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Regression Diagnostics meets Forecast Evaluation: Conditional Calibration, Reliability Diagrams, and Coefficient of Determination
Tilmann Gneiting, Johannes Resin
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Model diagnostics and forecast evaluation are two sides of the same coin. A common principle is that fitted or predicted distributions ought to be calibrated or reliable, ideally in the sense of auto-calibration, where the outcome is a random draw from the posited distribution. For binary responses, this is the universal concept of reliability. For real-valued outcomes, a general theory of calibration has been elusive, despite a recent surge of interest in distributional regression and machine learning. We develop a framework rooted in probability theory, which gives rise to hierarchies of calibration, and applies to both predictive distributions and stand-alone point forecasts. In a nutshell, a prediction - distributional or single-valued - is conditionally T-calibrated if it can be taken at face value in terms of the functional T. Whenever T is defined via an identification function - as in the cases of threshold (non) exceedance probabilities, quantiles, expectiles, and moments - auto-calibration implies T-calibration. We introduce population versions of T-reliability diagrams and revisit a score decomposition into measures of miscalibration (MCB), discrimination (DSC), and uncertainty (UNC). In empirical settings, stable and efficient estimators of T-reliability diagrams and score components arise via nonparametric isotonic regression and the pool-adjacent-violators algorithm. For in-sample model diagnostics, we propose a universal coefficient of determination, R^∗ = (DSC-MCB)/UNC, that nests and reinterprets the classical R² in least squares (mean) regression and its natural analogue R¹ in quantile regression, yet applies to T-regression in general, with MCB ≥0, DSC ≥0, and R^∗ ∈[0,1] under modest conditions.
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