This class computes the discrepancy for randomly shifted, then baker folded points of a set \(\mathcal{P}\). More...
Public Member Functions | |
| DiscShiftBaker1 (double[][] points, int n, int s) | |
| Constructor with the \(n\) points points[i] in \(s\) dimensions, with all the weights \(\gamma_r = 1\). | |
| DiscShiftBaker1 (double[][] points, int n, int s, double[] gamma) | |
| Constructor with the \(n\) points points[i] in \(s\) dimensions, with weights \(\gamma_r = \) gamma[r-1]. | |
| DiscShiftBaker1 (int n, int s, double[] gamma) | |
| The number of points is \(n\), the dimension \(s\), and the. | |
| DiscShiftBaker1 (PointSet set) | |
| Constructor with the point set set. | |
| DiscShiftBaker1 () | |
| Empty constructor. | |
| double | compute (double[][] points, int n, int s) |
Computes the discrepancy ( baker1 ) for the \(s\)-dimensional points of set points, containing. | |
| double | compute (double[][] points, int n, int s, double[] gamma) |
Computes the discrepancy ( baker1 ) for the first \(n\) points of points in dimension \(s\) and with weight \(\gamma_r = \) gamma[r-1]. | |
| double | compute (double[] T, int n) |
Computes the discrepancy ( baker1dim1 ) for the first \(n\) points of \(T\) in 1 dimension, with weight \(\gamma= 1\). | |
| double | compute (double[] T, int n, double gamma) |
Computes the discrepancy ( baker1dim1 ) for the first \(n\) points of \(T\) in 1 dimension, with weight \(\gamma=\) gamma. | |
| 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 (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, then baker folded points of a set \(\mathcal{P}\).
It is given by
[81] (eq. 15)
\begin{align} [\mathcal{D}(\mathcal{P})]^2 & = -1 + \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \prod_{r=1}^s \left(1 - \frac{4\gamma_r^2}{3} \left[B_4(\{x_{ir} - x_{jr}\}) - B_4(\{\{x_{ir} - x_{jr}\}-1/2\})\right]\right. \nonumber \\ & \qquad\qquad{} - \frac{\gamma_r^4}{9} \left[7B_4(\{x_{ir} - x_{jr}\}) - 2B_4(\{\{x_{ir} - x_{jr}\}-1/2\})\right] \tag{baker1} \\ & \qquad\qquad{} \left. {} - \frac{16\gamma_r^4}{45} \left[B_6(\{x_{ir} - x_{jr}\}) -B_6(\{\{x_{ir} - x_{jr}\}-1/2\})\right] \right)\nonumber, \end{align}
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
\begin{align} [\mathcal{D}(\mathcal{P})]^2 & = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \left[ - \frac{4\gamma^2}{3} \left[B_4(\{x_i - x_j\}) - B_4(\{\{x_i - x_j\}-1/2\})\right]\right. \nonumber \\ & \qquad\qquad{} - \frac{\gamma^4}{9} \left[7B_4(\{x_i - x_j\}) - 2B_4(\{\{x_i - x_j\}-1/2\})\right] \tag{baker1dim1} \\ & \qquad\qquad{} \left. {} - \frac{16\gamma^4}{45} \left[B_6(\{x_i - x_j\}) -B_6(\{\{x_i - x_j\}-1/2\})\right] \right]\nonumber. \end{align}
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
\[ K(\mathbf{x},\mathbf{y}) = \prod_{r=1}^s \left[ - \frac{\gamma_r^4}{4!} B_4(\{x_r-y_r\}) + \sum_{\alpha=0}^2 \frac{\gamma_r^{2\alpha}}{(\alpha!)^2} B_{\alpha}(x_r)B_{\alpha}(y_r) \right], \]
The norm of the vectors in \(\mathcal{H}\) is defined by
\[ \| f\|^2 = \sum_{u \subseteq S}\sum_{v \subseteq u}\gamma_u^{-2}\gamma_v^{-2} \int_{[0,1]^v}d\mathbf{x}_v\left[\int_{[0,1]^{S-v}} \frac{\partial^{|u| + |v|}f}{\partial\mathbf{x}_u\partial\mathbf{x}_v} d\mathbf{x}_{S-v} \right]^2, \]
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 80 of file DiscShiftBaker1.java.
Constructor & Destructor Documentation
◆ DiscShiftBaker1() [1/5]
| umontreal.ssj.discrepancy.DiscShiftBaker1.DiscShiftBaker1 | ( | double | points[][], |
| int | n, | ||
| int | s ) |
Constructor with the \(n\) points points[i] in \(s\) dimensions, with all the weights \(\gamma_r = 1\).
points[i][r] is the r-th coordinate of point i. Indices i and r start at 0.
Definition at line 107 of file DiscShiftBaker1.java.
◆ DiscShiftBaker1() [2/5]
| umontreal.ssj.discrepancy.DiscShiftBaker1.DiscShiftBaker1 | ( | double | points[][], |
| int | n, | ||
| int | s, | ||
| double[] | gamma ) |
Constructor with the \(n\) points points[i] in \(s\) dimensions, with weights \(\gamma_r = \) gamma[r-1].
points[i][r] is the r-th coordinate of point i. Indices i and r start at 0.
Definition at line 116 of file DiscShiftBaker1.java.
◆ DiscShiftBaker1() [3/5]
| umontreal.ssj.discrepancy.DiscShiftBaker1.DiscShiftBaker1 | ( | int | n, |
| int | s, | ||
| double[] | gamma ) |
The number of points is \(n\), the dimension \(s\), and the.
\(s\) weight factors are gamma[ \(r\)], \(r = 0, 1, …, (s-1)\). The \(n\) points will be chosen later.
Definition at line 126 of file DiscShiftBaker1.java.
◆ DiscShiftBaker1() [4/5]
| umontreal.ssj.discrepancy.DiscShiftBaker1.DiscShiftBaker1 | ( | PointSet | set | ) |
Constructor with the point set set.
All the points are copied in an internal array.
Definition at line 134 of file DiscShiftBaker1.java.
◆ DiscShiftBaker1() [5/5]
| umontreal.ssj.discrepancy.DiscShiftBaker1.DiscShiftBaker1 | ( | ) |
Empty constructor.
One must set the points, the dimension, and the weight factors before calling any method.
Definition at line 142 of file DiscShiftBaker1.java.
Member Function Documentation
◆ compute() [1/4]
| double umontreal.ssj.discrepancy.DiscShiftBaker1.compute | ( | double[] | T, |
| int | n ) |
Computes the discrepancy ( baker1dim1 ) for the first \(n\) points of \(T\) in 1 dimension, with weight \(\gamma= 1\).
Reimplemented from umontreal.ssj.discrepancy.Discrepancy.
Reimplemented in umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.
Definition at line 216 of file DiscShiftBaker1.java.
◆ compute() [2/4]
| double umontreal.ssj.discrepancy.DiscShiftBaker1.compute | ( | double[] | T, |
| int | n, | ||
| double | gamma ) |
Computes the discrepancy ( baker1dim1 ) for the first \(n\) points of \(T\) in 1 dimension, with weight \(\gamma=\) gamma.
Reimplemented from umontreal.ssj.discrepancy.Discrepancy.
Reimplemented in umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.
Definition at line 225 of file DiscShiftBaker1.java.
◆ compute() [3/4]
| double umontreal.ssj.discrepancy.DiscShiftBaker1.compute | ( | double | points[][], |
| int | n, | ||
| int | s ) |
Computes the discrepancy ( baker1 ) for the \(s\)-dimensional points of set points, containing.
\(n\) points. All weights \(\gamma_r = 1\).
Reimplemented from umontreal.ssj.discrepancy.Discrepancy.
Reimplemented in umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.
Definition at line 151 of file DiscShiftBaker1.java.
◆ compute() [4/4]
| double umontreal.ssj.discrepancy.DiscShiftBaker1.compute | ( | double | points[][], |
| int | n, | ||
| int | s, | ||
| double[] | gamma ) |
Computes the discrepancy ( baker1 ) for the first \(n\) points of points in dimension \(s\) and with weight \(\gamma_r = \) gamma[r-1].
Reimplemented from umontreal.ssj.discrepancy.Discrepancy.
Reimplemented in umontreal.ssj.discrepancy.DiscShiftBaker1Lattice.
Definition at line 161 of file DiscShiftBaker1.java.
The documentation for this class was generated from the following file:
- src/main/java/umontreal/ssj/discrepancy/DiscShiftBaker1.java
