Local Sensitivity Analysis (LSA)

Local Sensitivity Analysis (LSA) evaluates how uncertainty in model output is apportioned to specific sources of uncertainty in the model input, focused around a nominal parameter value. This analysis is vital for determining factor prioritization—quantifying the contribution of model inputs to output uncertainty—and factor fixing, which identifies non-influential parameters that can be treated deterministically.

Methodology

The framework utilizes partial derivatives as local sensitivity measures. To allow for direct comparison between parameters with different physical units and magnitudes, indices are normalized as follows:

\[\overline{S}_{E',q_{i}} = \frac{q_{i}^{0}}{E'(q^{0};x)} \cdot \frac{\partial E'(q^{0};x)}{\partial q_{i}}\]

\[\overline{S}_{E'',q_{i}} = \frac{q_{i}^{0}}{E''(q^{0};x)} \cdot \frac{\partial E''(q^{0};x)}{\partial q_{i}}\]

Where \(q_{i}^{0}\) represents the realized baseline value of the \(i\)-th parameter.

Complex Modulus Magnitude

To simplify prioritization by considering both storage and loss moduli simultaneously, the magnitude of the normalized local sensitivity index for the complex modulus is calculated:

\[|\overline{S}_{E^{*},q_{i}}| = \frac{q_{i}^{0}}{|E^{*}(q^{0};x)|} \cdot \sqrt{\left(\frac{\partial E'(q^{0};x)}{\partial q_{i}}\right)^{2} + \left(\frac{\partial E''(q^{0};x)}{\partial q_{i}}\right)^{2}}\]

Quantitative Assessment via Norms

Because sensitivity indices vary across the frequency domain, we employ \(L_1\) norm to evaluate total parameter significance:

\[\|\overline{S}_{E',q_{i}}\|_{L_{1}} = \int |\overline{S}_{E',q_{i}}| d \log(a_{T}\omega)\]

Parameter Prioritization Results

Based on the local sensitivity analysis, parameters are prioritized by their impact on model variability:

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