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Global Sensitivity Analysis (GSA)

Global Sensitivity Analysis (GSA) explores model output variability by systematically altering all uncertain input parameters across their entire range. This framework employs the variance-based Sobol' method, which is a standard approach for identifying influential factors without constraints on the model form.

Sobol' Sensitivity Indices

The framework evaluates parameter influence using two primary indices derived from the Analysis of Variances (ANOVA) decomposition.

First-Order Index (\(S_i\))

The first-order index measures the isolated main effect contribution of each input factor to the total variance of the model output:

\[S_{i} = \frac{\mathbb{V}_{q_{i}}(\mathbb{E}_{q_{\sim i}}(y|q_{i}))}{\mathbb{V}(y)}\]

Total-Order Index (\(S_{Ti}\))

The total-order index accounts for the first-order effect along with all higher-order interaction effects between the parameter and the rest of the model inputs:

\[S_{Ti} = \frac{\mathbb{E}_{q_{\sim i}}(\mathbb{V}_{q_{i}}(y|q_{\sim i}))}{\mathbb{V}(y)} = 1 - \frac{\mathbb{V}_{q_{\sim i}}(\mathbb{E}_{q_{i}}(y|q_{\sim i}))}{\mathbb{V}(y)}\]

\(S_{Ti} = 0\) is a necessary and sufficient condition for a parameter to be considered non-influential. In the present study, however, \(S_{Ti} \approx S_i\), indicating that the interaction effects are absolute minimal. Additionally, by considering practical implications for determining the least influential parameters, \(S_i \approx 0\), or \(S_i < 0.1\) is assumed to be the threshold distinguishing between influential and non-influential parameters.

Computation and Tools

SALib Integration

For the computationally efficient evaluation of these indices, the SALib Python library is utilized. This package provides robust implementations of sensitivity analysis algorithms, allowing for precise estimation of Sobol' indices via low-discrepancy sampling sequences.

Factor Prioritization and Fixing

GSA results for the FMM-FMG and FMG-FMG models identify the following parameters:

Model Least influential parameters Most influential parameters
FMG-FMG \(\tau_{c_{2}}\)
\(\tau_{c_{1}}\)		 \(E_{c_{1}}\)
\(\alpha_{1}\)
FMM-FMG \(\tau_{c_{2}}\)
\(\beta_{1}\)
\(\tau_{c_{1}}\)		 \(E_{c_{1}}\)
\(\alpha_{1}\)

  • Influential Parameters: \(E_{c_1}\), \(\alpha_1\), and \(E_{c_2}\) consistently demonstrate high sensitivity indices, requiring careful calibration.

  • Non-Influential Parameters: \(\tau_{c_2}\), \(\tau_{c_1}\), and \(\beta_1\) exhibit negligible effects on output variance, allowing them to be fixed deterministically to reduce problem dimensionality.