Empirical Dynamic Quantiles for Visualization of High-Dimensional Time Series

Abstract

The empirical quantiles of independent data provide a good summary of the underlying distribution of the observations. For high-dimensional time series defined in two dimensions, such as in space and time, one can define empirical quantiles of all observations at a given time point, but such time-wise quantiles can only reflect properties of the data at that time point. They often fail to capture the dynamic dependence of the data. In this article, we propose a new definition of empirical dynamic quantiles (EDQ) for high-dimensional time series that mitigates this limitation by imposing that the quantile must be one of the observed time series. The word dynamic emphasizes the fact that these newly defined quantiles capture the time evolution of the data. We prove that the EDQ converge to the time-wise quantiles under some weak conditions as the dimension increases. A fast algorithm to compute the dynamic quantiles is presented and the resulting quantiles are used to produce summary plots for a collection of many time series. We illustrate with two real datasets that the time-wise and dynamic quantiles convey different and complementary information. We also briefly compare the visualization provided by EDQ with that obtained by functional depth. The R code and a vignette for computing and plotting EDQ are available at https://github.com/dpena157/HDts/.