Uncertainty quantification: a minimum variance unbiased (joint) estimator of the non-normalized Sobol’ indices

Abstract

Often, uncertainty quantification is followed by the computation of sensitivity indices of input factors. Variance-based sensitivity analysis and multivariate sensitivity analysis (MSA) aim to apportion the variability of the model output(s) into input factors and their interactions. Sobol’ indices (first-order and total indices), which quantify the effects of input factor(s), serve as a practical tool to assess interactions among input factors, the order of interactions, and the magnitude of interactions. In this paper, we investigate a novel way of estimating both the first-order and total indices based on U-statistics, including the statistical properties of the new estimator. First, we provide a minimum variance unbiased estimator of the non-normalized Sobol’ indices as well as its optimal rate of convergence and its asymptotic distribution. Second, we derive a joint estimator of Sobol’ indices, its consistency and its asymptotic distribution, and third, we demonstrate the applicability of these results by means of numerical tests. The new estimator allows for improving the estimation of Sobol’ indices for some degrees of the kernel.

Appendices

Appendix A: An useful summary of the U-statistics theory

This section presents the main theorems about U-statistics used in this paper. While we consider only three independent samples, the general version of the theorems can be found in Ferguson (1996), Hoeffding (1948a, b) and Sugiura (1965).

Let $(X_1, \ldots , X_{n_1})$, $(Y_1, \ldots , Y_{n_2})$, and $(Z_1, \ldots , Z_{n_3})$ be three independent samples of size $n_1, n_2, n_3$ from the distribution $D_1$, $D_2$, and $D_3$ respectively. Suppose that our parameter of interest $\theta$ is defined as follows:

$$\begin{aligned} \theta =\mathbb {E}\left[ K\left( X_1, \ldots , X_{p}; Y_1, \ldots , Y_{q}; Z_1, \ldots , Z_{r} \right) \right] , \end{aligned}$$

(6.1)

where $K\left( \cdot \right)$ is a symmetric kernel w.r.t its first, second, and third arguments. It has $p+q+r$ arguments.

The U-statistic corresponding to $\theta$ is given by

$$\begin{aligned} \widehat{\theta } = \frac{1}{\left( {\begin{array}{c}n_1\\ p\end{array}}\right) \left( {\begin{array}{c}n_2\\ q\end{array}}\right) \left( {\begin{array}{c}n_3\\ r\end{array}}\right) } \sum _{\begin{array}{c} 1\le i_1< i_2< \cdots< i_p \le n_1 \\ 1\le j_1< j_2< \cdots<j_q \le n_2 \\ 1\le k_1< k_2< \cdots <k_r \le n_3 \end{array}} K\left( X_{i_1}, \ldots , X_{i_p}; Y_{j_1}, \ldots , Y_{j_q}; Z_{k_1}, \ldots , Z_{k_r} \right) . \end{aligned}$$

(6.2)

The estimator $\widehat{\theta }$ is unbiased, and its variance is given by

$$\begin{aligned} \mathbb {V}\left[ \widehat{\theta }\right] = \sum _{i=1}^{p} \sum _{j=1}^{q} \sum _{k=1}^{r} \frac{\left( {\begin{array}{c}p\\ i\end{array}}\right) \left( {\begin{array}{c}n_1-p\\ p-i\end{array}}\right) \left( {\begin{array}{c}q\\ j\end{array}}\right) \left( {\begin{array}{c}n_2-q\\ q-j\end{array}}\right) \left( {\begin{array}{c}r\\ k\end{array}}\right) \left( {\begin{array}{c}n_3-r\\ r-k\end{array}}\right) }{m \left( {\begin{array}{c}n_1\\ p\end{array}}\right) \left( {\begin{array}{c}n_2\\ q\end{array}}\right) \left( {\begin{array}{c}n_3\\ r\end{array}}\right) } \sigma _{i,j,k}^2, \end{aligned}$$

(6.3)

with $\sigma _{i,j,k}^2= \mathbb {V}\Big [ \mathbb {E}\Big [K\Big (X_{1},\ldots , X_{p}; Y_{1}, \ldots , Y_{q}; Z_{1}, \ldots , Z_{r} \Big ) | X_{1}, \ldots , X_{i}; Y_{1}, \ldots , Y_{j}; Z_{1}, \ldots , Z_{k} \Big ]\Big ]$.

The U-statistic $\widehat{\theta }$ is a function of ordered statistics, as it is symmetric w.r.t each of its three types of arguments. It is known that ordered statistics are sufficient and complete statistics for non-parametric families such as the class of distribution functions having finite fourth moments. By Lehmann–Scheff theorem, it follows that $\widehat{\theta }$ is the unique, uniformly minimum variance unbiased estimator of $\theta$, given $X_1, \ldots , X_{n_1}; Y_1, \ldots , Y_{n_2}; Z_1, \ldots , Z_{n_3}$. We also have the asymptotic normality of the estimator $\widehat{\theta }$.

The same results can be derived for the multivariate U-statistics, and all elements can be found in Ferguson (1996), Lehmann (1999), Hoeffding (1948a, b) and Sugiura (1965).

Appendix B: a generalized estimator of the total sensitivity index

This section recalls mainly Corollary 2 from Lamboni (2016a).

For $p\ge 2$, let $\mathcal {X}=(\mathbf {X}^{(1)}_{u},\ldots , \mathbf {X}^{(p)}_{u}$) be p i.i.d copies of $\mathbf {X}_{u}$ and $\mathcal {Y}=\mathbf {X}_{\sim u}$. We consider the following symmetric kernel.

$$\begin{aligned} K\left( \mathbf {X}_{u}^{(1)}, \ldots , \mathbf {X}_{u}^{(p)}, \mathbf {X}_{\sim u} \right)= & {} \frac{1}{p^2(p-1)}\sum _{k=1}^p \left( \sum _{\begin{array}{c} j=1 \\ j\ne k \end{array}}^p [f(\mathbf {X}_{u}^{(k)}, \mathbf {X}_{\sim u}) - f(\mathbf {X}_{u}^{(j)}, \mathbf {X}_{\sim u})] \right) ^2.\nonumber \\ \end{aligned}$$

(6.4)

It is known that the expectation of $K\left( \mathbf {X}_{u}^{(1)}, \ldots , \mathbf {X}_{u}^{(p)}, \mathbf {X}_{\sim u} \right)$ is the non-normalized total index, that is,

$$\begin{aligned} D_u^{\textit{tot}} =\mathbb {E}\left[ K\left( \mathbf {X}_{u}^{(1)}, \ldots , \mathbf {X}_{u}^{(p)}, \mathbf {X}_{\sim u} \right) \right] . \end{aligned}$$

(6.5)

Given two independent samples $\mathcal {X}_i= (\mathbf {X}^{(1)}_{i,u},\ldots , \mathbf {X}^{(p)}_{i,u}$) and $\mathcal {Y}_i =(\mathbf {X}_{i,\sim u})$, $i=1, 2, \ldots , m$, from $\mathcal {X}$ and $\mathcal {Y}$ respectively, the U-statistic corresponding to the kernel in (6.4 ) is given by

$$\begin{aligned} \widehat{D}_{u}^{\textit{tot}} = \frac{1}{m p^2(p-1)} \sum _{i=1}^m \sum _{k=1}^p \left( \sum _{\begin{array}{c} j=1\\ j\ne k \end{array}}^p [f(\mathbf {X}_{i,u}^{(k)}, \mathbf {X}_{i,\sim u}) - f(\mathbf {X}_{i,u}^{(j)}, \mathbf {X}_{i,\sim u})] \right) ^2. \end{aligned}$$

(6.6)

The properties of $\widehat{D_u^{\textit{tot}}}$ are given as follows:

(i):

a MVUE of $D_u^{\textit{tot}}$ is $\widehat{D_u^{\textit{tot}}}$ defined in (6.6);

(ii):

the variance of $\widehat{D}_{u}^{\textit{tot}}$ is given by

$$\begin{aligned} \mathbb {V}(\widehat{D}_{u}^{\textit{tot}}) = \frac{\sigma _{p,1}^2 }{m}, \end{aligned}$$

(6.7)

with $\sigma _{p,1}^2$ the variance of the kernel.

Moreover, we have

$$\begin{aligned} m\mathbb {E}\left( \widehat{D}_{u}^{\textit{tot}} - D_{u}^{\textit{tot}} \right) ^2 = \sigma _{p,1}^2; \end{aligned}$$

(6.8)

(iii):

if $m \rightarrow +\infty$, we have

$$\begin{aligned} \sqrt{m}\left( \widehat{D}_{u}^{\textit{tot}} - D_{u}^{\textit{tot}} \right) \xrightarrow {\mathcal {D}} \mathcal {N}\left( 0, \sigma _{p,1}^2 \right) . \end{aligned}$$

(6.9)