Kriging for Interpolation in Random Simulation

Whenever simulation requires much computer time, interpolation is needed. There are several interpolation techniques in use (for example, linear regression), but this paper focuses on Kriging.This technique was originally developed in geostatistics by D.G.Krige, and has recently been widely applied in deterministic simulation.This paper, however, focuses on random or stochastic simulation.Essentially, Kriging gives more weight to 'neighbouring' observations.There are several types of Kriging; this paper discusses - besides Ordinary Kriging - a novel type, which 'detrends' data through the use of linear regression.Results are presented for two examples of input/output behaviour of the underlying random simulation model: A perfectly specified detrending function gives the best predictions, but Ordinary Kriging gives quite acceptable results; traditional linear regression gives the worst predictions.simulation;statistics;stochastic processes;methodology;linear regression

Application-driven Sequential Designs for Simulation Experiments: Kriging Metamodeling

This paper proposes a novel method to select an experimental design for interpolation in simulation.Though 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; i.e., 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 a design with a prefixed sample size.experimental design;simulation;interpolation;sampling;sensitivity analysis;metamodels

Customized Sequential Designs for Random Simulation Experiments: Kriging Metamodelling and Bootstrapping

This paper proposes a novel method to select an experimental design for interpolation in random simulation.(Though the paper focuses on Kriging, this method may also apply to other types of metamodels such as linear regression models.)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 new method is tested through the classic M/M/1 queueing simulation.For this simulation the novel design indeed gives better results than a Latin Hypercube Sampling (LHS) with a prefixed sample of the same size.simulation;statistical methods;bootstrap

Kriging metamodeling for simulation

Many scientific disciplines use mathematical models to describe complicated real systems. Often, analytical methods are inadequate, so simulation is applied. This thesis focuses on computer intensive simulation experiments in Operations Research/Management Science. For such experiments it is necessary to apply interpolation. In this thesis, Kriging interpolation for random simulation is proposed and a novel type of Kriging - called Detrended Kriging - is developed. Kriging turns out to give better predictions in random simulation than classic low-order polynomial regression. Kriging is not sensitive to variance heterogeneity: i.e. Kriging is a robust method. Moreover, the thesis develops a novel method to select experimental designs for expensive simulation. This method is sequential, and accounts for the specific input/output function implied by the underlying simulation model. For deterministic simulation the designs are constructed through cross-validation and jackknifing, whereas for random simulation the customization is achieved through bootstrapping. The novel method simulates relatively more input combinations in the interesting parts of the input/output function, and gives better predictions than traditional Latin Hypercube Sample designs with prefixed sample sizes.

Monotonicity-Preserving Bootstrapped Kriging Metamodels for Expensive Simulations

Kriging (Gaussian process, spatial correlation) metamodels approximate the Input/Output (I/O) functions implied by the underlying simulation models; such metamodels serve sensitivity analysis and optimization, especially for computationally expensive simulations. In practice, simulation analysts often know that the I/O function is monotonic. To obtain a Kriging metamodel that preserves this known shape, this article uses bootstrapping (or resampling). Parametric bootstrapping assuming normality may be used in deterministic simulation, but this article focuses on stochastic simulation (including discrete-event simulation) using distribution-free bootstrapping. In stochastic simulation, the analysts should simulate each input combination several times to obtain a more reliable average output per input combination. Nevertheless, this average still shows sampling variation, so the Kriging metamodel does not need to interpolate the average outputs. Bootstrapping provides a simple method for computing a noninterpolating Kriging model. This method may use standard Kriging software, such as the free Matlab toolbox called DACE. The method is illustrated through the M/M/1 simulation model with as outputs either the estimated mean or the estimated 90% quantile; both outputs are monotonic functions of the traffic rate, and have nonnormal distributions. The empirical results demonstrate that monotonicity-preserving bootstrapped Kriging may give higher probability of covering the true simulation output, without lengthening the confidence interval.Queues

Constrained Optimization in Simulation: A Novel Approach

This paper presents a novel heuristic for constrained optimization of random computer simulation models, in which one of the simulation outputs is selected as the objective to be minimized while the other outputs need to satisfy prespeci¯ed target values. Besides the simulation outputs, the simulation inputs must meet prespeci¯ed constraints including the constraint that the inputs be integer. The proposed heuristic combines (i) experimental design to specify the simulation input combinations, (ii) Kriging (also called spatial correlation mod- eling) to analyze the global simulation input/output data that result from this experimental design, and (iii) integer nonlinear programming to estimate the optimal solution from the Krig- ing metamodels. The heuristic is applied to an (s, S) inventory system and a realistic call-center simulation model, and compared with the popular commercial heuristic OptQuest embedded in the ARENA versions 11 and 12. These two applications show that the novel heuristic outper- forms OptQuest in terms of search speed (it moves faster towards high-quality solutions) and consistency of the solution quality.

Kriging for Interpolation in Random Simulation

Whenever simulation requires much computer time, interpolation is needed. There are several interpolation techniques in use (for example, linear regression), but this paper focuses on Kriging.This technique was originally developed in geostatistics by D.G.Krige, and has recently been widely applied in deterministic simulation.This paper, however, focuses on random or stochastic simulation.Essentially, Kriging gives more weight to 'neighbouring' observations.There are several types of Kriging; this paper discusses - besides Ordinary Kriging - a novel type, which 'detrends' data through the use of linear regression.Results are presented for two examples of input/output behaviour of the underlying random simulation model: A perfectly specified detrending function gives the best predictions, but Ordinary Kriging gives quite acceptable results; traditional linear regression gives the worst predictions.

Application-driven Sequential Designs for Simulation Experiments:Kriging Metamodeling

This paper proposes a novel method to select an experimental design for interpolation in simulation.Though 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; i.e., 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 a design with a prefixed sample size