Probabilistic analyses¶

These analyses estimate the parameters of a probability distribution that describes the extreme-rainfall regime, and use it to extract the expected heights for assigned return periods \(T\).

All methods assume that the annual maxima for a fixed duration are independent realisations of the same random variable.

Gumbel (EV1)¶

The 2-parameter Gumbel distribution describes the maxima of large samples:

\[ F(h) = \exp\!\left[-\exp\!\left(-\frac{h - \mu}{\beta}\right)\right] \qquad h(T) = \mu - \beta \ln\!\left[-\ln\!\left(1 - \frac{1}{T}\right)\right] \]

Two available estimators:

Method of moments. Fast, robust. \(\beta = s \sqrt{6}/\pi\), \(\mu = \bar{h} - \gamma\,\beta\) (\(\gamma \approx 0.5772\) Euler-Mascheroni constant).
Maximum Likelihood (ML). Iterative, slightly more efficient.

GEV — Generalized Extreme Value (L-moments)¶

A family that generalises Gumbel: it includes a shape parameter \(\kappa\) allowing heavy tails (\(\kappa > 0\)) or bounded tails (\(\kappa < 0\)).

\[ F(h) = \exp\!\left[-\left(1 - \kappa\,\frac{h - \xi}{\alpha}\right)^{1/\kappa}\right] \]

The three parameters \((\xi, \alpha, \kappa)\) are estimated via L-moments (Hosking 1990) — more robust than classical moments for small samples.

Heights h(T):

\[ h(T) = \xi + \frac{\alpha}{\kappa}\!\left[1 - \left(-\ln(1 - 1/T)\right)^{\kappa}\right] \]

GEV vs Gumbel?

Prefer GEV when the series has few years and unusual extreme values. For series with >50 years and no "fat tails", Gumbel is usually enough and easier to justify in a report.

Pearson III (method of moments — Foster/Kite)¶

A 3-parameter distribution (\(\bar{h}, s, C_s\) = mean, standard deviation, skewness) historically popular in the Anglo-Saxon world:

\[ h(T) = \bar{h} + K_T(C_s) \cdot s \]

with \(K_T\) a tabulated frequency factor (Wilson-Hilferty's formula for analytical computation).

Useful when the series has marked skewness and Gumbel underestimates the tail.

TCEV — Two-Component Extreme Value (VA.PI. — Italy)¶

Regional model for Italy based on the VA.PI. programme (Valutazione delle Piene in Italia, CNR-GNDCI). Distinguishes ordinary and extraordinary events by admitting two superposed EV1 components.

\[ F(h) = \exp\!\left[-\Lambda_1 e^{-h/\theta_1} - \Lambda_2 e^{-h/\theta_2}\right] \]

The 4 parameters \((\Lambda_1, \theta_1, \Lambda_2, \theta_2)\) are hard to estimate pointwise: Runoff Lab NX implements the classical 4 regionalisation levels:

Level	What's fixed	What's estimated
0 — pointwise	no regional parameter	all 4 from the station (poor under 30 years)
1 — regional	\(\Lambda^* = \Lambda_2/\Lambda_1\), \(\Theta^* = \theta_2/\theta_1\)	the remaining 2 from the station
2	as level 1 + \(\Lambda_1\)	only \(\theta_1\)
3	all shape parameters regional; \(\theta_1\) from scale	non-dimensional + local scale

The VA.PI. dataset is loaded for known Italian regions. For regions or countries not covered, the TCEV option is hidden when the interface language is not Italian.

When TCEV makes sense

TCEV captures well the upper tail of catastrophic events typical of the Italian climate (convective floods on small basins). Many basin authorities recommend it for Hydraulic Vigilance and PAI plans.

Goodness-of-fit tests¶

For each analysis the software runs two tests:

Kolmogorov-Smirnov (KS): maximum distance between empirical and theoretical CDF. Compared to the 5% critical value.
Pearson χ²: comparison on frequency classes.

Both are not rejected at 5% if p-value > 0.05. A single distribution that passes both is preferable to two that each fail one.

Reference bibliography¶

Gumbel E.J. (1958), Statistics of Extremes, Columbia University Press.
Hosking J.R.M. (1990), L-moments: analysis and estimation of distributions using linear combinations of order statistics, J. Royal Stat. Soc. B 52.
Kite G.W. (1977), Frequency and Risk Analyses in Hydrology, Water Resources Publ.
Rossi F., Versace P. (1982), Criteri e metodi per l'analisi statistica delle piene, CNR-PFC, Italy.
VA.PI. programme — CNR-GNDCI, Valutazione delle Piene in Italia, regional reports.