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Bayesian Modelling of Nonstationary Extreme Values Using a Nonparametric Hawkes Process

This paper proposes a Bayesian nonparametric Hawkes process model coupled with a hierarchical Generalised Pareto Distribution to effectively forecast nonstationary extreme events by capturing self-exciting clustering patterns and varying magnitudes across clusters, demonstrating superior predictive performance on both simulated and real-world data.

Original authors: Gordon J. Ross, Dean Markwick

Published 2026-05-06

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Gordon J. Ross, Dean Markwick

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 when the next "big storm" will hit. In the world of data, these storms are extreme events: a sudden crash in the stock market, a massive earthquake, or a spike in terrorism fatalities.

For a long time, statisticians tried to predict these storms using a very rigid rulebook. They assumed that storms happen at a steady, predictable pace (like rain in a steady drizzle) and that every storm is roughly the same size. But as the authors of this paper point out, real life doesn't work like that. Storms often come in clusters (a sudden burst of heavy rain followed by a drought), and the size of the storms can change depending on the season or the current situation.

This paper introduces a new, smarter way to model these chaotic events. Think of it as upgrading from a simple weather vane to a sophisticated, self-learning weather radar.

Here is how their new system works, broken down into simple concepts:

1. The "Self-Exciting" Alarm Clock (The Hawkes Process)

Traditional models treat events as independent. If a stock market drops today, a traditional model might think, "Okay, that's done. The chance of a drop tomorrow is just the average chance."

The authors use something called a Hawkes Process. Imagine a room full of people. If one person sneezes, it might make others sneeze too. If a second person sneezes, it might trigger a whole chain reaction.

The Metaphor: Think of an earthquake. When the ground shakes (an event), it makes the ground unstable, increasing the chance of an aftershock (another event) happening soon after.
The Innovation: The authors realized that extreme events often behave like this sneeze-chain reaction. They built a model where one extreme event "excites" the system, making it more likely for another extreme event to happen immediately after. This explains why bad days in the stock market often come in clusters.

2. The Shape-Shifting Trigger (The Nonparametric Kernel)

In the old "sneeze" model, you had to guess how the sneeze spreads. Does it last for 5 minutes? 1 hour? You had to pick a fixed shape (like a bell curve) and hope it was right. If you picked the wrong shape, your predictions would be off.

The authors say, "Why guess? Let the data tell us the shape."

The Metaphor: Instead of forcing the sneeze to fit a pre-made cookie cutter, they use a moldable clay approach (called a Dirichlet Process). They let the data sculpt the shape of the "excitement" itself.
The Result: If the data shows that events cluster tightly for a few hours and then fade, the model learns that shape. If they cluster for days, the model learns that too. It's flexible enough to handle weird, unpredictable patterns without the user having to guess the rules in advance.

3. The "Team-Based" Storm Sizes (Hierarchical GPD Marks)

Once the model knows when a storm is likely to happen, it needs to guess how big it will be.

The Problem: In the old days, statisticians assumed every storm, whether it happened in a calm year or a chaotic year, was drawn from the same "bag of sizes."
The Innovation: The authors realized that storms in a "cluster" (a chaotic week) might be bigger than storms in a "calm" week. However, some clusters are so small they don't have enough data to guess the size accurately on their own.
The Metaphor: Imagine a sports league. If you have a team with only 3 players, you can't really tell how good they are. But if you look at all the teams in the league, you can get a general idea of the skill level.
- The authors use a hierarchical model. They let each "cluster" of events have its own average size (scale), but they force all these clusters to share a common "tail shape" (how extreme the worst-case scenarios can get).
- This allows the model to say, "This specific cluster of events is usually bigger than average," while still borrowing strength from the whole dataset to make sure the prediction for the worst possible event is accurate.

4. Putting It All Together: The Prediction Machine

The authors combined these three ideas into a single "Bayesian" model. They tested it on four very different real-world datasets:

Stock Market (S&P 500): Looking for massive drops in value.
Market Volatility (VIX): Looking for spikes in fear.
Wind Speeds: Looking for hurricane-force gusts.
Terrorism: Looking for days with high fatality counts.

The Results:
When they tested their new "flexible clay" model against the old "rigid cookie cutter" models, the new model won every time.

It was better at predicting when the next extreme event would happen because it understood that events come in clusters.
It was better at predicting how big those events would be because it understood that the size of the event depends on which "cluster" or time period it belongs to.

Summary

The paper argues that to predict rare, dangerous events, we need to stop pretending the world is calm and predictable. Instead, we need a model that understands:

Events trigger other events (clustering).
The rules of triggering change (flexible shapes).
The size of the event depends on the current chaos (hierarchical sizes).

By using this approach, the authors created a tool that is significantly better at forecasting the next "big storm" in finance, nature, and security, without needing to make rigid, often incorrect, assumptions about how the world works.

Technical Summary: Bayesian Modelling of Nonstationary Extreme Values Using a Nonparametric Hawkes Process

Problem Statement
Effective risk management requires estimating the probability of large events occurring within a specific timeframe. Traditional approaches often rely on Extreme Value Theory (EVT), specifically the Peaks-Over-Threshold (POT) method, which models exceedances over a threshold $u$ using a Generalised Pareto Distribution (GPD). However, standard POT methods assume stationarity: that exceedance times follow a homogeneous Poisson process and that exceedance magnitudes are independent and identically distributed (i.i.d.) draws from a single GPD.

Real-world data, such as financial market indices (e.g., S&P 500, VIX), often violate these assumptions. Extreme events frequently exhibit clustering (temporal nonstationarity) and structural changes in magnitude (mark nonstationarity). For instance, periods of high volatility may see both a higher frequency of extreme drops and larger magnitudes of those drops. Existing nonparametric Bayesian models often focus on retrospective smoothing of historical data rather than explicit modeling of the conditional intensity required for future prediction. Furthermore, parametric extensions for nonstationarity often require strong assumptions about the functional form of time-varying parameters, risking over- or under-fitting.

Methodology
The authors propose a Bayesian marked Hawkes POT model that addresses nonstationarity in both the time and mark domains through a hierarchical structure.

Nonparametric Hawkes Process for Exceedance Times:
- Instead of a homogeneous Poisson process, the occurrence times of exceedances are modeled using a self-exciting Hawkes process. This allows the conditional intensity $\lambda(t | H_t)$ to increase following an event, naturally capturing temporal clustering.
- The Hawkes intensity is defined as $\lambda(t | H_t) = \mu + \sum_{t_i < t} \kappa h(t - t_i)$ , where $\mu$ is the background intensity, $\kappa$ is the branching ratio, and $h(\cdot)$ is the triggering kernel.
- Crucially, the triggering kernel $h(\cdot)$ is modeled nonparametrically using a Dirichlet Process (DP) mixture. Specifically, $h(t)$ is represented as an infinite mixture of Lognormal distributions, where the mixing distribution $G$ follows a Dirichlet Process prior. This allows the model to learn complex excitation patterns (e.g., heavy tails, multimodality) directly from the data without imposing a fixed parametric form.
Hierarchical GPD for Exceedance Magnitudes:
- The magnitudes of exceedances ( $y_i$ ) are modeled using the Generalised Pareto Distribution (GPD).
- To handle nonstationarity in magnitudes, the authors utilize the latent branching structure of the Hawkes process. Events are grouped into clusters based on the time intervals between background (exogenous) events.
- A hierarchical model is applied to the GPD scale parameter $\sigma_k$ for each cluster $k$ : $\log \sigma_k = \log \sigma_0 + \tau_\sigma z_k$ , where $z_k \sim N(0, 1)$ .
- The shape parameter $\xi$ is shared globally across all clusters. This "partial pooling" approach allows the model to borrow strength across clusters, enabling accurate estimation of the scale parameter even in clusters with few observations, while maintaining a stable global estimate for the tail shape.
Posterior Inference:
- A Markov Chain Monte Carlo (MCMC) algorithm is developed to sample from the joint posterior distribution.
- The algorithm uses a modular factorization: the Hawkes process parameters (including the latent branching variables $B$ ) are updated based on exceedance times, and the GPD mark parameters are updated conditional on the current clustering structure.
- The nonparametric kernel $h(\cdot)$ is updated using a Chinese Restaurant Process sampler combined with a Metropolis–Hastings correction for the integrated hazard term.
- The GPD parameters are updated using Hamiltonian Monte Carlo (via Stan).

Key Contributions
The paper makes three primary contributions:

Predictive Focus: It develops a marked Hawkes POT model specifically designed for forecasting future threshold exceedances, distinguishing it from previous nonparametric Bayesian point process models that primarily smooth historical data.
Nonparametric Excitation: It introduces a Dirichlet process mixture prior for the Hawkes triggering kernel, allowing the temporal excitation pattern to be learned flexibly from data rather than assuming a specific parametric form (e.g., exponential).
Hierarchical Mark Modeling: It links the latent branching structure of the Hawkes process to a hierarchical GPD mark model. This allows the scale of exceedance magnitudes to vary across Hawkes-induced regimes while sharing information across clusters, avoiding the instability of fitting separate GPDs to small clusters.

Results
The model was evaluated through a simulation study and applied to four real-world datasets: S&P 500 daily log returns, VIX closing values, Bradfield wind speeds, and the RAND-MIPT Terrorism database.

Simulation Study: In a controlled $2 \times 2$ design (varying kernel type and mark distribution), the results confirmed that:
- When data were generated with a non-exponential kernel, the DP-Hawkes model outperformed the parametric Exponential-Hawkes model.
- When data exhibited cluster-specific mark variation, the hierarchical GPD model outperformed the i.i.d. GPD model.
- When the data-generating mechanism did not require the extra flexibility (e.g., true exponential kernel), the flexible models incurred little predictive penalty, demonstrating robustness.
Real Data Applications:
- Across all four datasets, the full model (DP-Hawkes kernel + Hierarchical GPD marks) achieved the best held-out predictive log scores.
- The nonparametric kernel improved the prediction of exceedance times, while the hierarchical mark model improved the prediction of exceedance magnitudes.
- The improvement was most substantial for financial and wind speed data, with smaller but positive gains for terrorism data.
- Posterior predictive checks showed that the model's predictive intervals for maximum future excesses were consistent with observed data, with the terrorism dataset showing a heavier tail (positive $\xi$ ) as expected.

Significance and Claims
The authors claim that their framework successfully relaxes the strict stationarity assumptions of standard EVT, providing a more realistic tool for forecasting nonstationary extreme events. The significance lies in the demonstration that Hawkes-induced clustering is useful not only for predicting the timing of extremes but also for modeling changes in their magnitudes.

By combining self-exciting point processes with hierarchical extreme value models, the paper suggests a promising approach for forecasting where both the rate of occurrence and the severity of events vary over time. The authors modestly note that while the shared-shape specification is a compromise between fully stationary and fully local models, it effectively balances the need to capture local magnitude variations with the stability required for tail-shape inference in small clusters. The work does not claim to solve all nonstationarity issues but offers a flexible, data-driven alternative to parametric regression frameworks for time-varying extreme value parameters.