mixle.analysis.max_stable module

Max-stable processes for spatial extremes: the Smith (Gaussian-storm) model.

Block maxima of a spatial field (annual flood peaks, peak seismic amplitude, extreme porosity) are spatially dependent, and that dependence has its own limit law – a max-stable process – which the ordinary GEV/GPD (treated independently per site) misses. The Smith model is the canonical one: Z(s) = max_i xi_i * phi_Sigma(s - U_i) over a Poisson storm process, giving unit-Frechet margins and a closed-form pairwise dependence. The extremal coefficient theta(h) in [1, 2] summarizes it: 1 = full dependence (extremes always co-occur), 2 = independence. Part of the earth-science/UQ work (Phase 6).

class SmithMaxStable(sigma)[source]

Bases: object

The Smith max-stable process with Gaussian storm-profile covariance sigma (d x d, SPD).

A spatial process, not an i.i.d. leaf distribution: its full likelihood is intractable, so it is not a SequenceEncodableProbabilityDistribution – it exposes the things that do have closed forms (extremal_coefficient, bivariate_cdf) plus a sampler, and is fitted by the module-level fit_smith_maxstable() (composite/madogram estimation), mirroring the functional fit style of the other non-leaf spatial models. Margins are unit Frechet; spatial dependence grows with sigma.

Parameters:

sigma (np.ndarray)

extremal_coefficient(h)[source]

theta(h) = 2 * Phi(a/2) with a the Mahalanobis lag length – 1 at h=0 (full dependence) rising to 2 as the lag grows (independence).

Parameters:

h (ndarray)

Return type:

float

bivariate_cdf(z1, z2, h)[source]

P(Z(s) <= z1, Z(s+h) <= z2) = exp(-V(z1, z2)) – the Smith bivariate distribution.

Parameters:
Return type:

float

sampler(locations, seed=None)[source]
Parameters:
Return type:

SmithMaxStableSampler

fit_smith_maxstable(locations, fields)[source]

Fit an isotropic Smith max-stable process (sigma = s^2 I) to replicated spatial extremes.

locations is (n_locations, d) and fields is (n_replicates, n_locations) of block maxima. Estimation matches the binned empirical extremal coefficient (from the F-madogram) to the model 2 Phi(|h| / (2 s)). Returns a SmithMaxStable.

Parameters:
Return type:

SmithMaxStable