Extreme values

  

Figure: The maximum daily precipitation in Oslo (St. Hanshaugen, 1880-1965) each year, rr_max, (a), the distribution function for rr_max and fitted Gumbel curve, and (c) the best-fit Gumbel cumulative distribution function ( c.d.f). The probability that the maximum daily rainfall will exceed 60mm any year is around 1/100. Also shown in is the $\xi$ threshold (vertical dashed lines). [stats_uib_4_2.m]

The largest or smallest observed value of a variable. Extreme values are often assumed to follow the Gumbel distribution (extreme-value distribution or Fisher-Tippett type 1 distribution). The p.d.f. for the Gumbel distribution is shown in Fig. 4.2b.

Data gathering: observations are taken over a relatively short time interval or an ensemble of observations (a number of station values or climate scenarios) to obtain a single extreme value. For example, maximum temperature in July, minimum snowfall in December, strongest wind speed over a year, highest daily rainfall in a year. Only the extreme value for a given period is retained for the analysis.

Moving window (365-day windows starting on 01-Jan, 02, Jan, 03-Jan,...) vs fixed windows (eg 01-Jan to 31-Dec): the fixed-window accumulations must be multiplied by an empirically derived constant.

Parametric models for extreme value distributions: Gumbel (Pearsons type I, or EV-1), Pearsons type II (EV-II), or Pearson type III (EV-III). Both normal and exponential distribution lie in the domain of the attraction of the Gumbel function (the latter lies closest). Other distributions: Generalised Extreme Value (GEV), Weibull, Pareto, and Wakeby distributions. Pearson curves: scatter plots of kurtosis=f(skewness). Model fitting methods: moments (simple and common), maximum likelihood (sensitive to outliers), probability weighted moments (more robust), L-moments (more robust).

Assumption of stationarity and ergodicity^4.1. The GEV c.d.f. is given by:

$$F(x)= \exp \left\{ - \left[ 1 + \xi \left(\frac{-(x-\xi)}{\beta}\right)\right]^{-1/\xi} \right\}.$$ (4.7)

$$f(x)=\frac{1}{\beta} \exp \left\{ - \exp \left[ \frac{-(x-\xi)}{\beta}\right] - \frac{(x-\xi)}{\beta} \right\},$$ (4.8)

$\xi$=location parameter, $\beta$=scale parameter (NB, there is a typo in Wilks [], p. 98, eq. 4.42: a ``minus'' is missing). The distribution is skewed towards higher values, and has a peak at $x=\xi$. Equation 4.8 is integrateable, and the cumulative Gumbel distribution function is

$$F(x)= \exp \left\{ - \exp \left[ -\frac{-(x-\xi)}{\beta}\right] \right\}.$$ (4.9)

Return values: thresholds that on average are exceeded once per return period. Upper quantiles of the fitted extreme value distribution. For a X-year return value, rr_(X),

$$P(rr > rr_{(X)})= \int_{rr_{(X)}}^{\infty} f(rr) dr = 1/X.$$ (4.10)

The X-interval-length (eg 10-year) return value is the point on the abscissa where the c.d.f (Figure 4.2c) equals (1 - 1/X). For example, the 10-year return value for the maximum daily precipitation in Oslo is 46 mm (shown as dash-dot line), whereas the 100-year return value is 63mm (dashed).

The estimators are:

$$\hat{\beta}= \frac{s \sqrt{6}}{\pi},$$

and

$$\hat{\xi}= \overline{x} - \gamma \hat{\beta}.$$

$\gamma = 0.57721...$ (Euler's constant).

The Gumbel distribution is also discussed on pp45-50 in Von Storch and Zwiers (1999) [].

A good reference on extreme value theory by Stuart Coles can be found on URL: http://www.maths.lancs.ac.uk/coless/.