Kronecker GPs with non-Gaussian likelihoods

Structured Gaussian processes on product domains (space × time, space × wavelength, etc.) paired with non-conjugate observation models. All notebooks lean on gaussx for Kronecker operator dispatch and non-conjugate ELBO helpers, and pyrox.gp for kernels and likelihood wrappers.

Notebooks¶

• 01_spain_extremes — annual maxima on a space × time grid over Spain, modelled with an additive space + time GP and a Generalized Extreme Value observation likelihood. Inference is a custom mean-field variational GP with closed-form KL + Gauss–Hermite ELL; the extreme-value angle gives us return-level maps for current and projected climates.
• 02_spain_multiplicative — upgrades the scalar amplification factor β to a spatial GP $\beta(s)$, replacing the additive Kronecker-sum prior with a gaussx.SumKronecker of two rank-1-in-time Kronecker products ($K_\mu \otimes J_T + K_\beta \otimes dd^\top$). Yields a spatially heterogeneous 2024 $\to$ 2050 warming map.
• 03_spain_nonstationary — promotes the GEV scale σ and shape ξ from globals to per-location spatial GPs, giving four parallel latent fields ($\mu, \tilde\beta, \tilde\sigma, \tilde\xi$) with a 4-way mean-field posterior. The ELL becomes a 3-D Gauss–Hermite tensor-product quadrature — same gaussx.GaussHermiteIntegrator API, just a 3-D GaussianState. Produces a spatially varying 100yr$-$25yr return-gap map driven by the local tail $\hat\xi(s), \hat\sigma(s)$.
• 04_spain_copula — adds a Gaussian copula with a two-range exponential dependogram $R_{ij} = c_0 e^{-d/c_1} + (1-c_0) e^{-d/c_2}$ on the per-year cross-station residuals, ported from jej_vc_snippets/extremes/models/temp_gevd_gp_copula.py into the variational scaffold via plug-in / IFM. Synthetic data now generated jointly via Sklar’s inverse ($z_t \sim \mathcal{N}(0, R^*) \to u_t = \Phi(z_t) \to y_t = F_{\mathrm{GEV}}^{-1}(u_t)$). Joint return-period probabilities for clusters of nearby stations come out 3+ orders of magnitude larger than under conditional independence.

Follow-up directions (planned, not yet written):

• Full-rank multiplicative $K_s \otimes K_t$ with a Kronecker-structured variational posterior (lets the time response be non-linear per location instead of tied to the GMST basis).
• Temporal non-stationarity in $\sigma, \xi$ (e.g. $\log\sigma(s, t) = \log\sigma_0 + \tilde\sigma(s) + \gamma(s) \cdot d(t)$) — same SumKronecker recipe, one more spatial GP.
• Tail-dependent copulas (Student-$t$, Gumbel) — the Gaussian copula is asymptotically tail-independent; non-Gaussian copulas capture the tendency of extreme heatwaves to “synchronise” across regions.