Fast calibrated additive quantile regression

We propose a novel framework for fitting additive quantile regression models, which provides well calibrated inference about the conditional quantiles and fast automatic estimation of the smoothing parameters, for model structures as diverse as those usable with probabilistic GAMs, while maintaining equivalent numerical efficiency and stability. The inferential and model fitting framework proposed here is at the same time statistically rigorous and computationally efficient, because it adopts the general belief updating framework of Bissiri et al. (2016) to loss based inference, but computes by adapting the stable fitting methods of Wood et al. (2016). We enable the use of computationally efficient methods by proposing a novel smooth generalisation of the pinball loss, which is the loss function traditionally used in quantile regression. The new loss is motivated by its relation to kernel quantile estimators, which have favourable statistical properties relative to empirical quantile estimators. Further, our inferential framework offers reliable uncertainty estimates for the fitted conditional quantile, which is achieved by coupling asymptotic posterior approximations with a novel calibration approach to selection of the learning rate. 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 on the Comprehensive R Archive Network (CRAN).