Motivated by the problem of modeling time-varying quantiles in a way that provides rich quantitative information, we consider a class of models to describe the dynamics of a specific quantile for both univariate and multivariate time series data. This prompts us to present several methodological and computational contributions to dynamic quantile modeling, and, more generally, non-Gaussian time-varying models.
We begin with a discussion of the existing quantile estimation literature and the scope to which our methods contribute to the statistical community. We also discuss background on methods for atmospheric river characterization, an application that in part motivated this work and resurfaces throughout.
In the second chapter, we develop a flexible dynamic quantile linear model (exDQLM) utilizing a recently developed family of parametric distributions for quantile regression. A simulation study illustrates our exDQLM to be more robust than the standard Bayesian parametric quantile regression approach for non-standard distributions, performing better in both quantile estimation and predictive accuracy. In addition to a Markov chain Monte Carlo (MCMC) algorithm, we develop an efficient importance sampling variational Bayes (ISVB) algorithm for fast approximate Bayesian inference which is found to produce comparable results to the MCMC in a fraction of the computation time.
In the third chapter, we apply the exDQLM to the analysis of the integrated water vapor transport (IVT) magnitude quantile threshold, a primary component of many atmospheric river detection schemes. In contrast to current estimation methods, our methodology enables versatile, structured, and informative estimation of the threshold. Further, we develop a transfer function extension to our exDQLM as a method for quantifying non-linear relationships between a quantile of a climatological response and an input. The utility of our transfer function exDQLM is demonstrated in capturing both the immediate and lagged effects of El Niño Southern Oscillation Longitude Index on the estimation of the 0.85 quantile IVT.
In the fourth chapter, we present the R package exdqlm as a tool for dynamic quantile regression. The main focus of the package is to provide a framework for Bayesian inference and forecasting of exDQLMs by implementing the methods detailed in the previous two chapters. Non-time-varying quantile regression models, which comprise a majority of the current statistical software, are discussed as a special case of our methods. The software provides the choice of two different algorithms, MCMC or ISVB, for posterior inference. Routines for estimation of a nonlinear relationship via a transfer function model are available as well as routines for forecasting and model evaluation. We illustrate the implementation of the functions and algorithms in the exdqlm package with a step-by-step guide for the analysis of several real data sets.
In the fifth chapter, we develop a multivariate extended dynamic quantile linear model to consider multiple time series simultaneously and jointly estimate a specified quantile for each series. To do this, we first develop a multivariate exAL distribution. We then present the details of multivariate MCMC and ISVB algorithms for exact and approximate posterior inference, respectively. The utility of the multivariate model is illustrated via application to two real datasets, including an IVT dataset spanning all of CA.
Finally, we conclude with a brief review of the methodological and computational contributions presented, and discuss possible future work.
