# What are the first-order sensitivity indices?

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You have a stochastic model $Y(\mathbf{X})$ that is driven by the vector $\mathbf{X}$ of uncertain model input parameters. If you need to establish a ranking of the input parameters according to their importance on the model output, the first-order sensitivity indices are a commonly applied sensitivity measure for that purpose.

## Definition

The "first-order sensitivity index" is also referred to as the "first-order Sobol index" or "main effect index". The first-order sensitivity index $S_i$ associated with model input parameter $X_i$ quantifies the effect of varying $X_i$ alone, averaged over the contributions of the other input parameters $\mathbf{X}_{-i}$, with $\mathbf{X}_{-i} = \left[X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_M\right]$. $S_i$ is defined as the first-order effect $V_i$ from the variance decomposition divided by the variance $\operatorname{Var}[Y]$ of the model output $Y$:
$$S_i = \frac{V_i}{\operatorname{Var}[Y]} = \frac{\operatorname{Var}_{X_i}\left[ \operatorname{E}_{\mathbf{X}_{\sim i}}\left[Y|X_i\right] \right]}{\operatorname{Var}[Y]} \;,$$ where $\operatorname{E}_{\mathbf{X}_{\sim i}}\left[Y|X_i\right]$ is the expected value of the model output with respect to all input variables except $X_i$ (which is fixed).

## Sum of Sobol-based sensitivity indices

Note that due to the link between the Sobol-based sensitivity indices and the variance decomposition, it must hold that:
$$\sum_{i=1}^M S_i + \sum_{i<j}^M S_{ij} + \ldots + S_{12\ldots M} = 1\;,$$ where a higher-order index $S_\mathbf{a}$ is defined as:
$$S_\mathbf{a} = \frac{V_\mathbf{a}}{\operatorname{Var}[Y]}\;.$$

Thus, $1-\sum_{i=1}^M S_i$ quantifies the fraction of the model output $Y$ that cannot be explained by first-order effects. Note that for additive models we have by definition: $\sum_{i=1}^M S_i=1$. Correspondingly, for non-additive models, we have $\sum_{i=1}^M S_i<1$.

## Interpretation of first-order sensitivity indices

The first-order sensitivity indices are well suited for factor prioritization; i.e., to establish a ranking of the input parameters according to their importance on the model output. More specifically, the first-order sensitivity indices identify the model parameters that on average, once fixed, would cause the largest reduction in variance [Saltelli, 2008].

## Illustrative example

### The problem

We model fatigue deterioration by the Palmgren-Miner rule with constant stress ranges in all cycles. The fatigue damage $D$ is evaluated as: $$D = n\cdot\frac{\Delta S^m}{K} \,.$$ The parameters in the equation above are:

• $n$: the number of stress cycles; is modeled as deterministic and set to $10^7$.
• $m$: a material parameter; is modeled as deterministic and set to $3.0$.
• $\Delta S$: the stress range (which is constant in all cycles); is modeled as a log-Normal distribution with mean $50 N/s^2$ and standard deviation $10 N/s^2$.
• $K$: a material parameter; is modeled as a log-Normal distribution with mean $4.5\cdot 10^{12} N^3/s^6$ and standard deviation $2.25\cdot 10^{12} N^3/s^6$.

Thus, the fatigue damage depends on the uncertain model parameters $\Delta S$ and $K$.

We perform a sensitivity analysis in order to answer the following question: Which model parameter would (on average) cause the largest reduction in variance if we could determine its value exactly (i.e., fully eliminate the uncertainty of the associated parameter)?

### The solution

The first-order sensitivity indices are $$S_{\Delta S} = 0.53 \,,$$ and $$S_{K} = 0.32 \,.$$

The uncertain model parameter $\Delta S$ is the most influential; i.e., $\Delta S$ would on average cause the largest reduction in variance if its value could be fixed. However, also the uncertain model parameter $K$ has a notable influence.

The fraction of the model output that cannot be explained by first-order effects is $$1-S_{\Delta S} -S_{K}=1-0.53-0.32=0.15\,.$$ ■

## References

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

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