##### Annual exceedance probability (AEP)

The probability that an event of that size or greater will occur in a given year.

Note that this does not preclude the possibility of multiple events of that size occurring in the same year—or occurring in successive years, for that matter. It’s possible to calculate the cumulative probability of an event occurring at some point over a certain number of years.

The cumulative probability of an event occurring at least once over a period of n years is 1 - (1 - \mathrm{AEP})^n.

Historically the annual maxima method has been used to derive annual exceedance probabilities. Conversion to equivalent average recurrence interval based on the peak over threshold method can be done using the inverse of Langbein’s formula (Langbein, 1949); i.e. \mathrm{ARI} = -1/\ln(1 - \mathrm{AEP}). This formula is approximately equal to the commonly-used equation \mathrm{ARI} = 1/\mathrm{AEP} above an ARI of 10 years.

##### Average recurrence interval (ARI)

The average time period between events of that size or greater. Also known as the *return period*.

Note that this does not preclude the possibility of multiple events of that size occurring in the same year—or occurring in successive years, for that matter. It’s possible to calculate the cumulative probability of an event occurring at some point over a certain number of years.

Historically the peak over threshold method has been used to derive average recurrence intervals. Conversion to equivalent annual exceedance probability based on the annual maxima method can be done using Langbein’s formula (Langbein, 1949); i.e. \mathrm{AEP} = 1 - \exp(-1/\mathrm{ARI}). This formula is approximately equal to the commonly-used equation \mathrm{AEP} = 1/\mathrm{ARI} above an ARI of 10 years.

##### Annual Maxima Series (AMS)

Method used to select events that will be used for fitting an extreme value distribution to.

The annual maxima method simply identifies the largest event in each year, with no need for a threshold. This is the method that XRain uses.

More generally known as the *block maxima* method with a block length of 1 year.

##### Peak over threshold (POT)

Method used to select events that will be used for fitting an extreme value distribution to. Also known as the *partial duration series*.

This method considers events above a certain threshold value; i.e. a threshold value is chosen in advance and any event in the time series that exceeds this value is selected.

The major difficulties with the peak over threshold method are a) choosing an appropriate threshold value, and b) assuring the independence of events (Bezak et al., 2014).

##### Extreme value distribution

A curve fitted to events identified by way of the annual maxima or peak over threshold methods. Extreme value analysis has been a topic of active research since at least 1928 and can be applied to many fields.

##### Generalized extreme value (GEV) distribution

Fisher and Tippet (1928) showed that there are three possible limiting distributions for extremes, which are now known as the Gumbel (type 1), Fréchet (type 2) and reversed Weibull (type 3). The generalized extreme value (GEV) distribution (von Mises, 1936) unifies these three distributions into a single expression. It has been used previously to analyse precipitation across the globe (Papalexiou and Koutsoyiannis, 2013).