Empirical Dynamic Quantiles for Visualization of High-Dimensional Time Series

<b>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 <i>dynamic</i> 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</b>https://github.com/dpena157/HDts/.