SSJ
3.2.1
Stochastic Simulation in Java
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This class provides the basic structures for storing and manipulating linear digital nets in base \(b\), for an arbitrary base \(b\ge2\). More...
Classes | |
class | DigitalNetIterator |
class | DigitalNetIteratorNoGray |
Public Member Functions | |
double | getCoordinate (int i, int j) |
PointSetIterator | iterator () |
DigitalNet () | |
Empty constructor. | |
double | getCoordinateNoGray (int i, int j) |
Returns \(u_{i,j}\), the coordinate \(j\) of point \(i\), the points being enumerated in the standard order (no Gray code). More... | |
PointSetIterator | iteratorNoGray () |
This iterator does not use the Gray code. More... | |
void | addRandomShift (int d1, int d2, RandomStream stream) |
Adds a random digital shift to all the points of the point set, using stream stream to generate the random numbers. More... | |
void | addRandomShift (RandomStream stream) |
Same as addRandomShift(0, dim, stream), where dim is the dimension of the digital net. More... | |
void | clearRandomShift () |
Erases the current digital random shift, if any. | |
String | toString () |
void | leftMatrixScramble (RandomStream stream) |
Applies a linear scramble by multiplying each \(\mathbf{C}_j\) on the left by a \(w\times w\) nonsingular lower-triangular matrix \(\mathbf{M}_j\), as suggested by Matoušek [169] and implemented by Hong and Hickernell [86] . More... | |
void | leftMatrixScrambleDiag (RandomStream stream) |
Similar to leftMatrixScramble except that all the off-diagonal elements of the \(\mathbf{M}_j\) are 0. More... | |
void | leftMatrixScrambleFaurePermut (RandomStream stream, int sb) |
Similar to leftMatrixScramble except that the diagonal elements of each matrix \(\mathbf{M}_j\) are chosen from a restricted set of the best integers as calculated by Faure [56] . More... | |
void | leftMatrixScrambleFaurePermutDiag (RandomStream stream, int sb) |
Similar to leftMatrixScrambleFaurePermut except that all off-diagonal elements are 0. More... | |
void | leftMatrixScrambleFaurePermutAll (RandomStream stream, int sb) |
Similar to leftMatrixScrambleFaurePermut except that the elements under the diagonal are also chosen from the same restricted set as the diagonal elements. More... | |
void | iBinomialMatrixScramble (RandomStream stream) |
Applies the \(i\)-binomial matrix scramble proposed by Tezuka [217] (see also [184] ). More... | |
void | iBinomialMatrixScrambleFaurePermut (RandomStream stream, int sb) |
Similar to iBinomialMatrixScramble except that the diagonal elements of each matrix \(\mathbf{M}_j\) are chosen as in leftMatrixScrambleFaurePermut. More... | |
void | iBinomialMatrixScrambleFaurePermutDiag (RandomStream stream, int sb) |
Similar to iBinomialMatrixScrambleFaurePermut except that all the off-diagonal elements are 0. More... | |
void | iBinomialMatrixScrambleFaurePermutAll (RandomStream stream, int sb) |
Similar to iBinomialMatrixScrambleFaurePermut except that the elements under the diagonal are also chosen from the same restricted set as the diagonal elements. More... | |
void | stripedMatrixScramble (RandomStream stream) |
Applies the striped matrix scramble proposed by Owen [184] . More... | |
void | stripedMatrixScrambleFaurePermutAll (RandomStream stream, int sb) |
Similar to stripedMatrixScramble except that the elements on and under the diagonal of each matrix \(\mathbf{M}_j\) are chosen as in leftMatrixScrambleFaurePermut. More... | |
void | rightMatrixScramble (RandomStream stream) |
Applies a linear scramble by multiplying each \(\mathbf{C}_j\) on the right by a single \(k\times k\) nonsingular upper-triangular matrix \(\mathbf{M}\), as suggested by Faure and Tezuka [56] (see also [86] ). More... | |
void | unrandomize () |
Restores the original generator matrices and removes the random shift. | |
void | resetGeneratorMatrices () |
Restores the original generator matrices. More... | |
void | eraseOriginalGeneratorMatrices () |
Erases the original generator matrices and replaces them by the current ones. More... | |
void | printGeneratorMatrices (int s) |
Prints the generator matrices in standard form for dimensions 1 to \(s\). | |
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int | getDimension () |
Returns the dimension (number of available coordinates) of the point set. More... | |
int | getNumPoints () |
Returns the number of points. More... | |
abstract double | getCoordinate (int i, int j) |
Returns \(u_{i,j}\), the coordinate \(j\) of the point \(i\). More... | |
PointSetIterator | iterator () |
Constructs and returns a point set iterator. More... | |
void | setStream (RandomStream stream) |
Sets the random stream used to generate random shifts to stream . More... | |
RandomStream | getStream () |
Returns the random stream used to generate random shifts. More... | |
void | randomize (PointSetRandomization rand) |
Randomizes the point set using the given rand . More... | |
void | addRandomShift (int d1, int d2, RandomStream stream) |
This method does nothing for this generic class. More... | |
void | addRandomShift (RandomStream stream) |
This method does nothing for this generic class. More... | |
void | addRandomShift (int d1, int d2) |
Similar to addRandomShift(d1, d2, stream) , with the current random stream. | |
void | addRandomShift () |
Similar to addRandomShift(0, d2, stream) with the current random stream and d2 the dimension of the current random shift. | |
void | clearRandomShift () |
Erases the current random shift, if any. | |
void | randomize (int fromDim, int toDim, RandomStream stream) |
By default, this method simply calls addRandomShift (fromDim, toDim, stream) , which does nothing. | |
void | randomize (RandomStream stream) |
By default, this method simply calls randomize(0, dim, stream). | |
void | randomize (int d1, int d2) |
By default, this method simply calls addRandomShift(d1, d2) . | |
void | randomize () |
By default, this method simply calls addRandomShift() . | |
void | unrandomize () |
By default, this method simply calls clearRandomShift() . | |
String | toString () |
Formats a string that contains information about the point set. More... | |
String | formatPoints () |
Same as invoking formatPoints(n, d) with \(n\) and \(d\) equal to the number of points and the dimension of this object, respectively. More... | |
String | formatPoints (int n, int d) |
Formats a string that displays the same information as returned by toString, together with the first \(d\) coordinates of the first \(n\) points. More... | |
String | formatPoints (PointSetIterator iter) |
Same as invoking formatPoints(iter, n, d) with \(n\) and \(d\) equal to the number of points and the dimension, respectively. More... | |
String | formatPoints (PointSetIterator iter, int n, int d) |
Same as invoking formatPoints(n, d), but prints the points by calling iter repeatedly. More... | |
String | formatPointsBase (int b) |
Similar to formatPoints(), but the points coordinates are printed in base \(b\). More... | |
String | formatPointsBase (int n, int d, int b) |
Similar to formatPoints(n, d), but the points coordinates are printed in base \(b\). More... | |
String | formatPointsBase (PointSetIterator iter, int b) |
Similar to formatPoints(iter), but the points coordinates are printed in base \(b\). More... | |
String | formatPointsBase (PointSetIterator iter, int n, int d, int b) |
Similar to formatPoints(iter, n, d), but the points coordinates are printed in base \(b\). More... | |
String | formatPointsNumbered () |
Same as invoking formatPointsNumbered(n, d) with \(n\) and \(d\) equal to the number of points and the dimension, respectively. More... | |
String | formatPointsNumbered (int n, int d) |
Same as invoking formatPoints(n,d), except that the points are numbered. More... | |
Protected Member Functions | |
int | intToDigitsGray (int b, int i, int numDigits, int[] bary, int[] gray) |
Protected Attributes | |
int | b = 0 |
int | numCols = 0 |
int | numRows = 0 |
int | outDigits = 0 |
int [][] | genMat |
int [][] | digitalShift |
double | normFactor |
double [] | factor |
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double | EpsilonHalf = 1.0 / Num.TWOEXP[55] |
int | dim = 0 |
int | numPoints = 0 |
int | dimShift = 0 |
int | capacityShift = 0 |
RandomStream | shiftStream |
Additional Inherited Members | |
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static final int | MAXBITS = 31 |
This class provides the basic structures for storing and manipulating linear digital nets in base \(b\), for an arbitrary base \(b\ge2\).
We recall that a net contains \(n = b^k\) points in \(s\) dimensions, where the \(i\)th point \(\mathbf{u}_i\), for \(i=0,…,b^k-1\), is defined as follows:
\begin{align*} i & = \sum_{\ell=0}^{k-1} a_{i,\ell} b^{\ell}, \\ \begin{pmatrix} u_{i,j,1} \\ u_{i,j,2} \\ \vdots \end{pmatrix} & = \mathbf{C}_j \begin{pmatrix} a_{i,0} \\ a_{i,1} \\ \vdots \\ a_{i,k-1} \end{pmatrix} , \\ u_{i,j} & = \sum_{\ell=1}^{\infty}u_{i,j,\ell} b^{-\ell}, \\ \mathbf{u}_i & = (u_{i,0},…,u_{i,s-1}). \end{align*}
In our implementation, the matrices \(\mathbf{C}_j\) are \(r\times k\), so the expansion of \(u_{i,j}\) is truncated to its first \(r\) terms. The points are stored implicitly by storing the generator matrices \(\mathbf{C}_j\) in a large two-dimensional array of integers, with \(srk\) elements. For general \(b\), the element \((l,c)\) of \(\mathbf{C}_j\) (counting elements from 0) is stored at position \([jk+c][l]\) in this array.
The points \(\mathbf{u}_i\) are enumerated using the Gray code technique as proposed in [8], [219] (see also [67], [86] ). With this technique, the \(b\)-ary representation of \(i\), \(\mathbf{a}_i = (a_{i,0}, …, a_{i,k-1})\), is replaced in Equation ( digital-Cj ) by a Gray code representation of \(i\), \(\mathbf{g}_i = (g_{i,0}, …, g_{i,k-1})\). The Gray code \(\mathbf{g}_i\) used here is defined by \(g_{i,k-1} = a_{i,k-1}\) and \(g_{i,\ell} = (a_{i,\ell} - a_{i,\ell+1}) \mod b\) for \(\ell= 0,…,k-2\). It has the property that \(\mathbf{g}_i = (g_{i,0}, …, g_{i,k-1})\) and \(\mathbf{g}_{i+1} = (g_{i+1,0}, …, g_{i+1,k-1})\) differ only in the position of the smallest index \(\ell\) such that \(a_{i,\ell} < b - 1\), and we have \(g_{i+1,\ell} = (g_{i,\ell}+1) \mod b\) in that position.
This Gray code representation permits a more efficient enumeration of the points by the iterators. It changes the order in which the points \(\mathbf{u}_i\) are enumerated, but the first \(b^m\) points remain the same for every integer \(m\). The \(i\)th point of the sequence with the Gray enumeration is the \(i’\)th point of the original enumeration, where \(i’\) is the integer whose \(b\)-ary representation \(\mathbf{a}_{i’}\) is given by the Gray code \(\mathbf{g}_i\). To enumerate all the points successively, we never need to compute the Gray codes explicitly. It suffices to know the position \(\ell\) of the Gray code digit that changes at each step, and this can be found quickly from the \(b\)-ary representation \(\mathbf{a}_i\). The digits of each coordinate \(j\) of the current point can be updated by adding column \(\ell\) of the generator matrix \(\mathbf{C}_j\) to the old digits, modulo \(b\).
One should avoid using the method getCoordinate(i, j) for arbitrary values of i
and j
, because this is much slower than using an iterator to access successive coordinates.
Digital nets can be randomized in various ways [169], [56], [124], [184] . Several types of randomizations specialized for nets are implemented directly in this class.
A simple but important randomization is the random digital shift in base \(b\), defined as follows: replace each digit \(u_{i,j,\ell}\) in ( digital-uij ) by \((u_{i,j,\ell} + d_{j,\ell}) \mod b\), where the \(d_{j,\ell}\)’s are i.i.d. uniform over \(\{0,…,b-1\}\). This is equivalent to applying a single random shift to all the points in a formal series representation of their coordinates [124], [153] . In practice, the digital shift is truncated to \(w\) digits, for some integer \(w\ge r\). Applying a digital shift does not change the equidistribution and \((t,m,s)\)-net properties of a point set [86], [122], [153] . Moreover, with the random shift, each point is uniformly distributed over the unit hypercube (but the points are not independent, of course).
A second class of randomizations specialized for digital nets are the linear matrix scrambles [169], [56], [86], [184] , which multiply the matrices \(\mathbf{C}_j\) by a random invertible matrix \(\mathbf{M}_j\), modulo \(b\). There are several variants, depending on how \(\mathbf{M}_j\) is generated, and on whether \(\mathbf{C}_j\) is multiplied on the left or on the right. In our implementation, the linear matrix scrambles are incorporated directly into the matrices \(\mathbf{C}_j\) (as in [86] ), so they do not slow down the enumeration of points. Methods are available for applying linear matrix scrambles and for removing these randomizations. These methods generate the appropriate random numbers and make the corresponding changes to the \(\mathbf{C}_j\)’s. A copy of the original \(\mathbf{C}_j\)’s is maintained, so the point set can be returned to its original unscrambled state at any time. When a new linear matrix scramble is applied, it is always applied to the original generator matrices. The method resetGeneratorMatrices removes the current matrix scramble by resetting the generator matrices to their original state. On the other hand, the method eraseOriginalGeneratorMatrices replaces the original generator matrices by the current ones, making the changes permanent. This is useful if one wishes to apply two or more linear matrix scrambles on top of each other.
Linear matrix scrambles are usually combined with a random digital shift; this combination is called an affine matrix scramble [184] . These two randomizations are applied via separate methods. The linear matrix scrambles are incorporated into the matrices \(\mathbf{C}_j\) whereas the digital random shift is stored and applied separately, independently of the other scramblings.
Applying a digital shift or a linear matrix scramble to a digital net invalidates all iterators for that randomized point, because each iterator uses a cached copy of the current point, which is updated only when the current point index of that iterator changes, and the update also depends on the cached copy of the previous point. After applying any kind of scrambling, the iterators must be reinitialized to the initial point by invoking PointSetIterator.resetCurPointIndex or reinstantiated by the #iterator method (this is not done automatically).
void addRandomShift | ( | int | d1, |
int | d2, | ||
RandomStream | stream | ||
) |
Adds a random digital shift to all the points of the point set, using stream stream
to generate the random numbers.
For each coordinate \(j\) from d1
to d2-1
, the shift vector \((d_{j,0},…,d_{j,k-1})\) is generated uniformly over \(\{0,…,b-1\}^k\) and added modulo \(b\) to the digits of all the points. After adding a digital shift, all iterators must be reconstructed or reset to zero.
stream | random number stream used to generate uniforms |
void addRandomShift | ( | RandomStream | stream | ) |
Same as addRandomShift(0, dim, stream), where dim
is the dimension of the digital net.
stream | random number stream used to generate uniforms |
void eraseOriginalGeneratorMatrices | ( | ) |
Erases the original generator matrices and replaces them by the current ones.
The current linear matrix scrambles thus become permanent. This is useful if we want to apply several scrambles in succession to a given digital net.
double getCoordinateNoGray | ( | int | i, |
int | j | ||
) |
Returns \(u_{i,j}\), the coordinate \(j\) of point \(i\), the points being enumerated in the standard order (no Gray code).
i | point index |
j | coordinate index |
void iBinomialMatrixScramble | ( | RandomStream | stream | ) |
Applies the \(i\)-binomial matrix scramble proposed by Tezuka [217] (see also [184] ).
This multiplies each \(\mathbf{C}_j\) on the left by a \(w\times w\) nonsingular lower-triangular matrix \(\mathbf{M}_j\) as in leftMatrixScramble, but with the additional constraint that all entries on any given diagonal or subdiagonal of \(\mathbf{M}_j\) are identical.
stream | random number stream used as generator of the randomness |
void iBinomialMatrixScrambleFaurePermut | ( | RandomStream | stream, |
int | sb | ||
) |
Similar to iBinomialMatrixScramble except that the diagonal elements of each matrix \(\mathbf{M}_j\) are chosen as in leftMatrixScrambleFaurePermut.
stream | random number stream used to generate the randomness |
sb | Only the first \(sb\) elements of \(F\) are used |
void iBinomialMatrixScrambleFaurePermutAll | ( | RandomStream | stream, |
int | sb | ||
) |
Similar to iBinomialMatrixScrambleFaurePermut except that the elements under the diagonal are also chosen from the same restricted set as the diagonal elements.
stream | random number stream used to generate the randomness |
sb | Only the first \(sb\) elements of \(F\) are used |
void iBinomialMatrixScrambleFaurePermutDiag | ( | RandomStream | stream, |
int | sb | ||
) |
Similar to iBinomialMatrixScrambleFaurePermut except that all the off-diagonal elements are 0.
stream | random number stream used to generate the randomness |
sb | Only the first \(sb\) elements of \(F\) are used |
PointSetIterator iteratorNoGray | ( | ) |
This iterator does not use the Gray code.
Thus the points are enumerated in the order of their first coordinate before randomization.
void leftMatrixScramble | ( | RandomStream | stream | ) |
Applies a linear scramble by multiplying each \(\mathbf{C}_j\) on the left by a \(w\times w\) nonsingular lower-triangular matrix \(\mathbf{M}_j\), as suggested by Matoušek [169] and implemented by Hong and Hickernell [86] .
The diagonal entries of each matrix \(\mathbf{M}_j\) are generated uniformly over \(\{1,…,b-1\}\), the entries below the diagonal are generated uniformly over \(\{0,…,b-1\}\), and all these entries are generated independently. This means that in base \(b=2\), all diagonal elements are equal to 1.
leftMatrixScramble
sont carrées et triangulaires inférieures. PL pense qu’il faut considérer la possibilité de rajouter des lignes à ces matrices pour pouvoir randomiser plus les derniers chiffres ou les derniers bits. stream | random number stream used to generate the randomness |
void leftMatrixScrambleDiag | ( | RandomStream | stream | ) |
Similar to leftMatrixScramble except that all the off-diagonal elements of the \(\mathbf{M}_j\) are 0.
stream | random number stream used to generate the randomness |
void leftMatrixScrambleFaurePermut | ( | RandomStream | stream, |
int | sb | ||
) |
Similar to leftMatrixScramble except that the diagonal elements of each matrix \(\mathbf{M}_j\) are chosen from a restricted set of the best integers as calculated by Faure [56] .
They are generated uniformly over the first sb
elements of array \(F\), where \(F\) is made up of a permutation of the integers \([1..(b-1)]\). These integers are sorted by increasing order of the upper bounds of the extreme discrepancy for the given integer.
stream | random number stream used to generate the randomness |
sb | Only the first \(sb\) elements of \(F\) are used |
void leftMatrixScrambleFaurePermutAll | ( | RandomStream | stream, |
int | sb | ||
) |
Similar to leftMatrixScrambleFaurePermut except that the elements under the diagonal are also chosen from the same restricted set as the diagonal elements.
stream | random number stream used to generate the randomness |
sb | Only the first \(sb\) elements of \(F\) are used |
void leftMatrixScrambleFaurePermutDiag | ( | RandomStream | stream, |
int | sb | ||
) |
Similar to leftMatrixScrambleFaurePermut except that all off-diagonal elements are 0.
stream | random number stream used to generate the randomness |
sb | Only the first \(sb\) elements of \(F\) are used |
void resetGeneratorMatrices | ( | ) |
Restores the original generator matrices.
This removes the current linear matrix scrambles.
void rightMatrixScramble | ( | RandomStream | stream | ) |
Applies a linear scramble by multiplying each \(\mathbf{C}_j\) on the right by a single \(k\times k\) nonsingular upper-triangular matrix \(\mathbf{M}\), as suggested by Faure and Tezuka [56] (see also [86] ).
The diagonal entries of the matrix \(\mathbf{M}\) are generated uniformly over \(\{1,…,b-1\}\), the entries above the diagonal are generated uniformly over \(\{0,…,b-1\}\), and all the entries are generated independently. The effect of this scramble is only to change the order in which the points are generated. If one computes the average value of a function over all the points of a given digital net, or over a number of points that is a power of the basis, then this scramble makes no difference.
stream | random number stream used as generator of the randomness |
void stripedMatrixScramble | ( | RandomStream | stream | ) |
Applies the striped matrix scramble proposed by Owen [184] .
It multiplies each \(\mathbf{C}_j\) on the left by a \(w\times w\) nonsingular lower-triangular matrix \(\mathbf{M}_j\) as in leftMatrixScramble, but with the additional constraint that in any column, all entries below the diagonal are equal to the diagonal entry, which is generated randomly over \(\{1,…,b-1\}\). Note that for \(b=2\), the matrices \(\mathbf{M}_j\) become deterministic, with all entries on and below the diagonal equal to 1.
stream | random number stream used as generator of the randomness |
void stripedMatrixScrambleFaurePermutAll | ( | RandomStream | stream, |
int | sb | ||
) |
Similar to stripedMatrixScramble except that the elements on and under the diagonal of each matrix \(\mathbf{M}_j\) are chosen as in leftMatrixScrambleFaurePermut.
stream | random number stream used as generator of the randomness |
sb | Only the first \(sb\) elements of \(F\) are used |