# 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.