We consider performing simulation experiments in the presence of covariates. Here, covariates refer to some input information other than system designs to the simulation model that can also affect the system performance. To make decisions, decision makers need to know the covariate values of the problem. Traditionally in simulation-based decision making, simulation samples are collected after the covariate values are known; in contrast, as a new framework, simulation with covariates starts the simulation before the covariate values are revealed, and collects samples on covariate values that might appear later. Then, when the covariate values are revealed, the collected simulation samples are directly used to predict the desired results. This framework significantly reduces the decision time compared to the traditional way of simulation. In this paper, we follow this framework and suppose there are a finite number of system designs. We adopt the metamodel of stochastic kriging (SK) and use it to predict the system performance of each design and the best design. The goal is to study how fast the prediction errors diminish with the number of covariate points sampled. This is a fundamental problem in simulation with covariates and helps quantify the relationship between the offline simulation efforts and the online prediction accuracy. Particularly, we adopt measures of the maximal integrated mean squared error (IMSE) and integrated probability of false selection (IPFS) for assessing errors of the system performance and the best design predictions. Then, we establish convergence rates for the two measures under mild conditions. Last, these convergence behaviors are illustrated numerically using test examples.
翻译:我们考虑在共变值存在的情况下进行模拟实验。 这里, 共变数是指系统设计以外的某些输入信息, 也可以影响系统性能的模拟模型。 为了做出决策, 决策者需要了解问题的共变值。 传统上, 在基于模拟的决策中, 模拟样本是在共变值为已知之后采集的; 相反, 作为新框架, 与共变数的模拟开始模拟, 在共变值被披露之前, 并收集比较变数值的样本。 然后, 当共变数值被披露时, 收集的模拟样本直接用于预测预期结果。 这个框架将大大缩短决定时间, 与传统的模拟方式相比。 在本文中, 我们遵循这个框架, 假设系统设计数量有限。 我们采用Stochacist kriging (SK) 的元模型, 并用它来预测每件设计的系统性能和最佳设计。 目标是研究预测误差与差异点相比, 预测误差减少的速度有多快, 采集的模拟样本样本直接用来预测结果。 这是在模拟中, 精确度中, 和精确度的精确度中, 我们用两个精确度的计算方法来量化的精确度 。