Nonparametric quantile regression for time series with replicated observations and its application to climate data

Journal: Statistical Science 

Abstract: In climate studies, often data are collected in a way where a variable of interest is observed many times, but the factors impacting it are recorded once for every time-point. For example, it is believed that the average sea-surface temperature has some effect on how cyclones appear in a particular year; and thus, the temperature is calculated as a yearly average while the number of cyclones can be anything within a year. Such datasets, in statistics, are known as time series data with replicated observations. 

In this paper, the authors work with the above setting and focus on the estimation of quantiles to understand how extreme behaviour of a variable can be analysed. They propose a model-free nonparametric estimator of conditional quantile of a time series regression model where the covariate vector is repeated many times for different values of the response. Although the use of quantile regression is standard in such studies, the opportunity to improve the results using the replicated nature of data is increasingly realized. The proposed method exploits this feature of the data and improves on the restrictive linear model structure of conventional quantile regression.  

Relevant asymptotic theories for the nonparametric estimators of the mean and variance function of the model are derived under a very general framework. Subsequently, via a detailed simulation study, the authors demonstrate the gain in efficiency of the proposed method over other benchmark models, especially when the true data generating process entails nonlinear mean function and heteroskedastic pattern with time-dependent covariates. It is established that he predictive accuracy of the nonparametric method is remarkably high compared to other approaches when attention is on the higher quantiles of the variable of interest. 

As mentioned above, this type of data abounds in climate studies. In fact, for application of the new method proposed in this paper, the authors take two interesting examples. First one is a well-known tropical cyclone wind-speed data, where higher quantiles of the wind-speed are connected to greater chances of cyclones. The authors analyse how sea-surface temperature and southern oscillation index may affect different levels of the wind-speed. In the second application, an air pollution data is studied in detail. Here, the pollutants’ information are collected at a daily level from various stations, thus rendering multiple observations per timepoint; whereas the covariates, such as trend and seasonality, are considered to be fixed for every timepoint. The analysis of both of these datasets ascertain the efficacy of the newly proposed technique, over other existing parametric or nonparametric approaches.