The Annals of Statistics

Principal component analysis for second-order stationary vector time series

Jinyuan Chang, Bin Guo, and Qiwei Yao

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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 analyzed 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 analyzing 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.

Article information

Ann. Statist., Volume 46, Number 5 (2018), 2094-2124.

Received: September 2016
Revised: July 2017
First available in Project Euclid: 17 August 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62M10: Time series, auto-correlation, regression, etc. [See also 91B84]
Secondary: 62H25: Factor analysis and principal components; correspondence analysis

$\alpha$-mixing autocorrelation cross-correlation dimension reduction eigenanalysis high-dimensional time series weak stationarity


Chang, Jinyuan; Guo, Bin; Yao, Qiwei. Principal component analysis for second-order stationary vector time series. Ann. Statist. 46 (2018), no. 5, 2094--2124. doi:10.1214/17-AOS1613.

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Supplemental materials

  • Supplement to “Principal component analysis for second-order stationary vector time series”. This supplement contains simulation studies and all technical proofs.