Choosing an extreme-value distribution#

Use the family name when constructing an extreme-value marginal in PySTRA. The Type I, II and III names describe the classical families; the maxima or minima convention determines which tail is bounded. State the reference period of the extreme, such as an annual maximum load, alongside its units and parameters.

Match the family and the tail#

Extreme-value names in PySTRA 2.0#
Classical family	PySTRA class	Tail convention	Former 1.x name
Type I, largest	`Gumbel`	Gumbel maxima; unbounded support	`TypeIlargestValue`
Type I, smallest	`GumbelMin`	Gumbel minima; reflected maxima	`TypeIsmallestValue`
Type II, largest	`Frechet`	Fréchet maxima; heavy upper tail	`TypeIIlargestValue`
Type III, smallest	`Weibull`	Weibull minima; finite lower bound	`TypeIIIsmallestValue`
Generalized extreme value, maxima	`GEV`	Shape selects Type I, II or III	`GEV`; `GEVmax` also remains valid
Generalized extreme value, minima	`GEVMin`	Reflected GEV maxima	`GEVmin`

GEVmax is a permanent alias for GEV: both names refer to the same class. The old type-numbered names and GEVmin have been replaced; see Migrating from 1.x to 2.0 for script conversion.

The ordinary Weibull distribution describes minima. A Type III distribution for maxima has a finite upper endpoint and is often called the reversed Weibull distribution. Construct that case with GEV(shape=...) using a negative shape. The Fréchet and Weibull names and their SciPy parameterizations are documented in SciPy’s invweibull reference and weibull_min reference.

Read the GEV shape convention#

Write \(\ell\) for loc, \(s>0\) for scale and \(\xi\) for shape. The maxima CDF is

\[F_{\max}(x) = \exp\!\left[-\left(1+\xi\frac{x-\ell}{s}\right)^{-1/\xi}\right], \qquad 1+\xi\frac{x-\ell}{s}>0.\]

At \(\xi=0\) the limiting expression is \(F_{\max}(x)=\exp[-\exp(-(x-\ell)/s)]\). PySTRA uses the opposite shape sign to SciPy’s genextreme: the corresponding SciPy argument is c=-shape.

Shape and support for GEV maxima#
Shape	Family	Support
\(\xi=0\)	Type I, Gumbel	All real values
\(0<\xi<0.5\)	Type II, Fréchet	Lower endpoint \(\ell-s/\xi\); unbounded above
\(\xi<0\)	Type III, reversed Weibull	Unbounded below; upper endpoint \(\ell-s/\xi\)

GEVMin(loc=ell, scale=s, shape=xi) is the distribution of \(-Y\) when \(Y\) has GEV(loc=-ell, scale=s, shape=xi). Reflection changes the CDF to

\[F_{\min}(x) = 1-\exp\!\left[-\left(1+\xi\frac{\ell-x}{s}\right)^{-1/\xi}\right].\]

Its support requires \(1+\xi(\ell-x)/s>0\). Thus negative shape gives a finite lower endpoint \(\ell+s/\xi\), and positive shape gives a heavy lower tail with finite upper endpoint \(\ell+s/\xi\). Zero shape gives GumbelMin.

Both GEV classes require shape < 0.5 for finite variance. Supply either mean and std, or loc and scale, together with shape. Location and scale have the variable’s units; shape is dimensionless. Location generally differs from the mean, and scale generally differs from the standard deviation.

Check equivalent parameterizations#

The following examples compare quantiles across each named family. For Fréchet scale \(a\) and shape \(k>2\), the equivalent GEV parameters are \((\ell,s,\xi)=(a,a/k,1/k)\). For Weibull lower bound \(b\), scale \(a\) and shape \(k>0\), the equivalent GEVMin parameters are \((\ell,s,\xi)=(b+a,a/k,-1/k)\). These relations follow by substitution into the CDFs; the family-specific shape is not the GEV shape.

import numpy as np
import pystra as ra

probabilities = np.array([0.01, 0.25, 0.5, 0.75, 0.99])
pairs = [
    (
        ra.Gumbel("Q", loc=10.0, scale=2.0),
        ra.GEV("Q", loc=10.0, scale=2.0, shape=0.0),
    ),
    (
        ra.GumbelMin("R", loc=10.0, scale=2.0),
        ra.GEVMin("R", loc=10.0, scale=2.0, shape=0.0),
    ),
    (
        ra.Frechet("Q", scale=12.0, shape=4.0),
        ra.GEV("Q", loc=12.0, scale=3.0, shape=0.25),
    ),
    (
        ra.Weibull("R", lower=2.0, scale=8.0, shape=4.0),
        ra.GEVMin("R", loc=10.0, scale=2.0, shape=-0.25),
    ),
]
for family, generalized in pairs:
    np.testing.assert_allclose(
        family.ppf(probabilities),
        generalized.ppf(probabilities),
        rtol=1e-12,
        atol=1e-12,
    )
assert ra.GEVmax is ra.GEV

Each pair represents the same marginal distribution. Choosing a family from data still requires checking the support, tail fit and reference period. For the exact maximum of a specified number of independent, identically distributed parent variables, see Maximum and its parent-CDF construction.

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