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:

\[\mathbb{E}[Y] = \int_{\Omega_Y} y\rho_Y(y)dy,\]

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

\[\mathbb{V}[Y] = \int_{\Omega_Y} {\left(y - \mathbb{E}[Y]\right)}^2\rho_Y(y)dy,\]

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:

\[x = \int_{-\infty}^{P_x}\rho_Y(y)dy.\]

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:

\[I_x = \left[P_{(x/2)}, P_{(1-x/2)}\right]. \label{eq:prediction}\]

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.