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67
QuasiRandom Sequences and Their Discrepancies
 SIAM J. Sci. Comput
, 1994
"... Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is meas ..."
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Cited by 92 (6 self)
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Quasirandom (also called low discrepancy) sequences are a deterministic alternative to random sequences for use in Monte Carlo methods, such as integration and particle simulations of transport processes. The error in uniformity for such a sequence of N points in the sdimensional unit cube is measured by its discrepancy, which is of size (log N) s N \Gamma1 for large N , as opposed to discrepancy of size (log log N) 1=2 N \Gamma1=2 for a random sequence (i.e. for almost any randomlychosen sequence). Several types of discrepancy, one of which is new, are defined and analyzed. A critical discussion of the theoretical bounds on these discrepancies is presented. Computations of discrepancy are presented for a wide choice of dimension s, number of points N and different quasirandom sequences. In particular for moderate or large s, there is an intermediate regime in which the discrepancy of a quasirandom sequence is almost exactly the same as that of a randomly chosen sequence...
Latin Supercube Sampling for Very High Dimensional Simulations
, 1997
"... This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and QuasiMonte Carlo (QMC). In LSS, the input variables ..."
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Cited by 84 (8 self)
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This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and QuasiMonte Carlo (QMC). In LSS, the input variables are grouped into subsets, and a lower dimensional QMC method is used within each subset. The QMC points are presented in random order within subsets. QMC methods have been observed to lose effectiveness in high dimensional problems. This paper shows that LSS can extend the benefits of QMC to much higher dimensions, when one can make a good grouping of input variables. Some suggestions for grouping variables are given for the motivating examples. Even a poor grouping can still be expected to do as well as LHS. The paper also extends LHS and LSS to infinite dimensional problems. The paper includes a survey of QMC methods, randomized versions of them (RQMC) and previous methods for extending Q...
Recent Advances In Randomized QuasiMonte Carlo Methods
"... We survey some of the recent developments on quasiMonte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a highdimensional inte ..."
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Cited by 78 (15 self)
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We survey some of the recent developments on quasiMonte Carlo (QMC) methods, which, in their basic form, are a deterministic counterpart to the Monte Carlo (MC) method. Our main focus is the applicability of these methods to practical problems that involve the estimation of a highdimensional integral. We review several QMC constructions and dierent randomizations that have been proposed to provide unbiased estimators and for error estimation. Randomizing QMC methods allows us to view them as variance reduction techniques. New and old results on this topic are used to explain how these methods can improve over the MC method in practice. We also discuss how this methodology can be coupled with clever transformations of the integrand in order to reduce the variance further. Additional topics included in this survey are the description of gures of merit used to measure the quality of the constructions underlying these methods, and other related techniques for multidimensional integration.
Monte Carlo variance of scrambled net quadrature
 SIAM Journal on Numerical Analysis
, 1997
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On the use of low discrepancy sequences in Monte Carlo methods
 MONTE CARLO METHODS AND APPLICATIONS
, 1996
"... Quasirandom (or low discrepancy) sequences are sequences for which the convergence to the uniform distribution on [0; 1) s occurs rapidly. Such sequences are used in quasiMonte Carlo methods for which the convergence speed, with respect to the N first terms of the sequence, is in O(N \Gamma1 ..."
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Cited by 40 (2 self)
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Quasirandom (or low discrepancy) sequences are sequences for which the convergence to the uniform distribution on [0; 1) s occurs rapidly. Such sequences are used in quasiMonte Carlo methods for which the convergence speed, with respect to the N first terms of the sequence, is in O(N \Gamma1 (ln N) s ), where s is the mathematical dimension of the problem considered. The disadvantage of these methods is that error bounds, even if they exist theoretically, are inefficient in practice. Nevertheless, to take advantage of these methods for what concerns their convergence speed, we use them as a variance reduction technique, which lead to great improvements with respect to standard Monte Carlo methods. We consider in this paper two different approaches which combine Monte Carlo and quasiMonte Carlo methods. The first one can use every low discrepancy sequence and the second one, called Owen's method, uses only Niederreiter sequences. We prove that the first approach has the same...
A MultiLevel CrossClassified Model for Discrete Response Variables
 Transportation Research Part B
, 2000
"... In many spatial analysis contexts, the variable of interest is discrete and there is spatial clustering of observations. This paper formulates a model that accommodates clustering along more than one dimension in the context of a discrete response variable. For example, in a travel mode choice conte ..."
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Cited by 32 (11 self)
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In many spatial analysis contexts, the variable of interest is discrete and there is spatial clustering of observations. This paper formulates a model that accommodates clustering along more than one dimension in the context of a discrete response variable. For example, in a travel mode choice context, individuals are clustered by both the home zone in which they live as well as by their work locations. The model formulation takes the form of a mixed logit structure and is estimated by maximum likelihood using a combination of Gaussian quadrature and quasiMonte Carlo simulation techniques. An application to travel mode choice suggests that ignoring the spatial context in which individuals make mode choice decisions can lead to an inferior data fit as well as provide inconsistent evaluations of transportation policy measures.
Quasirandom Simulation of Discrete Choice Models
 Transportation Research Part B
"... We describe the properties of (t,m, s)nets and Halton draws. Four types of (t,m, s)nets, two types of Halton draws, and independent draws are compared in an application of maximum simulated likelihood estimation of a mixed logit model. All of the quasirandom procedures are found to perform far be ..."
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Cited by 26 (9 self)
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We describe the properties of (t,m, s)nets and Halton draws. Four types of (t,m, s)nets, two types of Halton draws, and independent draws are compared in an application of maximum simulated likelihood estimation of a mixed logit model. All of the quasirandom procedures are found to perform far better than independent draws. The best performance is attained by one of the (t,m, s)nets. The properties of the nets imply that two of them should outperform the other two, and our results comfirm this expectation. The two moreaccurate nets perform better than both types of Halton draws, while the two lessaccurate nets perform worse than the Halton draws. 1
Faster Evaluation of Multidimensional Integrals
 Computers in Physics
, 1997
"... In a recent paper Keister proposed two quadrature rules as alternatives to Monte Carlo for certain multidimensional integrals and reported his test results. In earlier work we had shown that the quasiMonte Carlo method with generalized Faure points is very effective for a variety of high dimensiona ..."
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Cited by 25 (1 self)
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In a recent paper Keister proposed two quadrature rules as alternatives to Monte Carlo for certain multidimensional integrals and reported his test results. In earlier work we had shown that the quasiMonte Carlo method with generalized Faure points is very effective for a variety of high dimensional integrals occuring in mathematical finance. In this paper we report test results of this method on Keister's examples of dimension 9 and 25, and also for examples of dimension 60, 80 and 100. For the 25 dimensional integral we achieved accuracy of 10 \Gamma2 with less than 500 points while the two methods tested by Keister used more than 220; 000 points. In all of our tests, for n sample points we obtained an empirical convergence rate proportional to n \Gamma1 rather than the n \Gamma1=2 of Monte Carlo. 1 1 Introduction Keister [1] points out that multidimensional integrals arise frequently in many branches of physics. He rules out product rules of onedimensional methods becaus...
Adaptive Random Testing: the ART of Test Case Diversity
 JOURNAL OF SYSTEMS AND SOFTWARE
, 2010
"... Random testing is not only a useful testing technique in itself, but also plays a core role in many other testing methods. Hence, any significant improvement to random testing has an impact throughout the software testing community. Adaptive Random Testing (ART) has been proposed as an effective alt ..."
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Cited by 25 (4 self)
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Random testing is not only a useful testing technique in itself, but also plays a core role in many other testing methods. Hence, any significant improvement to random testing has an impact throughout the software testing community. Adaptive Random Testing (ART) has been proposed as an effective alternative to random testing. This paper presents a synthesis of the most important research results related to ART. In the course of our research and through further reflection, we have realized how the techniques and concepts of ART can be applied in a much broader context, which we present here. We believe such ideas can be applied in a variety of areas in software testing, and even beyond software testing. Amongst these ideas, we particularly note the fundamental role of diversity in test case selection strategies. We hope this paper serves to provoke further discussions and investigations of these ideas.
QuasiMonte Carlo Node Sets from Linear Congruential Generators
 and QuasiMonte Carlo Methods
, 1998
"... . In this paper we present a new approach to finding good lattice points. We employ the spectral test, a wellknown figure of merit for uniform random number generators. This concept leads to an assessment of lattice points g that is closely related to the classical BabenkoZaremba quantity ae(g; N ..."
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Cited by 18 (10 self)
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. In this paper we present a new approach to finding good lattice points. We employ the spectral test, a wellknown figure of merit for uniform random number generators. This concept leads to an assessment of lattice points g that is closely related to the classical BabenkoZaremba quantity ae(g; N ). The associated lattice rules are good uniformly over a whole range of dimensions. Our numerical examples suggest that this simple approach leads to quasiMonte Carlo node sets that perform very well in comparison to the best available (t; m; s)nets. 1 Introduction There is no contradiction in the title of this paper. We show how to employ concepts that belong to the field of random number generation to obtain excellent node sets for quasiMonte Carlo integration in high dimensions. Our approach uses linear congruential generators ("LCGs") and the spectral test to find good lattice points ("GLPs") of the Korobov type. LCGs are a classical method for generating uniform random numbers, see...