# What is sensitivity analysis?

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in short Let's assume we have a model $h:\, \mathbf{X} \to Y$ that takes input parameters $\mathbf{X}$ and evaluates model output $Y$ as a function of $\mathbf{X}$. In sensitivity analysis, we assess how the input parameters $\mathbf{X}$ and their uncertainties influence the output $Y$ of the model.

## Defining the purpose of the analysis

A difficulty is that "the sensitivity" of $Y$ with respect to $\mathbf{X}$ is not uniquely defined: There exists a large variety of approaches/methods to evaluate the sensitivity of $Y$ with respect to $\mathbf{X}$ — and the interpretation of the so-obtained sensitivities can differ depending on the approach taken. In order to select an appropriate approach, one must know the purpose of the analysis. Therefore, an integral part and first step of any sensitivity analysis is the definition of the goal/purpose/objective of the analysis. In other words, to get an appropriate answer, one must know the question to be answered. This identified purpose should also always be clearly communicated with the evaluated sensitivities.

In particular, the definition of the purpose should clearly state what the model output of interest is: Often the output of a model is multivariate. For example, if the model is a finite element model, stresses/displacements/forces are returned for each node or element. In this case, it is essential to clearly state what quantity at which position/node should be the focus of the analysis.

Moreover, the purpose should contain the motivation behind conducting the sensitivity analysis. Possible motivations are:

• Ranking: Establishing a ranking of the input parameters according to their importance on the model output. This is also referred to as factor prioritization.
• Screening: In order to simplify a model, one might want to fix the value of some of the uncertain model input parameters. Screening aims at identifying input parameters that are of minor importance with respect to the model output. This is also referred to as factor fixing.
• Mapping: Identifying areas in the joint domain of the input parameters that cause extreme values in the model output.
• Robustness: Understanding the robustness of a model with respect to assumptions made on the model input.

## Examples for purposes

Some examples for purposes underlying a sensitivity analysis are:

• Establish a ranking of the uncertain input parameters with respect to the variance of the largest stresses occurring in the structure.
• What uncertain input parameters can be set to a fixed value with only a negligible impact on the probability of failure of the structure?
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