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Bayesian Gaussian Process Modeling of Nonstationary Climate Extremes with Unknown Physical Drivers: A GP-GEV Framework for West African Rainfall

This study introduces a Bayesian Gaussian Process–Generalized Extreme Value (GP–GEV) framework that models nonstationary West African rainfall extremes using latent processes instead of uncertain physical covariates, demonstrating superior predictive accuracy and uncertainty calibration compared to traditional stationary and parametric-trend models.

Original authors: Silas Alebikenaba Ayariga, James Azuure Atambire, Nana Kena Frempong

Published 2026-07-31

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Silas Alebikenaba Ayariga, James Azuure Atambire, Nana Kena Frempong

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to predict the weather by looking at a single, frozen snapshot of the sky. That's essentially what old-school climate models have been doing for decades: they assume that the rules of the game never change. They act like a librarian who believes that if it rained heavily in 1990, it will rain exactly the same amount in 2090, because the "average" hasn't shifted. But we all know the world isn't a static painting; it's a living, breathing movie where the plot twists, the characters change, and the weather is getting wilder. This is the world of Extreme Value Theory, a branch of statistics dedicated to understanding the "once-in-a-lifetime" events—like the biggest flood or the hottest heatwave. The tricky part is that these events are rare, and the climate is nonstationary, meaning the rules are changing over time. Scientists have been struggling to figure out why these rules are changing. Is it the ocean getting warmer? Is it pollution? Or is it a complex mix of invisible forces we can't quite name yet? If we can't name the drivers, how can we build bridges, dams, and houses that won't get washed away?

Enter a new approach from a team of researchers who decided to stop guessing the "why" and start mastering the "what." Instead of trying to force the climate data to fit a specific story about ocean temperatures or wind patterns, they built a mathematical detective tool called a Bayesian Gaussian Process. Think of this tool as a super-smart, flexible rubber band. Instead of snapping into a straight line (like a simple trend) or a rigid curve (like a complex formula), this rubber band stretches and bends to fit the actual shape of the data, no matter how wiggly or weird it gets. They applied this to the West African Sahel, a region that has seen dramatic swings from drought to deluge. Their goal was to see if this "shape-shifting" model could predict future rainfall extremes better than the old, rigid methods, especially when we don't know exactly what is driving the changes.

The paper, titled "Bayesian Gaussian Process Modeling of Nonstationary Climate Extremes with Unknown Physical Drivers," proposes a framework called GP-GEV. In simple terms, they combined the "Generalized Extreme Value" (GEV) distribution—which is the standard tool for measuring rare, extreme events—with a "Gaussian Process" (GP), which is a way of modeling how things change over time without needing a specific reason for the change. Imagine you are trying to draw the path of a drunk person walking home. A traditional model might try to guess if they are walking faster because they are happy or slower because they are tired (the "physical drivers"). The GP-GEV model, however, just looks at the wobbly line they left on the ground and says, "Okay, I see the pattern is getting more erratic, so I'll draw a path that accounts for that wobble without needing to know why they are wobbly."

The researchers tested this idea in two ways. First, they ran a massive simulation lab. They created eight different "fake worlds" with different types of weather chaos—some with simple straight-line trends, some with wild, multi-layered oscillations, and some with hidden, nonlinear drivers. They pitted their new GP-GEV model against the old stationary models and other "trend" models. The results were clear: when the weather was doing something simple and straight, the old models held their own. But as soon as the weather got complicated, wiggly, or unpredictable (which is exactly what the real world looks like), the GP-GEV model crushed the competition. It was far more accurate at predicting the size of the "once-in-50-years" storm and, crucially, it was much better at admitting when it wasn't sure. The old models often gave a single number and pretended they were 100% confident, while the GP-GEV model gave a range of possibilities that actually captured the truth about 93% of the time, compared to the old models which were wrong up to 35% of the time.

Then, the team took their tool to the real world, using 83 years of rainfall data from the West African Sahel (from 1940 to 2022). They looked at twelve different locations, from the coast of Guinea to the border of Niger and Mali. What they found was a bit alarming but very important. The old, stationary models suggested that a "50-year flood" (a flood so big it happens once every 50 years on average) would be about 7 to 8 millimeters smaller than what the new GP-GEV model predicted. That might not sound like much, but in the world of engineering and flood protection, that difference is huge. It means that if you build a dam or a levee based on the old math, you are essentially building it too low. The new model showed that the "return levels" (the height of the water we need to prepare for) have been creeping up in a way the old models completely missed. The GP-GEV model also revealed that the weather isn't just getting worse in a straight line; in some places, it's following an "S-shaped" curve, where it was flat during the great droughts of the 1970s and 80s, and then suddenly started accelerating again recently.

The study argues that we can no longer rely on models that assume the climate is static or that require us to know the exact "driver" of the change before we can make predictions. In a region like the Sahel, where the drivers are a messy mix of ocean temperatures, wind patterns, and human activity, waiting to understand the "why" before we plan for the "what" is a dangerous game. The authors suggest that this new, flexible, data-driven approach should become the standard for designing infrastructure in vulnerable regions. By using a model that learns the shape of the chaos directly from the data, rather than forcing the data into a pre-made box, we can get a much clearer, and safer, picture of the risks we face. The paper concludes that while the old methods aren't useless, they are systematically underestimating the danger, and it's time to upgrade our tools to match the complexity of a changing world.

Technical Summary: Bayesian Gaussian Process Modeling of Nonstationary Climate Extremes

Problem Statement
The West African Sahel is experiencing intensified extreme rainfall, rendering conventional stationary extreme value models inadequate for risk assessment. Existing nonstationary approaches typically rely on predefined physical covariates (e.g., ENSO, Atlantic SST) to drive changes in Generalized Extreme Value (GEV) parameters. However, in the Sahel, the relationships between these drivers and rainfall are uncertain, unstable, and subject to complex multi-scale interactions (e.g., reversed ENSO teleconnections since the 1970s). Consequently, current methods face a fundamental tension: they either assume stationarity (ignoring documented trends) or require analysts to specify physical drivers a priori, a task often impossible when drivers are contested or partially observed. This study addresses the gap in modeling temporal nonstationarity without explicit covariate specification, aiming to provide robust uncertainty quantification for infrastructure design and climate adaptation.

Methodology: The GP-GEV Framework
The authors propose a Bayesian Gaussian Process–Generalized Extreme Value (GP-GEV) framework. Unlike traditional methods, this approach treats the temporal evolution of GEV parameters as latent stochastic processes governed by Gaussian Process (GP) priors, learning nonstationarity directly from the data.

Model Structure: The framework is a three-level hierarchical model.
- Observation Level: Annual maximum daily rainfall ( $y_t$ ) follows a GEV distribution with time-varying location ( $\mu_t$ ) and scale ( $\sigma_t$ ) parameters, and a time-invariant shape parameter ( $\xi$ ).
- Process Level: The location parameter $\mu_t$ and the log-scale parameter $\log(\sigma_t)$ are modeled as independent GPs. Two variants are defined: GP-GEV- $\mu$ (only location evolves) and GP-GEV- $\mu\sigma$ (both location and scale evolve).
- Hyperparameter Level: GP priors utilize covariance kernels (Squared Exponential, Matérn-3/2, and Rational Quadratic) to capture various timescales of variability. The Rational Quadratic kernel is highlighted for its ability to model multi-scale nonstationarity (superposition of interannual and multi-decadal variability).
Inference: Due to the non-Gaussian nature of the GEV likelihood, the marginal likelihood is analytically intractable. The study employs a nested Laplace approximation (based on [3]) to approximate the posterior distribution of latent functions. This approach achieves accuracy comparable to full Hamiltonian Monte Carlo but reduces computation time by orders of magnitude, enabling regional analysis across hundreds of ERA5 grid points.
Priors: The study adopts Penalized Complexity (PC) priors to penalize unnecessary departures from stationarity. The shape parameter $\xi$ follows a Normal prior centered on empirical values for daily precipitation maxima.

Simulation Study
To validate the framework, the authors conducted a simulation study across eight Data-Generating Processes (DGPs) spanning stationary, linearly trending, smoothly nonlinear, multi-scale, and covariate-driven regimes.

Benchmarks: The GP-GEV models were compared against a stationary GEV, linear/quadratic trend models, covariate-based models, and spline-based models.
Metrics: Performance was evaluated using RMSE, Mean Absolute Error, Continuous Ranked Probability Score (CRPS), WAIC, and empirical coverage of 95% credible intervals for 50-year return levels.

Key Results

Superior Performance under Nonstationarity: Under nonstationary DGPs (3–8), the GP-GEV framework substantially outperformed all alternatives. Specifically, GP-GEV- $\mu$ achieved RMSE reductions of 35–59% relative to the stationary GEV and 15–40% relative to the best frequentist alternative (NS-Spline).
Uncertainty Calibration: The most significant finding was the calibration of uncertainty intervals. GP-GEV models maintained empirical 95% coverage within 2–5 percentage points of the nominal level across all scenarios. In contrast, stationary and parametric nonstationary models exhibited severe undercoverage (deficits of up to 35 percentage points) under latent-process nonstationarity.
Robustness to Stationarity: Under the stationary null (DGP-1), the GP-GEV models incurred only a negligible performance penalty (2% increase in RMSE) compared to the correctly specified stationary model, demonstrating that the approach does not overfit when nonstationarity is absent.
Two-Parameter Extension: The GP-GEV- $\mu\sigma$ model (modeling both location and scale) provided meaningful improvements (23.5% RMSE reduction) specifically when scale nonstationarity was present, without degrading performance in location-only scenarios.
Return Level Stability: The GP models maintained stable coverage across return periods (20, 50, and 100 years), whereas stationary models showed structural undercoverage that did not improve with longer return periods.

Empirical Application: West African Sahel
The framework was applied to ERA5 annual maximum daily rainfall data (1940–2022) across the West African Sahel domain.

Risk Underestimation: The analysis revealed that stationary models systematically underestimate 50-year return levels by approximately 7.0–7.8 mm compared to the GP-GEV model. The median uplift was 7.3 mm.
Spatial Heterogeneity: The GP-GEV latent trends captured complex, non-monotonic structures, such as an S-shaped trajectory in the Central Sahel (reflecting the drought period of 1970–1990 and subsequent recovery) that linear models failed to detect.
Infrastructure Implications: The study concludes that facilities designed based on stationary 50-year return levels would be underdesigned for current extreme event magnitudes across the region.

Significance and Claims
The paper claims four principal contributions:

Methodological Novelty: It introduces a fully Bayesian hierarchical model that learns temporal nonstationarity without specifying physical drivers, extending GP-based extreme value analysis from the spatial to the temporal domain.
Statistical Robustness: It provides simulation evidence that latent-process models achieve near-nominal uncertainty coverage where parametric and stationary alternatives fail, particularly under complex, multi-scale nonstationarity.
Practical Utility: It demonstrates that modeling both location and scale parameters yields tangible gains in predictive accuracy when heteroscedasticity is present.
Policy Relevance: It offers empirical evidence that stationary frequency analysis systematically underestimates flood risk in the Sahel. The authors argue that for regions with uncertain climatic drivers, latent-process Bayesian models offer a flexible, statistically robust alternative for resilient infrastructure design and climate adaptation policy.

The study concludes that while stationary models remain competitive under true stationarity, the GP-GEV framework is the superior choice for regions like the Sahel where drivers are uncertain and nonstationarity is likely. It recommends adopting nonstationary Bayesian latent-process models as the standard for extreme rainfall frequency analysis in such contexts.

Technical Summary: Bayesian Gaussian Process Modeling of Nonstationary Climate Extremes

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