This class computes the discrepancy in [84] (eq. More...
Public Member Functions | |
| DiscShift2 (double[][] points, int n, int s) | |
Constructor with the \(n\) points \(P_i = \) points[i] in dimension \(s\), with all weights \(\gamma_j =1\). More... | |
| DiscShift2 (double[][] points, int n, int s, double[] gamma) | |
Constructor with the \(n\) points \(P_i = \) points[i] in dimension \(s\), with the weights \(\gamma_j = \) gamma[j-1], \(j = 1, …, s\). More... | |
| DiscShift2 (int n, int s, double[] gamma) | |
The number of points is \(n\), the dimension \(s\), and the \(s\) weight factors are gamma[ \(j\)], \(j = 0, 1, …, (s-1)\). More... | |
| DiscShift2 (PointSet set) | |
Constructor with the point set set. More... | |
| DiscShift2 () | |
| Empty constructor. More... | |
| double | compute (double[][] points, int n, int s) |
Computes the discrepancy ( shift2 ) for the first \(n\) points of set points in dimension \(s\). More... | |
| double | compute (double[][] points, int n, int s, double[] gamma) |
Computes the discrepancy ( shift2 ) for the first \(n\) points of set points in dimension \(s\) and with weight \(\gamma_r = \) gamma[r-1]. | |
| double | compute (double[] T, int n) |
| Computes the discrepancy ( shift2dim1 ) 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 ( shift2dim1 ) for the first \(n\) points of \(T\) in 1 dimension, with weight \(\gamma=\) gamma. | |
 Public Member Functions inherited from Discrepancy | |
| Discrepancy (double[][] points, int n, int s) | |
Constructor with the \(n\) points points[i] in \(s\) dimensions. More... | |
| 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[ \(j\)], \(j = 0, 1, …, (s-1)\). More... | |
| Discrepancy (int n, int s, double[] gamma) | |
The number of points is \(n\), the dimension \(s\), and the \(s\) weight factors are gamma[ \(j\)], \(j = 0, 1, …, (s-1)\). More... | |
| Discrepancy (PointSet set) | |
Constructor with the point set set. More... | |
| Discrepancy () | |
| Empty constructor. More... | |
| 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, int n, int s, double[] gamma) |
Computes the discrepancy of the first n points of points in dimension s with weights gamma. | |
| abstract double | compute (double[][] points, int n, int s) |
Computes the discrepancy of the first n points of points in dimension s with weights \(=1\). | |
| double | compute (double[][] points) |
Computes the discrepancy of all the points of points in maximum dimension. More... | |
| double | compute (double[] T, int n) |
Computes the discrepancy of the first n points of T in 1 dimension. More... | |
| double | compute (double[] T) |
Computes the discrepancy of all the points of T in 1 dimension. More... | |
| 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. More... | |
| 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 \(s\). More... | |
| void | setPoints (double[][] points) |
Sets the points to points. More... | |
| 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. | |
Static Protected Member Functions | |
| static double [] | setC (double gam) |
| static void | setC (double[] C1, double[] C2, double[] gam, int s) |
| Static Protected Member Functions inherited from Discrepancy | |
| static void | setONES (int s) |
Additional Inherited Members | |
| Static Public Member Functions inherited from Discrepancy | |
| static double [][] | toArray (PointSet set) |
Returns all the \(n\) points ( \(s\)-dimensional) of umontreal.ssj.hups.PointSet set as an array points[ \(n\)][ \(s\)]. | |
| static DoubleArrayList | sort (double[] T, int n) |
| Sorts the first \(n\) points of \(T\). More... | |
| Protected Member Functions inherited from Discrepancy | |
| void | appendGamma (StringBuffer sb, double[] gamma, int s) |
| Protected Attributes inherited from Discrepancy | |
| double [] | gamma |
| double [][] | Points |
| int | dim |
| int | numPoints |
| Static Protected Attributes inherited from Discrepancy | |
| static double [] | ONES = { 1 } |
| Static Package Attributes inherited from Discrepancy | |
| static final double | UNSIX = 1.0/6.0 |
| static final double | QUARAN = 1.0/42.0 |
| static final double | UNTRENTE = 1.0 / 30.0 |
| static final double | DTIERS = 2.0 / 3.0 |
| static final double | STIERS = 7.0 / 3.0 |
| static final double | QTIERS = 14.0 / 3.0 |
Detailed Description
This class computes the discrepancy in [84] (eq.
15) for the randomly shifted points of a set \(\mathcal{P}\). It is given by
\[ [\mathcal{D}(\mathcal{P})]^2 = -1 + \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \prod_{r=1}^s \left[1 + \frac{\gamma_r^2}{2} B_2(\{x_{ir} - x_{jr}\}) - \frac{\gamma_r^4}{12}B_4(\{x_{ir} - x_{jr}\}) \right], \tag{shift2} \]
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). The notation \(\{x\}\) means the fractional part of \(x\), defined as \(\{x\} = x \bmod1\). In one dimension, the formula simplifies to
\[ [\mathcal{D}(\mathcal{P})]^2 = \frac{1}{n^2} \sum_{i=1}^n \sum_{j=1}^n \left[\frac{\gamma^2}{2} B_2(\{x_i - x_j\}) - \frac{\gamma^4}{12}B_4(\{x_i - x_j\}) \right]. \tag{shift2dim1} \]
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 \(\gamma_u = \prod_{r\in u} \gamma_r\).
Constructor & Destructor Documentation
◆ DiscShift2() [1/5]
| DiscShift2 | ( | double | points[][], |
| int | n, | ||
| int | s | ||
| ) |
Constructor with the \(n\) points \(P_i = \) points[i] in dimension \(s\), with all weights \(\gamma_j =1\).
points[i][j] is the j-th coordinate of point i. Both indices i and j starts at 0.
◆ DiscShift2() [2/5]
| DiscShift2 | ( | double | points[][], |
| int | n, | ||
| int | s, | ||
| double [] | gamma | ||
| ) |
Constructor with the \(n\) points \(P_i = \) points[i] in dimension \(s\), with the weights \(\gamma_j = \) gamma[j-1], \(j = 1, …, s\).
points[i][j] is the j-th coordinate of point i. Both indices i and j starts at 0.
◆ DiscShift2() [3/5]
| DiscShift2 | ( | 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.
◆ DiscShift2() [4/5]
| DiscShift2 | ( | PointSet | set | ) |
Constructor with the point set set.
All the points are copied in an internal array.
◆ DiscShift2() [5/5]
| DiscShift2 | ( | ) |
Empty constructor.
One must set the points, the dimension, and the weight factors before calling any method.
Member Function Documentation
◆ compute()
| double compute | ( | double | points[][], |
| int | n, | ||
| int | s | ||
| ) |
Computes the discrepancy ( shift2 ) for the first \(n\) points of set points in dimension \(s\).
All weights \(\gamma_r = 1\).
The documentation for this class was generated from the following file:
- DiscShift2.java
