Adaptive Wavelet Domain Principal Component Analysis for Nonstationary Time Series

High-dimensional multivariate nonstationary time series, that is, data whose second order properties vary over time, are common in many scientific and industrial applications. In this article we propose a novel wavelet domain dimension reduction technique for nonstationary time series. By constructing a time-scale adaptive principal component analysis of the data, our proposed method is able to capture the salient dynamic features of the multivariate time series. We also introduce a new time and scale dependent <i>cross-coherence</i> measure to quantify the extent of association between a multivariate nonstationary time series and its proposed wavelet domain principal component representation. Theoretical results establish that our associated estimation scheme enjoys good bias and consistency properties when determining wavelet domain principal components of input data. The proposed method is illustrated using extensive simulations and we demonstrate its applicability on a real-world dataset arising in a neuroscience study. Supplementary materials, with proofs of theoretical results, additional simulations and code, are available online.