# Uncertainty quantification¶

The goal of an uncertainty quantification is to describe the unknown distribution of the model output \(\rho_Y\) through statistical metrics. The two most common statistical metrics used in this context are the mean \(\mathbb{E}\) (also called the expectation value) and the variance \(\mathbb{V}\). The mean is defined as:

and tells us the expected value of the model output \(Y\). The variance is defined as:

and tells us how much the output varies around the mean.

Another useful metric is the \((100\cdot x)\)-th percentile \(P_x\) of \(Y\), which defines a value below which \(100 \cdot x\) percent of the simulation outputs are located. For example, 5% of the simulations of a model will give an output lower than the \(5\)-th percentile. The \((100\cdot x)\)-th percentile is defined as:

We can combine two percentiles to create a prediction interval \(I_x\), which is a range of values such that a \(100\cdot x\) percentage of the outputs \(Y\) occur within this range:

The \(90\%\) prediction interval gives us the interval within \(90\%\) of the \(Y\) outcomes occur, which also means that \(5\%\) of the outcomes are above and \(5\%\) below this interval.