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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 69 (7 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...
Valuation of Mortgage Backed Securities Using Brownian Bridges to Reduce Effective Dimension
, 1997
"... The quasiMonte Carlo method for financial valuation and other integration problems has error bounds of size O((log N) k N \Gamma1 ), or even O((log N) k N \Gamma3=2 ), which suggests significantly better performance than the error size O(N \Gamma1=2 ) for standard Monte Carlo. But in hig ..."
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Cited by 68 (13 self)
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The quasiMonte Carlo method for financial valuation and other integration problems has error bounds of size O((log N) k N \Gamma1 ), or even O((log N) k N \Gamma3=2 ), which suggests significantly better performance than the error size O(N \Gamma1=2 ) for standard Monte Carlo. But in high dimensional problems this benefit might not appear at feasible sample sizes. Substantial improvements from quasiMonte Carlo integration have, however, been reported for problems such as the valuation of mortgagebacked securities, in dimensions as high as 360. We believe that this is due to a lower effective dimension of the integrand in those cases. This paper defines the effective dimension and shows in examples how the effective dimension may be reduced by using a Brownian bridge representation. 1 Introduction Simulation is often the only effective numerical method for the accurate valuation of securities whose value depends on the whole trajectory of interest Mathematics Departmen...
Computer Experiments
, 1996
"... Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, a ..."
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Cited by 67 (5 self)
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Introduction Deterministic computer simulations of physical phenomena are becoming widely used in science and engineering. Computers are used to describe the flow of air over an airplane wing, combustion of gasses in a flame, behavior of a metal structure under stress, safety of a nuclear reactor, and so on. Some of the most widely used computer models, and the ones that lead us to work in this area, arise in the design of the semiconductors used in the computers themselves. A process simulator starts with a data structure representing an unprocessed piece of silicon and simulates the steps such as oxidation, etching and ion injection that produce a semiconductor device such as a transistor. A device simulator takes a description of such a device and simulates the flow of current through it under varying conditions to determine properties of the device such as its switching speed and the critical voltage at which it switches. A circuit simulator takes a list of devices and the
Monte Carlo Variance of Scrambled Net Quadrature
 SIAM J. Numer. Anal
, 1997
"... . Hybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. This paper studies the variance of one such hybrid, scrambled nets, by applying a multidimensional multiresolution (w ..."
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Cited by 28 (1 self)
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. Hybrids of equidistribution and Monte Carlo methods of integration can achieve the superior accuracy of the former while allowing the simple error estimation methods of the latter. This paper studies the variance of one such hybrid, scrambled nets, by applying a multidimensional multiresolution (wavelet) analysis to the integrand. The integrand is assumed to be measurable and square integrable but not necessarily of bounded variation. In simple Monte Carlo, every nonconstant term of the multiresolution contributes to the variance of the estimated integral. For scrambled nets, certain lowdimensional and coarse terms do not contribute to the variance. For any integrand in L 2 , the sampling variance tends to zero faster under scrambled net quadrature than under Monte Carlo sampling, as the number of function evaluations n tends to infinity. Some finite n results bound the variance under scrambled net quadrature by a small constant multiple of the Monte Carlo variance, uniformly ove...
Snobfit  Stable Noisy Optimization by Branch and Fit
"... this paper produces a userspeci ed number of suggested evaluation points in each step; proceeds by successive partitioning of the box (branch) and building local quadratic models ( t); combines local and global search and allows the user to determine which of both should be emphasized; h ..."
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Cited by 25 (5 self)
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this paper produces a userspeci ed number of suggested evaluation points in each step; proceeds by successive partitioning of the box (branch) and building local quadratic models ( t); combines local and global search and allows the user to determine which of both should be emphasized; handles local search from the best point with the aid of trust regions; allows for hidden constraints and assigns to such points a function value based on the function values of nearby feasible points
Monte Carlo Extension Of QuasiMonte Carlo
 Proceedings of the 1998 Winter Simulation Conference
, 1998
"... This paper surveys recent research on using Monte Carlo techniques to improve quasiMonte Carlo techniques. Randomized quasiMonte Carlo methods provide a basis for error estimation. They have, in the special case of scrambled nets, also been observed to improve accuracy. Finally through Latin super ..."
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Cited by 15 (0 self)
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This paper surveys recent research on using Monte Carlo techniques to improve quasiMonte Carlo techniques. Randomized quasiMonte Carlo methods provide a basis for error estimation. They have, in the special case of scrambled nets, also been observed to improve accuracy. Finally through Latin supercube sampling it is possible to use Monte Carlo methods to extend quasiMonte Carlo methods to higher dimensional problems. 1 INTRODUCTION The problem we consider is the estimation of an integral I = Z [0;1] d f(x)dx: (1) Standard manipulations can be applied to express integrals over domains other than the unit cube or with respect to nonuniform measures in the form (1). Similarly, the integrand f in (1) subsumes weighting functions from importance sampling or periodization. We are especially interested in cases where the dimension d is large, and some of the methods considered here apply to the case d = 1. The focus of this article is on ways of combining Monte Carlo and quasiMo...
Estimating Mean Dimensionality
, 2003
"... The mean dimension of a function is defined as a certain weighted combination of its ANOVA mean squares. This paper presents some new identities relating this mean dimension, and some similarly defined higher moments, to the variable importance measures of Sobol' (1993). The methods are applied to s ..."
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Cited by 7 (1 self)
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The mean dimension of a function is defined as a certain weighted combination of its ANOVA mean squares. This paper presents some new identities relating this mean dimension, and some similarly defined higher moments, to the variable importance measures of Sobol' (1993). The methods are applied to study some functions arising in the bootstrap, in computational finance, and in extreme value theory.
Visualization And Exploration Of HighDimensional Functions Using The Functional Anova Decomposition
, 1995
"... In recent years the statistical and engineering communities have developed many highdimensional methods for regression (e.g. MARS, feedforward neural networks, projection pursuit). Users of these methods often wish to explore how particular predictors affect the response. One way to do so is by dec ..."
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Cited by 6 (0 self)
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In recent years the statistical and engineering communities have developed many highdimensional methods for regression (e.g. MARS, feedforward neural networks, projection pursuit). Users of these methods often wish to explore how particular predictors affect the response. One way to do so is by decomposing the model into loworder components through a functional ANOVA decomposition and then visualizing the components. Such a decomposition, with the corresponding variance decomposition, also provides information on the importance of each predictor to the model, the importance of interactions, and the degree to which the model may be represented by first and secondorder components. This manuscript develops techniques for constructing and exploring such a decomposition. It begins by suggesting techniques for constructing the decomposition either numerically or analytically, proceeds to describe approaches to plotting and interpreting effects, and then develops methodology for rough infer...
Detecting Near Linearity in High Dimensions
, 1998
"... This paper presents a quasiregression method for determining the degree of linearity in a function. Quasiregression estimates regression coefficients without matrix inversion. For a given number n of observations, quasiregression is usually less efficient than ordinary regression. But for function ..."
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Cited by 3 (2 self)
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This paper presents a quasiregression method for determining the degree of linearity in a function. Quasiregression estimates regression coefficients without matrix inversion. For a given number n of observations, quasiregression is usually less efficient than ordinary regression. But for functions of d variables, the cost of linear regression grows as O(nd
A MULTIVARIATE CENTRAL LIMIT THEOREM FOR RANDOMIZED ORTHOGONAL ARRAY SAMPLING DESIGNS IN COMPUTER EXPERIMENTS
, 2007
"... Let f: [0,1) d → R be an integrable function. An objective of many computer experiments is to estimate ∫ [0,1) d f(x)dx by evaluating f at a finite number of points in [0,1) d. There is a design issue in the choice of these points and a popular choice is via the use of randomized orthogonal arrays. ..."
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Cited by 3 (0 self)
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Let f: [0,1) d → R be an integrable function. An objective of many computer experiments is to estimate ∫ [0,1) d f(x)dx by evaluating f at a finite number of points in [0,1) d. There is a design issue in the choice of these points and a popular choice is via the use of randomized orthogonal arrays. This article proves a multivariate central limit theorem for a class of randomized orthogonal array sampling designs [Owen (1992a)] as well as for a class of OAbased Latin hypercubes [Tang (1993)].