# What is derivative-based sensitivity analysis?

subject area
in short
prerequisite knowledge

Derivative-based sensitivity measures belong to the class of local sensitivity analysis; i.e., the derivatives are evaluated at a specific point $\mathbf{x}_0$.

## Deterministic model parameters

If the model parameter of interest is deterministic (i.e., not uncertain), a common measure to quantify the relative importance with respect to other (deterministic) parameters is:
$$S^p_{x_i} = \frac{\partial Y}{\partial x_i}\;,$$ where $Y$ is the model output of interest (which can also be an expected value if the model contains uncertain parameters) and $x_i$ is the $i$th coefficient of vector $\mathbf{x}_0$.

The measure above quantifies how much $Y$ changes locally at $\mathbf{x}_0$ with respect to changes in the $i$the parameter $x_i$.

## Uncertain model parameters

The local sensitivity measure $S^p_{x_i}$ is inept to quantify the sensitivity of the model output for uncertain model parameters, as it does not account for the uncertainty about the targetted model parameter. For derivative-based local sensitivity analysis of uncertain model parameters $X_i$, the above measure is usually augmented by the ratio of input-output standard deviations:
$$S^\sigma_{X_i} = \frac{\sigma_{X_i}}{\sigma_Y} \frac{\partial Y}{\partial X_i}\;,$$ where $\sigma_{X_i}$ is the standard deviation of $X_i$ and $\sigma_Y$ is the standard deviation of the model output.

However, remember that for nonlinear models, the interpretation of the obtained (local) sensitivities is only valid around $\mathbf{x_0}$.