Global Sensitivity Analysis (GSA)¶

Global sensitivity analysis (GSA) aims at quantifying the sensitivity of the model response or quantity of interest (QoI) with respect to the variation of the uncertain parameters and inputs. In other words, the influence of each of the parameters in the propagated uncertainty in the QoI is measured. In contrast to the local sensitivity analysis, in GSA all the parameters are allowed to vary simultaneously over their admissible space, [Smith13]. In UQit, the Sobol sensitivity indices [Sobol01] are computed to measure GSA.

Sobol Sensitivity Indices¶

Theory¶

The following short description has been taken from [Rezaeiravesh20]. For more in depth review, the reader is referred to [Sobol01], Chapter 15 in [Smith13], and [Ghanem17].

By analysis of variance (ANOVA) or Sobol decomposition [Sobol01], a model function is decomposed as,

\[f(\mathbf{q}) = f_0 +\sum_{i=1}^p f_i(q_i) + \sum_{1\leq i<j\leq p} f_{ij}(q_i,q_j)+\cdots\,, \label{eq:anova_f}\tag{1}\]

where, \(f_0\) is the mean of \(f(\mathbf{q})\), \(f_i(q_i)\) specify the contribution of each parameter, \(f_{ij}(q_i,q_j)\) denote effects of interaction between each pair of parameters, and so on for other interactions. These contributors are defined as,

\[\begin{split}\begin{eqnarray*} f_0 &=& \mathbb{E}_\mathbf{q}[f(\mathbf{q})] \,, \\ f_i(q_i) &=&\mathbb{E}_\mathbf{q}[f(\mathbf{q})|q_i] - f_0 \,, \\ f_{ij}(q_{i},q_j) &=& \mathbb{E}_\mathbf{q}[f(\mathbf{q})|q_i,q_j] -f_i(q_i) -f_j(q_j) - f_0 \,. \end{eqnarray*}\end{split}\]

Here, \(\mathbb{E}_\mathbf{q}[f(\mathbf{q})|q_i]\), for instance, denotes the expected value of \(f(\mathbf{q})\) conditioned on fixed values of \(q_i\). Similar to Eq. eqref{eq:anova_f}, the total variance of \(f(\mathbf{q})\), denoted by \(D\), is decomposed as,

\[\begin{equation} \mathbb{V}_\mathbf{q}[f(\mathbf{q})] = D=\sum_{i=1}^p D_i + \sum_{1\leq i<j\leq p} D_{ij} + \cdots \,, \end{equation}\]

where, \(D_i=\mathbb{V}_\mathbf{q}[f_i(q_i)]\), \(D_{ij}=\mathbb{V}_\mathbf{q}[f_{ij}(q_i,q_j)]\), and so on. The main Sobol indices are eventually defined as the contribution of each of \(D_i\), \(D_{ij}\), … in the total variance \(D\):

\[\begin{equation} S_i=D_i/D\,,\quad S_{ij}=D_{ij}/D \,,\, \ldots \,, \quad i,j=1,2,\cdots,p \label{eq:sobol} \tag{2} \end{equation}\]

Example¶

Given samples q with associated pdf and response values f, the Sobol indices in UQit are computed by,

sobol_=sobol(q,f,pdf)
Si=sobol_.Si     #1st-order main indices
STi=sobol_.STi   #1st-order total indices
Sij=sobol_.Sij   #2nd-order interactions
SijName=sobol_.SijName  #Names of Sij

Implementation¶

Parameters are assumed to be independent from each other
Parameters can have any arbitrary distribution. The discrete PDF should be imported to sobol().
The samples generated for each of the parameters must be UNIFORMLY-spaced.
Up to second-order interactions are currently implemented.

class sobol.sobol(q, f, pdf)¶

Computes Sobol sensitivity indices for a p-D parameter (p>1).

Assumptions:

Parameters are independent from each other
Up to second-order interactions are considered.

Args:

q: A list of length p: q=[q1,q2,…,qp] where qi: 1D numpy array of size ni containing uniformly-spaced parameter samples
f: 1d numpy array of size (n1*n2*…*np): Response values at samples q. The tensor product with ordering=’F’ (Fortran-like) is considered.
pdf: List of length p of 1D numpy arrays: The i-th array in the list contains the values of the PDF of q_i, where i=1,2,…,p

Methods:

compute():: Computes the Sobol indices

Attributes:

Si: A 1D numpy array of size p: First-order main Sobol indices
Sij: A 1D numpy array: Second-order Sobol indices
‘SijName’: A list: Containing the name of the Sobol indices represented by Sij
STi: A 1D numpy array of size p: Total Sobol indices

comp()¶: Computes Sobol indices based on HDMR

doubleInteg(g, x1, x2)¶

Numerical double integration \(\int_{x2} \int_{x1} g(x1,x2) dx1 dx2\)

Args:

g: numpy array of shape (n1,n2): Values of integrand over the grid of x1-x2
x1,`x2`: 1D numpy arrays of sizes n1 and n2: Integrated variables

Returns:

T: scalar: Integral value

dualInteractTerm(fij_, fi_, fj_)¶: Computes 2nd-order interaction terms in the HDMR decomposition based on \(f_{ij}(q_i,q_j)=\int f(q)dq_{\sim{ij}}-f_i(q_i)-f_j(q_j)-f_0\)

hdmr()¶: HDMR decomposition

hdmr_0_1()¶: Zeroth- and first-order HDMR decomposition

hdmr_2()¶: 2nd-order HDMR decomposition

partVariance()¶: Partial variances and Sobol indices

Notebook¶

Try this GSA Notebook to see how to use UQit to compute Sobol indices. The provided examples can also be seen as a way to validate the implementation of the methods in UQit.