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October 2018 Principal component analysis for second-order stationary vector time series
Jinyuan Chang, Bin Guo, Qiwei Yao
Ann. Statist. 46(5): 2094-2124 (October 2018). DOI: 10.1214/17-AOS1613

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 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.

Citation

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Jinyuan Chang. Bin Guo. Qiwei Yao. "Principal component analysis for second-order stationary vector time series." Ann. Statist. 46 (5) 2094 - 2124, October 2018. https://doi.org/10.1214/17-AOS1613

Information

Received: 1 September 2016; Revised: 1 July 2017; Published: October 2018
First available in Project Euclid: 17 August 2018

zbMATH: 06964327
MathSciNet: MR3845012
Digital Object Identifier: 10.1214/17-AOS1613

Subjects:
Primary: 62M10
Secondary: 62H25

Keywords: $\alpha$-mixing , Autocorrelation , cross-correlation , Dimension reduction , eigenanalysis , high-dimensional time series , weak stationarity

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.46 • No. 5 • October 2018
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