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.