Nonparametric and quantile regression approaches for estimating spatiotemporal climate trends

Richard Chandler, UCL

This work, carried out jointly with Ken Liang and Bryson Bates, was motivated by the need to characterise long-term trends in rainfall over a relatively large region of southwest Western Australia. Over the last 40 years, some parts of the area have experienced a severe rainfall decline whereas others have remained relatively unaffected: a realistic characterisation of the trends must therefore accommodate spatial variation. Moreover, the structure in the data does not lend itself obviously to a parametric representation of the variation, either spatially or temporally. It is therefore natural to consider nonparametric regression methods. In the first instance, a fairly standard application of generalised additive models (GAMs), using tensor product spline bases for three-dimensional smoothing (two dimensions of space and one of time) and with inference adjusted for residual spatial dependence, is demonstrated as a compelling way to visualise the underlying structure in the data. This analysis led directly to a decision by the State Water Supply planning officers to double the capacity of a seawater desalination plant, at a cost of AUS$450 million. Subsequently, the work has been extended using quantile regression, to examine trends in different parts of the rainfall distributions. This leads to some interesting statistical and computational challenges: existing nonparametric quantile regression algorithms struggle to cope with smoothing in more than a couple of dimensions, and inference - even in the absence of residual dependence - is notoriously complicated due to the nondifferentiability of the quantile regression objective function. We get around these problems by working with a smooth approximation to the objective function, and by using a carefully chosen iterative weighted least squares algorithm for the estimation. This approach simultaneously regularises the inference (at the cost of some asymptotically negligible approximations) and casts the estimation problem into a form that can be solved using off-the-shelf GAM routines - thus providing the opportunity for smoothing in higher dimensions than can be achieved using existing quantile regression software. Simulation studies indicate that the approach performs extremely well. It is then applied to the Australian rainfall data to answer questions such as: is the decline due to a decrease in the frequency of particularly wet winters (corresponding to the upper quantiles of the winter rainfall distribution); or are dry years getting preferentially drier (lower quantiles); or is it simply that the entire distribution is shifting downwards? The answers to these questions can provide insights into the physical mechanisms that might be driving the rainfall decline.