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Electronic Theses, Treatises and Dissertations
The Graduate School
2004
Scrambled Quasirandom Sequences and
Their Applications
Hongmei Chi
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THE FLORIDA STATE UNIVERSITY
COLLEGE OF ARTS AND SCIENCE
SCRAMBLED QUASIRANDOM SEQUENCES AND THEIR
APPLICATIONS
By
HONGMEI CHI
A Dissertation submitted to the
Department of Computer Science
in partial fulfillment of the
requirements for the degree of
Doctor of Philosophy
Degree Awarded:
Summer Semester, 2004
The members of the Committee approve the dissertation of Hongmei Chi defended on
June 4, 2004.
Michael Mascagni
Professor Directing Dissertation
Sam Huckaba
Outside Committee Member
Mike Burmester
Committee Member
Robert van Engelen
Committee Member
Ashok Srinivasan
Committee Member
The Office of Graduate Studies has verified and approved the above named committee members.
ii
To Changyun and Judy . . .
iii
ACKNOWLEDGEMENTS
First and foremost, I would like to express my sincere gratitude to my major advisor
Dr. Michael Mascagni for his valuable support, infinite patience and research guidance
throughout the course of my graduate study.
I would also like to express my deep
appreciation to the other committee members, Dr. Mike Burmester, Dr. Ashok Srinivasan,
Dr. Robert van Engelen and Dr. Sam Huckaba, for their valuable time, helpful discussions
and suggestions. Without their help, it is impossible for me to complete my dissertation.
Special thanks are due to the past and current members of Dr. Mascagni’s research group:
Dr. Aneta Karaivanova, Dr. Nikolai Simonov, and Mr. Chuck Fleming. I have enjoyed
working in CSIT– a very supportive environment. Special thanks for Ms. Mimi Burbanks
for her LaTex technique support during my writing dissertation period.
I would also like to thank my beloved parents, who have encouraged me to make my
dreams come true and have provided me with love and spiritual support. I am deeply
grateful for my husband and daughter, who have given me unlimited support during my
graduate study. Finally, many thanks are due to everyone who helped me during my time
at Florida State University.
iv
TABLE OF CONTENTS
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
1.
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Randomized Quasi-Monte Carlo Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Quasirandom Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 The Koksma-Hlawka Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Scrambled Quasirandom Seqences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Derandomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.6 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.7 Paper Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.8 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.
MEASURES OF IRREGULARITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Theoretical Bounds on Discrepancy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Other Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Orthogonal Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.2 Practical Integral Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12
12
15
15
15
17
3.
THE SCRAMBLED HALTON SEQUENCE . . . . . . . . . . . . . . . . . . . . . . . .
3.1 The Halton Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Methods to Break Correlations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4 A Scrambled Halton Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.5 Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Linear Scrambling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Optimal Halton Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
18
18
21
23
25
26
27
28
30
4.
THE SCRAMBLED AND OPTIMAL FAURE SEQUENCE . . . . . . . . . .
4.1 The Scrambled Faure Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Generalized Faure (GFaure) Sequences . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 I-binomial Scrambling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 The Optimal Faure Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Geometric Asian Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
32
33
34
35
37
v
4.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.
THE SCRAMBLED SOBOĹ SEQUENCE . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 The Soboĺ Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1.1 Initial Direction Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Scrambling Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 An Algorithm for Scrambling the Soboĺ sequence . . . . . . . . . . . . . . . . . . . . . .
5.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
41
42
43
45
46
48
6.
RANDOMIZATION OF LATTICE POINTS . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 The Methods of Good Lattice Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Criteria for Good Generating Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4 Randomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.5 Infinite Lattice Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
49
50
51
53
54
54
7.
APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.1 Automatic Error Estimates for QMC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Parallel Quasirandom Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.1 A Parallel and Distributed Library . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2.2 Testing Parallel Quasirandom Sequences . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Derandomization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
55
55
56
58
59
59
60
8.
CONCLUSIONS AND FUTURE WORK . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
A. PARALLEL PSEUDORANDOM NUMBER GENERATORS . . . . . . . . . 63
B. LCGS WITH SOPHIE-GERMAIN MODULI . . . . . . . . . . . . . . . . . . . . . . . 65
C. LINEAR SCRAMBLING AND DERANDOMIZATION . . . . . . . . . . . . . . 68
D. ADDITIONAL NUMERICAL EXPERIMENTS . . . . . . . . . . . . . . . . . . . . . 75
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
vi
LIST OF TABLES
3.1 Optimal values for Wp for the first 40 dimensions of the Halton Sequence . . . . 29
1
1
3.2 Estimates of the Integral 0 . . . 0 Πsi=1 |4xi − 2|dx1 . . . dxs = 1 by using Halton
sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.1 Parameters Used for Numerical Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Pricing Geometric Asian Options Using Parameters in Table 3.1 . . . . . . . . . . . 38
B.1 The Sophie-Germain (S-G) Primes Closest to but Less Than 2q . . . . . . . . . . . . 66
D.1 Estimates of I1 (f ) in (D.1) by using Halton sequences . . . . . . . . . . . . . . . . . . . 79
D.2 Estimates of I2 (f ) in (D.2) with parameters ai = 0 by using Halton sequences
vii
80
LIST OF FIGURES
1.1 Left figure: 2000 pseudorandom numbers; right figure: 2000 Soboĺ (quasirandom) numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 L2 -discrepancy for 8 dimensional Soboĺ sequence.
3
. . . . . . . . . . . . . . . . . . . . . . 14
2.2 Left figure: 1024 points of the Halton sequence; right figure: 1024 points of the
Faure sequence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1 Poor 2-D projections were studied in several papers. For example, left top: was
included in Braaten’s paper [1], right top: in Morokoff’s paper [2], left bottom:
in Kocis’s paper [3], right bottom: random-Start sequence [4]. . . . . . . . . . . . . . 20
4.1 Left: The original Faure sequence, right: an optimal Faure sequence . . . . . . . . 33
4.2 Left figure: geometric mean of 3 stock prices; right figure: geometric mean of
50 stock prices. Here the label “Faure” refers to the original Faure sequence
[5], while “dFaure” refers to my optimal Faure sequence. . . . . . . . . . . . . . . . . . . 39
5.1 Left: 4096 points of the original Soboĺ sequence and the initial direction
numbers are from Bratley and Fox’s paper [6]; right: 4096 points of the
scrambled version of the Soboĺ sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.2 Left: 4096 points of the original Soboĺ sequence with all initial direction
numbers ones [7], right: 4096 points of the scrambled version of the Soboĺ
sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.3 Left figure: geometric mean of 10 stock prices; right figure: geometric mean of
30 stock prices. Here the label “Sobol” refers to the original Soboĺ sequence
[6], while “DSobol” refers to my optimal Soboĺ sequence. . . . . . . . . . . . . . . . . . 47
6.1 An example of a lattice point set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
D.1 Estimates of the integral I1 (f ) in (D.1) by using various Halton sequences. . . . 81
D.2 Estimates of the integral I2 (f ) in (D.2) with parameters ai = 0 by using various
Halton sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
D.3 Estimates of the integral I2 (f ) in (D.2) with parameters ai = i by using various
Halton sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
D.4 Estimates of the integral I2 (f ) in (D.2) with parameters ai = i2 by using various
Halton sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
D.5 Estimates of the integral I1 (f ) in (D.1) by using various Faure sequences. . . . . 85
viii
D.6 Estimates of the integral I2 (f ) in (D.2) with parameters ai = 0 by using various
Faure sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
D.7 Estimates of the integral I2 (f ) in (D.2) with parameters ai = i by using various
Faure sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
D.8 Estimates of the integral I2 (f ) in (D.2) with parameters ai = i2 by using various
Faure sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
D.9 Estimates of the integral I1 (f ) in (D.1) by using various Soboĺ sequences. . . . . 89
D.10 Estimates of the integral I2 (f ) in (D.2) with parameters ai = 0 by using various
Soboĺ sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
D.11 Estimates of the integral I2 (f ) in (D.2) with parameters ai = i by using various
Soboĺ sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
D.12 Estimates of the integral I2 (f ) in (D.2) with parameters ai = i2 by using various
Soboĺ sequences. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
ix
ABSTRACT
Quasi-Monte Carlo methods are a variant of ordinary Monte Carlo methods that employ
highly uniform quasirandom numbers in place of Monte Carlo’s pseudorandom numbers.
Monte Carlo methods offer statistical error estimates; however, while quasi-Monte Carlo has
a faster convergence rate than normal Monte Carlo, one cannot obtain error estimates from
quasi-Monte Carlo sample values by any practical way. A recently proposed method, called
randomized quasi-Monte Carlo methods, takes advantage of Monte Carlo and quasi-Monte
Carlo methods. Randomness can be brought to bear on quasirandom sequences through
scrambling and other related randomization techniques in randomized quasi-Monte Carlo
methods, which provide an elegant approach to obtain error estimates for quasi-Monte
Carlo based on treating each scrambled sequence as a different and independent random
sample. The core of randomized quasi-Monte Carlo is to find an effective and fast algorithm
to scramble (randomize) quasirandom sequences.
This dissertation surveys research on
algorithms and implementations of scrambled quasirandom sequences and proposes some
new algorithms to improve the quality of scrambled quasirandom sequences.
Besides obtaining error estimates for quasi-Monte Carlo, scrambling techniques provide
a natural way to parallelize quasirandom sequences. This scheme is especially suitable for
distributed or grid computing.
By scrambling a quasirandom sequence we can produce a family of related quasirandom
sequences. Finding one or a subset of optimal quasirandom sequences within this family
is an interesting problem, as such optimal quasirandom sequences can be quite useful for
quasi-Monte Carlo. The process of finding such optimal quasirandom sequences is called
the derandomization of a randomized (scrambled) family. We summarize aspects of this
technique and propose some new algorithms for finding optimal sequences from the Halton,
Faure and Soboĺ sequences. Finally we explore applications of derandomization.
x
CHAPTER 1
INTRODUCTION
1.1
Randomized Quasi-Monte Carlo Methods
Monte Carlo (MC) methods are based on the simulation of stochastic processes whose
expected values are equal to computationally interesting quantities. MC methods offer
simplicity of construction, and are often designed to mirror some process whose behavior
is only understood in a statistical sense. However, there are a wide class of problems
where MC methods are the only known computational method of solution. Despite the
universality of MC methods, a serious drawback is their slow convergence, which is based
on the O(N −1/2 ) behavior of the size of statistical sampling errors. One generic approach
to improving the convergence of MC methods has been the use of highly uniform random
numbers in place of the usual pseudorandom numbers. While pseudorandom numbers are
constructed to mimic the behavior of truly random numbers, these highly uniform numbers,
called quasirandom numbers (or low-discrepancy sequences), are constructed to be as evenly
distributed as is mathematically possible. The use of quasirandom numbers in MC leads to
quasi-Monte Carlo (QMC) methods [8, 9]. Indeed, pseudorandom numbers are scrutinized
via batteries of statistical tests that check for statistical independence in a variety of different
ways. In addition, these tests check for uniformity of distribution, but not with excessively
stringent requirements. Thus, one can think of computational random numbers as either
those that possess considerable independence, pseudorandom numbers, or those that possess
considerable uniformity, quasirandom numbers.
Quasirandom numbers are constructed to minimize a measure of their deviation from
uniformity called discrepancy (the definition will be given in the next section). While
quasirandom numbers do improve the convergence of applications like numerical integration,
1
it is by no means trivial to enhance the convergence of all MC methods. In order to improve
the situation for MC and especially QMC methods, the analysis and use of randomized
quasirandom sequences has been undertaken. The core idea behind randomizing QMC
(RQMC) [10] is to apply an effective and fast randomization (scrambling) algorithm to
existing quasirandom sequences.
The purpose of scrambling in QMC is threefold.
Primarily, it provides a practical
method to obtain error estimates for QMC based on treating each scrambled sequence as a
different and independent random sample from a family of randomly scrambled quasirandom
numbers [11]. Thus, RQMC overcomes the main disadvantage of QMC while maintaining
the favorable convergence rate of QMC. Secondarily, scrambling gives us a simple and unified
way to generate quasirandom numbers for parallel, distributed, and grid-based computational
environments. Finally, RQMC provides many more choices of quality quasirandom sequences
for QMC applications, and perhaps even optimal choices as a result of derandomization.
Thus, a careful exploration of scrambling and derandomization methods coupled with
library-level implementations will play a central role in the continued development and use
of RQMC techniques. This dissertation explores these issues and summarizes research on
algorithms and implementations of scrambled quasirandom sequences and proposes new
algorithms to improve the quality of scrambled quasirandom sequences.
1.2
Quasirandom Sequences
A quasirandom sequence, sometimes called a low-discrepancy sequence, is normally
generated in the unit s-dimensional hypercube, I s = [0, 1)s , and attempts to fill the
hypercube as uniformly as possible. The original construction of quasirandom sequences
was related to the van der Corput sequence, which is a one-dimension quasirandom sequence
based on digital inversion.
This digital inversion method is a central idea behind the
construction of many current quasirandom sequences in arbitrary bases and dimensions.
Following that, Halton [12] generalized the van der Corput sequence to s dimensions, and
Soboĺ [13, 14] introduced the sequences that now bear his name. A significant generalization
of these methods was proposed by Faure [15] to the sequences that bear his name. Later,
Niederreiter [16] generalized the existing construction of the Soboĺ and Faure sequences to
2
2000 2-D pseudorandom points(mlfg)
2000 2-D quasirandom points (sobol)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0.2
0.4
0.6
0.8
0
1
0.2
0.4
0.6
0.8
1
Figure 1.1. Left figure: 2000 pseudorandom numbers; right figure: 2000 Soboĺ (quasirandom) numbers.
arbitrary bases. These are now called Niederreiter sequences. Tezuka [17] further generalized
Niederreiter sequences by using the polynomial arithmetic analogue of Halton sequences.
From Fig. (1.1), I can see that pseudorandom numbers tend to cluster while quasirandom
numbers are uniformly distributed.
Before I give the definition of a low-discrepancy sequence, I must define a common
measure of uniformity, called discrepancy. Discrepancy is a measure of the lack of uniformity
or equidistribution of points placed in a set, usually in the unit hypercube, [0, 1)s . The most
widely studied discrepancy measures are based on the Lp norms (p = 2, or p = ∞). With
p = 2 this discrepancy is called the L2 -discrepancy. When p = ∞, that discrepancy is called
∗
the star-discrepancy, DN
, and its definition is the following [2]:
(1)
(2)
(s)
Definition 1 For any sequence {xn } ∈ [0, 1)s with N elements, define xi = (xi , xi , . . . , xi ),
∗
and J(ν) = [0, ν1 ) × [0, ν2 ) × · · · × [0, νs ), then the star-discrepancy of this sequence, DN
, is
given by:
where 0 ≤ νj ≤ 1.
s
1
∗
DN
= sup #{xi ∈ J(ν)} −
νj ,
0≤νj <1 N
j=1
3
(1.1)
While seemingly complicated, for a one dimensional point set the star-discrepancy is
the Kolmogorov-Smirnov statistic based on the uniform distribution. The construction of
quasirandom sequences is based on minimizing their discrepancy. Quasirandom sequences
aim to have the fraction of their points within any subset J(ν) = [0, ν1 ) × · · · × [0, νs ) as
close as possible to the subset’s volume fraction. Based on star-discrepancy, the definition
of a low-discrepancy sequence in [0, 1)s is expressed as:
Definition 2 For any N > 1, and sequence {xi }, let {xi }1≤i≤N denote the first N points of
the sequence {xi }. If I have
∗
≤ Cs
DN
(log N)s
,
N
(1.2)
where the constant c(s) depends only on the dimension, s, then the sequence {xi }, is called
a low-discrepancy sequence.
1.3
The Koksma-Hlawka Inequality
The discrepancy of a quasirandom sequence enters into QMC via the famous KoksmaHlawka inequality. Assume that an integrand, f , is defined over the s-dimensional unit cube,
[0, 1)s , and that I(f ) is defined as:
I(f ) =
f (x)dx =
Is
0
1
...
1
f (x(1) , . . . , x(s) )dx(1) . . . dx(s) .
(1.3)
0
Then the s-dimensional integral, I(f ), in Equation (1.3) may be approximated by QN (f ) [8]:
QN (f ) =
N
ωi f (xi ),
(1.4)
i=1
where xi is in [0, 1)s , and the ωi′ s are weights. If {x1 , . . . , xN } is chosen randomly, and ωi =
1
,
N
then QN (f ) becomes the standard Monte Carlo integral estimate, whose statistical error can
be estimated using the Central Limit Theorem. If {x1 , . . . , xN } are a set of quasirandom
numbers, then QN (f ) is a quasi-Monte Carlo estimate.
In fact, the Koksma-Hlawka
inequality is essentially the only theoretical tool for estimating the accuracy of such a QMC
4
estimate, and was motivated by numerical integration. If a function, f , in [0, 1)s is of bounded
variation, then I have the Koksma-Hlawka inequality:
Theorem 1 For any sequence {xn }1≤n≤N and any function f with variation in the sense of
Hardy-Krause, V (f ), bounded, I have
N
1 ∗
f (xi ) − I(f ) ≤ V (f )DN
,
N
i=1
(1.5)
∗
is the star-discrepancy of point set {x1 , . . . , xN }.
where DN
While this is a very fundamental result in QMC, Caflisch [18] gives a surprisingly simple and
elegant proof of this inequality that elucidates how basic it really is in QMC. The following
proof is based on Caflisch’s paper. Define the variation of f , a function of a single variable,
as
V (f ) =
1
0
df dx.
dx In s dimensions, the variation in the sense of Hardy-Krause is defined as
s
s
(1)
f
∂
(i)
dx ...dx(s) +
V (f ) =
V (f1 )
(1)
(s)
∂x
...∂x
[0,1)s
i=1
(i)
where f1 is the restriction of f to the boundary xi = 1.
If I introduce the notation R(J(ν)) as Calfisch [18]. for a sequence of N points {xn } in
the unit cube I s = [0, 1)s , I can define
RN (J(ν)) =
s
i=1
νi −
1
#{xn ∈ J(ν)},
N
and
∗
DN
= sup |RN (J(ν))| .
ν∈I s
Note that in the unit cube, I s = [0, 1)s ,
N
1 R(x) = 1 −
δ(x − xi ) dx,
N i=1
5
(1.6)
where R(x) = RN (J(x)) as defined in equation (1.6), and consider a function, f , that vanishes
on the boundary of the unit cube, I s .
N
N
1
1
f (xi ) = f (x)dx −
f (xi )
I(f ) −
Is
N i=1
N i=1
N
δ(x − xi )]f (x)dx
= [1 −
Is
i=1
= R(x)df (x)
Is
≤ (sup R(x))
|df (x)|
=
x
∗
DN V
Is
(f ).
(1.7)
For f that is nonzero on the boundary of the unit cube, the terms from this intgration by
parts are bounded by the boundary terms in V (f ).
I should note that in the Koksma-Hlawka inequality both V (f ) and the star-discrepancy,
∗
DN
, are difficult to calculate in practice, and so the Koksma-Hlawka inequality is actually not
very useful in practical QMC error estimation [9]. Thus I have to find a practical method to
calculate the QMC integration error. Fortunately, scrambled quasirandom sequences provide
a very good procedure for estimating QMC integration error, as I shall see.
1.4
Scrambled Quasirandom Seqences
Randomness can be brought to bear on quasirandom sequence through various scrambling
techniques. By using random numbers to scramble the order of the quasirandom numbers
or their digits, one randomizes quasirandom sequences. Thus by the term “scrambling” I
are referring more generally to the randomization of quasirandom numbers. By scrambling
a quasirandom sequence, one can produce a family of related quasirandom sequences. This
family can be used in generating parallel quasirandom sequencs. Also, a group of optimal
quasirandom sequences within this family can be quite useful for enhancing the performance
of ordinary QMC.
Finding fast and effective scrambling algorithms is the main task in this dissertation.
After I studied various scrambling methods, I found that these fall into two basic categories.
One is based on randomized shifting [19], which has the form
6
zn = xn + r
(mod 1),
(1.8)
where xn is a quasirandom number in [0, 1)s , and r is a single s-dimensional pseudorandom
number. With different pseudorandom numbers, r, I have a different scrambled version {zn }
of the original quasirandom numbers {xn }.
(1)
(2)
(s)
The other method [11] is based on digit permutations. Let xn = (xn , xn , . . . , xn ) be a
(1)
(2)
(s)
quasirandom number in [0, 1)s , and zn = (zn , zn , . . . , xn ) be the scrambled version of the
(j)
(j)
(j) (j)
(j)
point xn . Suppose each xn can be represented in base b as xn = 0.xn1 xn2 ...xnK ... with K
being the number of digits to be scrambled. Then I define
zn(j) = σ(x(j)
n ), for j = 1, 2, .., s,
(j)
(1.9)
(j)
where zni = πi (xni ), for i = 1, 2, ..., K, and σ = {π1 , π2 , . . . , πK }, and each πi is a
permutation of the digits, {0, ..., b − 1}.
There are various versions of scrambling methods based on digital permutation, and
the differences among those methods are based on the definitions of the πi ’s.
These
include Owen’s nested scrambling [11, 20], Tezuka’s GFaure [7], and Matousek’s linear
scrambling [21].
Whenever scrambled methods are applied, pseudorandom numbers are the “scramblers”.
Therefore, it is important to find a good pseudorandom number generator (PRNG) to act
as a scrambler so that I can obtain well scrambled quasirandom sequences. Also, since one
major application of scrambled quasirandom sequences is for parallelization, I will also need
a good parallel PRNG. Please see Appendix A and B for more details.
1.5
Derandomization
Scrambled sequences have good performance in practice [22]. Since scrambling produces
a stochastic family of quasirandom sequences, searching and specifying optimal quasirandom sequences that achieve theoretically and empirically optimal results is an important
problem for QMC. The process of finding such optimal quasirandom sequences is called
derandomization.
7
It is not a new idea to find an optimal quasirandom sequence. Since poor two-dimensional
projections were first found in the Halton sequence [1], Braaten and Weller [1] searched for
an optimal Halton sequence by searching an optimal permutation from among all possible
permutations. A similar example appears in searching for a good set of lattice points.
Any set of good lattice points is completely determined by its generating vector (g1 , ..., gs ).
Korobov [23] suggested considering a particular form such as (g1 , ..., gs ) = (1, a, a2 , ..., as−1 )
instead of all possible vectors, which turn out to be more efficient.
Recently, FINDER [24, 25], a software package for computing financial derivatives by
using quasirandom numbers, successfully used derandomization to find very good instances
of the Soboĺ sequence and the generalized Faure sequence, GFaure. The derandomization
of quasirandom numbers in FINDER was obtained empirically, not theoretically. Tezuka’s
i-binomial scrambling [26] is a special case of GFaure, which gives us a specific search
criterion and smaller space in which to search for an optimal GFaure sequence. In fact,
there are very few theoretical or practical results for derandomizing quasirandom sequences.
Thus, an important open question is how to provide a theoretical basis for derandomization.
In any case, there has been some theoretical progress in derandomization. Faure [27]
proved that it is possible to obtain a better star-discrepancy for the Halton sequence by
using good permutations in one dimension. Also, Atanassov [28] proved that there exist
digit permutations of scrambled Halton sequences which give better convergence rates in
terms of discrepancy bounds than the original Halton sequences. These theoretical and
practical results give us hope that the derandomization of quasirandom sequences will be
generally possible, and will give us a way to improve the quality of quasirandom sequences
used in QMC. An interesting point is that derandomization is not the only way to obtain
better quasirandom sequences from a family, but is also a way to increase the convergence
rate of QMC [29].
Before derandomization, I have to choose a scrambling space. Theoretically, Owen’s
nested scrambling [11] is powerful, but because of too much bookkeeping in the various
implementations of Owen’s original scrambling, modified and simplified Owen’s scrambling
methods were explored [?, 30] and linear scrambling is considered to be suitable choices
for scrambling methods. From the implementational point-of-view, linear scrambling is
the simplest and most effective scrambling method to improve the quality of quasirandom
8
sequences. Therefore I will facus on the linear scrambling space to search for optimal
quasirandom sequences. In this dissertation, I will illustrate the reasons of choosing linear
scrambling and propose some new methods for derandomizing scrambled quasirandom
sequences in linear scrambling space.
More details about derandomization and linear
scrambling can be found in Appendix C.
1.6
Applications
QMC has been successfully applied to computer graphics [31], computational physics [32],
computational finance [25], linear algebra [33], and Bayesian networks [34]. Although QMC is
more accurate than MC, a disadvantage of QMC is that it is hard to use the Koksma-Hlawka
inequality as a practical tool for computing error bounds. In fact, the common practice in
MC, of using a predetermined error as termination criterion, is almost impossible to realize
in QMC without extra technology. Scrambled quasirandom sequences are such a technology,
as they allows us to obtain error estimates for QMC in a practical way. This is because I
may think of each uniquely scrambled version of a given quasirandom sequence of prescribed
quality as coming from a statistical distribution. One may then average estimates taken
from these different scrambled sequences and combine the results statistically. In this way,
one may recover confidence interval bounds that are common in MC.
Besides obtaining error estimates, one of the important applications of scrambled
quasirandom sequence is to produce parallel quasirandom numbers. Scrambled quasirandom
sequences provide us with a family of similar quality quasirandom sequences, which in turn
gives us a natural way to implement parallel quasirandom numbers. This is because one may
assign a different scrambled sequence to each process requiring quasirandom numbers.
1.7
Paper Organization
The remainder of this dissertation is organized as follows. In Chapter 2, I present a
review of theoretical and practical measures for quasirandom sequences. These measures are
widely used to judge the quality of a quasirandom sequence in practice.
In Chapter 3, scrambled and optimally scrambled Halton sequences are presented. In
this chapter, to study the phenomenon of poor two-dimensional projections, I propose a
9
particular scrambling method as a solution. The correlation coefficients between the different
dimensions in the Halton sequence are obtained in general, and I show that the standard
Halton sequence has poor two-dimensional projections.
Scrambled and optimally scrambled Faure sequences are discussed in Chapter 4. A new
algorithm for finding one or a set of optimal Faure sequences is proposed. Also an approach
to optimize error estimation by using the optimal Faure sequences is studied.
In Chapter 5, I analyze how the choices of initial direction numbers affect the quality of the
Soboĺ sequence. Based on this analysis, a new algorithm for scrambling the Soboĺ sequences
is studied. Implementation issues are addressed and finally an algorithm for finding optimal
Soboĺ sequence is proposed.
Chapter 6 reviews lattice point methods and their randomization. The advantages of
lattice rules lie in their simplicity and their power as integration nodes for a wide class of
periodic integrands. The scrambling methods [30, 19] used for lattice points can also be
applied to the Halton, Faure and Soboĺ sequences.
In Chapter 7, I consider applications of scrambled quasirandom sequences. I describe
the layout of a parallel and distributed library, which includes automatic error estimation,
parallel quasirandom sequences generated by the algorithms I proposed in this dissertation,
and practical high-dimensional integral problems for use as testing benchmarks. Conclusions
and future work are given in Chapter 8, the final section of the dissertation.
1.8
Contributions
The contributions of this dissertation are listed below:
1. I give a detailed analysis for the correlations among dimensions in the standard Halton
sequence. I analyze the result of them, and compute the correlation coefficients in
section 3.2. Based on this analysis, I propose a new and simpler modified scrambling
algorithm for the Halton sequence in section 3.3. A new algorithm for searching for
this optimal Halton sequence is also proposed. This optimal Halton sequence is then
numerically tested and shown empirically to be far superior to the original sequence in
section 3.7. Two publications [35, 36] resulted from this works. One [36] is published
and the other [35] is submitted.
10
2. I propose a new algorithm for obtaining optimal Faure sequences based on i-binomial
scrambling in section 4.2. This algorithm is a natural extension of the algorithm above
for finding optimal Halton sequences. I apply this optimal Faure sequence to evaluate
a high-dimension integral in computational finance in section 4.3. Two papers [37, 38]
based on this work were published by us.
3. I develop a new scrambling algorithm for the Soboĺ sequence, called the multi-digit
scrambling algorithm in section 5.3. Most proposed scrambling methods randomize a
single digit at a time. In contrast, my scheme randomizes many digits in a single point
at a time, and is very efficient when using standard pseudorandom number generators
as scrambler. One paper [39] has published.
4. Appendix B is a part of our paper [40] to appear Parallel Computing.
5. A review paper based on the previous version of this dissertation (my prospectus) has
been submitted to SIAM Review.
11
CHAPTER 2
MEASURES OF IRREGULARITY
This chapter presents a review of the most relevant literature related to measuring
the uniformity of a quasirandom sequence. The uniformity of a sequence is theoretically
measured via discrepancy. In addition, other measures, such as two-dimensional projections
and high-dimensional integral problems, are often used to evaluate a quasirandom sequence
in practice.
The Halton, Faure and Soboĺ sequences are studied in this dissertation. Therefore, a
review of these measures for these sequences are presented in this chapter. Bounds on the
discrepancy of these sequences, as well as other analytical properties, have been presented in
Niederreiter’s book [8]. Morokoff and Caflisch [2] have an excellent survey on quasirandom
sequences and their discrepancy.
2.1
Theoretical Bounds on Discrepancy
The Halton, Faure and Soboĺ sequences all have star discrepancy bounds of the same
form, and N is the number of points and s is the number of dimensions.
∗
≤ Cs
DN
(log N)s
(log N)s−1
+ O(
).
N
N
Let CsH , CsF , CsS denote the Cs coefficient for the Halton, Faure and Soboĺ sequences
respectively, then
CsH =
s
pj − 1
,
2
log
p
j
j=1
where the dimensional bases pj ’s are pairwise coprime. In practice, I always use the first s
primes as the bases. For the Faure sequence, the coefficient can be written as
CsF =
1 ps − 1 s
(
),
s! 2 log ps
12
where ps is the base for the Faure sequence. It is known that the coefficient CsF has the
desired property that lims→∞ CsF = 0.
CsS =
2ts
.
s!(log 2)s
The bound for ts is given in Soboĺ [13] as
K
s log s
s log s
≤ ts ≤
+ O(s log log s),
log log s
log 2
which shows that ts grows superexponentially with s, like in the Halton sequence. In addition,
Faure’s paper [15] shows that CsF is smaller than both CsS and CsH . So the Faure sequence
is best in the terms of asymptotic discrepancy bound of these sequences.
However, this fact does not mean that the Faure sequence is superior to the Halton
and Soboĺ sequences. In practice, an asymptotic discrepancy bound is not that useful to
compare these sequences. First, the difficulty is due to the fact that the star-discrepancy
is hard to calculate. There seem to be no effective and fast algorithms for computing
star-discrepancy [21, 2].
The L2 -discrepancy, TN , is computationally more tractable than the star-discrepancy.
∗
∗
≤ TN . Two
since DN
In practice, the L2 -discrepancy is frequently used to replace DN
algorithms [41, 42] are available for computing L2 -discrepancy. The fastest current algorithms for computing discrepancies are those for computing TN . Warnock [41] proposed an
algorithm for TN with complexity O(sN 2 ). Heinrich [42] improved that complexity bound
with a new algorithm to compute TN -discrepancy with complexity O(Bs N(log2 N)s ), where
Bs is a constant that grows with s. Thus, Heinrich’s algorithm is efficient for small numbers
of dimensions. The formula used to compute TN is as follows:
N
s
N
N
s
1
1 2 (k) 2
(k)
(k)
(TN ) = s −
(1 − (xi ) ) + 2
(1 − max(xi , xj )),
s
3
N ∗ 2 i=1 k
N i=1 j=1 k
2
(k)
where xi
(1)
(2.5)
(s)
denotes the kth coordinate of xi , namely, xi = (xi , . . . , xi ).
However, the calculated value of TN for quasirandom sequences may be misleading if the
number of points, N, is relatively small. For small N, nearly the best possible TN value is
obtained by a set all whose points are clustered near the corner (0,...,0).
13
Figure 2.1. L2 -discrepancy for 8 dimensional Soboĺ sequence.
In addition, the constant CsF is part of an asymptotic discrepancy bound. In practice,
the number of points used, N, might not be so large as to reach this asymptotic region for
this bound. Also, other measures for the quality of these sequences should be considered,
such as two-dimensional projections.
Certain scrambling techniques do not affect the asymptotic discrepancy of these sequences [11]. Although scrambled quasirandom sequences improve the quality of quasirandom sequences, that improvement cannot be seen directly in the calculation of L2
discrepancy.
Fig. (2.1) compares TN between the unscrambled Soboĺ sequence and the mean of
10 scrambled Soboĺ sequences. This picture does not reveal any advantage of scrambled
quasirandom sequences with respect to discrepancy.
14
There has been some theoretical progress [28, 27] in optimal quasirandom sequences,
which improve the theoretical bounds on discrepancy for one-dimensional Halton sequences.
2.2
2.2.1
Other Measures
Orthogonal Projections
Besides using TN to measure quasirandom sequences, orthogonal projections are another
approach to check uniformity for high dimensional quasirandom sequence. In other words,
L2 -discrepancy can be regarded as a measure of all dimensions of an s-dimensional quasirandom sequence, while orthogonal projections may be regarded as checking the quality of any
µ-dimensional quasirandom sequence based on projections of a s-dimensional quasirandom
sequence, where µ < s.
However, it is hard to analyze more than three-dimensional
projections, and two-dimensional projections are commonly used in practice.
The idea behind this approach is simple: if a sequence is uniformly distributed in [0, 1)s ,
the any two-dimensional projection should also be uniformly distributed. However, even if
a sequence has poor two-dimensional projections, it may still be fairly uniform in [0, 1)s .
Poor projections is not a sufficient condition for a sequence to not be uniformly distributed
in [0, 1)s . It is important to understand this potential problem when using a sequence with
poor two-dimensional projections.
Poor two-dimensional projections can be seen for any quasirandom sequence. From
Fig. (2.2), two-dimensional Faure sequence has poor two-dimensional projections even from
first dimension and second dimension. The Halton and Soboĺ sequences have the better
performance in two-dimensional projections for first 5-dimensional sequences. The quality
of the Soboĺ sequence heavily depends on the choices of its initial direction numbers. So it
is important to be aware of the potential problems these quasirandom sequence may have. I
note that scrambling quasirandom sequences definitely improves this behavior with respect
to two-dimensional projections.
2.2.2
Practical Integral Problems
High-dimensional integral problems are always a good way to test the quality of
quasirandom sequences. A published set of test integrands [43], computational finance
15
1024 points of Halton sequence
1024 points of Faure sequence
1
0.8
0.8
Dimension 2
Dimension 2
1
0.6
0.6
0.4
0.4
0.2
0.2
0
0.2
0.4
0.6
Dimension 1
0.8
0
1
0.2
0.4
0.6
Dimension 1
0.8
1
Figure 2.2. Left figure: 1024 points of the Halton sequence; right figure: 1024 points of the
Faure sequence.
problems [44], approximate inference problems in Bayesian networks [45], Bayesian statistics
problems [46], and finding weak repetitive pattern in bioinformatics [47] can be chosen to
test the effectiveness of quasirandom sequences.
Numerical methods are used for a variety of purposes in modern finance [44, 25]. These
includes risk analysis, the valuation of securities, and the stress testing of portfolios. The
Monte Carlo approach has proved to be a valuable computational tool in modern finance.
However, for many applications in computational finance, the use of quasirandom sequences,
seems to provide a faster rate of convergence than random sequences. Thus, the generation
of appropriate high-quality quasirandom sequences is important to the quasi-Monte Carlo
approach to many problems in computational finance [48].
Bayesian networks are gaining popularity as a modelling tool for complex problems involving reasoning under uncertainty. However, approximate inference to any desired precision has
been shown to be an NP-hard problem [49]. Besides stochastic sampling methods [50], some
deterministic algorithms have been proposed, such as system sampling [45, 51], and Latin
hypercube sampling. Cheng and Druzdzel [34] investigated applications of quasi-Monte Carlo
methods to Bayesian networks, and pointed out that approximate inference in Bayesian net16
works is an excellent test-bed for studying the properties of quasirandom sequences. Cheng
and Druzdzel ’s experimental results show that quasirandom sequences significantly improve
the performance of simulation algorithms in Bayesian networks compared to ordinary Monte
Carlo methods.
In this dissertation, I use the test integrals discussed in [43, 5] and an Asian option from
computational finance to test my scrambled and optimal quasirandom sequences.
2.3
Conclusion
The major goal of this chapter is to give a comprehensive review of the theoretical and
practical measures used to judge the quality of quasirandom sequences. In summary, it is
hard to show one of quasirandom sequences is superior to the others. Therefore, all of the
Halton, Faure and Soboĺ sequences are used in practice for QMC applications. I have to
consider all of them in this dissertation.
The first use of RQMC is in error estimation. Thus, for this use, one needs a fast
scrambling algorithm for all the sequences obtained [11, 48]. Although scrambling [29] does
not change the theoretical bounds on discrepancy of these sequences, scrambling methods
do improve the measures of two-dimensional projections and evaluation of high-dimensional
integrals.
In addition, theoretically it is impossible to prove that one of scrambled
quasirandom sequences has better performance than the others so far.
Therefore, in the following three Chapters, new algorithms for scrambling and finding
optimal sequences are proposed. Measures of two-dimensional projections and practical
integral problems are used to assess the scrambled sequences. In this dissertation, I focus on
proposing effective and fast scrambling algorithms for widely used sequences: the Halton,
Faure, and Soboĺ sequences.
17
CHAPTER 3
THE SCRAMBLED HALTON SEQUENCE
The Halton sequence is one of the standard (along with (t, s)-sequences and lattice points)
low-discrepancy sequences, and thus is widely used in QMC applications. However, the
original Halton sequence suffers from correlations between radical inverse functions with
different bases used for different dimensions. These correlations result in poorly distributed
two-dimensional projections. A standard solution to this is to use a randomized (scrambled)
version of the Halton sequence. Here, I analyze the correlations in the standard Halton
sequence, and based on this analysis propose a new and simpler modified scrambling
algorithm in this chapter.
3.1
The Halton Sequence
A classical family of low-discrepancy sequences is the Halton sequence [12]. One of its
important advantages is that the Halton sequence is easy to implement due to its definition
via the radical inverse function.
φp (n) ≡
bm
b0 b1
+ 2 + ... + m+1 ,
p
p
p
(3.1)
where p is a prime number, and the p-ary expansion of n is given as n = b0 + b1 p + ... + bm pm ,
with integers 0 ≤ bj < p. The Halton sequence, Xn , in s-dimensions is then defined as
Xn = (φp1 (n), φp2 (n), ..., φps (n)),
(3.2)
where the dimensional bases p1 , p2 , ..., ps are pairwise coprime. In practice, I always use the
first s primes as the bases.
In comparison to other low-discrepancy sequences, the Halton sequence is much easier
to implement due to the ease of implementation of the radical inverse function.
18
The
radical inverse function simply reverses the digit expansion of n, and places it to the right
of the “decimal” point. Moreover, moving from φp (n) to φp (n + 1) can be implemented
with rightward-carry addition of 1/p, and thus is very efficiently implemented. However,
a problem with the Halton sequence arises from correlations between the radical inverse
functions for different dimensions. These correlations cause the Halton sequence to have poor
two-dimensional projections for some pairs of dimensions. For example, the two-dimensional
projections of the 7th and 8th [1], 28th and 29th [2], and 39th and 40th [3] dimensions are very
poor. Fig. 3.1 illustrates the poor projections in these cases. In addition, Section 3.3 has a
more detailed analysis of these correlations.
The poor two-dimensional projections are caused by the fact that the difference between
the two primes bases corresponding to the different dimensions is very small relative to the
base size. Fox [5] coded the first 40-dimension Halton sequence, and the first 40 primes were
used for the bases. Among these 40 primes, there are eight pairs of twin primes greater than
10. The list is as follows: (11,13), (17,19), (41,43), (59,61), (71,73), (101,103), (107,109),
(149,151). All of them have poor two-dimensional projections.
The following anaysis is based on my published paper [36]. To study this phenomenon,
consider one base, p and another base, p + α, where the difference, α, can be thought of as
being relatively small. Let n be a positive integer, then n = a0 + a1 (p + α) + ... + am (p + α)m .
The formula for φp (n) is given in equation (3.1), and the formula for φp+α(n) is :
a0
a1
am
+
+
...
+
p + α (p + α)2
(p + α)m+1
a1
am
a0
+ 2
+
...
+
.
=
p(1 + α/p) p (1 + α/p)2
pm+1 (1 + α/p)m+1
φp+α (n) =
(3.3)
From Equations (3.1) and (3.3), I can see that correlation between φp (n) and φp+α (n) is due
to the fact that when α is small compared to p, then (1 + αp ) is close to 1. Thus one would
expect the worst problems when α is the smallest possible, say α = 2, the case of twin primes
for p and p + α.
However, good two-dimensional projections for the Halton sequence may be obtained if
the number of points is equal to the product of the bases. This is due to the fact that the
least significant digit in the p-ary expansion of n is b0 , and so it repeats every p; similarly,
a0 repeats every p + α. Since the uniformity is dictated most by this digit, I should get a
uniform two-dimensional projection by using p(p + α) points. According to this reasoning,
19
100 points of Halton sequence
1
0.8
Dimension 8, p=19
Dimension 28, p=109
0.8
0.6
0.6
0.4
0.4
0.2
0
1
0.2
0.2
0.4
0.6
Dimension 7, p=17
0
0.8
2000 points of Halton sequence
0.2
0.4
0.6
0.8
Dimension 27, p=107
1
512 points of Random-Start Halton sequence
1
0.8
Dimension 13, p=43
Dimension 40, p=173
0.8
0.6
0.6
0.4
0.4
0.2
0
4096 points of Halton sequence
0.2
0.2
0.4
0.6
0.8
Dimension 39, p=167
0
1
0.2
0.4
0.6
0.8
Dimension 13, p=41
1
Figure 3.1. Poor 2-D projections were studied in several papers. For example, left top: was
included in Braaten’s paper [1], right top: in Morokoff’s paper [2], left bottom: in Kocis’s
paper [3], right bottom: random-Start sequence [4].
the Halton sequence should have good two-dimensional projection if I choose the number of
points to be the product of bases. For example, if I plan to use the 39th and 40th dimension
of the Halton sequence, the bases are 167 and 173 respectively, and 167 × 173 = 28891 points
should be well distributed in these two dimensions.
20
3.2
Correlations
The following anaysis is based on my submitted paper [35].
The original Halton
sequence suffers from correlations between radical inverse functions with different bases used
for different dimensions. These correlations result in poorly distributed two-dimensional
projections among other things. In this section, I will calculate the correlation coefficient
between two radical inverse functions, φp (n) and φp+α(n). The calculations will provide some
insight into the correlations between dimensions in the Halton sequence and show that the
original Halton sequence is weak. Based on this analysis, an effective scrambling algorithm
will be proposed in the next section.
In Fig. 3.1, I can see similarities in the poor two-dimensional projections. For example
there are two clusters of lines parallel to the line y = x. A more careful analysis would reveal
that the number of parallel lines in each cluster is almost equal to the ceiling of the number
of points divided by the prime base. At the end of this section, I will give explanations for
the above observations.
The main point of this section is to compute the correlation coefficient between φp (n)
and φp+α(n). I have the p-ary and the p + α-ary expansion of n given as
n ≡ b0 + b1 p + ... + bm pm = a0 + a1 (p + α) + ... + am (p + α)m .
(3.4)
Let us consider only the first two most significant digits, i.e., m = 1. Then after truncating
at m = 1, I obtain the following relation from equation (3.4):
b1 = a1 + ⌊ ap0 ⌋
b0 = a0 + αa1 (mod p).
(3.5)
The period of both a0 and b0 is p ∗ (p + α), so I only consider the range of n between 1 to
p ∗ (p + α). Therefore, φp+α (n) can be expressed as follows:
a0
a1
φp+α(n) =
+
+ O(p−2)
2
p + α (p + α)
(3.6)
By combining equations (3.1) and (3.5), φp (n) can thus be expressed in terms of the
(p + α)-ary expansion of n as:
a0 + αa1
(mod p)
a1 + ⌊ ap0 ⌋
+
+ O(p−2).
(3.7)
p
p2
For 1 ≤ n ≤ p(p+α), I partition this interval to p+α parts, namely kp+1 ≤ n ≤ (k +1)p−1
φp (n) =
for k = 0, 1, 2, . . . , p + α − 1. Then I calculate Rk , the correlation coefficient between φp (n)
21
and φp+α(n) with kp ≤ n < (k + 1)p. However, Rk can be obtained from R0 , the correlation
coefficient between φp (n) and φp+α (n) with 1 ≤ n < p − 1. This is due to the fact that the
second most significant digits, a1 and b1 , will not change until the most significant digits
have cycled. Thus, the correlation between φp (n) and φp+α (n) is primarily based on the
correlation of their most significant digits, b0 and a0 . Therefore, computing the correlation
coefficient between φp (n) and φp+α(n) with kp ≤ n < (k + 1)p translates into computing the
correlation coefficient between
b0
p+α
and
b0 +αb1
p
with 1 ≤ n < p.
I now define the formula for the correlation coefficient, R, between any two sequences
{xi }1≤i≤N and {yi }1≤i≤N . Let x̄ and ȳ denote the average of the two sequences respectively,
then the formula for the correlation coefficient is defined as
where Sxy =
(xi − x̄)(yi − ȳ), Sxx
i
and yi
p+α
1
1
+ O( p+α ) and ȳ
2
In my case xi =
is small, x̄ =
Sxy
R= ,
Sxx Syy
= (xi − x̄)2 , and Syy = (yi − ȳ)2 .
(3.8)
= pi , and I take those i’s such that 1 ≤ i ≤ p − 1. Thus as α
= 21 . Then pieces of the correlation coefficient between φp (n)
and φp+α(n), for n = 1, ..., p − 1, can be calculated by the following formula:
Sxy ≈
p−1
i=1
2
(
i
1 i
1
− )( − )
p+α 2 p 2
p + (4 − 3α)
12(p + α)
α
p
+ O( ).
=
12
p
=
(3.9)
Then Sxx and Syy can be calculated as follows:
Sxx Syy
p−1
p
1 i
1
i
− )2
(
=
( − )2
p + α 2 i=0 p 2
i=0
p(p2 + 3p + 2 − α(1 − α)) p2 + 2
)(
)
12(p + α)2
12p
p
α2
= ( )2 + O(
).
12
(p + α)2
= (
(3.10)
p 2
) . Let R0
Using the same assumption above, I can approximate Equation (3.10) by ( 12
denote the correlation coefficient between φp (n) and φp+α(n), for n = 1, 2, ..p − 1, then
p
p
(3.11)
R0 ≈ / ( )2 = 1.
12
12
22
One can calculate that Rk ≈ 1 for kp + 1 ≤ n ≤ (k + 1)p − 1 with b1 = k. This explains
why the poor two-dimensional projections of the Halton sequence in Fig. 3.1 look like lines
parallel to the line y = x, since all pairs of points approximately fall on the lines y = Rk x + c
with c = b1 or c = a1 . This is based on a common interpretation of the correlation coefficient
[52].
Now, let us compute the number of parallel lines seen for these poor projections of the
Halton sequence. Every time, b1 or a1 changes, the line y = Rk x + c wraps. For n points, b1
n
will change ⌈ np ⌉ times and a1 will change ⌈ p+α
⌉. Thus, the total number of lines for any n
points may be computed as
n
n
⌉.
⌈ ⌉+⌈
p
p+α
3.3
Methods to Break Correlations
There are at least two possible ways to break the correlations I have seen in the Halton
sequence. One is by increasing the difference between the bases for any pair of dimensions;
the other is to scramble the Halton sequence.
The first method is only useful when the number of dimensions is small. When p is
large, p + α has to be much larger if I want to break the correlations. Let α = ep where
e > 0, and then from equations (3.9) and (3.10), I see that the correlation coefficient will
be approximately
1
.
e+1
Thus I can increase e until the correlation coefficient is sufficiently
small. However, e = p is normally considered to be sufficient to ensure small correlation. This
implies prime pairs of the form p and p+p2 , but increasing α also raises a problem in the upper
bound of the star-discrepancy for the Halton sequence. For N points in the s-dimensional
Halton sequence, the upper bound of the star-discrepancy satisfies the inequality:
∗
DN
≤ C(p1 , .., ps )
where C(p1 , .., ps ) ≈
increase as
pj
logpj
pj −1
j=1 2logpj .
s
(logN)s−1
(logN)s
+ O(
),
N
N
With pj increasing, this constant in the upper bound will
is an increasing function of pj .
23
The other method to break the correlations is to scramble the Halton sequence. The
first 4-dimensions of the Halton sequence gives us a hint for obtaining better quality
high-dimensional sequences. If one can reorder or shuffle the digits in each point of the
Halton sequence for different dimensions, the correlations between different dimensions can
be made very small. This is due to the fact that there are gaps between the most significant
digits of φ2 (n), φ3 (n), φ5 (n), and φ7 (n), which have good two-dimensional projections with
p < 10. However, when p > 10, there are no gaps for the most significant base 10 digits of
φp (n) and φp+α (n). From Fig. 3.2, it is easy to see that the most significant digits for φ17 (n)
and φ19 (n) go from 1 to 9 without jumps. However, the most significant digits for φ5 (n) and
φ7 (n) jump.
φ17 (n)
0.117647
0.176471
0.235294
0.294118
0.352941
0.411765
0.470588
0.529412
0.588235
0.647059
0.705882
0.764706
0.823529
0.882353
0.941176
φ19 (n)
φ5 (n)
0.105263
0.157895
0.210526
0.263158
0.315789
0.368421
0.421053
0.473684
0.526316
0.578947
0.631579
0.684211
0.736842
0.789474
0.842105
0.400000
0.600000
0.800000
0.040000
0.240000
0.440000
0.640000
0.840000
0.080000
0.280000
φ7 (n)
0.285714
0.428571
0.571429
0.714286
0.857143
0.020408
0.163265
0.306122
0.448980
0.591837
Fig. 3.2: A list of φ17 (n) and φ19 (n) with 2 ≤ n ≤ 16, and φ5 (n) and φ7 (n) with 2 ≤ n ≤ 11.
Scrambling the Halton sequence can break these cycles, and the correlations created with
the radical inverse function. It is clear that the correlations seen are due to the shadowing
of the most significant digits of the Halton sequence.
24
3.4
A Scrambled Halton Sequence
The Halton sequence uses different primes as bases for the inverse radical functions for
different dimensions, and suffers from the correlation between the radical inverse functions.
In order to improve this situation, many scrambling procedures have been proposed. A
less elaborate, but easier to implement scrambling technique was proposed by Morokoff
and Caflisch [2]: after obtaining N elements of the s-dimension Halton sequence, permute
this block.
This procedure maintains low-discrepancy and gives good two-dimensional
projections. Morokoff and Caflisch scramble the Halton sequence independently in each
dimension: if N points in [0, 1)s are required, then s sequences of N random numbers are
generated and sorted from the smallest to the largest. The mapping of the original position
in the sequence to final position is then used to permute the Halton sequence. This method
is called dimensional permutation.
The other proposed scrambling methods are based on digital permutations. A digital
permutation of the Halton sequence is defined as
φp (n; π) ≡
π0 (a0 ) π1 (a1 )
πm (am )
+
+ ... + m+1 ,
2
p
p
p
(3.12)
where π(.) is a permutation of the integers 1,2,3,..., p − 1.
Several digital permutation methods have been proposed:
• Braaten and Weller[1] improved the Halton sequence by picking permutation πp (.)
that minimizes the one dimensional discrepancy of the set { πpp(1) , . . . , πpp(j) , πp (j+1)
}.
p
This procedure does not specify a unique permutation, so a permutation table up to
dimension 16 is given.
• Tuffin [19] extended Braaten and Weller’s work and created permutations for high
dimensions.
• Kocis and Whiten[3] propose two methods for improving the Halton sequence. They
first proposed an algorithm for modifying Braaten-Weller’s permutation function and
secondly considered leaped Halton sequences.
– Kocis and Whiten developed modified permutations π(ai (j, n)) of the Halton
sequence by reverse permuting the ai (j, n) in base two, and removing any values
25
that are too large. In this way, the cycles (correlation) of the Halton sequence are
broken.
– Leaped Halton sequences use only every Lth Halton number subject to the
condition that L is a prime different from all bases.
Compared to digital permutations, dimensional permutations of the Halton sequence are
much less computationally costly, since each point in the s-dimensional version of Morokoff’s
scrambling method needs s permutations. Meanwhile, digital permutations of the Halton
sequence need at least sm different digital permutations, where m is the number of digits
scrambled.
Yet another scrambling approach is that of Wang and Hickernell [4] called the randomstart Halton sequence, which is the original Halton sequence but started at different random
integers ni for each dimension:
Xn = (φp1 (n + n1 ), φp2 (n + n2 ), ..., φps (n + ns )).
(3.13)
This method does not break the two-dimensional correlations. The correlation coefficient is
not changed by random starting, as the period of the radical inverse function is unchanged.
In my method, I will modify Morokoff’s procedure, by permuting every other dimension.
Considering any pair of dimensions, as long as one of them breaks the cycles, the correlation
between the two dimensions will not be significant.
3.5
Implementation Issues
Based on the analysis of the correlation coefficient between any pair of dimensions,
the method of dimensional permutation is preferable because it is simpler and faster while
maintaining its effectiveness. One implementation of the Halton sequence [53, 5] uses the
formula xn+1 = xn ⊕p 1p , where ⊕p is defined as rightward carry addition1 . This algorithm
provides a fast generation of the Halton sequence as rightward carry addition steps from
n to n + 1 in the radical inverse function. Digital permutation, however, cannot use this
algorithm for generating a scrambled Halton sequence.
1
Normal addition is leftward carry.
26
I thus seek a method of scrambling Halton sequences without sacrificing the speed of
generating them. One drawback of dimensional permutation is that I have to know the
total number of points before I can scramble the sequence. In order to take advantage of
Fox and Halton’s generation algorithm [53, 5] and overcome the disadvantage of dimensional
permutation, I can permute the sequence according to each dimension over certain period (p
or p2 ). The period of the most significant digits for each point in each dimension is its base
(p). Permuting the most significant digit of each point is the same as permuting each p-long
block in {φp (1), φp (2), . . . , φp (p − 1)}, {φp(p), φp (p + 1), . . . , φp (2p − 1)}, . . . . The advantage
of my procedure is that I do not need to know the total number of the Halton points in
advance. In addition, omitting the first few Halton points in practice leads to a smaller
discrepancy.
3.6
Linear Scrambling
Linear scrambling is the simplest and most effective scrambling method to break this
correlation. This is the reason why we focus on linear scrambling and try to look for the
“best” in the linear space in the next section.
Many scrambling methods [1, 21, 4, 41] have been proposed for the Halton sequence to
break such correlation between dimensions. Most of them are based on digital permutation,
and its definition is as follows:
φπpi (n) ≡
πpi (b0 ) πpi (b1 )
πpi (bm )
,
+
...
+
+
pi
p2i
pm+1
i
(3.14)
where πpi is a permutation of the set {0, 1, 2, 3, ..., pi − 1}.
Before we start to search for the optimal Halton sequence, we must decide which
permutation functions can be chosen for πpi . In other words, we are trying to find a function
f (x) from many permutations of a given form, x ∈ {0, 1, 2, 3, ..., pi − 1}, such that f (x)
is a permutation of the set {0, 1, 2, 3, ..., pi − 1}. There are two simple functions, which
conveniently defined a subset of the pi ! permutations [54]: one is f (x) = wx + c (mod pi ),
and it is the “best” in some sense among a subset of the p! possible permutations. The other
is f (x) = xk (mod pi ), and gcd(k, pi − 1) = 1. From the implementational point-of-view,
the linear scrambling, f (x) = wx + c (mod pi ), is quite effective in comparison to other
scrambling methods.
27
We choose the linear function, f (x) = wx + c (mod pi ), with c = 0 as our πpi to scramble
the Halton sequence.
πpi (bj ) = wi bj
(mod pi ),
(3.15)
where 1 ≤ wi ≤ pi −1 and 0 ≤ j ≤ s. The reason for considering c = 0 is that we want not to
permute zero. The idea of not permuting zero is to keep the sequence unbiased. Permuting
zero (assuming an infinite string of trailing zeros) leads to a biased sequence. This linear
scrambling gives us a stochastic family of the scrambled Halton sequences, which includes
(p1 − 1)(p2 − 1) . . . (ps−1 − 1)(ps − 1) sequences for the s-dimensional Halton sequence. The
main goal of this paper is to find an optimal sequence from this scrambled family. The
algorithm for finding the optimal sequence is described in the next section.
3.7
Optimal Halton Sequences
In this section, I search for the optimal Halton sequence in the linear scrambling space
with cj = 0.
Therefore, my goal focuses on searching for the best wi for the linear
congruential generator πpi (bj ) = wi bj (mod pi ) to find the best permutation on the set
bj ∈ {0, 1, 2, . . . , pi − 1}.
There are several theoretical procedures to make this assessment: the spectral test
and discrepancy are commonly used criteria. Since the modulus is small in my case, the
(2)
spectral test [55] is not suitable. Instead I consider using the L2 -discrepancy, DN . For a
(2)
prime modulus, p, and a primitive root, W , modulo p as multiplier, the discrepancy, DN ,
satisfies [56]
(p −
(2)
1)Dp−1
≤ 2+
q
ai ,
(3.16)
i=1
where ai is the ith digit in the continued fraction expansion of
W
p
with aq = 1. My job is
now reduced to finding a primitive root Wp modulo p such that Wp has the smallest sum of
continued fraction expansion digits with
Wp
p
= [a1 , a2 , ..., aq ] and aq = 1. In Table 3.1, I list
the results of my search for the best primitive root modulo p for the first 40 dimensions of
the Halton sequence. Wp is the best primitive root modulo p based on this criterion, and p
is the prime for the base at dimension s.
28
Table 3.1. Optimal values for Wp for the first 40 dimensions of the Halton Sequence
s
p
Wp
s
p
Wp
s
p
Wp
s
1
2
3
4
5
6
7
8
9
10
2
3
5
7
11
13
17
19
23
29
1
2
3
3
8
11
12
14
7
18
11
12
13
14
15
16
17
18
19
20
31
37
41
43
47
53
59
61
67
71
12
13
17
18
29
14
18
43
41
44
21
22
23
24
25
26
27
28
29
30
73
79
83
89
97
101
103
107
109
113
40
30
47
65
71
28
40
60
79
89
31
32
33
34
35
36
37
38
39
40
Table 3.2. Estimates of the Integral
sequences
1
0
...
1
0
p
Wp
127 56
131 50
137 52
139 61
149 108
151 56
157 66
163 63
167 60
173 66
Πsi=1 |4xi − 2|dx1 . . . dxs = 1 by using Halton
Generators
N
s = 13 s = 20 s = 25
Halton
DHalton
1000
1000
1.171
0.875
2.324
0.601
34.513
0.612
681382.379
0.311
Halton
DHalton
2000
2000
1.091
0.922
1.444
0.952
17.450
0.846
340691.207
0.255
Halton
DHalton
3000
3000
1.091
0.908
1.362
0.869
12.178
0.769
227127.541
0.515
Halton
DHalton
5000
5000
0.978
0.942
1.140
0.985
7.811
1.979
136276.627
0.419
Halton
DHalton
7000
7000
0.922
0.942
0.998
1.216
5.782
1.742
97340.706
0.489
Halton
DHalton
30000
30000
0.979
0.988
0.888
1.097
2.13
1.171
22713.137
1.276
Halton
DHalton
40000
40000
0.974
1.014
0.889
1.118
1.796
1.381
17035.076
1.118
Halton
DHalton
50000
50000
0.984
1.006
0.903
1.116
1.568
1.289
13628.735
1.034
29
s = 40
To empirically verify optimality, I evaluate the test integral discussed in [43, 5], namely,
1
1
|4xi − 2| + ai
...
Πsi=1
dx1 . . . dxs = 1.
(3.17)
1 + ai
0
0
The accuracy of quasi-Monte Carlo integration depends not simply on the dimension of the
integrands, but on their effective dimension. The test function in Equation (3.17) is among
the most difficult cases for high-dimensional numerical integration. I have estimated the
values of these test integrals in dimension 20 < s < 40, with a1 = a2 = · · · = as = 0. In
this integral, all the variables are equally important, and Wang and Fang [57] calculated the
effective dimension is approximately the same as the real dimension of the integrand. Thus,
I may expect improvements from quasi-Monte Carlo by using the derandomized sequences.
The result for a1 = a2 = · · · = as = 0, is listed in Table 3.2. As shown in [5], the
errors of the numerical results for over 20 dimensions become quite large. However, after
derandomization, I found that the integral is reasonably well approximated in dimensions
over 20.
In Table 3.2, the label Halton refers to the original Halton sequence provided by Fox [5],
while DHalton refers to my derandomized Halton sequence.
3.8
Conclusion
A major problem with the Halton sequence comes from the correlations between the
radical inverse functions for different dimensions. The significance of these correlations
becomes apparent for medium and large dimension. Scrambling can improve the performance
of the Halton sequence.
I provide a number theoretic criterion to choose the optimal
scrambling from among a large family of random scramblings. Based on this criterion, I
have found the optimal scrambling for up to 60 dimensions for the Halton sequence. This
derandomized Halton sequence is then numerically tested and shown empirically to be far
superior to the original sequence
In this Chapter, I gave a review of the Halton sequence and scrambled Halton sequence
and explored the reasons for the poor two-dimensional projections of the Halton sequence.
Various scrambling methods were studied and compared based my quantitative analysis. In
practice, effective scrambling methods for the Halton sequence were presented. I presented
a new algorithm for searching for an optimal Halton sequence. This was shown to be very
30
important for practical quasi-Monte Carlo applications through the example of a difficult
high-dimensional integral. Even though it is well-known that the distribution of the Halton
sequence in high dimensions is not good, scrambling or optimally scrambling the Halton
sequence can often improve the quality. Thus the scrambled and optimal Halton sequence
can be widely applied in quasi-Monte Carlo applications.
31
CHAPTER 4
THE SCRAMBLED AND OPTIMAL FAURE
SEQUENCE
The Faure sequence is one of the most widely used quasirandom sequences in QMC. I
summarize aspects of scrambling techniques for the Faure sequence and present a modified
scrambling algorithm. In additon, I propose a new efficient algorithm for finding optimal
Faure sequences, and use the optimal Faure sequence to evaluate a particular derivative
security from computational finance. Numerical results show that this optimal sequence
give promising results even for high dimensions.
4.1
The Scrambled Faure Sequence
The original construction of quasirandom sequences was related to the van der Corput
sequence, which is a one-dimension quasirandom sequence based on digital inversion.
This digital inversion method is a central idea behind the construction of many current
quasirandom sequences in arbitrary bases and dimensions. Following the construction of
the van der Corput sequence, a significant generalization of this method was proposed
by Faure [15] to the sequences that now bear his name. Later, Tezuka [7] proposed the
generalized Faure sequence, GFaure, which forms a family of randomized Faure sequences.
The Faure sequence is based on the radical inverse function, φb (n), and a generator
matrix, C. Let b ≥ 2 be prime, and n = (n0 , n1 , ..., nm−1 )T be an integer vector with its
elements the b-adic expansion of the integer n. Then the radical inverse function, φb (n), is
defined as
φb (n) =
nm−1
n0 n1
+ 2 + ... + m .
b
b
b
The Faure sequence defines a different generator matrix for each dimension. The generator
matrix of the jth dimension for an s-dimensional Faure sequence is denoted as C (j) = P j−1
32
4096 Optimal Faure points
4096 Faure points
0.8
0.8
Dimension 100
1
Dimension 100
1
0.6
0.6
0.4
0.4
0.2
0
0.2
0.2
0.4
0.6
Dimension 99
0.8
0
1
0.2
0.4
0.6
Dimension 99
0.8
1
Figure 4.1. Left: The original Faure sequence, right: an optimal Faure sequence
for (1 ≤ j ≤ s), where P , the Pascal matrix, is defined as follows:
r−1
j−1
=
(j − 1)(r−k) (mod b), k ≥ 1, r ≥ 1.
P
k−1
(4.1)
(1)
(2)
(s)
Above k is the row index, and r is the column index. Thus let xn = (xn , xn , ..., xn ) be
(j)
the nth Faure point, then xn , can be represented as follows:
(j)
x(j)
n = φb (C n),
(4.2)
and so
(φb (P 0 n), φb (P 1 n), ..., φb (P s−1n))
gives the s-dimensional Faure sequence.
4.1.1
Generalized Faure (GFaure) Sequences
Tezuka’s GFaure has the jth dimension generator matrix as C (j) = A(j) P j−1, where A(j)
is a random nonsingular lower triangular matrix and can be expressed as follows:
33
A(j)

h11 0
0
 g21 h22 0

 g31 g32 h33
=
 ·
·
·

 ·
·
·
·
·
·
0
0
0
·
·
·
...
...
...
...
...
...

0
0

0

,
.

.
. m×m
where hii is uniformly distributed on the set {1, 2, ..., b − 1}, gij is uniform on the set
{0, 1, 2, ..., b−1}, and m is the number of digits to be scrambled. Thus GFaure is a stochastic
family of the Faure sequence, and this family has as many as b
m2
2
different sequences. An
interesting problem is finding one or a subset of optimal Faure sequence within such a large
family.
4.1.2
I-binomial Scrambling
Tezuka [58] proposed an algorithm to reduce the number of sequences in this GFaure
family while maintaining the original quality of Faure sequence. A subset of GFaure is called
“GFaure with the i-binomial property”

h1
 g2

 g3

(j)
A =
 g4
·

·
·
[58], with A(j) defined to be Toeplitz:

0 0 0 ... 0
h1 0 0 ... 0

g2 h1 0 ... 0

g3 g2 h1 ... 0
,


·
·
· ... . 
·
·
· ... . 
·
·
· ... . m×m
(4.3)
where h1 is uniformly distributed on the set {1, 2, ..., b − 1}, and gi , 2 ≤ i ≤ m, is uniformly
on the set {0, 1, 2, ..., b − 1}. For each A(j) , there will be a different random matrix in the
above form.
2 /2
I-binomial scrambling reduces the scrambling space from O(bm
) to O(bm ).
This
reduction makes searching for the optimal Faure sequence computationally tractable. Based
on i-binomial scrambling, Owen [20] proposed another form for A(j) :
34
A(j)

h1 0 0
h1 h2 0

h1 h2 h3
=
·
·
·

·
·
·
·
·
·
0
0
0
·
·
·
...
...
...
...
...
...

0
0

0

.
.

.
. m×m
(4.4)
The matrix in (4.4) for A(j) makes the same reduction as the matrix in (4.3) from GFaure.
However, this form is not suitable for my algorithm to find an optimal sequence within the
GFaure family.
Following the lead of i-binomial scrambling proposed by Tezuka [58], I try to find an
optimal Faure sequence from a relatively smaller space, rather than the whole GFaure.
4.2
The Optimal Faure Sequence
In this section, I provide a number theoretic criterion to choose an optimal scrambling
from among a large family of possible (random) scramblings of the Faure sequence. Based on
this criterion, I have found the optimal scramblings for any dimension. This derandomized
Faure sequence is then numerically tested and shown empirically to be far superior to the
original unscrambled sequence.
There have been various scrambling methods proposed for the Faure sequence to obtain
better uniformity for quasirandom sequences in high dimensions. Among these scrambling
algorithms, the simplest and most effective is linear matrix scrambling [20]. GFaure and
i-binomial scrambling are good examples of linear matrix scrambling.
In the rest of this section, my algorithm for searching for an optimal Faure sequence
within GFaure with the i-binomial property is described. The diagonal element, h1 , of A(j)
in (4.3) scrambles all digits of each original Faure point. The element g2 scrambles all but
first digit of that Faure point. Most importantly, the two-most significant digits of the Faure
point are only scrambled by h1 and g2 . Thus the choice of these two elements is crucial for
producing optimally scrambled Faure sequences. I focus on finding the best and simplest
values for h1 and g2 so that an optimal Faure sequence can be obtained. This reduces the
search to a space of size b(b − 1). In my example, I consider a simple form for A(j) within
i-binomial scrambling:
35
A(j)
 j−1
h1
0
0
0
 g2 h1j−1
0
0

j−1
 0
g2 h1
0

j−1
=
0
g2 h1
 0
 ·
·
·
·

 ·
·
·
·
·
·
·
·
...
...
...
...
...
...
...

0
0

0

.
0


.
.
. m×m
(4.5)
The idea behind this reducation of matrix is becuase the error for reducation is at most
1
.
b2
In another words, the maximum difference between this scrambled sequence by using matrix
A(j) in (4.5) and one completely scrambled with matrix A(j) in (4.3) is
1
.
b2
For example,
b = 53 for a 50 dimensional Faure sequence with two digits scrambled gives an error at most
1
532
≈ 0.00035. Hence, the issue of the reducation of matrix A(j) becomes much less of a
concern for high dimensional Faure sequence. Thus, I search for the optimal Faure sequence
in this reduced i-binomial scrambling space. My goal focuses on searching for the best h1 as
the multiplier in a linear congruential generator, πp (nj ) = h1 nj (mod b), so that I can find
the best permutation on the set nj ∈ {1, 2, . . . , b − 1}.
There are several theoretical procedures to make this assessment, and the spectral test
and discrepancy are commonly used criteria. Assume that b is small, and thus the spectral
(2)
test [55] is not suitable. Instead consider using the L2 -discrepancy, DN . For a prime modulus
(2)
b, and a primitive root h1 modulo b as multiplier, I have that the discrepancy, DN , of the
associated linear congruential generator satisfies [56]
(2)
(b − 1)Db−1 ≤ 2 +
q
ai ,
(4.6)
i=1
where ai is the ith digit in the continued fraction expansion of
h1
b
with aq = 1. my job
is reduced to finding a primitive root h1 modulo b such that h1 has the smallest sum of
continued fraction expansion digits with
h1
b
= [a1 , a2 , ..., aq ] and aq = 1. A table for the best
primitive root modulo b based on this criterion is listed in [36]. Then g2 is chosen to be
a primitive root modulo b such that g1 has the second smallest sum of continued fraction
expansion
g2
b
= [a1 , a2 , ..., aq ] and aq = 1. In Figure 4.1, the right figure is an optimal Faure
sequence with h1 = 28 and g2 = 83, and the left figure is the original Faure sequence.
36
In addition, error estimation can be obtained by using several scrambled optimal Faure
sequences, and each scrambled optimal Faure sequence can be produced by assigning gi , for
3 ≤ i ≤ m, in A(j) in (4.5) to numbers randomly chosen from the set {0, 1, 2, .., b − 1}.
4.3
Geometric Asian Options
In this section, I examine the valuation of a complex option for which there is a simple
analytical solution. The popular example for such problems is a European call option on the
geometric mean of several assets, sometimes called a geometric Asian option. Let K be the
strike price at the maturity date, T . Then the geometric mean of N assets is defined as
N
1
G = ( Si ) N ,
i=1
where Si is the ith asset price. Thus the payoff of this call option at maturity can be
expressed as
max(0, G − K).
Boyle [59] proposed an analytical solution for the price of a geometric Asian option. The basic
idea is that the product of lognormally distributed variables is also lognormally distributed.
This is due to the fact that the behavior of an asset price, Si , follows geometric Brownian
motion [60]. The formula for using the Black-Scholes equation
[61, 60] to evaluate a
European call option can be represented by:
CT = S ∗ Norm(d1 ) − K ∗ e−r(T −t) ∗ Norm(d2 ),
ln(S/K) + (r + σ 2 )(T − t)
√
,
with d1 =
σ T −t
√
d2 = d1 − σ T − t,
(4.7)
where t is the current time, and r is risk-free rate of interest that is constant in the
Black-Scholes world. Norm(x) is the cumulative normal distribution. Since there exists
an analytical solution for a geometric Asian option, this offers us a benchmark to compare
my simulation results. The parameters used for my numerical studies are listed in Table 4.1.
37
Table 4.1. Parameters Used for Numerical Studies
Number of assets
Initial asset prices, Si (0)
Volatilities, σi
Correlations, ρij
Strike price, K
Risk-free rate, r
Time to maturity, T
N
100, for i = 1, 2, ..., N
0.3
0.5, for i < j
100
10%
1 year
Table 4.2. Pricing Geometric Asian Options Using Parameters in Table 3.1
N
K
Analytic Solution
3 100
50 100
13.771
12.223
The formula to compute the analytic solution for a geometric Asian option is computed
by a modified Black-Scholes formula. Using the Black-Scholes formula, the call price can be
computed by equation (4.8) with the modified parameters, S and σ 2 , as follows:
S = Ge(−A/2+σ
N
1 2
σ
A =
N i=1 i
σ2 =
2 /2)T
N
N
1 ρij σi σj .
N 2 i=1 i=j
I follow the above formula and compute the prices for different values of N and list the
results in Table 4.2.
4.4
Numerical Results
For each simulation, I have an analytical solution, so I compute the relative error between
that and my simulated solution with the formula
|pqmc − p|
,
p
where p is the analytical solution in Table 3.1 and pqmc is the price obtained by simulation.
For different N, the pqmc is obtained by simulating the asset price fluctuations using geometric
38
Figure 4.2. Left figure: geometric mean of 3 stock prices; right figure: geometric mean
of 50 stock prices. Here the label “Faure” refers to the original Faure sequence [5], while
“dFaure” refers to my optimal Faure sequence.
Brownian motion. The results are shown in Figure 4.2, where the label “Faure” refers to the
original Faure sequence [5], while “dFaure” refers to my optimal Faure sequence.
From equation (4.8), I can see that I have to use random variables sampled from a normal
distribution. Each Faure point must be transformed into a normal variable. The favored
transformation method for quasirandom numbers is the inverse of the cumulative normal
distribution function. The inverse normal function provided by Moro [62] is used in my
numerical studies.
From Figure (4.2), it is easily seen that the optimal Faure sequence and the original Faure
sequence have the same performance when the number of dimensions is low 3. However, when
the number of dimensions increases to 50, the optimal Faure sequence has better performance
than the original Faure sequence.
39
4.5
Conclusion
For many applications in computational finance, the use of quasirandom sequences seems
to provide a faster rate of convergence than pseudorandom sequences. Unfortunately, at
present there are only a few types of quasirandom sequences widely available. By scrambling
a quasirandom sequence I can produce a family of related sequences. Derandomization
provides more choices by which to find suitable quasirandom sequences. In this chapter, I
focused on finding the optimal Faure sequence within GFaure. Based on Tezuka’s i-binomial
scrambling, I proposed an algorithm and found an optimal Faure sequence within the family.
I applied this sequence to evaluate a complex security and found promising results even for
high dimensions.
40
CHAPTER 5
THE SCRAMBLED SOBOĹ SEQUENCE
The Soboĺ sequence is the most popular quasirandom sequence because of its simplicity
and efficiency in implementation [13, 14]. I summarize aspects of the scrambling technique
applied to Soboĺ sequences and propose a new simpler modified scrambling algorithm, called
the multi-digit scrambling scheme. Most proposed scrambling methods randomize a single
digit at each iteration. In contrast, my multi-digit scrambling scheme randomizes one point
at each iteration, and therefore is more efficient. After the scrambled Soboĺ sequence is
produced, I use this sequence to evaluate a particular derivative security, and found that
when this sequence is numerically tested, it is shown empirically to be far superior to the
original unscrambled sequence.
5.1
The Soboĺ Sequence
The construction of the Soboĺ sequence uses linear recurrence relations over the finite
field, F2 , where F2 = {0, 1}. Let the binary expansion of the nonnegative integer n be given
by n = n1 20 + n2 21 + ... + nw 2w−1 . Then the nth element of the jth dimension of the Soboĺ
(j)
sequence, xn , can be generated by
(j)
(j)
(j)
x(j)
n = n1 ν1 ⊕ n2 ν2 ⊕ ... ⊕ nw νw .
(j)
where νi
(5.1)
is a binary fraction called the ith direction numbers in the jth dimension. These
direction numbers are generated by the following q-term recurrence relation
(j)
νi
(j)
(j)
(j)
(j)
(j)
= a1 νi−1 ⊕ a2 νi−2 ⊕ ...aq νi−q+1 ⊕ νi−q ⊕ (νi−q /2q ).
(5.2)
I have i > q, and the bit, ai , comes from the coefficients of a degree-q primitive polynomial
over F2 . Note that one should use a different primitive polynomial to generate the Soboĺ
41
(j)
is to use the
(j)
becomes the
direction numbers in each different dimension. Another representation of νi
(j)
integer mi
(j)
= νi
∗ 2i . Thus, the choice of q initial direction numbers νi
(j)
problem of choosing q odd integers mi
(j)
< 2i . The initial direction numbers, νi
(j)
=
mi
2i
,
(j)
in the recurrence, where i ≤ q, can be decided by the mi ’s, which can be arbitrary odd
integers less than 2i .
The Gray code is widely used in implementations [63, 6, 64] of the Soboĺ sequence. A
Gray code, a permutation of integers, is a function of the integer n, and let G(n) be the nth
Gray code. The binary representation of any G(n) and G(n + 1) differ in exactly one bit.
The algorithm for generating Gray code is simple by the operation bitwise exclusive-or(⊕)
of n and integer part of n/2: G(n) = n ⊕ n2 . The advantage of this implementation is that
the Soboĺ sequence can be generated recursively. Instead of using ni , binary expansion of n
in Equation (5.1), Antonov and Saleev [63] first used the expansion of the Gray code, G(n)
instead of n. Then equation (5.1) can be replaced by the following recursive equation:
(j)
(j)
xn+1 = x(j)
n ⊕ νc ,
(5.3)
where c is determined by the rightmost zero-bit in binary representation of n. Of course, the
order of the Soboĺ points is different when Gray code is used to replace n. However, Gray
code permutes the order of integers from 0 to 2n − 1. This order of the Soboĺ sequence does
not effect its discrepancy if I all the numbers of this power-of-two.
5.1.1
Initial Direction Numbers
The direction numbers in Soboĺ sequences come recursively from a degree-q primitive
polynomial; however, the first q direction numbers can be arbitrarily assigned for the
above recursion (equation (5.2)). Selecting them is crucial for obtaining high-quality Soboĺ
sequences. The left pictures in both figures (5.1) and (5.2) show that different choices of
initial direction numbers can make the Soboĺ sequence quite different. The initial direction
numbers for the left picture in figure (5.1) is from Bratley and Fox’s paper [6]; while left
picture in figure (5.2) results when the initial direction numbers are all ones.
Soboĺ [14] realized the importance of initial direction numbers, and published an
additional property (called Property A) for direction numbers to produce more uniform
Soboĺ sequences; but implementations [64] of Soboĺ sequences showed that Property A is
42
not really that useful in practice. Cheng and Druzdzel [34] developed an empirical method
(j)
to search for initial direction numbers, mi , in a restricted space. Their search space was
limited because they had to know the total number of quasirandom numbers, N, in advance
(j)
to use their method. Jackel [65] used a random sampling method to choose the initial mi
(j)
with a uniform random number uij , so that mi
(j)
condition that mi
is odd.
= ⌊uij × 2i−1 ⌋ for 0 < i < q with the
Owing to the arbitrary nature of initial direction numbers of the sequence, poor twodimensional projections frequently appear in the Soboĺ sequence. Morokoff and Caflisch
[2] noted that poor two-dimensional projections for the Soboĺ sequence can occur anytime
because of the improper choices of initial direction numbers. The bad news is that I do not
know in advance which initial direction numbers cause poor two-dimensional projections. In
other words, poor two-dimensional projections are difficult to prevent by trying to effectively
choose initial direction numbers. Fortunately, scrambling Soboĺ sequences [66, 11] can help
us improve the quality of the Soboĺ sequence having to pay attention to the proper choice of
the initial direction numbers.
5.2
Scrambling Methods
Many methods [1, 21, 4, 41] have been proposed for scrambling quasirandom sequences.
Some scrambling methods [66, 22, 46] were designed specifically for the Soboĺ sequence.
Recall that this sequence is defined over the finite field [54], F2 . Digit permutation is
commonly thought effective in the finite field, Fp . When digit permutation is used to scramble
a quasirandom point over Fp , the zero is commonly left out. One reason is that zero is never
the most significant bit, and when zero is added in the permutation, bias is introduced. The
other is that permuting zero (assuming an infinite string of trailing zeros) leads to a biased
sequence in the sense that zero can be added to the end of any sequence while no other digit
can. So this strategy for pure digital permutation, where zero is not changed, is not suitable
for the Soboĺ sequence because the Soboĺ sequence is over F2 .
The linear permutation [66] is also not a proper method for scrambling the Soboĺ
(1)
(2)
(s)
sequence. Let xn = (xn , xn , . . . , xn ) be any quasirandom number in [0, 1)s , and zn =
(1)
(2)
(s)
(j)
(zn , zn , . . . , zn ) be the scrambled version of the point xn . Suppose that each xn has a
43
0.8
0.8
Dimension 28,
1
Dimension 28,
1
0.6
0.6
0.4
0.4
0.2
0
0.2
0.2
0.4
0.6
Dimension 27
0.8
0
1
0.2
0.4
0.6
Dimension 27
0.8
1
Figure 5.1. Left: 4096 points of the original Soboĺ sequence and the initial direction
numbers are from Bratley and Fox’s paper [6]; right: 4096 points of the scrambled version
of the Soboĺ sequence
(j)
(j) (j)
(j)
b-ary representation as xn = 0.xn1 xn2 ...xnK ..., where K defines the number of digits to be
scrambled in each point. Then I define
zn(j) = c1 x(j)
n + c2 , for j = 1, 2, .., s,
(5.4)
where c1 ∈ {1, 2, ..., b − 1} and c2 ∈ {0, 1, 2, ..., b − 1}. Since the Soboĺ sequence is built over
F2 , one must assign 1 to c1 and 0 or 1 to c2 . Since the choice of c1 is crucial to the quality
of the scrambled Soboĺ sequence, this linear scrambling method is not suitable for the Soboĺ
sequence or any sequence over F2 .
As stated previously, the quality of the Soboĺ sequence depends heavily on the choices
of initial direction numbers.
The correlations between different dimensions are due to
improper choices of initial direction numbers [34]. Many methods [34, 65] to improve the
Soboĺ sequence focus on placing more uniformity into the initial direction numbers; but this
approach is difficult to judge by any measure. I concentrate on improving the Soboĺ sequence
independent of the initial direction numbers. This idea motivates us to find another approach
to obtain high-quality Soboĺ sequences by means of scrambling each point.
44
5.3
An Algorithm for Scrambling the Soboĺ sequence
I provide a new approach for scrambling the Soboĺ sequence, and measure the effectiveness
of this approach with the number theoretic criterion that I have used before [?]. Using this
new approach, I can now scramble the Soboĺ sequence in any number of dimensions.
The idea of our algorithm is to scramble k bits of the Soboĺ sequence instead of scrambling
one digit at a time. Assume xn is nth Soboĺ point, and I want to scramble first k bits of xn .
Let zn be the scrambled version of xn . My procedure is described as follows:
1. yn = ⌊xn ∗ 2k ⌋, is the k most-significant bits of xn , to be scramble.
2. yn∗ = ayn (mod m) and m ≥ 2k − 1, is the linear scrambling, applied to this integer.
3. zn =
∗
yn
2k
+ (xn −
yn
),
2k
is the reinsertion of these scrambled bits into the Soboĺ point.
The key step of this approach is based on using Linear Congruential Generators (LCGs)
as scramblers. LCGs with both power-of-two and prime moduli are common pseudorandom
number generators. When the modulus of an LCG is a power-of-two, the implementation
is cheap and fast due to the fact that modular addition and multiplication are just
ordinary computer arithmetic when the modulus corresponds to a computer word size. The
disadvantage, in terms of quality, is hard to obtain the desired quality of pseudorandom
numbers when using a power-of-two as modulus. More details are given in [67]. So LCGs
with prime moduli are chosen in this chapter.
The rest of my job is to search for a suitable and reliable LCGs as my scrambler. When
the modulus of a LCG is prime, implementation is more expensive. A special form of prime,
such as a Merssene
1
or a Sophie-Germain prime 2 , can be chosen so that the costliest part
of the generation, the modular multiplication, can be minimized [67].
To simplify the scrambling process, I look to LCGs for guidance. Consider the following
LCG:
yn∗ = ayn
(mod m),
(5.5)
where m is chosen to be a Merssene, 2k −1, or Sophie-Germain prime in the form of 2k+1 −k0 ,
k is the number of bits needed to “scramble”, and a is a primitive root modulo m [55, 68].
1 q
2 − 1 and q are primes, and 2q − 1 is a Merssene prime.
2
2q + 1 and q are primes, and 2q + 1 is a Sophie-Germain prime.
45
I choose the modulus to be a Merssene or Sophie-Germain [67, 40] because of the existence
of a fast modular multiplication algorithms for these primes. For more details please refer
to Appendix B. The optimal a should generate the optimal Soboĺ sequence, and the optimal
a’s for modulus 231 − 1 are tabulated in [68]. A proposed algorithm for finding such optimal
primitive root modulus m, a prime, is described in Chapter 3.
The purpose of my algorithm is twofold. Primarily, it provides a practical method to
obtain a family of scrambled Soboĺ sequences. Secondarily, it gives us a simple and unified
way to generate an optimal Soboĺ sequence from this family.
FINDER [24, 25], a commercial software system which uses quasirandom sequences to solve
problems in finance, is an example of the successful use of derandomization. A modified
Soboĺ sequence is included in FINDER. Although the creators of FINDER pointed out that the
major improvements in their modified Soboĺ sequence were achieved via optimized initial
direction numbers for dimension up to 360, the method they used for this improvement was
not revealed, and FINDER was patented.
However, using my algorithm, I can begin with the worse choices for initial direction
numbers in the Soboĺ sequence: all initial direction numbers are ones. The results are
showed in figure (5.3). The only unscrambled portion is a straight line in both pictures. The
reason is that the new scrambling algorithm cannot change the point with the same elements
into a point with different elements.
5.4
Numerical Results
Here, I present the valuation of a complex option which has a simple analytical solution.
The popular example for such problems is a European call option on the geometric mean of
several assets, sometimes called a geometric Asian option.
I followed the above formula in section 3.3 of Chapter 3, computed the prices for different
values of N = 10 and N = 30, with K = 100, I computed p = 12.292 and p = 12.631
respectively.
From Figure 5.3, it is easily seen that the optimal Soboĺ sequence performs much better
than the original Soboĺ sequence when the number of dimensions is as low as 10 and the
number of dimensions increases to 30.
46
0.8
0.8
Dimension 28,
1
Dimension 28,
1
0.6
0.6
0.4
0.4
0.2
0
0.2
0.2
0.4
0.6
Dimension 27
0.8
0
1
0.2
0.4
0.6
Dimension 27
0.8
1
Figure 5.2. Left: 4096 points of the original Soboĺ sequence with all initial direction
numbers ones [7], right: 4096 points of the scrambled version of the Soboĺ sequence
Figure 5.3. Left figure: geometric mean of 10 stock prices; right figure: geometric mean
of 30 stock prices. Here the label “Sobol” refers to the original Soboĺ sequence [6], while
“DSobol” refers to my optimal Soboĺ sequence.
47
5.5
Conclusion
A new algorithm for scrambling the Soboĺ sequence is proposed. This approach can avoid
avoiding the consequences of improper choices of initial direction numbers that negatively
impact the quality of this sequence. Therefore, my approach can enhance the quality of the
Soboĺ sequence without worrying about the choices of initial direction numbers.
In addition, I proposed an algorithm and found an optimal Soboĺ sequence within the
scrambled family. I applied this sequence to evaluate a complex security and found promising
results even for high dimensions. I have shown the performance of the Soboĺ sequence
generated by my new algorithm empirically to be far superior to the original sequence. The
promising results prompt us to use more applications to test the sequences, and to reach for
more general scrambling techniques for the Soboĺ sequence.
48
CHAPTER 6
RANDOMIZATION OF LATTICE POINTS
6.1
Introduction
There are three important families of quasirandom sequences: Halton sequences, lattice
points, and (t, s)-sequences. The scrambled versions of Halton sequences and (t, s)-sequences
have been discussed in previous chapters. In this chapter, I will focus on lattice points,
sometimes called number-theoretic methods, and their randomization.
This chapter is
included for completeness but contains no new results of ours.
Let {x1 , x2 , . . . , xN } denote a quasirandom sequence. There are several methods for
constructing the point set: either depending on N or not. If each point, xn , in the set
{x1 , x2 , . . . , xN } is constructed independently of N, this is called the open quasi-Monte
Carlo method. Halton sequences and (t, s)-nets are examples of the open quasi-Monte Carlo
method. The other construction is when the point, xn , depends on N: lattice points are an
example of this so-called closed quasi-Monte Carlo method.
When a sequence is related to N, the disadvantage is that it is impossible to obtain
useful error estimates for closed quasi-Monte Carlo by repeating the calculation with an
increasing number of points. However, randomization give us an approach for obtaining
error estimates in this case. Another approach is that of Hickernell [69], which is based on
constructing infinite lattice sequences, therefore N is not needed in advance.
The advantages of lattice rules lie in their simplicity and their power as integration nodes
for a wide class of periodic integrands, especially those whose multiple Fourier expansions
have coefficients tending rapidly to zero. Meanwhile, the motivation for randomizing lattice
rules is to find a practical error estimate when lattice rules are used for numerical integration.
49
Figure 6.1. An example of a lattice point set.
The main scrambling technique for lattice points is shifting, which is defined as xn + u
(mod 1), where xn is a lattice point and u is a random number uniformly chosen in [0, 1)s .
6.2
The Methods of Good Lattice Points
Lattice rules can be viewed as multidimensional analogues of the one-dimensional
trapezoidal rule for periodic integrands.
Definition 1 Let (N; g1 , g2 , ..., gs ) be a vector of integers satisfying 1 ≤ gj ≤ N, and gk = gj
for k = j and 1 ≤ k, j ≤ s. Then
xij =
igj
N
(mod 1), 1 ≤ j ≤ s, i = 1, 2, . . . ,
(6.1)
where the point xn = (xn1 , . . . , xns ). Then the set {xn , 1 ≤ n ≤ N} is called the set of lattice
points, and (N; g1 , g2 , ..., gs ) is called the generating vector.
The simplest and most effective form for lattice rules is the method of good lattice points.
Similar to the definitions of the Halton sequence and (t, s)-sequence, an s-dimensional good
lattice point rule can be expressed as [8, 70]:
50
s
∗
= Θ( (logNN ) ),
Definition 2 If the lattice point set {xn , 1 ≤ n ≤ N} has star-discrepancy DN
then the set {xn , 1 ≤ n ≤ N} is called a set of good lattice points.
Any set of good lattice points is completely determined by its generating vector
(N; g1 , g2 , ..., gs ). In practice, N is given, and I have to find a suitable (g1 , g2 , ..., gs ). It
is normally a very computationally costly task to find the best generating vector, i. e. a
vector which has the smallest discrepancy among all possible sets of lattice points. Therefore
Korobov [23] suggests considering {g1, ..., gs } to be of the form
g = {g1 , ..., gs } = {1, a, a2 , ..., as−1 }
(mod N),
(6.2)
with 0 < a < N an integer.
For given values of N and s, a computer search can be implemented to find the vector, g,
which minimizes an appropriate figure of merit. A table of generating vectors (s < 19) can
be found in [70, 46]. The Korobov form for good lattice points is a good choice on theoretical
as well as practical reasons.
6.3
Criteria for Good Generating Vectors
Generating vectors for lattices are usually obtained by minimizing some measure of the
discrepancy of the lattice. Classically, two criteria are widely used to measure the lattices:
one is denoted as Pα (g, N) [71], the other is R(g, N) [56].
Let Eα (c) be the class of functions, f , whose Fourier coefficients satisfy
ˆ
|f(z)|
≤
c
,
(z¯1 z¯2 ...z¯s )α
(6.3)
where c > 0, z̄ = max(1, |z|), and α > 1 is fixed.
Pα is defined as
Pα (g, N) =
gz≡0
(mod N )
(z¯1 z¯2 ...z¯s )−α − 1,
(6.4)
with the α coming from Eα (c). The quantity Pα measures the quality of the lattice point
set. Those readers familiar with the spectral test should recognize a similarity here to that
51
well-known figure of merit from random number testing [55].
Sloan and Wozniakowski [72] modify Pα with weights γ as follows,
Pα,γ (g, N) = −1 +
(β̄1 (z1 )...β̄s (zs ))−α .
(6.5)
gz≡0
Here βi (zi ) = zi /γi, and the nonincreasing weights γi > 0, for 1 ≤ i ≤ s, are associated with
the successive coordinate directions.
Let{x1 , x2 , . . . , xN } denote a good lattice point set, and I(f ) = [0,1]s f (x)dx denote the
true integral value of f , f ∈ Eα (c). Let N1 N
i=1 f (xi ) be an approximation to the integral of
the function, f . Then the error between this and I(f ) can be expressed:
Theorem 1 For any real number α > 1 and C > 0, any g ∈ Z s , and any integer N ≥ 1, I
have
N
1 f (xi ) − I(f ) ≤ CPα (g, N),
(6.6)
N
i=1
where the function f ∈ Eα (c), and {x1 , . . . , xN } is any good lattice point set.
Another criterion that can be used to assess lattice points is the quantity R(g, N) [56],
defined by
R(g, N) =
gz≡0
where W = {z ∈ Z s : − N2 ≤ zi ≤
N
,1
2
(mod N ),z∈W
1
,
z¯1 z¯2 ...z¯s
(6.7)
≤ i ≤ s}.
∗
is
The relation among Pα (g, N), R(g, N)), and DN
∗
DN
≤
1
s
+ R(g, N), and
N
2
Pα (g, N) ≤ R(g, N)α + O(N −α ).
(6.8)
(6.9)
∗
, the Koksma-Hlawka inequality (equaSince R(g, N) is related to the star-discrepancy, DN
tion 1.3) can be expressed in term of R(g, N).
52
6.4
Randomization
In general, a lattice point set is related to N, and increasing N cannot help us obtain
error estimates since the above quality measures tend to fluctuate erratically. I note that
the reason for this is that different point sets related to different values of N, in general,
have nothing in common. However, randomization of lattice rules can allow the calculation
of practical error estimates.
The Cranley-Patterson [73] randomization can be expressed as
xij =
igj
+ ∆j
N
(mod 1),
(6.10)
for j = 1, 2, ..., s, i = 1, 2, ..., N and where g = (g1 , . . . , gs ) is the generating vector of a good
lattice points set, and ∆ = (∆1 , ..., ∆s ) is a random vector from [0, 1)s . Two possible choices
for ∆ are considered. The first is where ∆ is chosen from a multivariate uniform distribution,
and the second where it is chosen by systematic sampling. In this paper, I only consider the
first choice. xkij =
igj
N
+ ∆jk (mod 1) for k = 1, 2, . . . , r forms a stochastic family. Taking r
replicates of these N points, confidence intervals for the error can be obtained.
Let I(f ) be defined as above. Then I define Qf (g, N, ∆k ) as
N −1
1 f (xki ),
Qf (g, N, ∆k ) =
N i=0
(6.11)
where xki = ( igN1 +∆k , igN2 +∆k , . . . , igNs +∆k ) (mod 1). Suppose ∆k has a multivariate uniform
distribution in [0, 1)s , then I have
E(Qf (h, N, ∆k )) = I(f ),
(6.12)
where E(.) is expectation with respect to ∆k .
For any integer, r, let ∆1 , . . . , ∆r be independent random vectors [71] chosen from a
multivariate uniform distribution in [0, 1)s , then the estimate
r
1
Q(f ) (g, N, ∆k )
Q̂(f ) (g, N) =
r
k=1
is an unbiased estimate of I(f ).
53
(6.13)
6.5
Infinite Lattice Sequences
Hickernell [69] extended the idea of thinking of lattices via their underlying (t, m, s)-nets,
and in this way obtained infinite lattice sequences. The basic idea is to replace N, the
number of lattice points, with bm . In addition, Joe [74] gave an explicit expression for a
mean L2 -discrepancy for shifted infinite lattice point sets.
Hickernell’s approach goes as follows. The ith term of a good lattice is
then the term i/N
ih
.
N
Let N = bm ,
= i/bm , i = 0, 1, ..., N − 1, may be replaced by using the digital
inverse function φb (i) = i1 b−1 + i2 b−2 + · · · = (0.i1 i2 ...)b , from the van der Corput sequence.
Therefore, a shifted infinite good lattice sequence in base b with generating vector g and
shift ∆ can be defined as
P = {φb (i)g + ∆
(mod 1) |i = 0, 1, 2, ...}.
(6.14)
Similar to the definition for (t, s)-sequences, every run has bm points. For example, the lth
run is the set {xn | lbm ≤ n < (l + 1)bm }. Suppose that Q is the set consisting of the (l + 1)st
run of the bm terms of the infinite lattice sequence, then Q is defined as
Q = {φb (lbm + i)h + ∆
(mod 1)}.
(6.15)
For i = 0, 1, 2, ..., bm , l = 0, 1, 2, ..., Q can be expressed as
Q = {φb (i)g + φb (l)b−m−1 g + ∆
(mod 1)},
(6.16)
for i = 0, 1, 2, ..., bm − 1.
6.6
Conclusion
Lattice point sets are simple to code, yet their difficulty lies in finding a good generator
value, g, given N.
When lattice point sets are used to approximate an integral in
s-dimensions, the integrand, f , must be smooth, periodic, and f ∈ Eα (c).
It is very demanding computational work to find the best generating vector {g1 , g2 , ..., gs } ∈
Rs . In practice, Korobov’s form [70] is considered, and tables of good values for N and g
can be found in [70] for 1 < s < 19. However, for dimension s > 19, one should still search
for his own optimal g and N. After g and N are chosen, the implementation of lattice rules
and their randomizations is straight forward.
54
CHAPTER 7
APPLICATIONS
Three aspects of applications for scrambled quasirandom sequences are addressed here:
(1) obtaining automatic statistical error estimates for quasi-Monte Carlo; (2) generating
parallel quasirandom sequences that are especially good for distributed or grid computing;
(3) and providing more optimal quasirandom sequences for quasi-Monte Carlo. This chapter
explores all three aspects below in detail.
7.1
Automatic Error Estimates for QMC
The original motivation [73, 1, 11] for scrambled quasirandom sequences was to obtain
automatic statistical error estimates for QMC and improve the quality of quasirandom
sequences. Methods for obtaining unbiased error estimates for QMC will be presented in
this section.
1
The convergence rate for Monte Carlo methods is asymptotically O(N − 2 ), yet quasiMonte Carlo methods can have an error bound which behaves as well as O((logN)s N −1 ).
The Koksma-Hlawka inequality (1.5) gives us the deterministic error bound for QMC.
However, it is very difficult to compute the total variation of the integrand, V (f ), and
∗
the star-discrepancy, DN
, in practice. The Koksma-Hlawka inequality is actually not very
useful in practical QMC error estimation. In [75], the authors pointed out that “Quasi-Monte
Carlo methods will come into their own only when improved error estimates are available.”
Therefore, it is important to find practical ways to obtain direct, a posteriori error estimates
in QMC.
Scrambled quasirandom sequences play a central role in providing a statistical method
for error estimation in QMC. The problem I consider here is estimating an integral in [0, 1)s :
I(f ) =
f (x)dx.
(7.1)
[0,1)s
55
QMC computes an approximate value of equation (7.1) by
N
1 ˆ
I(f ) =
f (xi ).
N i=1
(7.2)
There are several proposed methods to estimate the accuracy of equation (7.2).
• Replication method [76]
By using randomized QMC, I can take r independent replicates of a scrambled net.
The corresponding estimates Iˆ1 ,...,Iˆr are unbiased estimates of I. I then calculate the
overall estimate of I by
r
¯ )= 1
I(f
Iˆk .
r
(7.3)
k=1
¯ ).
The error of numerical integration is estimated by using the variance of I(f
r
1
¯ ))2 .
(Iˆk − I(f
σ̂ =
r(r − 1) k=1
2
(7.4)
• Partition method
The partition method [76] is similar to the replication method except that it uses r
partitions of a single net, each of size n, instead of r scrambled nets.
• Multipartition method
Synder [77] proposed a modified partition method reducing the bias which may be
introduced by the partition method in some cases. If a single net of size n is partitioned
into b sets of n/b, the sample variance for sets of size n/b is estimated using a statistic
evaluated from these partitions.
Randomized QMC not only provides us with unbiased estimators of the error, but also
allows us to utilize certain variance reduction techniques [78].
7.2
Parallel Quasirandom Sequences
One advantage of QMC is that it is easy to parallelize applications, and so producing
high quality parallel quasirandom sequences is important. Scrambling provides a natural
56
way to parallelize quasirandom sequences, because scrambled quasirandom sequences form
a stochastic family which can be assigned to different processes in a parallel computation.
This scheme is different from other proposed schemes such as leap-frog [79] and blocking [80],
which split up a single quasirandom sequence.
MC applications are often readily parallelized, and one would expect the same for QMC
applications. Parallel computations using QMC require a source of quasirandom sequences,
which are distributed among the individual processing units. In contrast to the study of
parallel pseudorandom numbers, there are very few papers on using quasirandom sequences
for parallel computing. Schmid [80, 81] pointed out “Only a little amount of work has
been done using (t, s)-sequences for parallel numerical integration.” Bromley [79] describes
a leap-frog parallelization technique to break up Soboĺ sequences into interleaved subsets.
Schmid extends and generalizes Bromley’s work by the use of blocking and leap-frogging
for all types of binary digital (t, s)-sequences. They find that blocking is more robust than
leap-frogging. Li and Mullen [82] proposed a parallel algorithm for (t, m, s)-nets for use
in financial problems, and then use it on (t, m, s)-nets in valuating derivatives and other
securities.
Similar to the methods for parallelizing pseudorandom number sequences, there are three
basic ways to parallelize quasirandom number sequences:
• Leap-frog - The sequence is partitioned in turn among the processors like a deck of
cards dealt to card players.
• Sequence splitting or blocking- The sequence is partitioned by splitting it into nonoverlapping contiguous subsections.
• Independent sequences - Each processor has its own independent sequence.
The first and second schemes produce numbers from a single quasirandom sequence. Meanwhile, the third scheme needs a family of quasirandom sequences. Scrambling techniques can
generate such a stochastic family of quasirandom sequences from one original quasirandom
sequence. Thus each scrambled sequence in the family is independent and can be assigned
to each processor. Ökten and Srinivasan [83] first used scrambled Halton sequences as a
method for parallelizing quasirandom sequences. Scrambling methods provide a natural
57
way to parallelize quasirandom sequences, and scrambling itself depends on permutations or
pseudorandom numbers. Hence, different permutations will lead to different quasirandom
sequences (or sets). Each scrambled variant of a parent stream can be considered as another
parallel stream of quasirandom numbers.
Blocking and leapfrog use a single quasirandom sequence and assign subsequences of
this quasirandom sequence to different processes. The idea behind blocking and leapfrog is
to assume that any subsequence of quasirandom sequence has the same uniformity as the
parent quasirandom sequence. This is an assumption that is often false. In comparison
to blocking and leapfrog, each scrambled sequence can be thought of as an independent
sequence and assigned to a processor, and under certain circumstances it can be proven that
the scrambled sequences are as uniform as the parent. Since the quality (small discrepancy) of
quasirandom sequences is a collective property of the entire sequence, forming new sequences
from parts is potentially troublesome. Therefore, scrambled quasirandom sequences provide
a very appealing alternative for parallel quasirandom sequences, especially where a single
quasirandom sequence is scrambled to provide all the parallel streams. Such a scheme would
also be very useful for providing QMC support to the computational grid [84].
7.2.1
A Parallel and Distributed Library
QMC applications have high degrees of parallelism, can tolerate large latencies, and usually require considerable computational effort, making them extremely well suited to parallel,
distributed, and even Grid-based computational environments. In these environments, a
large QMC problem is broken up into many small subproblems. These subproblems are then
scheduled on the parallel, distributed, or Grid-based environment. In a more traditional
instantiation, these environments are usually a workstation cluster connected by a local-area
network where the computational workload is cleverly distributed. Recently, peer-to-peer or
Grid computing, the cooperative use of geographically distributed resources unified to act
as a single powerful computer, has been investigated as an appropriate computational environment for MC applications [84]. There, the computational infrastructure developed was
based on the existence of a high-quality tool for parallel pseudorandom number generation,
58
the Scalable Parallel Random Number Generators (SPRNG) library [85]. The extension of
this technology to quasirandom numbers would be very useful.
7.2.2
Testing Parallel Quasirandom Sequences
As mentioned above, discrepancy is the standard measure of uniformity for quasirandom
sequences. Besides that, the Kolmogorov-Smirnov (KS) and the Cramer-von Mises (CS)
type statistics are used for testing multivariate distributions. Let F denote the theoretical
cumulative distribution of the distribution being tested which must be a continuous distribution, and Fn denote the empirical distribution, then these statistics are well known and
can be defined as:
KS = sup |Fn (x) − F (x)| ,
(7.5)
x∈[0,1)s
and
CM = n
[0,1)s
|Fn (x) − F (x)|2 dx.
(7.6)
A connection between the discrepancy and KS and CM statistics is: let P denote a
set of all sampling points. When the points in P are random samples, and F (x) has a
uniform distribution in [0, 1)s , then star-discrepancy reduces to a KS-type statistic, and the
L2 -discrepancy reduces to a CM-type statistic [86].
Testing uniformity in (0, 1]s is reviewed in [87]. Liang et al. [88] obtained new statistics
to test multivariate uniformity. These statistics come from generalized discrepancy [89] and
are easy to calculate in high dimensions. The best way of testing a parallel library is to solve
practical problems. High-dimensional integral problems for computational finance, Bayesian
networks, and a published set of test integrands can be chosen to test the effectiveness
of scrambled quasirandom sequences. However, for parallel computing, I have to consider
interactions between different streams, which can be thought of as a kind of correlation. Thus
I must resort to empirical testing in an attempt to discover such properties. Bot statistical
tests and high-dimensional integral problems play a critical part in empirical testing.
7.3
Derandomization
I have proposed optimal algorithms for the Halton, Faure and Soboĺ sequence in previous
chapters. Here, I give a summary of derandomization.
59
As the utility of QMC has developed, many researchers [73, 1] have not been satisfied
with the quality of existing quasirandom sequences. As such, many methods have been used
to attempt to improve the quality of quasirandom sequences. Scrambled sequences have
good performance in practice. Thus, the practice is now to incorporate such randomized
sequences routinely in applications.
If one proposes scrambling algorithm with random scramblings, this produces a stochastic family of quasirandom sequences.
The process of searching and specifying optimal
quasirandom sequences that achieve theoretically and empirically optimal results is an
important problem in QMC. The process of finding such optimal quasirandom sequences
is commonly called “derandomization.” For example, GFaure [7] is a family of scrambled
Faure sequences that has been successfully used in computational finance [25]. Tezuka’s
i-binomial scrambling [26] is a special case of GFaure and reduces the scrambling space from
O(K 2 ) to O(K), and gives one a specific search criterion and smaller space in which to
find optimal GFaure sequences. In fact, there are very few theoretical or practical results for
derandomizing quasirandom sequences. One important and open question is how to provide a
theoretical basis for derandomization and also to provide practical derandomizations [29, 20].
In addition, derandomizing quasirandom sequences not only aims to finding an optimal
sequence within this scrambled family, but also finding a set of optimal sequences. Thus,
error estimation can be obtained by using several scrambled optimal sequences. In such
cases, the accuracy of error estimation is expected to keep steady and smaller.
7.4
Conclusion
In this chapter, applications of scrambled quasirandom sequences were introduced.
Scrambled quasirandom sequences contribute to QMC in obtaining automatic error estimates
and generating parallel quasirandom sequences. Possible methods for achieving automatic
error estimates were presented as were approaches for parallelizing quasirandom sequences
from a family of scrambled quasirandom sequences. Derandomization provides us not only
the optimal quasirandom sequence used in QMC, but also the possibility of reducing the
error in conjunction with automatic error estimation.
60
CHAPTER 8
CONCLUSIONS AND FUTURE WORK
Quasirandom sequences are important in practice. This is because QMC methods offer
the hope of fast convergence for problems where MC is important. However, one has only
a few choices of quasirandom sequences: the Halton sequence, (t, s)-sequences, and lattice
points. Fortunately, scrambled quasirandom sequences provide us more choices, and the use
of derandomization can potentially lead to finding optimal quasirandom sequences for use
in applications.
In this dissertation, I detailed various scrambling methods for quasirandom sequences
and proposed some new scrambling algorithms for Halton, Faure and Soboĺ sequences. I
show that my scrambled quasirandom sequences do improve the quality of unscrambled
sequences through measures of two-dimensional projections and high-dimensional integration
problems. I also presented applications for scrambled sequences that included automatic
error estimation and the generation of high-quality parallel quasirandom numbers. I further
introduced the notion of derandomizing a family of scrambled quasirandom sequences to find
optimal sequences. The schemes for derandomizing quasirandom sequences are explored in
this dissertation. I limit my search space in the linear scrambling for searching for the
optimal quasirandom numbers. I feel that as randomization technology in quasirandom
number generation becomes more widespread, the usefulness of QMC will grow, and more
people will be tempted to see if their MC application can be accelerated with quasirandom
numbers.
Based on my research into effective scrambling methods and outlines in Chapter 7, I
could integrate a highly effective library for parallel and distributed quasirandom number
generation. Such a library would be modelled after the SPRNG [90] library, and also provide
tools for automatic error estimation in QMC, and in particular offer error estimation for
automatic integration [43].
61
Due to the relative simplicity of the Halton sequence and lattice points, randomization of
them are fast and relatively simple to implement. However, constructing (t, s)-sequences
is complicated, and so their randomization is more complicated. Also, straightforward
implementation of nested scrambling [11, 76, 22] (Owen scrambling) is too complicated
in practice. Tezuka scrambling [91, 26] was only designed for (0, s)-sequences. Finding
an effective scrambling method suitable for (t, s)-sequences and keeping the generality of
Owen’s nested scrambling is still ongoing. Producing a quasirandom sequence should not
occupy too much CPU time and computer memory in practice. In another words, it should
be effective and fast to produce quasirandom numbers in any application for QMC. The
idea behind Tezuka scrambling is to scramble the generator matrix, which is common for
constructing every (t, s)-sequence. Following this lead, there may exist a way to scramble
(t, s)-sequence with t > 0.
Derandomization plays the same important role as scrambling in QMC. Derandomization
has been successfully used in practice in several cases. For a certain application, existing
quasirandom sequences may not be satisfactory, and derandomization can provide us with
the important alternatives. Since derandomization is still a relatively new technique, there
are many open questions that need to be answered, I list just a few of them as below.
• Finding an effective and easy implementation of a general method to derandomize a
scrambled family of quasirandom sequences.
• Searching for a family of scrambled quasirandom sequences among various scrambled
families. In another words, finding a derandomization method that works independently of the scrambling method.
• Theoretical and empirical criteria for measuring the quality of derandomization that
are computationally tractable.
62
APPENDIX A
PARALLEL PSEUDORANDOM NUMBER
GENERATORS
Pseudorandom number generators can be used as scramblers for quasirandom numbers.
The main algorithms used for sequential pseudorandom number generators are the following:
• Linear congruential generators (LCGs),
• Lagged-Fibonacci generators (LFGs),
• Shift-register generators (SRGs),
• Inversive congruential generators (ICGs),
• Combinations of the above generators.
Among these generators, LCGs are the best known and have the best developed theory.
Therefore, most current implementations of parallel random number generators are based
on LCGs. For an overview, please see [92].
In general, there are three techniques to parallelize random numbers
• Leap-frog [92]: the sequence is partitioned in turn among the processors as are cards
around a card table. If each processor leap-frogs by L in the sequence of random
numbers, {Xn }, then processor Pi will generate a random sequence with numbers
Xi , Xi+L , Xi+2L , . . . ,
• Blocking [92]: the sequence is partitioned by splitting it into non-overlapping contigu-
ous sections. If the length of each section is L in the random number sequence, {Xn },
then processor Pi will be assigned the random sequence with numbers
XiL , XiL+1 , XiL+2 , . . . ,
63
• Parameterization [85]: the initial seed or other parameters in a generator can be
carefully chosen in such a way so as to produce long period independent sequences
for each processor. For example, an LCG can be expressed as
Xn+1 = aXn + b (mod p).
(A.1)
There are two ways to parameterize the LCG in Eq.(A.1). One is the judicious choice
of different additive constants, b, to form different sequences with power-of-two moduli,
and the other is to choose different multipliers, a, with prime moduli [67].
However, both the parallelization methods of leap-frog and blocking may suffer from
long-range correlations [93]. One must be aware of those correlations before one chooses the
size of blocking or the leap-frog jump. In addition, the period of the parent generator has
to be long enough so that one has enough random numbers to assign to each processor. In
comparison to blocking and leapfrog, each parameterized sequence can be thought of as an
independent sequence and assigned to a processor.
Good random number generators are hard to find, and high-quality, efficient algorithms
for pseudorandom number generation on parallel computers are even more difficult to find.
There are a few available software packages for random number generators, such as the SPRNG
library [85], the Mersenne twister [94] and the nonlinear inversive congruential generator from
pLab [95]. We can choose widely tested and relatively reliable PRNGs as our scrambler by
using such packages.
64
APPENDIX B
LCGS WITH SOPHIE-GERMAIN MODULI
Linear Congruential Generators (LCGs) with both power-of-two and prime moduli have
been used in implementations of scrambling the Soboĺ sequence.
Here we give a brief
introduction of LCGs with Sophie-Germain moduli.
If the modulus of the LCGs is a properly chosen Sophie-Germain prime, the LCG can
achieve roughly the same generation speed as LCGs with Mersenne prime moduli. Modular
multiplication within the generator PMLCG (Prime Modulus LCG) in the SPRNG (Scalable
Parallel Random Number Generators) library [90] is replaced by integer addition. We will
use the same trick for LCGs with Sophie-Germain prime moduli.
After a brief search, we have found suitable Sophie-Germain primes to use as the moduli
for our implementations. There seem to always be many Sophie-Germain primes close to 2q .
In Table B.1, we list the Sophie-Germain primes that are just below the power-of-two for
exponents 15 through 64. There may be “closer” Sophie-Germain primes slightly larger than
these powers-of-two, but they would require one more bit in their binary representation. We
do not think that using such primes will gain benefit in practice.
B.1
Parallelization
Since we wish to utilize the same algorithm on every processor, we cannot choose to
parameterize the modulus when we work so hard to optimize modular reduction based on the
modulus. In addition, if we consider modular parameterization, the period of the sequences
will be different, and the theoretical measure of interprocessor correlation via exponential
sums is analytically intractable.
To understand the computational efficiency of the choice of Sophie-Germain primes in
parameterized LCGs, we implemented a SGMLCG (Linear Congruential Generator with
Sophie-Germain Modulus) using the same structure as the PMLCG library in SPRNG [85].
65
Table B.1. The Sophie-Germain (S-G) Primes Closest to but Less Than 2q
2q
264
262
260
258
256
254
252
250
248
246
244
242
240
238
236
234
232
230
228
226
224
222
220
218
216
S-G primes (2q − k)
264 -1469
262 -10565
260 -3677
258 -137
256 -2249
254 -4805
252 -473
250 -161
248 -5297
246 -857
244 -1493
242 -2201
240 -437
238 -401
236 -137
234 -641
232 -209
230 -1385
228 -437
226 -677
224 -317
222 -17
220 -233
218 -17
216 -269
2q
263
261
259
257
255
253
251
249
247
245
243
241
239
237
235
233
231
229
227
225
223
221
219
217
215
S-G primes (2q − k)
263 -4569
261 -2373
259 -18009
257 -3993
255 -789
253 -1269
251 -465
249 -2709
247 -1485
245 -573
243 -741
241 -1965
239 -381
237 -45
235 -849
233 -9
231 -69
229 -189
227 -405
225 -633
223 -321
221 -9
219 -45
217 -285
215 -165
The performance comparisons are presented in table 2. There we tabulate the initialization
and generation times for PMLCG and SGMLCG generators.
The two generators we used were specifically the following:
• PMLCG
The generator is defined by the following relation:
xn = axn−1
(mod 261 − 1).
(B.1)
where the multiplier, a, differs for each process. The multiplier is chosen to be certain
powers of 37, a primitive root modulo 261 − 1, that give maximal period cycles of
66
acceptable quality. The period of this generator is 261 − 2 and the number of distinct
streams available is roughly 258 .
• SGMLCG
The generator is defined by the following relation:
xn = axn−1
(mod 264 − 21017).
(B.2)
where the modulus, m = 264 − 21017 is a 64-bit Sophie-Germain prime, and the
multiplier, a, differs for each stream. The multiplier is chosen to be certain powers of
13, a primitive root modulo 264 − 21017, that give maximal period cycles of acceptable
quality. The period of this generator is (264 − 21017) − 1, and the number of distinct
streams available is 263 − 10509. This generator passed both the DIEHARD [96]
statistical tests of randomness and the extensive randomness test suite in SPRNG [85].
If we carefully choose a Sophie-Germain prime as modulus, m, in the form 2q − k, we can
implement an LCG with both a fast modular multiplication and the fast calculation of the
kth integer relatively prime to m − 1.
Besides fast modular multiplication and fast calculation of the kth number relatively
prime to m − 1, we have more choices for Sophie-Germain primes (in Table (B.1)). There
are only four Mersenne primes in the interval (215 , 264 ): 217 − 1, 219 − 1, 231 − 1, and 261 − 1,
and the next Mersenne prime is 289 − 1. By contrast, we expect there to be approximately
1.23 × 1016 Sophie-Germain primes in this same interval.
Since SGMLCG gives us both a smaller initialization time and a competitive generation
speed, LCGs with Sophie-Germain prime moduli are the better choice.
SGMLCG is currently being incorporated into SPRNG.
67
To that end,
APPENDIX C
LINEAR SCRAMBLING AND
DERANDOMIZATION
Randomizing quasirandom sequences produces a stochastic family of quasirandom sequences, which can be used in a number of settings. It is a natural question, therefore, to
ask how to choose an optimal quasirandom sequence from this family. Derandomization
techniques provide us a way to find an optimal sequence from such a family of quasirandom sequences. The process of finding such optimal quasirandom sequences is called the
derandomization of a randomized (scrambled) family. Before derandomization, one has to
choose a scrambling space. In this appendix, I will give an overview of scrambling methods
and a motivation for derandomization of the Halton, Faure and Soboĺ sequences in linear
scrambling spaces. In addition, I describe the derandomization process in detail.
C.1
Scrambling Methods
The purpose of scrambling is two-fold. The original motivation of scrambling [1, 2, 46]
aims toward obtaining more uniformity for quasirandom sequences in high dimensions, which
can be checked via two-dimensional projections. Secondly, Owen scrambling [11, 46], called
nested scrambling, was developed to provide a practical error estimate for QMC.
Now, I outline the background of various scrambling methods for (t, s)-sequences. The
first scrambling technique was proposed by Warnock [41] in order to calculate the L2 discrepancy of Halton sequences. In that paper, Warnock tried to minimize the discrepancy
of the Halton sequence by considering scrambled versions. Later, Braaten and Weller [1]
used permutations to further minimize the discrepancy and improve the behavior of any
pair of coordinates of the Halton sequence.
68
In 1988, Shaw [46] proposed a scrambling technique which combines shifting and
permutations, and applied these scrambled quasirandom sequences to compute posterior
distributions in Bayesian statistics. Shaw first applied his scrambling to Soboĺ sequences, and
pointed out that scrambling can be used to assess the overall accuracy of integration. After
Niederreiter sequences [16] were proposed, Owen [11] and Tezuka [7] in 1994 independently
developed two powerful scrambling methods for (t, s)-sequences. Owen also explicitly pointed
out that scrambling can be used to provide error estimates for QMC.
Although many other methods for scrambling (t, s)-sequences have been proposed [21,
48, 97, 98], most of them are really modified or simplified Owen and Tezuka schemes. Owen’s
scheme is theoretically powerful for (t, s)-sequences. Tezuka’s algorithm was proved to be
efficient for (0, s)-sequences.
C.1.1
Owen Nested Scrambling
(1)
(2)
(s)
Let xn = (xn , xn , . . . , xn ) be a quasirandom number in [0, 1)s , and let zn =
(1)
(2)
(s)
(j)
(zn , zn , . . . , xn ) be the scrambled version of the point xn . Suppose each xn can be
(j)
(j) (j)
(j)
represented in base b as xn = (0.xn1 xn2 ...xnK ...)b with K being the number of digits to be
scrambled. Then nested scrambling proposed by Owen [11, 20] can be defined as follows:
(j)
(j)
(j)
(j)
zn1 = π• (xn1 ), and zni = π•x(j) x(j) ...x(j) (xni ), with independent permutations π•x(j) x(j) ...x(j)
n1
n2
n1
ni−1
n2
ni−1
for i ≥ 2. Of course, (t, m, s)-net remains (t, m, s)-net under nested scrambling. However,
nested scrambling requires bi−1 permutations to scramble the ith digit. Owen scrambling
(nested scrambling), which can be applied to all (t, s)-sequences, is powerful; however,
from the implementation point-of-view, nested scrambling or so-called path dependent
permutations requires a considerable amount of bookkeeping, and leads to more problematic
implementation.
C.1.2
Tezuka Scrambling
Tezuka [17, 29] proposed generator matrix scrambling methods: GFaure and NFaure for
(0, s)-sequences.
Generalized Faure Sequences: The idea behind GFaure is to use a random matrix
to permute the generator matrix to obtain a stochastic family of Faure sequences. Source
69
code for GFaure is available at [7]. GFaure’s generator matrix for ith coordinate is defined as
C (i) = A(i) P i−1 , where P is the usual Pascal matrix, and A(i) is a random nonsingular lower
triangular matrix. With a randomized generator matrix, I produce scrambled quasirandom
numbers directly. If all the elements of the lower triangular matrix, A(i) , below the upper
triangle are non-zero, then GFaure sequences are a special case of Owen scrambling. This
property comes from the fact that multiplication by a non-zero element of Fb is a particular
permutation of elements in Fb .
If the matrices, A(i) , are randomly chosen, I obtain an approach to Tezuka and
Owen scrambling suggested by Hong and Hickernell. GFaure, which is available in the
FINDER [24, 25] software package, is based on carefully selected matrices, A(i) . FINDER has
some well chosen A(i) , that are fixed in its implementation.
A New Generalization of the Faure Sequences: (NFaure) Let b be a prime power
with b ≥ s. For an arbitrary nonsingular upper triangular (NUT) infinite matrix, U, and
arbitrary elements γi ∈ Fb (with 1 ≤ i ≤ b), define U (i) by
U (i) = γiU.
(C.1)
Then, the sequences produced by the generator matrices C (i) = P i−1 U (i) are (0, s)-sequences
in the prime power base b ≥ s.
• These (0, s)-sequences are a new generalization of Faure sequences. Their generator
matrices are NUT, while the generator matrices of GFaure are not triangular, only the
A(i) ’s.
• The matrix U and the number γi can be chosen randomly. However, for best results
they should be chosen empirically in order to obtain a good quality generator.
C.1.3
Linear Scrambling
Since the straightforward implementation of nested scrambling is very hard and inefficient
in practice, Morohosi and Fushimi [30] gave a practical comparison among these scrambling
methods in implementation and error estimation. They also pointed out that implementation
of Owen scrambling is time-consuming in practice and has the same performance in error
70
estimation. Inspired by Tezuka’s scrambling, several researchers [21, 48, 66] gave proofs and
implementations in terms of linear scrambling for (t, s)-sequences.
When the Halton sequence is studied, the best way to break the correlations between
dimensions is linear scrambling. The efficient way to scramble the Faure sequence is GFaure.
Therefore, linear scrambling is the simplest and most effective scrambling method to improve
the quality of quasirandom sequences. This is the reason why I focus on linear scrambling and
try to look for the “best” scrambling in this linear space. In addition, linear permutations
are easily implemented.
C.2
Derandomization
The main goal of the derandomization is to reduce the size of the scrambling space
and to find a set of sequences with good quality and then use them in QMC or error
estimation. Derandomization is not new idea since a version was proposed when poor 2-D
projections in the Halton sequence were first reported [1]. Fixing these problems was not
called derandomization then. Recently, FINDER [24, 25], a commercial software system which
uses quasirandom sequences to solve problems in finance, is an example of the successful use
of derandomization. There are two types of quasirandom sequences included in FINDER.
One is GFaure and the other is a modified Soboĺ sequence. Although the creators of FINDER
pointed out that the major improvements in their modified Soboĺ sequence were achieved via
optimized initial direction numbers for dimension up to 360, the method they used for this
improvement was not revealed, and FINDER was patented. As for GFaure in FINDER, recall
that GFaure has a generator matrix which can be expressed as C (j) = A(j) P j−1, (1 ≤ j ≤ b),
where A(j) is an arbitrary nonsingular lower triangular matrix over Fb , and P is the Pascal
matrix. FINDER empirically chooses A(j) to optimize the simulation results. This is a typical
example of applying derandomization. Thus one may think of derandomization as using some
means, empirical or theoretical, to choose optimal parameters in a quasirandom number
generator. Optimality is usually based on uniformity measures, but may be related to
performance in a particular application.
71
C.2.1
Reasons to Derandomize
Quasirandom sequences are deterministically generated and are constructed to be highly
uniformly distributed. Although the use of quasirandom numbers in QMC leads to a faster
convergence rate [9], it is by no means trivial to provide practical error estimates in QMC
due to the fact that the only rigorous error bounds, provided via the Koksma-Hlawka
inequality [8], are very hard to utilize in practice. In fact, the common procedure in MC
of using a predetermined error criterion as a deterministic termination condition, is almost
impossible to achieve in QMC without extra technology. The solution to this problem is to
add randomness into quasirandom sequences by using various scrambling techniques.
Unlike pseudorandom numbers, there are only a few common choices for quasirandom
number generators. Randomizing quasirandom sequences gives us a stochastic family of
sequences. Finding one optimal quasirandom sequence within this family can be quite useful
for enhancing the performance of ordinary QMC.
In addition, derandomizing quasirandom sequences not only seeks to find an optimal
sequence within a scrambled family, but also at finding a set of such optimal sequences.
These optimal families are useful in automatic error estimation. Randomized QMC provides
an elegant approach to obtain error estimates for quasi-Monte Carlo based on treating each
scrambled sequence as a different and independent random sample. When error estimation
is used in practice, one just has to randomly choose several scrambled sequences from the
whole scrambled family. The idea to derandomize is that one can find a set of optimal
sequences within a family of scrambled sequence family, and use sequences within this set
for error estimation. While I have seen the utility of this for automatic error estimation in
QMC, the process of searching and specifying optimal quasirandom sequences that achieve
theoretically and empirically optimal results is also an important problem in QMC.
C.2.2
Examples
The final goal of derandomization is to find optimal sequences within a family of
scrambled sequences. Therefore, a criterion to measure this optimality has to be both computationally tractable and easy to implement. To better illustrate the ideas of derandomization,
72
I use examples to demonstrate that the derandomization of Owen scrambling is impractical,
while the derandomization of GFaure and linear scrambling families is admissible.
Owen scrambling is theoretically powerful for any (t, s)-sequence. However, the derandomization of Owen scrambling is inadmissible. This is due to two reasons: (1) the huge
nested space implicit in Owen scrambling, and (2) the previously mentioned lack of an
effective measurement criterion. For each scrambled digit, I have p! possible permutations,
where p is the base. If one wants to scramble K digits of each quasirandom point, pK (p!)
permutations have to be stored. Finally, Owen [20] pointed out that direct implementation
of his nested scrambling is not practical for high dimensions. Therefore, nested scramblings
are only used for comparison in this dissertation.
The GFaure family provides a successful example of reducing scrambling space and finding
optimal sequences.
Derandomization has been done successfully with GFaure and now
with Tezuka’s i-binomial scrambling [26]. Thus, I present an example based on finding
optimal Faure sequences from the GFaure family. As described to Chapter 4, i-binomial
scrambling [58] is an algorithm which considers only a reduced number of sequences within
the GFaure family, while maintaining the original overall quality of the Faure sequence.
Tezuka’s i-binomial scrambling [26] is a special case of GFaure and reduces the scrambling
space from K 2 p to Kp, where K is the number of bits to be scrambled, and gives one
a specific search criterion and smaller space in which to find optimal GFaure sequences.
Following this lead, I focus on finding optimal Faure sequences from among the i-binomial
scramblings instead of the entire GFaure family.
The GFaure family is a good example of finding optimal scramblings within the linear
scrambling space, as in this dissertation, linear scrambling is under consideration. We
focus on finding optimal scramblings for the Halton, Faure and Soboĺ sequences in a linear
scrambling space rather than the whole scrambling space. Linear scrambling is easy to
implement and the size of this scrambling space is tractably small. The most important
fact for my use of linear scrambling space is that I have a theoretical criterion, the extreme
discrepancy based on a sample of continued fraction expansion[56], to measure optimal
scramblings in this context. Note that in many other cases, there are no criteria to measure
optimal scramblings.
73
The advantage of linear scrambling is to reduce the permutation for each digit from
a set of p! permutations to a set of only p. This reduction makes the linear scrambling
space smaller and easier to search for optimal scramblings. In this dissertation, I present
new algorithms for finding the optimal Halton, Faure and Soboĺ sequences within this linear
scrambling space. Numerical results of comparisons are presented in Figures D.1 -D12. From
Tables 3.2 and Figures D.1–D.12, it is easily to see that linearly optimal sequences are stable
with increasing dimension and number of samples and have better performance than the
original sequences. Also, the original Faure and Soboĺ sequences have their own favorable
number of points [6] based on powers of their bases. In contrast, the optimal sequences do
not have these limitations.
C.3
Conclusion
There are only a few types of commonly used quasirandom sequences.
However,
derandomization techniques allow us to find optimal quasirandom sequences having similar
attributes. Scrambled quasirandom sequences form a large stochastic family of quasirandom
sequences. It is usually impossible to conduct an exhaustive search to find an optimal
quasirandom sequence within such a family. However, it is often useful to find a group of
sequences having smaller L2 -discrepancy or better uniformity. In practice, one tries to reduce
the search space before undertaking such a computation. We find a criterion based on the
extreme discrepancy to justify optimal scramblings from among the linear scrambling space.
We use these optimal scrambling in a published set of test integrands described in the next
appendix. These derandomized sequences are numerically tested and shown empirically to
be far superior to the original sequences as well as randomly chosen scrambled sequences.
74
APPENDIX D
ADDITIONAL NUMERICAL EXPERIMENTS
The purpose of this appendix is twofold: to provide numerical experiments among
different scrambling methods and to empirically verify optimality. Consider all scrambled
Halton (Faure or Soboĺ) sequences as the comparison space of all possible sequences. Before
one can randomly choose a scrambled Halton sequence, one has to choose a scrambling
method. Linear scrambling is the main scrambling method used in this dissertation. In
order to compare linear scrambling methods with the rest of the scrambling space, two other
scrambling methods, randomized shifting and nested scrambling, are chosen for comparison.
Note that these two scrambling methods are not linear scrambling methods. For each type of
sequence, I chose ten sequences within each type and these sequences distributed for different
dimensions and different integral functions. Clearly, there are other parts of the scrambled
Halton space that are not considered here. However, without specific scrambling methods,
I am not aware of how to “randomly” generate sequences in other parts of the scrambled
Halton space, thus I restrict myself to compare with scrambling methods above.
D.1
Test Functions
High-dimensional integral problems are always a good way to test the quality of
quasirandom sequences. A published set of test integrands [43, 99, 64, 100, 57] is a good
way to test different scrambled sequences. Consider a class of test functions:
I1 (f ) =
1
...
0
I2 (f ) =
1
...
0
1
0
1
0
s
π
i=1
2
(sin πx)dx1 . . . dxs = 1.
s
|4xi − 2| + ai
i=1
1 + ai
75
dx1 . . . dxs = 1.
(D.1)
(D.2)
where ai are parameters. Such functions allow an automatic tuning of the relative importance
of the variables, as well as of their interactions, by appropriate choices of ai . The effective
dimension can be computed and tabulated in [57]. The effective dimension is closely related
to sensitivity indices [101]. Three choices of the parameters will be considered:
1. a1 = a2 = · · · = as = 0
2. a1 = a2 = · · · = as = i, for 1 ≤ i ≤ s
3. a1 = a2 = · · · = as = i2 , for 1 ≤ i ≤ s
For the first choice of ai = 0, all variables are equally important. The effective dimension
is approximately the real dimension, s. For the last two choices of ai , the importance of the
successive variables is decreasing. The effective dimension is 10 for ai = i and 5 for ai = i2 .
In general, when ai becomes bigger, the variables are decreasing quickly in importance and
the effective dimension becomes smaller.
D.2
Numerical Results
For comparison purposes, I present numerical results for original, linearly optimal and two
types of scrambled sequences, which are randomly chosen from among randomized shifting
and nested scrambled families.
D.2.1
Halton Sequnences
Besides original Halton and optimal Halton sequence from the linear scrambling family,
I choose randomized shifted Halton [4], and nested scrambled Halton sequences [3] for
comparison. The numerical results are listed in Figures D.1 –D.4 and Tables D.1–D.2 for
each integral function.
• Halton refers to the original Halton sequence provided by Fox [5]
• DHalton refers to the derandomized Halton sequence proposed in Section 3.7.
• S1Halton refers to a randomly chosen sequence from random-start Halton sequences.
76
• S2Halton refers to a randomly chosen sequence from permuted Halton sequences
In Figures D.1 and D.2, estimated values by using the original Halton sequence in dimension
s = 25 and s = 40 are not included since these values are too large compared to 1. All these
estimated values are listed in Tables D.1 and D.2.
In Figure D.1, it is shown that derandomized Halton sequence in s = 40 (DHalton)
performs a much better than randomly chosen scrambled sequences. However, DHalton
has the same or a little better performance in Figure D.2 for s = 40. In Figures D.3 and
D.4, I only show one picture for s = 40 since the other cases have the similar performance
because the effective dimension of the integral function in (D.2) for ai = i and ai = i2 is low.
In general, the derandomized Halton sequence tends to have better performance when the
effective dimension is high.
D.2.2
Faure Sequences
Besides the original Faure and optimal Faure sequence from the linear scrambling family,
I choose a scrambled sequence based on randomized shifting [66] and a nested scrambled
Faure sequence [48]. The numerical results are listed in Figures D.5 –D.8 for each integral
function.
• Faure refers to the original Faure sequence provided by Fox [5]
• DFaure refers to the derandomized Faure sequence proposed in Section 4.2.
• S1Faure refers to a randomly chosen sequence from the randomized shifting family.
• S2Faure refers to a randomly chosen sequence from the nested scrambled Faure
sequences
In Figure D.5, it is shown that derandomized Faure sequence in s = 40 (DFaure) has
a better convergence rate than randomly chosen scrambled sequences. DFaure is stable in
Figure D.6 for s = 40. In Figures D.7 and D.8, I only show one picture for s = 40 since
the other cases have similar performance because the effective dimension of the integrand in
(D.2) for ai = i and ai = i2 is low. In general, derandomized Faure sequences tend to have
a faster convergence rate when the effective dimension is high.
77
D.2.3
Soboĺ Sequences
Besides the original Soboĺ and optimal Soboĺ sequence from the linear scrambling family,
I choose one scrambled Soboĺ sequence from among randomized shifts [66], and one from
nested scrambled Soboĺ sequences [102]. The numerical results are listed in Figures D.9
–D.12 for each integral function.
• Sobol refers to the original Soboĺ sequence provided by Fox [6]
• DSobol refers to the derandomized Soboĺ sequence proposed in Section 5.3.
• S1Sobol refers to a randomly chosen sequence from randomized shifted Soboĺ family.
• S2Sobol refers to a randomly chosen sequence from nested scrambled Soboĺ sequences.
In Figure D.9, it is shown that derandomized Soboĺ sequence in s = 40 (DSobol) has
the same convergence rate as randomly chosen scrambled sequences. In Figure D.10 the
derandomized Soboĺ sequence appears to have better performance than the other sequences.
In Figures D.11 and D.12, I only show one picture for s = 40 since the other cases have the
similar performance because the effective dimension of the integrand in (D.2) for ai = i and
ai = i2 is low. As a genaral rule, the derandomized Soboĺ sequence tends to have a faster
convergence rate when the effective dimension is high.
D.3
Conclusion
Clearly, effective dimension plays an important role when analyzing the performance of
various quasirandom sequences. When effective dimension is low (below 10), I can not see
any advantage to optimal sequences. However, whenever effective dimension is high (greater
than 20), the merit of optimal sequences is relatively stable and provides faster convergence.
78
Table D.1. Estimates of I1 (f ) in (D.1) by using Halton sequences
Generators N
s = 13
s = 20 s = 25 s = 40
Halton
DHalton
S1Halton
S2Halton
500
500
500
500
0.785
0.873
0.685
0.890
0.548
0.656
0.498
0.687
0.346
0.489
0.399
0.434
0.239
0.481
0.576
0.493
Halton
DHalton
S1Halton
S2Halton
1000
1000
1000
1000
0.893
0.937
0.903
0.931
0.634
0.768
0.589
0.703
0.273
0.542
0.503
0.657
0.149
0.903
0.759
0.692
Halton
DHalton
S1Halton
S2Halton
7000
7000
7000
7000
0.943
0.977
0.935
0.967
1.028
1.058
0.993
0.965
0.960
1.095
0.967
1.210
0.708
0.815
0.678
0.756
Halton
DHalton
S1Halton
S2Halton
20,000
20,000
20,000
20,000
0.984
0.989
0.967
1.023
1.023
1.014
0.967
1.034
0.893
1.084
1.201
1.190
0.859
0.943
0.830
1.202
Halton
DHalton
S1Halton
S2Halton
40,000
40,000
40,000
40,000
0.988
0.988
0.976
0.958
0.971
1.081
0.992
0.987
0.955
1.098
1.094
1.103
3.5260
0.940
0.760
1.326
Halton
DHalton
S1Halton
S2Halton
100,000
100,000
100,000
100,000
0.996
0.995
1.02
1.09
0.982
1.013
1.034
1.092
0.984
1.055
1.11
1.003
2.03
0.978
0.876
1.230
79
Table D.2. Estimates of I2 (f ) in (D.2) with parameters ai = 0 by using Halton sequences
Generators N
s = 13 s = 20 s = 25 s = 40
Halton
DHalton
S1Halton
S2Halton
500
500
500
500
1.303
0.799
0.641
0.578
3.129
0.786
0.702
0.698
68.549
0.498
0.401
0.456
1.363 × 106
0.432
0.201
0.445
Halton
DHalton
S1Halton
S2Halton
1000
1000
1000
1000
1.171
0.875
0.673
0.739
2.324
0.601
0.734
0.769
34.513
0.612
0.721
0.658
6.814 × 105
0.311
0.421
0.399
Halton
DHalton
S1Halton
S2Halton
7000
7000
7000
7000
0.922
0.942
0.953
0.902
0.998
1.216
0.934
0.899
5.782
1.742
0.711
0.659
9.734 × 104
0.489
0.789
0.403
Halton
DHalton
S1Halton
S2Halton
20,000
20,000
20,000
20,000
0.977
0.982
0.956
0.890
0.939
1.118
0.923
0.915
2.876
1.327
0.765
0.835
3.407 × 104
0.689
0.578
0.674
Halton
DHalton
S1Halton
S2Halton
40,000
40,000
40,000
40,000
0.974
1.014
1.032
0.941
0.889
1.183
1.121
0.984
1.796
1.380
1.203
1.120
1.704 × 104
0.732
0.654
1.309
Halton
DHalton
S1Halton
S2Halton
100,000
100,000
100,000
100,000
0.985
1.008
0.953
0.991
0.897
1.068
0.972
0.982
1.242
1.003
0.923
1.028
6.815 × 103
1.102
0.674
1.327
80
1.5
1.5
1
0.5
0
–0.5
Camparisons among Halton sequences (s=20)
2
Estimated value
Estimated value
Camparisons among Halton sequences (s=13)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
Halton
DHalton
S1Halton
S2Halton
Halton
DHalton
S1Halton
S2Halton
1.5
1.5
1
0.5
0
–0.5
Camparisons among Halton sequences (s=40)
2
Estimated value
Estimated value
Camparisons among Halton sequences (s=25)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
Halton
DHalton
S1Halton
S2Halton
Halton
DHalton
S1Halton
S2Halton
Figure D.1. Estimates of the integral I1 (f ) in (D.1) by using various Halton sequences.
81
1.5
1.5
1
0.5
0
–0.5
Camparisons among Halton sequences (s=20)
2
Estimated value
Estimated value
Camparisons among Halton sequences (s=13)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
Halton
DHalton
S1Halton
S2Halton
DHalton
S1Halton
S2Halton
Exactvalue
1.5
1.5
1
0.5
0
–0.5
Camparisons among Halton sequences (s=40)
2
Estimated value
Estimated value
Camparisons among Halton sequences (s=25)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
DHalton
S1Halton
S2Halton
Exactvalue
DHalton
S1Halton
S2Halton
Exactvalue
Figure D.2. Estimates of the integral I2 (f ) in (D.2) with parameters ai = 0 by using
various Halton sequences.
82
2
Camparisons among Halton sequences (s=40)
Estimated value
1.5
1
0.5
0
20000
40000
60000
80000
Number of simulations
100000
–0.5
Legend
Halton
DHalton
S1Halton
S2Halton
Exactvalue
Figure D.3. Estimates of the integral I2 (f ) in (D.2) with parameters ai = i by using various
Halton sequences.
83
2
Camparisons among Halton sequences (s=40)
Estimated value
1.5
1
0.5
0
20000
40000
60000
80000
Number of simulations
100000
–0.5
Legend
Halton
DHalton
S1Halton
S2Halton
Exactvalue
Figure D.4. Estimates of the integral I2 (f ) in (D.2) with parameters ai = i2 by using
various Halton sequences.
84
1.5
1.5
1
0.5
0
–0.5
Camparisons among Faure sequences (s=20)
2
Estimated value
Estimated value
Camparisons among Faure sequences (s=13)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
Faure
DFaure
S1Faure
S2Faure
Faure
DFaure
S1Faure
S2Faure
1.5
1.5
1
0.5
0
–0.5
Camparisons among Faure sequences (s=40)
2
Estimated value
Estimated value
Camparisons among Faure sequences (s=25)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
Faure
DFaure
S1Faure
S2Faure
Faure
DFaure
S1Faure
S2Faure
Figure D.5. Estimates of the integral I1 (f ) in (D.1) by using various Faure sequences.
85
1.5
1.5
1
0.5
0
–0.5
Camparisons among Faure sequences (s=20)
2
Estimated value
Estimated value
Camparisons among Faure sequences (s=13)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
Faure
DFaure
S1Faure
S2Faure
Faure
DFaure
S1Faure
S2Faure
1.5
1.5
1
0.5
0
–0.5
Camparisons among Faure sequences (s=40)
2
Estimated value
Estimated value
Camparisons among Faure sequences (s=25)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
Faure
DFaure
S1Faure
S2Faure
Faure
DFaure
S1Faure
S2Faure
Figure D.6. Estimates of the integral I2 (f ) in (D.2) with parameters ai = 0 by using
various Faure sequences.
86
2
Camparisons among Faure sequences (s=40)
Estimated value
1.5
1
0.5
0
20000
40000
60000
80000
Number of simulations
100000
–0.5
Legend
Faure
DFaure
S1Faure
S2Faure
ExactValue
Figure D.7. Estimates of the integral I2 (f ) in (D.2) with parameters ai = i by using various
Faure sequences.
87
2
Camparisons among Faure sequences (s=40)
Estimated value
1.5
1
0.5
0
20000
40000
60000
80000
Number of simulations
100000
–0.5
Legend
Faure
DFaure
S1Faure
S2Faure
ExactValue
Figure D.8. Estimates of the integral I2 (f ) in (D.2) with parameters ai = i2 by using
various Faure sequences.
88
1.5
1.5
1
0.5
0
–0.5
Camparisons among Sobol sequences (s=20)
2
Estimated value
Estimated value
Camparisons among Sobol sequences (s=13)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
Sobol
DSobol
S1Sobol
S2Sobol
Sobol
DSobol
S1Sobol
S2Sobol
1.5
1.5
1
0.5
0
–0.5
Camparisons among Sobol sequences (s=40)
2
Estimated value
Estimated value
Camparisons among Sobol sequences (s=25)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
Sobol
DSobol
S1Sobol
S2Sobol
Sobol
DSobol
S1Sobol
S2Sobol
Figure D.9. Estimates of the integral I1 (f ) in (D.1) by using various Soboĺ sequences.
89
1.5
1.5
1
0.5
0
–0.5
Camparisons among Sobol sequences (s=20)
2
Estimated value
Estimated value
Camparisons among Sobol sequences (s=13)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
Sobol
DSobol
S1Sobol
S2Sobol
Sobol
DSobol
S1Sobol
S2Sobol
1.5
1.5
1
0.5
0
–0.5
Camparisons among Sobol sequences (s=40)
2
Estimated value
Estimated value
Camparisons among Sobol sequences (s=25)
2
1
0.5
20000 40000 60000 80000 100000
Number of simulations
0
–0.5
Legend
20000 40000 60000 80000 100000
Number of simulations
Legend
Sobol
DSobol
S1Sobol
S2Sobol
Sobol
DSobol
S1Sobol
S2Sobol
Figure D.10. Estimates of the integral I2 (f ) in (D.2) with parameters ai = 0 by using
various Soboĺ sequences.
90
2
Camparisons among Sobol sequences (s=40)
Estimated value
1.5
1
0.5
0
20000
40000
60000
80000
Number of simulations
100000
–0.5
Legend
Sobol
DSobol
S1Sobol
S2Sobol
Exactvalue
Figure D.11. Estimates of the integral I2 (f ) in (D.2) with parameters ai = i by using
various Soboĺ sequences.
91
2
Camparisons among Sobol sequences (s=40)
Estimated value
1.5
1
0.5
0
20000
40000
60000
80000
Number of simulations
100000
–0.5
Legend
Sobol
DSobol
S1Sobol
S2Sobol
Exactvalue
Figure D.12. Estimates of the integral I2 (f ) in (D.2) with parameters ai = i2 by using
various Soboĺ sequences.
92
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99
BIOGRAPHICAL SKETCH
Hongmei Chi
Hongmei Chi was born in Dalian, China. She remained there and received a high school
diploma in 1985. She earned a Bachelor of Science in Statistics from Nankai University
in 1989 and earned a Master of Science in Computer Science from Dalian University of
Technique in 1992. She received a Ph.D. degree in Computer Science from Florida State
University in 2004.
Her research interests span many areas related to scientific computing. Her current
research focuses on the areas of quasi-Monte Carlo methods, computational finance and
distributed or grid computing.
100