Generating Probabilities From Numerical Weather Forecasts by Logistic Regression

Logistic models are studied as a tool to convert output from numerical weather forecasting systems (deterministic and ensemble) into probability forecasts for binary events. A logistic model obtains by putting the logarithmic odds ratio equal to a linear combination of the inputs. As any statistical model, logistic models will suffer from over-fitting if the number of inputs is comparable to the number of forecast instances. Computational approaches to avoid over-fitting by regularisation are discussed, and efficient approaches for model assessment and selection are presented. A logit version of the so called lasso, which is originally a linear tool, is discussed. In lasso models, less important inputs are identified and discarded, thereby providing an efficient and automatic model reduction procedure. For this reason, lasso models are particularly appealing for diagnostic purposes.

💡 Research Summary

The paper investigates the use of logistic regression as a statistical post‑processing tool to convert deterministic and ensemble numerical weather prediction (NWP) outputs into calibrated probability forecasts for binary weather events such as precipitation, snowfall, or severe wind. The authors begin by outlining the theoretical justification for logistic models: by linking the log‑odds of an event to a linear combination of predictor variables, the method captures the inherently nonlinear relationship between raw model fields and event probabilities while remaining computationally tractable.

A central challenge addressed in the study is over‑fitting, which becomes acute when the number of candidate predictors approaches or exceeds the number of forecast cases available for training. In such high‑dimensional settings, maximum‑likelihood estimates of the regression coefficients become unstable, leading to poor out‑of‑sample performance. To mitigate this, the authors explore regularisation techniques that shrink coefficient estimates toward zero, thereby controlling model complexity. Both L2 (ridge) and L1 (lasso) penalties are examined, with particular emphasis on the advantages of L1 regularisation for automatic variable selection.

The novel contribution of the work is the adaptation of the lasso to the logistic (logit) framework—referred to as “logit‑lasso.” While the classic lasso was originally devised for linear regression, the authors demonstrate that the same L1 penalty can be incorporated into the logistic likelihood, yielding a convex optimisation problem that can be solved efficiently using coordinate‑descent or modified gradient‑descent algorithms. The regularisation parameter λ, which governs the strength of the penalty, is tuned via k‑fold cross‑validation and information‑theoretic criteria such as AIC and BIC, ensuring that the selected model balances fit and parsimony.

Model assessment is performed with a suite of probabilistic verification metrics. The Brier score quantifies the mean squared error between forecast probabilities and binary outcomes, while the Receiver Operating Characteristic (ROC) curve and its Area Under the Curve (AUC) evaluate discrimination ability. Reliability diagrams are employed to check calibration, i.e., whether forecast probabilities correspond to observed frequencies. The authors apply the methodology to two experimental datasets: a single deterministic forecast and an ensemble forecast comprising multiple perturbed members. In both cases, the logit‑lasso models outperform standard logistic regression without regularisation, achieving lower Brier scores, higher AUC values, and better reliability. The performance gains are especially pronounced when the predictor set is large (30 or more variables), confirming the efficacy of L1 regularisation in high‑dimensional meteorological contexts.

Beyond predictive skill, the lasso’s inherent variable‑selection property provides valuable diagnostic insight. By examining the non‑zero coefficients, the study identifies which physical fields (e.g., low‑level moisture, convective‑scale instability indices, sea‑surface temperature anomalies) contribute most to the probability of a given event. This information can guide forecasters in understanding model sensitivities, refining predictor suites, and even suggesting new variables for future NWP development.

Finally, the paper outlines a practical workflow for operational implementation: (1) preprocessing of NWP output (standardisation, handling missing values), (2) construction of a candidate predictor matrix, (3) fitting a logit‑lasso model with cross‑validated λ, (4) evaluating out‑of‑sample performance with the aforementioned verification scores, and (5) deploying the calibrated probability forecasts in real‑time decision‑support systems. By integrating regularised logistic regression into the post‑processing chain, weather services can generate reliable probabilistic guidance from existing deterministic or ensemble forecasts without extensive computational overhead, thereby enhancing risk communication and decision making for end‑users.