# Methodological information

Methodological information and interpretation of the extreme value analyses and intensity diagram using the example of stations Genève-Cointrin and Lugano.

## Methods used

According to Extreme Value Theory, the largest values of a quantity (say daily precipitation or one-second wind gusts) taken each over intervals of the same size can be used to infer this quantity’s extreme behavior, i.e. at a level we have not yet experienced. These maxima are said to follow a Generalized Extreme Value distribution (GEV). The observations behind the extreme value analyses presented on this web site are therefore seasonal and yearly maxima.

## Interpretation of the extreme value analysis at Genève-Cointrin

### Return level plot Return level plot of 1-day precipitation at station Genève-Cointrin (return level estimate blue, 95% confidence interval green).

Return level plots display the probability that a given value is exceeded. The probability is expressed in terms of years. Thus a value that has a probability of 1% of being exceeded in any given year is - in the (very) long term average - expected to be exceeded once in a 100 years. Then, the value, or "return level'', is said to have a "return period'' of 100 years. In fact, return levels are extreme quantiles: the 100-year return value has a probability of 0.99 of not being exceeded, and therefore corresponds to the customary 0.99-quantile (cf. further information).

The extremal behavior of daily precipitation at the station Genève-Cointrin can be deduced from the return level plot: The return level estimate (blue line) is slightly negatively curved, which suggests that beyond a given daily precipitation amount, the probability that an amount should be exceeded is zero. The strong positive curvature of the upper confidence bound (green line), on the other hand, reveals that this extremal behavior is subject to uncertainty: It is possible that any amount of daily precipitation could also have a non-zero probability of being exceeded, and this probability would decrease rather slowly (i.e., polynomially). This means that very severe events would have a non-negligible probability of being exceeded.

## Reliability of results of the extreme value analysis is poor – What to do?

Each extreme value analysis is tested for its reliability, and the verdict is stated on the main page. This verdict does not pertain to the inherent uncertainty of the estimation due to the limited sample size. Rather, it describes the possibility that the assumptions motivating the choice of the statistical model may not be fulfilled.

If the reliability is poor, for instance, the model represents the observed extreme values badly. Three degrees of reliability have been defined: poor, questionable, and good. Their meaning and what to do in each case is shown below.

Reliability of extreme value analyses
reliability interpretation action Observed extreme values are badly represented by the model Do not use this statistical model. Use largest observed events or closest reliable station instead. Observed extreme values are not well represented by the model,  the underlying assumptions might be violated. Careful assessment necessary.Use visual guides (tech. report, p. 32) to decide. If points line up – proceed as usual; if not – same action as for poor fits. Observed extreme values are well represented by the model. The statistical model can be used.

## Interpretation of the intensity diagram of Lugano

### Intensity diagram

The intensity diagram reveals at a glance the behavior of very intense precipitation at different durations. It is purposely presented in a form similar to the Intensity-Duration-Frequency (IDF) diagrams familiar in the field of hydrology, but differ substantially in content: the return levels represented here were estimated for each duration separately, based on precipitation data measured at hourly intervals. No relation is assumed between precipitation intensities at different durations, and therefore no dependence of intensity on duration was assumed to infer return levels at short durations (see Technical Report of MeteoSwiss 255 for details).

## What to do when the reliability of the intensity diagram is poor?

Each precipitation intensity extreme value analysis is tested for its reliability, and the verdict is stated on the main page. This verdict does not pertain to the inherent uncertainty of the estimation due to the limited sample size. Rather, it describes the possibility that the assumptions motivating the choice of the statistical model may not be fulfilled.

If the reliability is questionable, for instance, the model does not represent well the observed extreme values of precipitation intensities for some of the durations. Three degrees of reliability have been defined: poor, questionable, and good. Their meaning and what to do about them is shown below.

Summarized reliability of extreme value analyses for multiple intensities
reliability interpretation action For some durations, the observed extreme values are badly represented by the model. Do not use return level estimates. Use the boxplots of largest observed events for a first impression; consult statistics of the annual maxima of precipitation sums if they are available and reliable. If some durations show no problems, the estimates can be used tentatively. Otherwise use the closest reliable station. For some durations, the observed extreme values are not well represented by the model. Careful assessment necessary. Check the quality of the fit with the presented visual guides. If only single intensities are of interest:  check if the duration of interest shows apparent problems. If it doesn’t, proceed as usual; if it does, same action as for poor fits. For most durations, the observed extreme values are well represented by the model. The return level estimates can be used. Overall, annual maximum precipitation intensities across durations are well represented by the estimated distributions.