Estimating Mean Dimensionality of Analysis of Variance Decompositions

Liu, Ruixue; Owen, Art B.

doi:10.1198/016214505000001410

Cited by 133 publications

(119 citation statements)

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“…The reason why RQMC might still work for large s is because f can often be well approximated by a sum of lowdimensional functions, and RQMC with a well-chosen point set can integrate these low-dimensional functions with small error. A standard way of formalizing this is as follows [44,62].…”

Section: Anova Decompositionmentioning

confidence: 99%

Randomized Quasi-Monte Carlo: An Introduction for Practitioners

L’Ecuyer

2018

Springer Proceedings in Mathematics &Amp; Statistics

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We survey basic ideas and results on randomized quasi-Monte Carlo (RQMC) methods, discuss their practical aspects, and give numerical illustrations. RQMC can improve accuracy compared with standard Monte Carlo (MC) when estimating an integral interpreted as a mathematical expectation. RQMC estimators are unbiased and their variance converges at a faster rate (under certain conditions) than MC estimators, as a function of the sample size. Variants of RQMC also work for the simulation of Markov chains, for function approximation and optimization, for solving partial differential equations, etc. In this introductory survey, we look at how RQMC point sets and sequences are constructed, how we measure their uniformity, why they can work for high-dimensional integrals, and how can they work when simulating Markov chains over a large number of steps.

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Section: Anova Decompositionmentioning

confidence: 99%

Randomized Quasi-Monte Carlo: An Introduction for Practitioners

L’Ecuyer

2018

Springer Proceedings in Mathematics &Amp; Statistics

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“…In this paper we follow [5] and [6] and work with the unnormalized versions of the indices since the estimation of the overall variance is equal for all methods that we will consider. There are several extensions to the indices given in (5).…”

Section: A Quick Overview Of Fanova Decomposition and Sobol Indicesmentioning

confidence: 99%

“…By analogy with the total sensitivity index, we consider the socalled total interaction index (TII), that measures the influence of a pair of variables together with all its interactions. The TII is a particular case of superset importance, a sensitivity index investigated in Hooker [5] and Liu and Owen [6]. If the TII of a pair of variable {X i , X j } is zero, then there is no interaction term containing simultaneously X i and X j , which leads to the elimination of the pair {X i , X j } from the list of possible interactions.…”

Section: Introductionmentioning

confidence: 99%

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Total interaction index: A variance-based sensitivity index for second-order interaction screening

Fruth¹,

Roustant²,

Kuhnt³

2014

Journal of Statistical Planning and Inference

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Sensitivity analysis aims at exploring which of a number of variables have an impact on a certain response. Not only are the individual variables of interest but also whether they interact or not. By analogy with the total sensitivity index, used to detect the most influential variables, a screening of interactions can be done efficiently with the so-called total interaction index (TII), defined as the superset importance of a pair of variables. Our aim is to investigate the TII, with a focus on statistical inference. At the theoretical level, we derive its connection to total and closed sensitivity indices. We present several estimation methods and prove the asymptotical efficiency of the Liu and Owen estimator. We also address the question of estimating the full set of TIIs, with a given budget of function evaluations. We observe that with the pick-and-freeze method the full set of TIIs can be estimated at a linear cost with respect to the problem dimension. The different estimators are then compared empirically. Finally, an application is given aiming at discovering a block-additive structure of a function, where no prior knowledge either about the interaction structure or about the blocks is available.

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“…The notions of effective dimension in the truncation and superposition sense were introduced in [CMO97]. Further, Owen added the notion of "average dimension" which is more practical from the computational point of view [LO06]. Definitions and evaluations of effective dimensions are based on the knowledge of Sobol' sensitivity indices.…”

Section: Introductionmentioning

confidence: 99%

Pricing and Risk Management with High-Dimensional Quasi-Monte Carlo and Global Sensitivity Analysis

Bianchetti¹,

Kucherenko

Scoleri³

2015

Wilmott

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We review and apply Quasi Monte Carlo (QMC) and Global Sensitivity Analysis (GSA) techniques to pricing and risk management (greeks) of representative financial instruments of increasing complexity. We compare QMC vs standard Monte Carlo (MC) results in great detail, using high-dimensional Sobol' low discrepancy sequences, different discretization methods, and specific analyses of convergence, performance, speed up, stability, and error optimisation for finite differences greeks. We find that our QMC outperforms MC in most cases, including the highest-dimensional simulations and greeks calculations, showing faster and more stable convergence to exact or almost exact results. Using GSA, we are able to fully explain our findings in terms of reduced effective dimension of our QMC simulation, allowed in most cases, but not always, by Brownian bridge discretization. We conclude that, beyond pricing, QMC is a very promising technique also for computing risk figures, greeks in particular, as it allows to reduce the computational effort of high-dimensional Monte Carlo simulations typical of modern risk management.

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Estimating Mean Dimensionality of Analysis of Variance Decompositions

Cited by 133 publications

References 1 publication

Randomized Quasi-Monte Carlo: An Introduction for Practitioners

Randomized Quasi-Monte Carlo: An Introduction for Practitioners

Total interaction index: A variance-based sensitivity index for second-order interaction screening

Pricing and Risk Management with High-Dimensional Quasi-Monte Carlo and Global Sensitivity Analysis

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