, Int32, Int32)	Creates a Faure sequence.

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Methods

	Name	Description
	ComputeParameters	Compute needed parameters.
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	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.)
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	NextDouble	Returns the first value of the next point in the sequence.
	NextPoint	Returns the next point in the sequence.
	NextPrime	Returns the smallest prime greater than or equal to n.
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	Base	The base.
	Dimension	Returns the dimension of the sequence.
	NumberOfProcessors	Perform the parallel calculations with the maximum possible number of processors set to NumberOfProcessors.
	Skip	Returns the number of points skipped at the beginning of the sequence.

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Remarks

Discrepancy measures the deviation from uniformity of a point set.

The discrepancy of the point set $x_1,\ldots,x_n \in [0,1]^d,\, d \ge 1$ , is

$D_n^{(d)} = \sup_E \left| \frac{A(E;n)}{n} - \lambda(E) \right|,$

where the supremum is over all subsets of

of the form

$E = \left[0,t_1\right) \times \cdots \times \left[0,t_d\right), \,\, 0 \le t_j \le 1, \, 1 \le j \le d,$

$\lambda$ is the Lebesque measure, and A(E;n) is the number of the

contained in E.

The sequence $x_1, x_2, \ldots$ of points in is a low-discrepancy sequence if there exists a constant c(d), depending only on d, such that

$D_n^{(d)} \le c(d) \frac{(\log n)^d}{n}$

for all $n \gt 1$ .

Generalized Faure sequences can be defined for any prime base $b \ge d$ . The lowest bound for the discrepancy is obtained for the smallest prime $b \ge d$ , so the base defaults to the smallest prime greater than or equal to the dimension.

The generalized Faure sequence $x_1, x_2, \ldots$ , is computed as follows:

Write the positive integer n in its b-ary expansion,

$n=\sum_{i=0}^\infty a_i(n)b^i$

where

are integers, $0 \le a_j(n) \lt b$ .

The j-th coordinate of is

$x_n^{(j)} = \sum_{k=0}^\infty \sum_{d=0}^\infty c_{kd}^{(j)} a_d(n) b^{-k-1}, \,\, 1 \le j \le d$

The generator matrix for the series, $c_{kd}^{(j)}$ , is defined to be

$c_{kd}^{(j)} = j^{d-k} c_{kd}$

and $c_{kd}$ is an element of the Pascal matrix,

$c_{kd} = \left\{ \begin{array}{ll} \frac{d!}{c! (d-c)!} & k \le d \\ 0 & k \gt d \end{array} \right.$

It is faster to compute a shuffled Faure sequence than to compute the Faure sequence itself. It can be shown that this shuffling preserves the low-discrepancy property.

The shuffling used is the b-ary Gray code. The function G(n) maps the positive integer n into the integer given by its b-ary expansion. The sequence computed by this function is $\vec{x}(G(n))$ , where $\vec{x}$ is the generalized Faure sequence.

Reference

Imsl.Stat Namespace

Other Resources

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