The Q‐Score: A Magnitude‐Weighted Goodness‐of‐Fit Score for Earthquake Forecasting

Julia Jansson Valter, Alejandra Arjon, Francesco Serafini, & Frederic Schoenberg

Published July 14, 2026, SCEC Contribution #14995

Accurate forecasting of large earthquakes is of great importance, yet most current earthquake forecast evaluation metrics, such as the log-likelihood score, do not give additional weight to large-magnitude events. To this end, magnitude-weighted goodness-of-fit scores for earthquake forecasting have recently been introduced, such as potency weighted log-likelihood and the Q-score. In this paper, we investigate properties of the Q-score, which is a quotient emphasizing model fit for the largest 5% of earthquakes. We explore the theoretical properties of the Q-score, demonstrating that under certain null conditions, the expectations of the numerator and denominator are equal and thus the expectation of Q is 1 in a ratio sense. Additionally, the score satisfies a law of large numbers. We evaluated the Q-score of 21 next-day gridded earthquake forecasts for California, provided by the Collaborative for the Study of Earthquake Predictability (CSEP) for the years 2012, 2014, and 2017. We also calculate the log-likelihood scores of the forecasts to obtain a more comprehensive evaluation on how well different models perform, both for predicting the largest events and also in terms of overall model fit.

Citation
Valter, J., Arjon, A., Serafini, F., & Schoenberg, F. (2026). The Q‐Score: A Magnitude‐Weighted Goodness‐of‐Fit Score for Earthquake Forecasting. Environmetrics, 37(5). https://doi.org/10.1002/env.70113.