Sigma Normalization

The aim of this section is to show how important it can be to sigma-normalize the Elementary Effects or the derived statistics thereof. A code example is given by:

# Compute sigma-normalized statistics of Elementary Effects.
measures_sigma_norm = hms.compute_measures(ees_list, sd_qoi, sd_inputs, sigma_norm=True)

Let \(g(X_1, ..., X_k) = \sum_{i = 1}^{k} c_i X_i\) be an arbitrary linear function. Let \(\rho_{i,j}\) be the linear correlation between \(X_i\) and \(X_j\). Then, for all \(i \in 1, ..., k\), I expect

\[\begin{split}d_i^{u,*} = c_i,\\ d_i^{c,*} = \sum_{j = 1}^{k} \rho_{i,j} c_{j}.\end{split}\]

These results correspond to the intuition provided by the example in [Saltelli.2008], p. 123. Both equations state that, conceptually, the result does not depend on the sampling scheme.

Let us consider the case without any correlations between the inputs. Additionally, let \(c_i = \{3,2,1\}\) and \(\sigma^2_{X_i}=\{1,4,9\}\) for \(i \in \{1,2,3\}\). The following results are derived from [Saltelli.2008]. Let us first compute the Sobol’ indices. As \(g\) does not include any interactions, \(S_i^T = S_i\). Additionally, we have \(\text{Var}(Y)=\sum_{i=1}^k c_i^2 \sigma_{X_i}^2\) and \(\text{Var}_{\pmb{X_{\sim i}}}\big( E_{X_{\sim i}}[Y|\pmb{X_{\sim i}}] \big) = c_i^2 \sigma_{X_i}^2\). Table 1 compares three different sensitvity measures. These are the total Sobol’ indices, \(S_i^T\) (Measure I, the mean absolute EE, \(\gamma_i^*\) (Measure II), and the squared sigma-normalized mean absolute EE, \((\mu_i^* \frac{\sigma_{X_i}}{\sigma_Y})^2\) (Measure III).

Table 1: Importance measures for parametric uncertainty

Parameter	Measure I	Measure II	Measure III
X_1	9	3	9
X_2	8	2	8
X_3	9	1	9

In context of screening, \(S_i^T\) is the objective measure that we would like to predict approximately. We observe that \(\gamma_i^*\) ranks the parameters incorrectly. The reason is that \(\gamma_i^*\) is only a measure of the influence of \(X_i\) on \(Y\) and not of the influence of the variation and the level of \(X_i\) on the variation of \(Y\). We also see that \((\mu_i^* \frac{\sigma_{X_i}}{\sigma_Y})^2\) is an exact predictor for \(S_i^T\) as it does not only generate the correct ranking but also the right effect size. Importantly, this result is specific to a linear function without any interactions and correlations. However, it underlines the point that \(\gamma_i^*\) alone is not sufficient for screening. Following Ge.2017, one approach would be to additionally consider the EE variation, \(\sigma_i\). However, analysing two measures at once is difficult for models with a large number of input parameters. Table 1 indicates that \((\mu_i^* \frac{\sigma_{X_i}}{\sigma_Y})^2\) and also \(\mu_i^* \frac{\sigma_{X_i}}{\sigma_Y}\) can be an appropriate alternative. The actual derivative version of this measure is also recommended by guidelines of the Intergovernmental Panel for Climate Change ([IPCC.1999]).