gev_return_level#

gpjax.distributions.gev_return_level(return_period, loc, scale, concentration)[source]#

The GEV return level for a return period.

The return level \(z_T\) is the level that the annual maximum exceeds with probability \(1/T\), that is, on average once in \(T\) years:

\[z_T = \mu - \frac{\sigma}{\xi}\left[1 - \{-\log(1 - 1/T)\}^{-\xi}\right], \qquad z_T = \mu - \sigma \log\{-\log(1 - 1/T)\} \quad (\xi = 0).\]

The function is accurate and has finite gradients for all \(\xi\), including \(\xi = 0\), and for large \(T\). All arguments broadcast with the NumPy rules. Thus a user who has samples of \((\mu, \sigma, \xi)\) from any source gets samples of the return level.

Example

>>> import jax.numpy as jnp
>>> from gpjax.distributions import gev_return_level
>>> periods = jnp.array([10.0, 100.0])
>>> levels = gev_return_level(periods, loc=40.0, scale=12.0,
...                           concentration=0.1)
>>> levels.shape
(2,)
Parameters:

  • return_period (float | int | Float[jaxlib._jax.Array, '...'] | Float[ndarray, '...'] | Int[jaxlib._jax.Array, '...'] | Int[ndarray, '...']) – The return period \(T > 1\), in blocks (for example, years for annual maxima). \(T = 1\) gives the lower end point and \(T = \infty\) the upper end point; \(T < 1\) gives NaN.

  • loc (float | int | Float[jaxlib._jax.Array, '...'] | Float[ndarray, '...'] | Int[jaxlib._jax.Array, '...'] | Int[ndarray, '...']) – The location \(\mu\).

  • scale (float | int | Float[jaxlib._jax.Array, '...'] | Float[ndarray, '...'] | Int[jaxlib._jax.Array, '...'] | Int[ndarray, '...']) – The scale \(\sigma > 0\).

  • concentration (float | int | Float[jaxlib._jax.Array, '...'] | Float[ndarray, '...'] | Int[jaxlib._jax.Array, '...'] | Int[ndarray, '...']) – The shape \(\xi\). \(\xi > 0\) is the heavy-tailed (Fréchet) type and \(\xi < 0\) the bounded (Weibull) type. This is the climate convention; scipy.stats.genextreme uses \(c = -\xi\).

Returns:

The return levels, with the broadcast shape of the arguments.

Return type:

Float[jaxlib._jax.Array, ‘…’] | Float[ndarray, ‘…’]

Expand for references to gpjax.distributions.gev_return_level

gev_return_level