umontreal.ssj.discrepancy.DiscShift1 Class Reference

This class computes the discrepancy for randomly shifted points of a set \(\mathcal{P}\) [81]  (eq. More...

Inheritance diagram for umontreal.ssj.discrepancy.DiscShift1:

Public Member Functions
	DiscShift1 (double[][] points, int n, int s)
	Constructor with the \(n\) points points[i] in \(s\) dimensions and with all weights \(\gamma_r =1\).
	DiscShift1 (double[][] points, int n, int s, double[] gamma)
	Constructor with the \(n\) points points[i] in \(s\) dimensions, and with the weights \(\gamma_r = \) gamma[r-1],.
	DiscShift1 (int n, int s, double[] gamma)
	The number of points is \(n\), the dimension \(s\), and the.
	DiscShift1 (PointSet set)
	Constructor with the point set set.
	DiscShift1 ()
	Empty constructor.
double	compute (double[][] points, int n, int s)
	Computes the discrepancy ( shift1 ) for the first \(n\) points of set points in dimension \(s\).
double	compute (double[][] points, int n, int s, double[] gamma)
	Computes the discrepancy ( shift1 ) in dimension \(s\) with \(\gamma_r = \) gamma[r-1].
double	compute (double[] T, int n)
	Computes the discrepancy ( shift1dim1 ) for the 1-dimensional set of \(n\) points \(T\).
Public Member Functions inherited from umontreal.ssj.discrepancy.Discrepancy
	Discrepancy (double[][] points, int n, int s)
	Constructor with the \(n\) points points[i] in \(s\) dimensions.
	Discrepancy (double[][] points, int n, int s, double[] gamma)
	Constructor with the \(n\) points points[i] in \(s\) dimensions and the \(s\) weight factors gamma[ \(j\)], \(j = 0, 1, …, (s-1)\).
	Discrepancy (int n, int s, double[] gamma)
	The number of points is \(n\), the dimension \(s\), and the.
	Discrepancy (PointSet set)
	Constructor with the point set set.
	Discrepancy ()
	Empty constructor.
double	compute ()
	Computes the discrepancy of all the points in maximal dimension (dimension of the points).
double	compute (int s)
	Computes the discrepancy of all the points in dimension \(s\).
double	compute (double[][] points)
	Computes the discrepancy of all the points of points in maximum dimension.
double	compute (double[] T)
	Computes the discrepancy of all the points of T in 1 dimension.
double	compute (double[] T, int n, double gamma)
	Computes the discrepancy of the first n points of T in 1 dimension with weight gamma.
double	compute (PointSet set, double[] gamma)
	Computes the discrepancy of all the points in set in the same dimension as the point set and with weights gamma.
double	compute (PointSet set)
	Computes the discrepancy of all the points in set in the same dimension as the point set.
int	getNumPoints ()
	Returns the number of points \(n\).
int	getDimension ()
	Returns the dimension of the points \(s\).
void	setPoints (double[][] points, int n, int s)
	Sets the points to points and the dimension to \(s\).
void	setPoints (double[][] points)
	Sets the points to points.
void	setGamma (double[] gam, int s)
	Sets the weight factors to gam for each dimension up to \(s\).
double[]	getGamma ()
	Returns the weight factors gamma for each dimension up to \(s\).
String	toString ()
	Returns the parameters of this class.
String	formatPoints ()
	Returns all the points of this class.
String	getName ()
	Returns the name of the Discrepancy.

Additional Inherited Members
Static Public Member Functions inherited from umontreal.ssj.discrepancy.Discrepancy
static double[][]	toArray (PointSet set)
	Returns all the \(n\) points ( \(s\)-dimensional) of.
static DoubleArrayList	sort (double[] T, int n)
	Sorts the first \(n\) points of \(T\).

Detailed Description

This class computes the discrepancy for randomly shifted points of a set \(\mathcal{P}\) [81]  (eq.

15). It is given by

\[ [\mathcal{D}(\mathcal{P})]^2 = -1 + \frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n \prod_{r=1}^s \left[1 + \gamma_r^2 B_2(\{x_{ir} - x_{jr}\})\right], \tag{shift1} \]

where \(n\) is the number of points of \(\mathcal{P}\), \(s\) is the dimension of the points, \(x_{ir}\) is the \(r\)-th coordinate of point \(i\), and the \(\gamma_r\) are arbitrary positive weights. The \(B_{\alpha}(x)\) are the Bernoulli polynomials [1]  (chap. 23) of degree \(\alpha\) (see umontreal.ssj.util.Num.bernoulliPoly in class util/Num), and the notation \(\{x\}\) means the fractional part of \(x\), defined here as \(\{x\} = x \bmod1\). In one dimension, the formula simplifies to

\[ [\mathcal{D}(\mathcal{P})]^2 = \frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n B_2(\{x_i - x_j\}), \tag{shift1dim1} \]

where \(z_i\) is the point \(i\).

The discrepancy represents a worst-case error criterion for the approximation of integrals, when the integrands have a certain degree of smoothness and lie in a Hilbert space \(\mathcal{H}\) with a reproducing kernel \(K\) given by

\[ K(\mathbf{x},\mathbf{y}) = \prod_{r=1}^s \left[ \frac{\gamma_r^2}{2} B_2(\{x_r-y_r\}) + \sum_{\alpha=0}^1 \gamma_r^{2\alpha} B_{\alpha}(x_r)B_{\alpha}(y_r) \right], \]

The norm of the vectors in \(\mathcal{H}\) is defined by

\[ \| f\|^2 = \sum_{u \subseteq S} \gamma_u^{-2} \int_{[0,1]^u}d\mathbf{x}_u\left[\int_{[0,1]^{S-u}} \frac{\partial^{|u|}f}{\partial\mathbf{x}_u}d\mathbf{x}_{S-u} \right]^2, \]

where \(S= \{1, …, s\}\) is a set of coordinate indices, \(u \subseteq S\), and \(\gamma_u = \prod_{r\in u} \gamma_r\).

Definition at line 68 of file DiscShift1.java.

Constructor & Destructor Documentation

DiscShift1() [1/5]

umontreal.ssj.discrepancy.DiscShift1.DiscShift1	(	double	points[][],
		int	n,
		int	s )

Constructor with the \(n\) points points[i] in \(s\) dimensions and with all weights \(\gamma_r =1\).

points[i][j] is the j-th coordinate of point i. Indices i and j start at 0.

Definition at line 80 of file DiscShift1.java.

DiscShift1() [2/5]

umontreal.ssj.discrepancy.DiscShift1.DiscShift1	(	double	points[][],
		int	n,
		int	s,
		double[]	gamma )

Constructor with the \(n\) points points[i] in \(s\) dimensions, and with the weights \(\gamma_r = \) gamma[r-1],.

\(r = 1, …, s\). points[i][j] is the j-th coordinate of point i. Indices i and j start at 0.

Definition at line 91 of file DiscShift1.java.

DiscShift1() [3/5]

umontreal.ssj.discrepancy.DiscShift1.DiscShift1	(	int	n,
		int	s,
		double[]	gamma )

The number of points is \(n\), the dimension \(s\), and the.

\(s\) weight factors are gamma[ \(j\)], \(j = 0, 1, …, (s-1)\). The \(n\) points will be chosen later.

Definition at line 101 of file DiscShift1.java.

DiscShift1() [4/5]

umontreal.ssj.discrepancy.DiscShift1.DiscShift1

(

PointSet

set

)

Constructor with the point set set.

All the points are copied in an internal array.

Definition at line 109 of file DiscShift1.java.

DiscShift1() [5/5]

umontreal.ssj.discrepancy.DiscShift1.DiscShift1

(

)

Empty constructor.

One must set the points, the dimension, and the weight factors before calling any method.

Definition at line 117 of file DiscShift1.java.

Member Function Documentation

compute() [1/3]

double umontreal.ssj.discrepancy.DiscShift1.compute	(	double[]	T,
		int	n )

Computes the discrepancy ( shift1dim1 ) for the 1-dimensional set of \(n\) points \(T\).

Reimplemented from umontreal.ssj.discrepancy.Discrepancy.

Reimplemented in umontreal.ssj.discrepancy.DiscShift1Lattice.

Definition at line 169 of file DiscShift1.java.

compute() [2/3]

double umontreal.ssj.discrepancy.DiscShift1.compute	(	double	points[][],
		int	n,
		int	s )

Computes the discrepancy ( shift1 ) for the first \(n\) points of set points in dimension \(s\).

All weights \(\gamma_r = 1\).

Reimplemented from umontreal.ssj.discrepancy.Discrepancy.

Reimplemented in umontreal.ssj.discrepancy.DiscShift1Lattice.

Definition at line 125 of file DiscShift1.java.

compute() [3/3]

double umontreal.ssj.discrepancy.DiscShift1.compute	(	double	points[][],
		int	n,
		int	s,
		double[]	gamma )

Computes the discrepancy ( shift1 ) in dimension \(s\) with \(\gamma_r = \) gamma[r-1].

Reimplemented from umontreal.ssj.discrepancy.Discrepancy.

Reimplemented in umontreal.ssj.discrepancy.DiscShift1Lattice.

Definition at line 134 of file DiscShift1.java.

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The documentation for this class was generated from the following file:

• src/main/java/umontreal/ssj/discrepancy/DiscShift1.java