Customized sequential designs for random simulation experiments: Kriging metamodeling and bootstrapping

This paper proposes a novel method to select an experimental design for interpolation in random simulation, especially discrete event simulation. (Though the paper focuses on Kriging, this design approach may also apply to other types of metamodels such as non-linear regression models and splines.) Assuming that simulation requires much computer time, it is important to select a design with a small number of observations (or simulation runs). The proposed method is therefore sequential. Its novelty is that it accounts for the specific input/output behavior (or response function) of the particular simulation at hand; i.e., the method is customized or application-driven. A tool for this customization is bootstrapping, which enables the estimation of the variances of predictions for inputs not yet simulated. The method is tested through two classic simulation models, namely the expected steady-state waiting time of the M/M/1 queuing model, and the mean costs of a terminating (s, S) inventory simulation. For these two simulation models the novel design indeed gives better results than a popular alternative design, namely Latin Hypercube Sampling (LHS) with a prefixed sample.     

Robustness of Kriging when interpolating in random simulation with heterogeneous variances: Some experiments

This paper investigates the use of Kriging in random simulation when the simulation output variances are not constant. Kriging gives a response surface or metamodel that can be used for interpolation. Because Ordinary Kriging assumes constant variances, this paper also applies Detrended Kriging to estimate a non-constant signal function, and then standardizes the residual noise through the heterogeneous variances estimated from replicated simulation runs. Numerical examples, however, suggest that Ordinary Kriging is a robust interpolation method.     

On random interpolation polynomials

Translated from Matematicheskie Zametki, Vol. 52, No. 1, pp. 148–150, July, 1992.     

Kriging metamodeling in simulation: A review

This article reviews Kriging (also called spatial correlation modeling). It presents the basic Kriging assumptions and formulas—contrasting Kriging and classic linear regression metamodels. Furthermore, it extends Kriging to random simulation, and discusses bootstrapping to estimate the variance of the Kriging predictor. Besides classic one-shot statistical designs such as Latin Hypercube Sampling, it reviews sequentialized and customized designs for sensitivity analysis and optimization. It ends with topics for future research.     

Application-driven sequential designs for simulation experiments: Kriging metamodelling

This paper proposes a novel method to select an experimental design for interpolation in simulation. Although the paper focuses on Kriging in deterministic simulation, the method also applies to other types of metamodels (besides Kriging), and to stochastic simulation. The paper focuses on simulations that require much computer time, so it is important to select a design with a small number of observations. The proposed method is therefore sequential. The novelty of the method is that it accounts for the specific input/output function of the particular simulation model at hand; that is, the method is application-driven or customized. This customization is achieved through cross-validation and jackknifing. The new method is tested through two academic applications, which demonstrate that the method indeed gives better results than either sequential designs based on an approximate Kriging prediction variance formula or designs with prefixed sample sizes.     

Multivariate versus univariate Kriging metamodels for multi-response simulation models

To analyze the input/output behavior of simulation models with multiple responses, we may apply either univariate or multivariate Kriging (Gaussian process) metamodels. In multivariate Kriging we face a major problem: the covariance matrix of all responses should remain positive-definite; we therefore use the recently proposed “nonseparable dependence” model. To evaluate the performance of univariate and multivariate Kriging, we perform several Monte Carlo experiments that simulate Gaussian processes. These Monte Carlo results suggest that the simpler univariate Kriging gives smaller mean square error.     

Generating correlated random variables for a simulation model

In most simulation textbooks, a great deal of attention is given to generating independent random variables. The topic of generating correlated random variables is either omitted or given only a cursory analysis. The purpose of this note is to illustrate how correlated random variables were generated in a simulation model for analysing a firm's ability to meet the demand for its product.     

Some new perspectives in best approximation and interpolation of random data

A new notion of universally optimal experimental design is introduced, relevant from the perspective of adaptive nonparametric estimation. It is demonstrated that both discrete and continuous Chebyshev designs are universally optimal in the problem of fitting properly weighted algebraic polynomials to random data. The result is a direct consequence of the well-known relation between Chebyshev’s polynomials and the trigonometric functions. Optimal interpolating designs in rational regression proved particularly elusive in the past. The question can be effectively handled using its connection to elliptic interpolation, in which the ordinary circular sinus, appearing in the classical trigonometric interpolation, is replaced by the Abel-Jacobi elliptic sinus sn(x, k) of a modulus k. First, it is demonstrated that — in a natural setting of equidistant design — the elliptic interpolant is never optimal in the so-called normal case k ∈ (?1, 1), except for the trigonometric case k = 0. However, the equidistant elliptic interpolation is always optimal in the imaginary case k ∈ i?. Through a relation between elliptic and rational functions, the result leads to a long sought optimal design, for properly weighted rational interpolants. Both the poles and nodes of the interpolants can be conveniently expressed in terms of classical Jacobi’s theta functions.     

Special algorithms for the simulation of homogeneous random fields

In this paper, two new algorithms for the simulation of homogeneous random fields are proposed. Both algorithms are based on the well-known algorithm of rows and columns for the simulation of Gaussian fields with special correlation functions. The algorithms make it possible to efficiently simulate homogeneous random fields with nonconvex correlation functions.     

Imprecise random variables,random sets,and Monte Carlo simulation

The paper addresses the evaluation of upper and lower probabilities induced by functions of an imprecise random variable. Given a function g and a family $X_{\lambda }$ of random variables, where the parameter λ ranges in an index set Λ, one may ask for the upper/lower probability that $g(X_{\lambda })$ belongs to some Borel set B. Two interpretations are investigated. In the first case, the upper probability is computed as the supremum of the probabilities that $g(X_{\lambda })$ lies in B. In the second case, one considers the random set generated by all $g(X_{\lambda })$, $\lambda \in \mathrm{\Lambda }$, e.g. by transforming $X_{\lambda }$ to standard normal as a common probability space, and computes the corresponding upper probability. The two results are different, in general. We analyze this situation and highlight the implications for Monte Carlo simulation. Attention is given to efficient simulation procedures and an engineering application is presented.     

On simulation of tempered stable random variates

Various simulation methods for tempered stable random variates with stability index greater than one are investigated with a view towards practical implementation, in particular cases of very small scale parameter, which correspond to increments of a tempered stable Lévy process with a very short stepsize. Methods under consideration are based on acceptance-rejection sampling, a Gaussian approximation of a small jump component, and infinite shot noise series representations. Numerical results are presented to discuss advantages, limitations and trade-off issues between approximation error and required computing effort. With a given computing budget, an approximative acceptance-rejection sampling technique Baeumer and Meerschaert (2009) [11] is both most efficient and handiest in the case of very small scale parameter and moreover, any desired level of accuracy may be attained with a small amount of additional computing effort.     

The proper interpolation space for multivariate Birkhoff interpolation

Multivariate Birkhoff interpolation problem has many important applications, such as in finite element method. In this paper two algorithms are given to compute the basis of the minimal interpolation space and the lower interpolation space respectively for an arbitrary given node set and the corresponding interpolation conditions on each node. We can get the monomial basis, Newton-type basis as well as Lagrange-type basis. The interpolation polynomial can be derived from the basis directly.     

Efficient simulation and conditional functional limit theorems for ruinous heavy-tailed random walks

The contribution of this paper is to introduce change of measure based techniques for the rare-event analysis of heavy-tailed random walks. Our changes of measures are parameterized by a family of distributions admitting a mixture form. We exploit our methodology to achieve two types of results. First, we construct Monte Carlo estimators that are strongly efficient (i.e. have bounded relative mean squared error as the event of interest becomes rare). These estimators are used to estimate both rare-event probabilities of interest and associated conditional expectations. We emphasize that our techniques allow us to control the expected termination time of the Monte Carlo algorithm even if the conditional expected stopping time (under the original distribution) given the event of interest is infinity–a situation that sometimes occurs in heavy-tailed settings. Second, the mixture family serves as a good Markovian approximation (in total variation) of the conditional distribution of the whole process given the rare event of interest. The convenient form of the mixture family allows us to obtain functional conditional central limit theorems that extend classical results in the literature.     

Truncated stable random variables: characterization and simulation

Kanter (Ann Probab 3(4):697–707, 1975) and Chambers et al. (J Am Stat Assoc 71(354):340–344, 1976) developed a method for characterizing and simulating stable random variables, X, using nonlinear transformations involving two independent uniform random variables. Their method is scrutinized to provide a characterization and then develop a method for simulating random variables with distribution P(X ≤ x| X  > a), called here truncated stable random variables. Our characterization is rigorous when the characteristic exponent α ≠ 1. We extend our method to the case that α → 1.