Analyzing Simulation Experiments with Common Random Numbers

An approach to handle stationary colored noise in least-squares problems in the presence of data gaps is presented. The presence of colored noise implies that a non-diagonal covariance matrix has to be inverted. The problem is reduced to the solution of systems of linear equations with the covariance matrix as equation matrix. Each system is solved using a pre-conditioned conjugate gradient method. An ARMA representation of the colored noise is used both to design an efficient pre-conditioner and to compute the product of the covariance matrix with a vector. This results in an algorithm that has a numerical complexity of O(N) operations, where N is the number of observations. This makes the algorithm particularly suited for large least-squares problems. It is shown that the approach works perfectly in the sense that no edge effects show up and no observations have to be thrown away.

Parametric sensitivity of biochemical networks is an indispensable tool for studying system robustness properties, estimating network parameters, and identifying targets for drug therapy. For discrete stochastic representations of biochemical networks where Monte Carlo methods are commonly used, sensitivity analysis can be particularly challenging, as accurate finite difference computations of sensitivity require a large number of simulations for both nominal and perturbed values of the parameters. In this paper we introduce the common random number (CRN) method in conjunction with Gillespie's stochastic simulation algorithm, which exploits positive correlations obtained by using CRNs for nominal and perturbed parameters. We also propose a new method called the common reaction path (CRP) method, which uses CRNs together with the random time change representation of discrete state Markov processes due to Kurtz to estimate the sensitivity via a finite difference approximation applied to coupled reaction paths that emerge naturally in this representation. While both methods reduce the variance of the estimator significantly compared to independent random number finite difference implementations, numerical evidence suggests that the CRP method achieves a greater variance reduction. We also provide some theoretical basis for the superior performance of CRP. The improved accuracy of these methods allows for much more efficient sensitivity estimation. In two example systems reported in this work, speedup factors greater than 300 and 10,000 are demonstrated.

ABSTRACTThe central composite design (CCD) is often used to estimate the second-order linear model. This paper uses a correlation induction strategy of common random numbers (CRN) in simulation experiment and utilizes the induced correlations to obtain better estimates for the second-order linear model. This strategy assigns the CRN to all design points in the CCD. An appropriate selection of the axial points in CCD makes the weighted least squares (WLS) estimator be equivalent to ordinary least squares (OLS) estimator in estimating the linear model parameters of CCD. We analytically investigate the efficiency of this strategy in estimation of model parameters. Under certain conditions, this correlation induction strategy yields better results than independent random number strategy in estimating model parameters except intercept. The simulation experiment on a selected model supports such results. We expect a suggested random number assignment is useful in application of CCD in simulation experiments.

We present Q-SASS, a quasi-Newton method for unconstrained stochastic optimization that does not rely on common random numbers. Most existing quasi-Newton approaches leverage common random numbers to construct second-order updates. However, motivated by challenges in variational quantum algorithms—where such coordination is not possible—we consider the setting in which function values and gradients are accessible only through noisy probabilistic zeroth- and first-order oracles, and no common random numbers can be exploited. We derive high-probability tail bounds on the iteration complexity of our algorithm for nonconvex, convex, and strongly convex (more generally, those satisfying the PL condition) objective functions. Finally, we demonstrate the empirical benefits of our quasi-Newton updating scheme on both synthetic and quantum chemistry problems.

We consider noisy optimization and some traditional variance reduction techniques aimed at improving the convergence rate, namely (i) common random numbers (CRN), which is relevant for population-based noisy optimization and (ii) stratified sampling, which is relevant for most noisy optimization problems. We present artificial models of noise for which common random numbers are very efficient, and artificial models of noise for which common random numbers are detrimental. We then experiment on a desperately expensive unit commitment problem. As expected, stratified sampling is never detrimental. Nonetheless, in practice, common random numbers nonetheless provided, by far, most of the improvement.

We consider the problem of stochastic simulation optimization with common random numbers over a numerical search domain. We propose the Knowledge Gradient for Common Random Numbers (KG-CRN) sequential sampling algorithm, a simple elegant modification to the Knowledge Gradient that incorporates the use of correlated noise in simulation outputs with Gaussian Process meta-models. We compare this method against the standard Knowledge Gradient and a more recently proposed variation that allows for pairwise sampling. Our method significantly outperforms both baselines under identical laboratory conditions while greatly reducing computational cost compared to pairwise sampling.

Common random numbers in multivariate simulations

The article describes some recently-proposed procedures that identify the best simulated system when common random numbers are used. The procedures are based on a Bayesian average-case analysis, rather than a worst-case indifference zone formulation. The procedures allow decision makers to focus on reducing either the expected opportunity cost loss associated with potentially selecting an inferior system, or the probability of incorrect selection. Numerical experiments indicate that the new procedures outperform two existing procedures with respect to several criteria for a well-known selection problem.

Traditionally, exactness, numerical stability and speed are the three main criteria for evaluating algorithms for random variate generation. However, it is sometimes required that the algorithms provide correlation between generated variates for the purpose of inducing dependence among the output of simulation runs. The inverse transformation, which produces optimal correlation induction, often performs poorly in terms of the first three criteria. Algorithms based on composition, rejection, and special properties which often excel in terms of the first three criteria, tend to scramble the use of random numbers, causing many attempts at common random numbers, antithetic variates and external control variates to fail. The concept of obtaining correlation via algorithms other than the inverse transformation is examined here. To demonstrate feasibility, previously developed algorithms for Poisson and binomial random variate generation are modified to obtain both positive and negative correlation between runs....

Dual-energy computed tomography (DECT) is a recent advancement in CT technology, which can potentially reduce artifacts and provide accurate quantitative information for diagnosis. Recently, statistical iterative reconstruction (SIR) methods were introduced to DECT for radiation dose reduction. The statistical noise modeling of measurement data plays an important role in SIR and impacts on the image quality. Contrary to the conventional CT projection data, of which noise is independent from ray to ray, in spectral CT the basis material sinogram data has strong correlations. In order to analyze the image quality improvement by applying correlated noise model, we compare the effects of two different noise models (i.e., correlated noise model and independent model by ignoring correlations) by analyzing the bias and variance trade-off. The results indicate that in the same bias level, the correlated noise modeling results in up to 20.02% noise reduction compared to the independent noise model. In addition, their impacts to different numerical are also evaluated. The results show that using the non-diagonal covariance matrix in SIR is challenging, where some numerical algorithms such as a direct application of separable paraboloidal surrogates (SPS) cannot converge to the correct results.

We consider optimization via simulation over a finite set of alternatives. We employ a Bayesian value-ofinformation approach in which we allow both correlated prior beliefs on the sampling means and correlated sampling. Correlation in the prior belief allow us to learn about an alternative’s value from samples of similar alternatives. Correlation in sampling, achieved through common random numbers, allows us to reduce the variance in comparing one alternative to another. We allow for a more general combination of both types of correlation than has been offered previously in the Bayesian ranking and selection literature. We do so by giving an exact expression for the value of information for sampling the difference between a pair of alternatives, and derive new knowledge-gradient methods based on this valuation. 1

This paper addresses discrete optimization via simulation. We show that allowing for both a correlated prior distribution on the means (e.g., with discrete Kriging models) and sampling correlation (e.g., with common random numbers, or CRN) can significantly improve the ability to quickly identify the best alternative. These two correlations are brought together for the first time in a highly sequential knowledge-gradient sampling algorithm, which chooses points to sample using a Bayesian value of information (VOI) criterion. We provide almost sure convergence guarantees as the number of samples grows without bound when parameters are known and provide approximations that allow practical implementation including a novel use of the VOI’s gradient rather than the response surface’s gradient. We demonstrate that CRN leads to improved optimization performance for VOI-based algorithms in sequential sampling environments with a combinatorial number of alternatives and costly samples.

Bayesian estimation of dynamic asset pricing models with informative observations

A widely applied method of experimental design in stochastic simulation is to use the same pseudo random-number stream for each of the systems or alternatives to be compared. If there are only two alternatives, there is a well-known simple method of statistical analysis. This paper presents Graybill's3 extension of that analysis technique to the case of several systems or alternatives and points out its applicability in the simulation environment. The technique is illustrated using a stochastic simula tion model of a small inventory system. An empirical comparison with other methods of design and analysis is included.

Gives an overview of the main techniques for improving the statistical efficiency of simulation estimators. Efficiency improvement is typically (but not always) achieved through variance reduction. We discuss methods such as common random numbers, antithetic variates, control variates, importance sampling, conditional Monte Carlo, stratified sampling, and some others, as well as the combination of certain of those methods. We also survey the recent literature on this topic.