This class computes the Hickernell \(\mathcal{L}_2\)-star discrepancy in [78] (eq. More...
Public Member Functions | |
| 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. | |
| DiscL2Hickernell (double[][] points, int n, int s) | |
| Constructor with the \(n\) points points[i] in dimension. | |
| DiscL2Hickernell (int n, int s) | |
| Constructor with \(n\) points in dimension \(s\). | |
| DiscL2Hickernell (PointSet set) | |
| Constructor with the point set set. | |
| DiscL2Hickernell () | |
| Empty constructor. | |
| double | compute (double[][] points, int n, int s) |
| Computes the Hickernell. | |
| double | compute (double[] T, int n) |
| Computes the Hickernell. | |
| 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 Hickernell \(\mathcal{L}_2\)-star discrepancy in [78] (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 \(i\).
Definition at line 65 of file DiscL2Hickernell.java.
Constructor & Destructor Documentation
◆ DiscL2Hickernell() [1/4]
| umontreal.ssj.discrepancy.DiscL2Hickernell.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.
Definition at line 77 of file DiscL2Hickernell.java.
◆ DiscL2Hickernell() [2/4]
| umontreal.ssj.discrepancy.DiscL2Hickernell.DiscL2Hickernell | ( | int | n, |
| int | s ) |
Constructor with \(n\) points in dimension \(s\).
The \(n\) points will be chosen later.
Definition at line 85 of file DiscL2Hickernell.java.
◆ DiscL2Hickernell() [3/4]
| umontreal.ssj.discrepancy.DiscL2Hickernell.DiscL2Hickernell | ( | PointSet | set | ) |
Constructor with the point set set.
All the points are copied in an internal array.
Definition at line 93 of file DiscL2Hickernell.java.
◆ DiscL2Hickernell() [4/4]
| umontreal.ssj.discrepancy.DiscL2Hickernell.DiscL2Hickernell | ( | ) |
Empty constructor.
One must set the points and the dimension before calling any method.
Definition at line 101 of file DiscL2Hickernell.java.
Member Function Documentation
◆ compute() [1/3]
| double umontreal.ssj.discrepancy.DiscL2Hickernell.compute | ( | double[] | T, |
| int | n ) |
Computes the Hickernell.
\(\mathcal{L}_2\)-discrepancy ( hickD1 ) for the set of \(n\) 1-dimensional points T.
Reimplemented from umontreal.ssj.discrepancy.Discrepancy.
Definition at line 162 of file DiscL2Hickernell.java.
◆ compute() [2/3]
| double umontreal.ssj.discrepancy.DiscL2Hickernell.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.
Reimplemented from umontreal.ssj.discrepancy.Discrepancy.
Definition at line 114 of file DiscL2Hickernell.java.
◆ compute() [3/3]
| double umontreal.ssj.discrepancy.DiscL2Hickernell.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.
Reimplemented from umontreal.ssj.discrepancy.Discrepancy.
Definition at line 67 of file DiscL2Hickernell.java.
The documentation for this class was generated from the following file:
- src/main/java/umontreal/ssj/discrepancy/DiscL2Hickernell.java
