Some Methods for Analyzing Big Dependent Data

We consider an approach to analyze big data of time series. Big dependent data are first transformed into functional time series of densities via nonparametric density estimation. We then discuss some tools for exploratory data analysis of the resulting functional time series. The tools employed include K-means cluster analysis and tree-based classification. For modeling, we propose a threshold approximate-factor model and a Hellinger distance autoregressive model for functional time series of continuous densities. The latent factors of factor models are estimated by functional principal component analysis. Cross-validation and Hellinger distance are used to select the number of principal component functions. For prediction of high-dimensional time series, we use the results of cluster analysis to obtain parsimonious models. We demonstrate the proposed analysis by considering the demand of electricity, the behavior of daily U.S. stock returns, and U.S. income distributions.