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Extension of latin hypercube samples with correlated variables.
A procedure for extending the size of a Latin hypercube sample (LHS) with rank correlated variables is described and illustrated. The extension procedure starts with an LHS of size m and associated rank correlation matrix C and constructs a new LHS of size 2m that contains the elements of the original LHS and has a rank correlation matrix that is close to the original rank correlation matrix C. The procedure is intended for use in conjunction with uncertainty and sensitivity analysis of computationally demanding models in which it is important to make efficient use of a necessarily limited number of model evaluations
Latin hypercube sampling with dependence and applications in finance
In Monte Carlo simulation, Latin hypercube sampling (LHS) [McKay et al. (1979)] is a well-known variance reduction technique for vectors of independent random variables. The method presented here, Latin hypercube sampling with dependence (LHSD), extends LHS to vectors of dependent random variables. The resulting estimator is shown to be consistent and asymptotically unbiased. For the bivariate case and under some conditions on the joint distribution, a central limit theorem together with a closed formula for the limit variance are derived. It is shown that for a class of estimators satisfying some monotonicity condition, the LHSD limit variance is never greater than the corresponding Monte Carlo limit variance. In some valuation examples of financial payoffs, when compared to standard Monte Carlo simulation, a variance reduction of factors up to 200 is achieved. LHSD is suited for problems with rare events and for high-dimensional problems, and it may be combined with Quasi-Monte Carlo methods. --Monte Carlo simulation,variance reduction,Latin hypercube sampling,stratified sampling
Multiprocess parallel antithetic coupling for backward and forward Markov Chain Monte Carlo
Antithetic coupling is a general stratification strategy for reducing Monte
Carlo variance without increasing the simulation size. The use of the
antithetic principle in the Monte Carlo literature typically employs two strata
via antithetic quantile coupling. We demonstrate here that further
stratification, obtained by using k>2 (e.g., k=3-10) antithetically coupled
variates, can offer substantial additional gain in Monte Carlo efficiency, in
terms of both variance and bias. The reason for reduced bias is that
antithetically coupled chains can provide a more dispersed search of the state
space than multiple independent chains. The emerging area of perfect simulation
provides a perfect setting for implementing the k-process parallel antithetic
coupling for MCMC because, without antithetic coupling, this class of methods
delivers genuine independent draws. Furthermore, antithetic backward coupling
provides a very convenient theoretical tool for investigating antithetic
forward coupling. However, the generation of k>2 antithetic variates that are
negatively associated, that is, they preserve negative correlation under
monotone transformations, and extremely antithetic, that is, they are as
negatively correlated as possible, is more complicated compared to the case
with k=2. In this paper, we establish a theoretical framework for investigating
such issues. Among the generating methods that we compare, Latin hypercube
sampling and its iterative extension appear to be general-purpose choices,
making another direct link between Monte Carlo and quasi Monte Carlo.Comment: Published at http://dx.doi.org/10.1214/009053604000001075 in the
Annals of Statistics (http://www.imstat.org/aos/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Validating Sample Average Approximation Solutions with Negatively Dependent Batches
Sample-average approximations (SAA) are a practical means of finding
approximate solutions of stochastic programming problems involving an extremely
large (or infinite) number of scenarios. SAA can also be used to find estimates
of a lower bound on the optimal objective value of the true problem which, when
coupled with an upper bound, provides confidence intervals for the true optimal
objective value and valuable information about the quality of the approximate
solutions. Specifically, the lower bound can be estimated by solving multiple
SAA problems (each obtained using a particular sampling method) and averaging
the obtained objective values. State-of-the-art methods for lower-bound
estimation generate batches of scenarios for the SAA problems independently. In
this paper, we describe sampling methods that produce negatively dependent
batches, thus reducing the variance of the sample-averaged lower bound
estimator and increasing its usefulness in defining a confidence interval for
the optimal objective value. We provide conditions under which the new sampling
methods can reduce the variance of the lower bound estimator, and present
computational results to verify that our scheme can reduce the variance
significantly, by comparison with the traditional Latin hypercube approach
Identification of quasi-optimal regions in the design space using surrogate modeling
The use of Surrogate Based Optimization (SBO) is widely spread in engineering design to find optimal performance characteristics of expensive simulations (forward analysis: from input to optimal output). However, often the practitioner knows a priori the desired performance and is interested in finding the associated input parameters (reverse analysis: from desired output to input). A popular method to solve such reverse (inverse) problems is to minimize the error between the simulated performance and the desired goal. However, there might be multiple quasi-optimal solutions to the problem. In this paper, the authors propose a novel method to efficiently solve inverse problems and to sample Quasi-Optimal Regions (QORs) in the input (design) space more densely. The development of this technique, based on the probability of improvement criterion and kriging models, is driven by a real-life problem from bio-mechanics, i.e., determining the elasticity of the (rabbit) tympanic membrane, a membrane that converts acoustic sound wave into vibrations of the middle ear ossicular bones
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