SSJ
3.2.1
Stochastic Simulation in Java
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This class implements digital nets or digital sequences in base 2 formed by the first \(n = 2^k\) points of a Sobol’ sequence [210], [211] . More...
Public Member Functions | |
SobolSequence (int k, int w, int dim) | |
Constructs a new digital net with \(n = 2^k\) points and \(w\) output digits, in dimension dim , formed by taking the first \(n\) points of the Sobol’ sequence. More... | |
SobolSequence (int n, int dim) | |
Constructs a Sobol point set with at least n points and 31 output digits, in dimension dim . More... | |
SobolSequence (String filename, int k, int w, int dim) | |
Constructs a new digital net using the direction numbers provided in file filename . More... | |
String | toString () |
void | extendSequence (int k) |
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abstract void | extendSequence (int k) |
Increases the number of points to \(n = 2^k\) from now on. More... | |
DigitalNetBase2 | toNet () |
Transforms this digital sequence into a digital net without changing the coordinates of the points. More... | |
DigitalNetBase2 | toNetShiftCj () |
Transforms this digital sequence into a digital net by adding one dimension and shifting all coordinates by one position. More... | |
PointSetIterator | iteratorShift () |
Similar to #iterator, except that the first coordinate of the points is \(i/n\), the second coordinate is obtained via the generating matrix \(\mathbf{C}_0\), the next one via \(\mathbf{C}_1\), and so on. More... | |
PointSetIterator | iteratorShiftNoGray () |
This iterator shifts all coordinates of each point one position to the right and sets the first coordinate of point \(i\) to \(i/n\), so that the points enumerated with this iterator have one more dimension. More... | |
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void | printGeneratorMatrices (int s) |
Prints the generator matrices as bit matrices in standard form for dimensions 1 to \(s\). | |
void | printGeneratorMatricesTrans (int s) |
Prints the generator matrices transposed in the form of integers for dimensions 1 to \(s\). More... | |
double | getCoordinate (int i, int j) |
double | getCoordinateNoGray (int i, int j) |
PointSetIterator | iterator () |
PointSetIterator | iteratorNoGray () |
This iterator does not use the Gray code. More... | |
String | toString () |
void | clearRandomShift () |
void | addRandomShift (RandomStream stream) |
void | addRandomShift (int d1, int d2, RandomStream stream) |
void | leftMatrixScramble (RandomStream stream) |
void | iBinomialMatrixScramble (RandomStream stream) |
void | stripedMatrixScramble (RandomStream stream) |
void | rightMatrixScramble (RandomStream stream) |
void | nestedUniformScramble (RandomStream stream, double[][] output) |
Same as nestedUniformScramble(stream, output, 0) . | |
void | nestedUniformScramble (RandomStream stream, double[][] output, int numBits) |
Apply Owen's nested uniform scrambling. More... | |
void | leftMatrixScrambleDiag (RandomStream stream) |
void | leftMatrixScrambleFaurePermut (RandomStream stream, int sb) |
void | leftMatrixScrambleFaurePermutDiag (RandomStream stream, int sb) |
void | leftMatrixScrambleFaurePermutAll (RandomStream stream, int sb) |
void | iBinomialMatrixScrambleFaurePermut (RandomStream stream, int sb) |
void | iBinomialMatrixScrambleFaurePermutDiag (RandomStream stream, int sb) |
void | iBinomialMatrixScrambleFaurePermutAll (RandomStream stream, int sb) |
void | stripedMatrixScrambleFaurePermutAll (RandomStream stream, int sb) |
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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 Attributes | |
int [] | poly_from_file |
int | minit_from_file [][] |
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int [] | genMat |
int [] | digitalShift |
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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 |
Static Protected Attributes | |
static final int | MAXDIM = 360 |
static final int | MAXDEGREE = 18 |
static final int [] | poly |
static final int | minit [][] |
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static final int | MAXBITS = 31 |
Additional Inherited Members | |
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int | intToDigitsGray (int b, int i, int numDigits, int[] bary, int[] gray) |
This class implements digital nets or digital sequences in base 2 formed by the first \(n = 2^k\) points of a Sobol’ sequence [210], [211] .
Values of \(n\) up to \(2^{30}\) are allowed.
In Sobol’s proposal, the generator matrices \(\mathbf{C}_j\) are upper triangular matrices defined by a set of direction numbers
\[ v_{j,c} = m_{j,c} 2^{-c} = \sum_{l=1}^c v_{j,c,l} 2^{-l}, \]
where each \(m_{j,c}\) is an odd integer smaller than \(2^c\), for \(c=1,…,k\) and \(j=0,…,s-1\). The digit \(v_{j,c,l}\) is the element \((l,c)\) of \(\mathbf{C}_j\), so \(v_{j,c}\) represents column \(c\) of \(\mathbf{C}_j\). One can also write
\[ m_{j,c} = \sum_{l=1}^c v_{j,c,l} 2^{c-l}, \]
so column \(c\) of \(\mathbf{C}_j\) contains the \(c\) digits of the binary expansion of \(m_{j,c}\), from the most to the least significant, followed by \(w-c\) zeros, where \(w\) is the number of output digits. Since each \(m_{j,c}\) is odd, the first \(k\) rows of each \(\mathbf{C}_j\) form a non-singular upper triangular matrix whose diagonal elements are all ones.
For each dimension \(j\), the integers \(m_{j,c}\) are defined by selecting a primitive polynomial over \(\mathbb F_2\) of degree \(c_j\),
\[ f_j(z) = z^{c_j} + a_{j,1}z^{c_j-1} + \cdots+ a_{j,c_j}, \]
and the first \(c_j\) integers \(m_{j,0},…,m_{j,c_j-1}\). Then the following integers \(m_{j,c_j}, m_{j, c_j+1}, …\) are determined by the recurrence
\[ m_{j,c} = 2 a_{j,1} m_{j,c-1} \oplus\cdots\oplus2^{c_j-1} a_{j,c_j-1}m_{j,c-c_j+1} \oplus2^{c_j} m_{j,c-c_j}\oplus m_{j,c-c_j} \]
for \(c\ge c_j\), or equivalently,
\[ v_{j,c,l} = a_{j,1} v_{j,c-1,l} \oplus\cdots\oplus a_{j,c_j-1} v_{j,c-c_j+1,l} \oplus v_{j,c-c_j,l}\oplus v_{j,c-c_j,l+c_j} \]
for \(l\ge0\), where \(\oplus\) means bitwise exclusive or (i.e., bitwise addition modulo 2). Sobol’ has shown [210] that with this construction, if the primitive polynomials \(f_j(z)\) are all distinct, one obtains a \((t,s)\)-sequence whose \(t\)-value does not exceed \(c_0 + \cdots+ c_{s-1} + 1 - s\). He then suggested to list the set of all primitive polynomials over \(\mathbb F_2\) by increasing order of degree, starting with \(f_0(z) \equiv1\) (whose corresponding matrix \(\mathbf{C}_0\) is the identity), and take \(f_j(z)\) as the \((j+1)\)th polynomial in the list, for \(j\ge0\).
This list of primitive polynomials, as well as default choices for the direction numbers, are stored in precomputed tables. The ordered list of primitive polynomials is the same as in [154] and was taken from Florent Chabaud’s web site, at http://fchabaud.free.fr/. Each polynomial \(f_j(z)\) is stored in the form of the integer \(2^{c_j} + a_{j,1}2^{c_j-1} + \cdots+ a_{j,c_j}\), whose binary representation gives the polynomial coefficients.
For the set of direction numbers, there are several possibilities based on different selection criteria. The original values proposed by Sobol’ and implemented in the code of Bratley and Fox [24] for \(j\le40\) were selected in terms of his properties \(A\) and \(A’\), which are equivalent to \(s\)-distribution with one and two bits of accuracy, respectively. The default direction numbers used in this class have been taken from [154] . For \(j\le40\), they are the same as in [24] . Other direction numbers can be used by invoking SobolSequence(String,int,int,int) with the name of a file that contains the parameters. Several files of parameters for Sobol sequences are given on F. Kuo’s Web site.
SobolSequence | ( | int | k, |
int | w, | ||
int | dim | ||
) |
Constructs a new digital net with \(n = 2^k\) points and \(w\) output digits, in dimension dim
, formed by taking the first \(n\) points of the Sobol’ sequence.
The predefined generator matrices \(\mathbf{C}_j\) are \(w\times k\). Restrictions: \(0\le k\le30\), \(k\le w\) and dim
\( \le360\). To use other direction numbers or to create points in higher dimensions, one should use SobolSequence(String,int,int,int) instead of this constructor.
k | there will be 2^k points |
w | number of output digits |
dim | dimension of the point set |
SobolSequence | ( | int | n, |
int | dim | ||
) |
Constructs a Sobol point set with at least n
points and 31 output digits, in dimension dim
.
Equivalent to SobolSequence (k, 31, dim)
with \(k = \lceil\log_2 n\rceil\).
dim | dimension of the point set |
n | minimal number of points |
SobolSequence | ( | String | filename, |
int | k, | ||
int | w, | ||
int | dim | ||
) |
Constructs a new digital net using the direction numbers provided in file filename
.
The net has \(n = 2^k\) points, \(w\) output digits and dimension dim
. The file can be either on the user’s host, or somewhere on the Internet: in that case, the full url address must be given using either the http or ftp protocol. For example:
net = new SobolSequence( "http://web.maths.unsw.edu.au/~fkuo/sobol/joe-kuo-6.16900", k, w, dim);
The file must have the following format (the first line is treated as a comment by the program and discarded):
dim | \(s\) | \(a\) | \(m_i\) |
2 | 1 | 0 | 1 |
3 | 2 | 1 | 1 3 |
4 | 3 | 1 | 1 3 1 |
5 | 3 | 2 | 1 1 1 |
6 | 4 | 1 | 1 1 3 3 |
7 | 4 | 4 | 1 3 5 13 |
\(\vdots\) | \(\vdots\) |
where dim is the dimension, \(s\) the degree of the polynomial, the binary representation of \(a\) gives the inner coefficients of the polynomial (the first and the last coefficients are always 1), and \(m_i\) are the direction numbers. Thus if \(a = (a_1 a_2 …a_{s-1})_2\) for a given \(s\), then the polynomial is \(x^s + a_1x^{s-1} + a_2x^{s-2} + \cdots+ a_{s-1} x + 1\). For example, if \(s=4\) and \(a=4 = 100_2\), then the polynomial is \(x^4 + x^3 +1\).
Several files of parameters for Sobol sequences in this format are given on F. Kuo’s Web site up to very high dimensions. The different files give parameters that were selected using different criteria. To avoid waiting for a file to download every time a SobolSequence object is created, one should download the desired files and store them locally for faster access by invoking this constructor with the name of a local file.
k | number of points is \(2^k\) |
w | number of output digits |
dim | dimension of the point set |
filename | file containing the direction numbers |