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The EEPAS Model Revisited: Statistical Formalism and a High-Performance, Reproducible Open-Source Framework

Szu-Chi Chung, Chien-Hong Cho, Strong Wen

TL;DR

This work formalizes medium- to long-term earthquake forecasting models EEPAS and PPE by casting them in the inhomogeneous Poisson point-process framework and deriving key components such as $\Delta(m)$ and $\eta(m)$. It integrates this formal foundation into a fully automated, high-performance Python framework (Numba, NumPy, and joblib) configured via modular JSON, with seamless pyCSEP and SeismoStats integration for standardized evaluation. The authors demonstrate reproducibility on the Italy HORUS dataset, achieving results within one hour under identical initialization and delivering an end-to-end pipeline from raw catalogs to parameter estimation and forecasting that improves log-likelihoods and passes comprehensive statistical tests. By situating the framework within an open-source seismology ecosystem and releasing MIT-licensed code, this work lowers barriers to reproducible, rigorous probabilistic seismic forecasting and enables broader adoption and experimentation.

Abstract

While short-term models such as the Short-Term Earthquake Probability (STEP) and Epidemic-Type Aftershock Sequence (ETAS) are well established and supported by open-source software, medium- to long-term models, notably the Every Earthquake a Precursor According to Scale (EEPAS) and Proximity to Past Earthquakes (PPE), remain under-documented and largely inaccessible. Despite outperforming time-invariant models in regional studies, their mathematical foundations are often insufficiently formalized. This study addresses these gaps by formally deriving the EEPAS and PPE models within the framework of inhomogeneous Poisson point processes and clarifying the connection between empirical $Ψ$-scaling regressions and likelihood-based inference. We introduce a fully automated, open-source Python implementation of EEPAS that combines analytical modeling with Numba JIT acceleration, NumPy vectorization, and joblib parallelization, all configured via modular JSON files for usability and reproducibility. Integration with pyCSEP enables standardized evaluation and comparison. When applied to the Italy HORUS dataset, our system reproduces published results within one hour using identical initialization settings. It also provides a comprehensive pipeline from raw catalog to parameter estimation, achieving improved log-likelihoods and passing strict consistency tests without manual $Ψ$ identification. We position our framework as part of a growing open-source ecosystem for seismological research that spans the full workflow from data acquisition to forecast evaluation. Our framework fills a key gap in this ecosystem by providing robust tools for medium- to long-term statistical modeling of earthquake catalogs, which is an essential but underserved component in probabilistic seismic forecasting.

The EEPAS Model Revisited: Statistical Formalism and a High-Performance, Reproducible Open-Source Framework

TL;DR

This work formalizes medium- to long-term earthquake forecasting models EEPAS and PPE by casting them in the inhomogeneous Poisson point-process framework and deriving key components such as and . It integrates this formal foundation into a fully automated, high-performance Python framework (Numba, NumPy, and joblib) configured via modular JSON, with seamless pyCSEP and SeismoStats integration for standardized evaluation. The authors demonstrate reproducibility on the Italy HORUS dataset, achieving results within one hour under identical initialization and delivering an end-to-end pipeline from raw catalogs to parameter estimation and forecasting that improves log-likelihoods and passes comprehensive statistical tests. By situating the framework within an open-source seismology ecosystem and releasing MIT-licensed code, this work lowers barriers to reproducible, rigorous probabilistic seismic forecasting and enables broader adoption and experimentation.

Abstract

While short-term models such as the Short-Term Earthquake Probability (STEP) and Epidemic-Type Aftershock Sequence (ETAS) are well established and supported by open-source software, medium- to long-term models, notably the Every Earthquake a Precursor According to Scale (EEPAS) and Proximity to Past Earthquakes (PPE), remain under-documented and largely inaccessible. Despite outperforming time-invariant models in regional studies, their mathematical foundations are often insufficiently formalized. This study addresses these gaps by formally deriving the EEPAS and PPE models within the framework of inhomogeneous Poisson point processes and clarifying the connection between empirical -scaling regressions and likelihood-based inference. We introduce a fully automated, open-source Python implementation of EEPAS that combines analytical modeling with Numba JIT acceleration, NumPy vectorization, and joblib parallelization, all configured via modular JSON files for usability and reproducibility. Integration with pyCSEP enables standardized evaluation and comparison. When applied to the Italy HORUS dataset, our system reproduces published results within one hour using identical initialization settings. It also provides a comprehensive pipeline from raw catalog to parameter estimation, achieving improved log-likelihoods and passing strict consistency tests without manual identification. We position our framework as part of a growing open-source ecosystem for seismological research that spans the full workflow from data acquisition to forecast evaluation. Our framework fills a key gap in this ecosystem by providing robust tools for medium- to long-term statistical modeling of earthquake catalogs, which is an essential but underserved component in probabilistic seismic forecasting.

Paper Structure

This paper contains 21 sections, 45 equations, 9 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Mathematical foundation
  3. Implementation for EEPAS
  4. PPE model fitting
  5. Computing weighting coefficients wi
  6. Fitting parameters in EEPAS model
  7. Optimization of EEPAS
  8. Forecasting via EEPAS model
  9. The Framework
  10. Data Preparation and Seismological Parameter Estimation
  11. Model Fitting and Acceleration
  12. Forecasting
  13. Tests for Grid-Based Forecasts
  14. Evaluations
  15. Reproducing the Results from Literature
  16. ...and 6 more sections

Figures (9)

  • Figure 1: Schematic illustration of the EEPAS model. The total earthquake rate consists of a baseline background rate and a sum of precursor rates. Each precursor rate is composed of three factorized density components (time, magnitude, and space). All earthquakes, whether background or precursor events, are constrained to follow the same magnitude distribution as defined by the Gutenberg–Richter (GR) law.
  • Figure 2: The proposed framework and software stack. (a) Perform exploratory data analysis and estimate seismological parameters. The framework integrates seamlessly with parameter estimation packages and is built upon widely used plotting libraries. (b) Fit the EEPAS and PPE models. Model fitting and forecasting are accelerated using vectorization, parallelization, and just-in-time (JIT) compilation. (c) Generate forecasts and conduct analysis and visualization. (d) The module can be seamlessly integrated with popular statistical testing tools.
  • Figure 3: (a) Example of catalog data. (b) Diagram illustrating warm-up, learning, and testing periods. (c) Definition of the cell-based or polygonal two-dimensional study region.
  • Figure 4: Accelerating event term computation using Numba
  • Figure 5: Parallelizing integrals using joblib
  • ...and 4 more figures