A randomized orthogonal array-based procedure for the estimation of first- and second-order Sobol' indices

In variance-based sensitivity analysis, the method of Sobol' [Sensitivity analysis for nonlinear mathematical models. Math Model Comput Exp. 1993;1:407–414] allows one to compute Sobol' indices (SI) using the Monte Carlo integration. One of the main drawbacks of this approach is that estimating SI requires a number of simulations which is dependent on the dimension of the model of interest. For example, estimating all the first- or second-order SI of a <i>d</i>-dimensional function basically requires or independent input vectors, respectively. Some interesting combinatorial results have been introduced to weaken this defect, in particular by Saltelli [Making best use of model evaluations to compute sensitivity indices. Comput Phys Commun. 2002;145:280–297] and more recently by Owen [Variance components and generalized Sobol' indices. SIAM/ASA J Uncertain Quantification. 2013;1:19–41], but the quantities they estimate still depend linearly on the dimension <i>d</i>. In this paper, we introduce a new approach to estimate all the first- and second-order SI by using only two input vectors. We establish theoretical properties of such a method for the estimation of first-order SI and discuss the generalization to higher order indices. In particular, we prove on numerical examples that this procedure is tractable and competitive for the estimation of all first- and second-order SI. As an illustration, we propose to apply this new approach to a marine ecosystem model of the Ligurian sea (northwestern Mediterranean) in order to study the relative importance of its several parameters. The calibration process of this kind of chemical simulators is well known to be quite intricate, and a rigorous and robust – that is, valid without strong regularity assumptions – sensitivity analysis, as the method of Sobol' provides, could be of great help. This article has supplementary material online.

在方差基敏感性分析（variance-based sensitivity analysis）中，索博尔'方法（Sobol'）[《Sensitivity analysis for nonlinear mathematical models》, Math Model Comput Exp, 1993;1:407–414]可借助蒙特卡洛积分（Monte Carlo integration）计算索博尔'指数（Sobol' indices, SI）。该方法的主要缺陷之一在于，估算索博尔'指数所需的模拟次数与所关注模型的维度息息相关。例如，估算一个d维函数的全部一阶或二阶索博尔'指数，原则上分别需要若干独立输入向量。已有学者提出若干有趣的组合学成果以缓解这一缺陷，其中尤以Saltelli的研究[《Making best use of model evaluations to compute sensitivity indices》, Comput Phys Commun, 2002;145:280–297]以及近年Owen的成果[《Variance components and generalized Sobol' indices》, SIAM/ASA J Uncertain Quantification, 2013;1:19–41]最为典型，但他们所估算的指标仍与维度d呈线性相关。本文提出了一种仅需两组输入向量即可估算全部一阶与二阶索博尔'指数的新方法。我们针对一阶索博尔'指数的估算，推导了该方法的理论性质，并探讨了其向更高阶指数的推广路径。具体而言，我们通过数值算例证明，该方法在估算全部一阶与二阶索博尔'指数时，不仅易于实现，且具备良好的竞争力。作为示例，我们将该新方法应用于利古里亚海（地中海西北部）的海洋生态系统模型，以探究其多个参数的相对重要性。众所周知，这类化学模拟器的校准过程极为复杂，而索博尔方法所提供的严谨且稳健——即无需强正则性假设即可成立——的敏感性分析，将能提供极大助力。本文配有在线补充材料。