Principal component analysis for second-order stationary vector time series

TL;DR

This paper extends principal component analysis to second-order stationary vector time series, enabling segmentation into uncorrelated subseries for improved analysis and forecasting of high-dimensional linear dynamical structures.

Contribution

It introduces a PCA-based method for segmenting vector time series into uncorrelated subseries, with asymptotic theory for both fixed and diverging dimensions, and demonstrates its effectiveness through simulations and real data.

Findings

Effective dimension reduction in high-dimensional time series analysis.

Improves forecasting by approximate segmentation when exact linear transformation does not exist.

Method applicable to volatility process segmentation.

Abstract

We extend the principal component analysis (PCA) to second-order stationary vector time series in the sense that we seek for a contemporaneous linear transformation for a $p$-variate time series such that the transformed series is segmented into several lower-dimensional subseries, and those subseries are uncorrelated with each other both contemporaneously and serially. Therefore those lower-dimensional series can be analysed separately as far as the linear dynamic structure is concerned. Technically it boils down to an eigenanalysis for a positive definite matrix. When $p$ is large, an additional step is required to perform a permutation in terms of either maximum cross-correlations or FDR based on multiple tests. The asymptotic theory is established for both fixed $p$ and diverging $p$ when the sample size $n$ tends to infinity. Numerical experiments with both simulated and real data sets indicate that the proposed method is an effective initial step in analysing multiple time series data, which leads to substantial dimension reduction in modelling and forecasting high-dimensional linear dynamical structures. Unlike PCA for independent data, there is no guarantee that the required linear transformation exists. When it does not, the proposed method provides an approximate segmentation which leads to the advantages in, for example, forecasting for future values. The method can also be adapted to segment multiple volatility processes.