Optimal dimension reduction for high-dimensional and functional time series. (English) Zbl 1450.62111

Summary: Dimension reduction techniques are at the core of the statistical analysis of high-dimensional and functional observations. Whether the data are vector- or function-valued, principal component techniques, in this context, play a central role. The success of principal components in the dimension reduction problem is explained by the fact that, for any \(K\leq p\), the \(K\) first coefficients in the expansion of a \(p\)-dimensional random vector \(\mathbf{X}\) in terms of its principal components is providing the best linear \(K\)-dimensional summary of \(\mathbf X\) in the mean square sense. The same property holds true for a random function and its functional principal component expansion. This optimality feature, however, no longer holds true in a time series context: principal components and functional principal components, when the observations are serially dependent, are losing their optimal dimension reduction property to the so-called dynamic principal components introduced by D. R. Brillinger [Time series. Data analysis and theory. Expand. ed. San Francisco etc.: Holden-Day, Inc. (1981; Zbl 0486.62095)] in the vector case and, in the functional case, their functional extension proposed by Hörmann, Kidziński and Hallin [S. Hörmann et al., J. R. Stat. Soc., Ser. B, Stat. Methodol. 77, No. 2, 319–348 (2015; Zbl 1414.62133)].


62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62R10 Functional data analysis
62H25 Factor analysis and principal components; correspondence analysis
62G08 Nonparametric regression and quantile regression


freqdom; freqdom.fda
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