# What is factor fixing (in sensitivity analysis)?

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In factor fixing, the purpose of a sensitivity analysis is to identify the "least important" model input parameters in order to decide whether these parameters can be fixed to deterministic values; i.e., the target of the analysis is to simplify the model by reducing the number of uncertain input parameters. Of course, "least important" can mean different things in different settings.

## Factor fixing in a variance-based sensitivity analysis

An often applied interpretation of "importance" is to define the "least influential model input parameter" as the one that if all other model input parameters are fixed to $\mathbf{x}_{\sim i}$, would not impact the model output (no matter what value) [Saltelli, 2008]. This corresponds to requiring that $\operatorname{Var}_{X_i}\left[Y|\mathbf{X}_{\sim i}=\mathbf{x}_{\sim i}\right]=0$ (for noninfluential model input parameter $X_i$), averaged over all viable states of $\mathbf{X}_{\sim i}$. The averaging is performed over the joint distribution of the model parameters $\{X_1,\ldots,X_{i-1},X_{i+1},X_M\}$; i.e., $\operatorname{E}_{\mathbf{X}_{\sim i}}\left[\operatorname{Var}_{X_i}\left[Y|\mathbf{X}_{\sim i}\right]\right]=0$ if model input parameter $X_i$ is truly non-influential. Note that the total-effect sensitivity index is proportional to $\operatorname{E}_{\mathbf{X}_{\sim i}}\left[\operatorname{Var}_{X_i}\left[Y|\mathbf{X}_{\sim i}\right]\right]$. Thus, the total-effect sensitivity index is an appropriate sensitivity measure for factor fixing.

Note that the above-stated interpretation of "importance" does not require the model input parameters to have a true, albeit unknown value (we try to identify parameters whose value does truly not have an impact, no matter what value). Thus, this measure is able to quantify both epistemic and aleatoric types of uncertainty.

## References

[Saltelli, 2008] Saltelli, Andrea, et al. Global sensitivity analysis: the primer. John Wiley & Sons, 2008.

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