Spatial quantile autoregression for season within year daily maximum temperature data

Regression is the most widely used modeling tool in statistics. Quantile regression offers a strategy for enhancing the regression picture beyond custom-ary mean regression. With time series data, we move to quantile autoregression and, finally, with spatially referenced time series, we move to space-time quantile regression. Here, we are concerned with the spatio-temporal evolution of daily maximum temperature, particularly with regard to extreme heat. Our motivating dataset is 60 years of daily summer maximum temperature data over Aragón in Spain. Hence, we work with time on two scales—days within summer season across years—collected at geo-coded station locations. For a specified quantile, we fit a very flexible mixed effects autoregressive model, introducing four spatial processes. We work with asymmetric Laplace errors to take advantage of the available conditional Gaussian representation for these distributions. Further, while the auto-regressive model yields conditional quantiles, we demonstrate how to extract marginal quantiles with the asymmetric Laplace specification. Thus, we are able to interpolate quantiles for any days within years across our study region.