计算机模型的惩罚投影核校正
Penalized Projected Kernel Calibration for Computer Models
摘要：
投影核校正是一种新提出的频率校正方法，具有渐近正态性和半参数性。它的损失函数通常被称为PK损失函数。本文证明了PK损失函数的一致收敛性，并证明了：（1）当样本量较大时，真实过程和计算机模型之间的$L_2$损失的任何局部极小点和局部极大点都是PK损失函数的局部极小点;（2）PK损失函数的所有局部极小点收敛于同一个值。这些理论结果意味着，它是非常困难的投影核校准，以确定全球最小的$L_2$损失，即校准参数的最佳值。针对这一问题，本文提出了一种基于频率论的投影核校正方法，并对其进行了详细的分析。我们证明了所提出的方法是有效的投影核校准方法。通过一组广泛的数值模拟，和一个真实世界的案例研究，我们表明，所提出的校准方法可以准确地估计校准参数。我们还表明，它的性能优于其他校准方法，无论样本量。
英文摘要：
Projected kernel calibration is a newly proposed frequentist calibration method, which is asymptotic normal and semi-parametric. Its loss function is usually referred to as the PK loss function. In this work, we prove the uniform convergence of PK loss function and show that (1) when the sample size is large, any local minimum point and local maximum point of the $L_2$ loss between the true process and the computer model is a local minimum point of the PK loss function; (2) all the local minima of the PK loss function converge to the same value. These theoretical results imply that it is extremely hard for the projected kernel calibration to identify the global minimum of the $L_2$ loss, i.e. the optimal value of the calibration parameters. To solve this problem, a frequentist method which we term penalized projected kernel calibration method is suggested and analyzed in detail. We prove that the proposed method is as efficient as the projected kernel calibration method. Through an extensive set of numerical simulations, and a real-world case study, we show that the proposed calibration method can accurately estimate the calibration parameters. We also show that its performance compares favorably to other calibration methods regardless of the sample size.
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