Fast calibrated additive quantile regression
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Quantile regression (QR) provides a flexible approach for modelling the impact of several covariates on the conditional distribution of the dependent variable, without making any parametric distributional assumption. Motivated by an electricity load forecasting application, we develop a computationally reliable framework for additive QR based on penalized splines and calibrated empirical Bayesian Belief Updating (BU). The key objectives of our proposal are to obtain well calibrated inference about the smooth components and coefficients of the model, combined with fast automatic estimation of smoothing and variance parameters, and to do this for additive model structures as diverse as those usable with GAMs targeting the mean, while maintaining equivalent numerical efficiency and stability. Traditional QR methods are based on the pinball loss or, equivalently, on the Asymmetric Laplace (AL) distribution, whose non-differentiability is an impediment to smoothing parameter estimation and computational efficiency. Hence, the approach proposed here is based on a novel smooth loss function, which corresponds to a generalization of the AL density. This loss is related to kernel quantile estimators, which have favourable statistical properties relative to empirical quantile estimators. By adopting this loss we are able to employ the general BU framework of Bissiri et al. (2016), but to compute by adapting the computationally robust and efficient methods of Wood et al. (2016), while obtaining reliable uncertainty estimates via a novel calibration approach to selection of the "learning rate" required by the BU framework. Our work was motivated by a probabilistic electricity load forecasting application, which we use here to demonstrate the proposed approach. The methods described in this paper are implemented by the qgam R package, available at https://github.com/mfasiolo/qgam.
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