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A lightweight framework for characterising extreme precipitation events in climate ensembles

This paper presents a lightweight framework developed by the "Ca' Foscari" team for the EVA 2025 Data Challenge, which characterizes extreme precipitation events by reducing spatial problems to univariate order statistics, modeling exceedances with Generalized Pareto distributions and Generalized Additive Models, and addressing temporal dependence through conditional extreme value models to estimate exceedance probabilities and persistent events.

Original authors: Dáire Healy, Isadora Antoniano-Villalobos, Claudia Collarin, Nathan Huet, Ilaria Prosdocimi, Emilia Siviero

Published 2026-03-19

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CC BY 4.0

Original authors: Dáire Healy, Isadora Antoniano-Villalobos, Claudia Collarin, Nathan Huet, Ilaria Prosdocimi, Emilia Siviero

Original paper licensed under CC BY 4.0 (http://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 you are trying to predict the worst possible weather scenarios for a city, but instead of having one weather station, you have 25 scattered across the region. Furthermore, you don't have real historical data for the "once-in-a-millennium" storms; you only have four different computer simulations of what the climate might look like.

This paper is a report from a team of statisticians (the "Ca' Foscari" team) who entered a competition to solve this exact puzzle. Their goal was to figure out how often extreme rain events happen, even ones that are so rare they haven't happened yet in the data they were given.

Here is the story of how they did it, explained without the heavy math jargon.

The Big Problem: Too Much Data, Too Little Time

Usually, to predict extreme weather across 25 different locations, you need a super-complex model that understands how rain in one spot affects rain in another. Think of it like trying to predict how a ripple in a pond spreads to every single corner of a lake. It's computationally expensive (it takes a supercomputer years to run) and very hard to get right.

The team decided to take a shortcut. Instead of modeling the whole pond, they decided to look at specific "summary snapshots" of the water level.

The Strategy: The "Class Rank" Analogy

Instead of tracking all 25 locations, the team asked: "What if we just looked at the worst (or 20th worst, or 23rd worst) rain event happening anywhere in the city on a given day?"

Question 1: They looked at the absolute worst rain event (the 1st place) across the whole city.
Question 2: They looked at the 20th worst event.
Question 3: They looked at the 23rd worst event.

By focusing on just one number per day (the "class rank" of the rain), they turned a massive, complicated 25-dimensional puzzle into a simple, one-dimensional line graph. This is like turning a 3D movie into a 2D sketch; you lose some depth, but you can draw it much faster and still understand the main story.

The Toolkit: The "Flood Gate" (Peaks Over Threshold)

Once they had their simple line graph, they needed to predict the "freak" events that go way off the charts.

They used a method called Peaks Over Threshold (POT). Imagine a flood gate set at a specific height (the threshold).

Setting the Gate: They didn't use a fixed height. They realized rain is seasonal (it rains more in winter), so they adjusted the gate height every month.
The Shape of the Water: For the water that spills over the gate, they used a mathematical shape called the Generalized Pareto Distribution. Think of this as a flexible mold that fits the shape of the "freak" rain events perfectly, whether they are just slightly above the gate or massively huge.
The "Clumping" Problem: Rain doesn't fall in isolated drops; if it storms today, it often storms tomorrow. This creates "clusters" of extreme days. To handle this, they used a technique called Declustering. Imagine you have a string of beads where some are glued together in clumps. They cut the string to keep only the biggest bead from each clump, analyzed those, and then mathematically "stitched" the clumps back together at the end to get the final answer.

The Special Case: The "Hangover" Effect (Question 3)

For the third question, they had to predict if a storm would last for two days in a row.

Standard models assume that if it rains hard today, it might rain hard tomorrow, but the connection gets weaker as the rain gets heavier.
To handle this, they used a Conditional Extreme Value Model. Think of it like a "hangover" model: If you drink a lot of alcohol today (an extreme event), what is the probability you will still be drinking tomorrow? They built a model specifically to track how the "intensity" of the storm decays or persists from one day to the next.

The Grand Finale: The Simulation Factory

Since they couldn't wait 1,000 years to see if their predictions were right, they built a virtual factory.

They took their mathematical models (the "blueprints").
They ran a computer simulation 10,000 times, generating fake weather data that looked exactly like the real climate models but stretched out into the future.
They counted how many times their "fake" storms hit the target thresholds.
The average of these counts became their final answer, and the spread of the results gave them a confidence interval (a safety margin).

The Results: A Mixed Bag

Question 1 (The Worst Rain): They nailed it. Their prediction was very close to the actual answer hidden in the competition data.
Question 2 (The 20th Worst Rain): They were a bit off. Their model underestimated how often this would happen. They realized their "mold" (the mathematical shape) was too simple for this specific type of rain.
Question 3 (Two-Day Storms): They were also a bit off, underestimating the frequency. This is likely because predicting how storms "hang over" to the next day is incredibly tricky.

The Takeaway

The team proved that you don't always need a super-complex, slow-moving machine to solve big problems. By simplifying the data (looking at "class ranks" instead of every single location) and using smart, lightweight statistical tools, they could get answers that were fast, interpretable, and surprisingly accurate.

They admitted that in the rush to be fast, they threw away a lot of information (the 24 other locations they ignored each day), but for the purpose of this competition, the "lightweight framework" was the winning strategy. It's a bit like using a sketch to plan a building: it's not the full architectural blueprint, but it's enough to tell you if the roof will hold.

1. Problem Statement

The paper addresses the EVA 2025 Data Challenge, which required estimating the expected occurrence rates of extreme precipitation events across multiple locations using data from four independent climate model runs (CESM2 Large Ensemble). The specific challenges included:

High-dimensional spatial dependence: Estimating probabilities where extreme events occur simultaneously at multiple sites (e.g., all 25 sites, or at least 6 sites).
Extrapolation: Estimating probabilities for thresholds higher than those observed in the provided training data.
Temporal persistence: Estimating the frequency of extreme events persisting over consecutive days.
Computational constraints: Traditional spatial extreme value models (e.g., max-stable processes) are computationally prohibitive for large ensembles and complex to implement. The authors sought a "lightweight" framework that balances statistical rigor with computational efficiency.

2. Methodology

The authors propose a framework that reduces the spatial problem to a univariate setting using order statistics, combined with a Peaks Over Threshold (POT) approach and a simulation-based ensemble inference strategy.

A. Dimensionality Reduction via Order Statistics

Instead of modeling the full spatial field, the team defined a summary statistic $X_i$ for each day $i$ based on the spatial order statistics of the 25 grid locations:

Question 1 (All 25 sites > 1.7): Modeled as the spatial minimum ( $k=1$ ).
Question 2 (At least 6 sites > 5.7): Modeled as the 20th order statistic ( $k=20$ ).
Question 3 (At least 3 sites > 5 for $\ge$ 2 days): Modeled as the 23rd order statistic ( $k=23$ ).

B. Distributional Modeling (POT & GAMs)

Threshold Selection: A time-varying threshold $u_i$ was selected using Quantile Regression (specifically the 0.95 quantile) modeled as a function of the month to account for strong seasonality. The conditional distribution of data below the threshold was modeled using an Asymmetric Laplace Distribution.
Tail Modeling: Exceedances above the threshold were modeled using the Generalized Pareto (GP) distribution.
Parameter Variation: Generalized Additive Models (GAMs) were used to allow GP parameters (scale $\sigma$ $σ$ and shape $\xi$ $ξ$ ) to vary as functions of the month.
- Q1: Both scale and shape varied by month.
- Q2 & Q3: Shape was kept constant (a common assumption in environmental extremes), while scale varied.

C. Handling Temporal Dependence

Declustering: To satisfy the independence assumption of classical extreme value theory, a "run declustering" method was applied. Clusters of exceedances were identified, and only the maximum of each cluster was retained.
Extremal Index ( $\theta$ ): The extremal index was estimated to correct probabilities from the declustered scale back to the original clustered scale.
Conditional Extreme Value (CEV) Model: For Question 3 (persistence), a semi-parametric CEV model (Heffernan and Tawn, 2004) was employed. This models the dependence between consecutive days ( $X_{i+1}$ given $X_i$ is extreme) by transforming margins to a Laplace distribution and fitting a regression model with non-Gaussian residuals.

D. Ensemble Inference via Simulation

Since the four climate model runs were treated as exchangeable (no single run was favored), the authors used a generative simulation approach:

Fit separate models to each of the four runs.
Simulate synthetic ensembles (50 runs) by randomly selecting one of the four fitted models for each synthetic run.
Generate extreme events based on the fitted GP distributions and temporal dependence structures.
Calculate the target probabilities empirically from the simulated data to derive point estimates and confidence intervals.

3. Key Contributions

Computational Efficiency: The framework avoids complex multivariate spatial modeling by reducing the problem to univariate order statistics, making it feasible to analyze large climate ensembles.
Generative Ensemble Emulation: The method treats climate model runs as exchangeable emulators, using simulation to aggregate inference without imposing restrictive joint distributional assumptions.
Flexible Thresholding: The use of quantile regression with GAMs allows for robust handling of seasonal variations in extreme precipitation thresholds.
Hybrid Dependence Modeling: The integration of standard POT for marginal extremes with the CEV framework for temporal persistence provides a tailored solution for different types of extreme event definitions.

4. Results

The authors provided point estimates and 95% confidence intervals for the three challenge questions (Table 1):

Question	Target Event	Point Estimate	95% CI	Performance
Q1	All 25 sites > 1.7	0.207	(0.146, 0.273)	Accurate: True value (0.24) within CI. Good extrapolation.
Q2	$\ge$ 6 sites > 5.7	0.072	(0.040, 0.108)	Underestimation: True value was higher. Model struggled with extreme tail behavior.
Q3	$\ge$ 3 sites > 5 for $\ge$ 2 days	0.067	(0.035, 0.105)	Underestimation: True value was higher. Marginal modeling limitations affected results.

Q1 Success: The model successfully extrapolated beyond observed data (max observed was 1.53; target was 1.7), demonstrating the utility of allowing the shape parameter to vary by month.
Q2 & Q3 Limitations: The models underestimated the true occurrence rates. The authors noted that allowing the shape parameter to vary in Q2 led to unstable, unrealistic estimates, forcing a trade-off between model complexity and stability.

5. Significance and Discussion

Pragmatic Trade-off: The paper demonstrates that sacrificing some spatial complexity for a "lightweight" order-statistic approach yields informative, interpretable, and asymptotically justified results for large ensembles.
Handling Extrapolation: The framework is effective for extrapolating beyond observed data, provided the underlying distributional assumptions (like the GP tail) are appropriate.
Limitations:
- Information Loss: By focusing on specific order statistics, the method discards a significant amount of spatial information (e.g., 24 out of 25 data points per day for Q1).
- Marginal Fit: The underestimation in Q2 and Q3 suggests that the marginal GP models may not fully capture the extreme tail behavior of the specific climate ensemble, particularly when the shape parameter is constrained.
- Rescaling Ambiguity: The authors discuss the theoretical uncertainty regarding how to perfectly rescale probabilities from declustered data back to the clustered scale in a POT context, noting this as an area for future research.

In conclusion, the paper presents a robust, computationally efficient methodology for characterizing extreme precipitation in climate ensembles, successfully addressing the challenge of spatial and temporal dependence through order statistics and conditional modeling, while highlighting the challenges of extreme tail extrapolation.