This class computes the Hickernell \(\mathcal{L}_2\)-star discrepancy in [81] (eq. More...
Public Member Functions | |
| double | compute (double[][] points, int n, int s, double[] gamma) |
| DiscL2Hickernell (double[][] points, int n, int s) | |
Constructor with the \(n\) points points[i] in dimension \(s\). More... | |
| DiscL2Hickernell (int n, int s) | |
| Constructor with \(n\) points in dimension \(s\). More... | |
| DiscL2Hickernell (PointSet set) | |
Constructor with the point set set. More... | |
| DiscL2Hickernell () | |
| Empty constructor. More... | |
| double | compute (double[][] points, int n, int s) |
Computes the Hickernell \(\mathcal{L}_2\)-discrepancy ( disc.hicks ) for the set of \(n\) \(s\)-dimensional points points. | |
| double | compute (double[] T, int n) |
Computes the Hickernell \(\mathcal{L}_2\)-discrepancy ( hickD1 ) for the set of \(n\) 1-dimensional points T. | |
 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. | |
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) |
| Static Protected Member Functions inherited from Discrepancy | |
| static void | setONES (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 Hickernell \(\mathcal{L}_2\)-star discrepancy in [81] (eq.
5.1c) for a point set. It is based on the reproducing kernel Hilbert space which consists of functions with square-integrable first-order derivatives, and whose reproducing kernel is given by
\[ K(\mathbf{x},\mathbf{y}) = \prod_{j=1}^s \eta_j(x_j,y_j), \]
where
\[ \eta_j(x,y) = 1 + \frac{|x-a_j| + |y - a_j| - |x-y|}{2}. \]
The inner product is given by
\[ \langle f, g \rangle= \sum_{\mathfrak u\subseteq S} \int_{[0, 1)^{|\mathfrak u|}} \frac{\partial^{|\mathfrak u|}f(\mathbf{x}_{\mathfrak u}, \boldsymbol a)}{\partial\mathbf{x_{\mathfrak u}}} \frac{\partial^{|\mathfrak u|}g(\mathbf{x}_{\mathfrak u}, \boldsymbol a)}{\partial\mathbf{x_{\mathfrak u}}} d\mathbf{x_{\mathfrak u}}, \]
where \(S\) is the set of coordinates indices \(\{1, 2, …, s\}\), \(|\mathfrak u|\) denotes the cardinality of \(\mathfrak u\), \(\mathbf{x_{\mathfrak u}}\) denotes the vector made of the components of \( \mathbf{x} \) whose indices are in \(\mathfrak u\), and \(\boldsymbol a \in[0, 1]^s\) is called the anchor.
The worst-case error for this function space is the \(\mathcal{L}_2\) version of the star discrepancy \(\mathcal{D}_2^*\). Choosing \(\boldsymbol a =\mathbf{1}\), Hickernell obtained the formula
\[ [\mathcal{D}_2^*(\mathcal{P})]^2 = \left(\frac{4}{3}\right)^s - \frac{2}{n} \sum_{i=1}^n \prod_{k=1}^s \left(\frac{3 - z_{ik}^2}{2}\right) + \frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n \prod_{k=1}^s \Bigl(2 - \max(z_{ik}, z_{jk})\Bigr), \tag{disc.hicks} \]
where \(n\) is the number of points of set \(\mathcal{P}\), \(s\) is the dimension of the points, and \(z_{ik}\) is the \(k\)-th coordinate of point \(i\).
In 1 dimension, the formula is equivalent to
\[ [\mathcal{D}_2^*(\mathcal{P})]^2 = \frac{1}{3} + \frac{1}{n} \sum_{i=1}^n {z_i^2} - \frac{1}{n^2} \sum_{i=1}^n\sum_{j=1}^n \max(z_i, z_j), \tag{hickD1} \]
where \(z_i\) is the coordinate of point \(i\).
Constructor & Destructor Documentation
◆ DiscL2Hickernell() [1/4]
| DiscL2Hickernell | ( | double | points[][], |
| int | n, | ||
| int | s | ||
| ) |
Constructor with the \(n\) points points[i] in dimension \(s\).
points[i][j] is the \(j\)-th coordinate of point \(i\). Both \(i\) and \(j\) start at 0.
◆ DiscL2Hickernell() [2/4]
| DiscL2Hickernell | ( | int | n, |
| int | s | ||
| ) |
Constructor with \(n\) points in dimension \(s\).
The \(n\) points will be chosen later.
◆ DiscL2Hickernell() [3/4]
| DiscL2Hickernell | ( | PointSet | set | ) |
Constructor with the point set set.
All the points are copied in an internal array.
◆ DiscL2Hickernell() [4/4]
| DiscL2Hickernell | ( | ) |
Empty constructor.
One must set the points and the dimension before calling any method.
The documentation for this class was generated from the following file:
- DiscL2Hickernell.java
