Extreme Value Theory: The GEV Distribution
The Fisher-Tippett-Gnedenko theorem is the EVT counterpart of the Central Limit Theorem: the limiting distribution of block maxima (e.g., annual maximum temperature) belongs to the Generalized Extreme Value (GEV) family regardless of the underlying distribution:
\[G(x;\mu,\sigma,\xi) = \exp\!\left\{-\left[1 + \xi\!\left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi}\right\}\]for $1 + \xi(x-\mu)/\sigma > 0$, where $\mu$ is the location, $\sigma > 0$ the scale, and $\xi$ the shape parameter (tail index).
| Shape $\xi$ | Type | Tail | Example variable |
|---|---|---|---|
| $\xi \to 0$ | Gumbel | Light (exponential) | Temperature maxima |
| $\xi > 0$ | Fréchet | Heavy (power law) | Precipitation, wind |
| $\xi < 0$ | Weibull | Bounded above | Relative humidity |
For precipitation extremes, $\xi \approx 0.1$–$0.3$ is typical, indicating a heavier tail than the Gumbel.
Return Periods and Return Levels
The return period (or recurrence interval) $T$ of an event exceeding level $z$ is:
\[T = \frac{1}{1 - G(z)} \approx \frac{1}{P(X > z)}\]The $T$-year return level $z_T$ (exceeded with probability $1/T$ in any year) is obtained by inverting the GEV:
\[z_T = \mu - \frac{\sigma}{\xi}\!\left[1 - (-\ln(1 - 1/T))^{-\xi}\right]\]For the Gumbel case ($\xi \to 0$): $z_T = \mu - \sigma \ln(-\ln(1-1/T))$.
Return levels are estimated by fitting GEV parameters to a record of annual maxima via maximum likelihood or L-moments. The $95\%$ confidence interval on $z_{100}$ (the 1-in-100-year level) is typically wide—often $\pm 30$–$50\%$—because long-return-period events are inherently data-scarce.
Peaks-Over-Threshold and the GPD
An alternative to block maxima uses all exceedances above a high threshold $u$. By the Pickands-Balkema-de Haan theorem, threshold excesses $(X - u \mid X > u)$ converge to the Generalized Pareto Distribution (GPD):
\[H(y;\sigma_u,\xi) = 1 - \left(1 + \frac{\xi y}{\sigma_u}\right)^{-1/\xi}, \quad y > 0\]This peaks-over-threshold (POT) approach uses more data than annual maxima but requires careful threshold selection: too low introduces bias (non-asymptotic behavior), too high increases variance. The mean excess plot $E[X-u \mid X>u]$ vs $u$ is used diagnostically — a linearly increasing relationship confirms the GPD family.
The POT return level for a rate-$\lambda$ Poisson process of exceedances is:
\[z_T = u + \frac{\sigma_u}{\xi}\!\left[(\lambda T)^\xi - 1\right]\]Climate Change Attribution
Event attribution asks: how has climate change altered the probability of an observed extreme? The standard framework compares two counterfactual worlds using large climate model ensembles:
- Factual world ($\mathcal{F}$): current climate with observed forcing
- Counterfactual world ($\mathcal{CF}$): pre-industrial climate (no anthropogenic forcing)
The probability ratio (PR) and fraction attributable risk (FAR) are:
\[\text{PR} = \frac{P(X > x \mid \mathcal{F})}{P(X > x \mid \mathcal{CF})}, \qquad \text{FAR} = 1 - \frac{1}{\text{PR}}\]A $\text{PR} = 5$ means the event is five times more likely in the current climate; $\text{FAR} = 0.8$ means $80\%$ of the risk is attributable to climate change.
Non-stationary GEV models allow direct estimation by making $\mu$ (and sometimes $\sigma$) a function of a climate covariate (e.g., global mean temperature $T_g$):
\[\mu(t) = \mu_0 + \alpha\, T_g(t)\]This covariate approach can be fitted to observations alone, without needing large model ensembles, though it assumes a linear relationship between the location parameter and forcing.
Clausius-Clapeyron Scaling
A key physical anchor for precipitation extremes is Clausius-Clapeyron (CC) scaling: the saturation vapor pressure increases at $\approx 7\%\ \text{K}^{-1}$:
\[\frac{d\ln e_s}{dT} = \frac{L_v}{R_v T^2} \approx 7\%\ \text{K}^{-1}\]where $L_v$ is the latent heat of vaporization and $R_v$ the gas constant for water vapor. If atmospheric moisture content controls extreme precipitation intensity, extreme rainfall should also intensify at $\sim 7\%$ per degree of warming. Observational analyses find scaling rates of $6$–$8\%\ \text{K}^{-1}$ for short-duration convective extremes, broadly consistent with CC, though dynamical changes can produce super-CC or sub-CC scaling regionally.