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

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$$ Q $$‐score. In this article, we investigate properties of the Q$$ Q $$‐score, which is a quotient emphasizing model fit for the largest 5% of earthquakes. We explore the theoretical properties of the Q$$ Q $$‐score, demonstrating that under certain null conditions, the expectations of the numerator and denominator are equal and thus the expectation of Q$$ Q $$ is 1 in a ratio sense. Additionally, the score satisfies a law of large numbers. We evaluated the Q$$ Q $$‐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 of how well different models perform, both for predicting the largest events and also in terms of overall model fit.