# Optimal dimension reduction for high-dimensional and functional time series

Marc Hallin (), Siegfried Hörmann and Marco Lippi ()
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Siegfried Hörmann: Université libre de Bruxelles

Statistical Inference for Stochastic Processes, 2018, vol. 21, issue 2, No 8, 385-398

Abstract: Abstract 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 $$K\le p$$ K ≤ p , the K first coefficients in the expansion of a p-dimensional random vector $$\mathbf{X}$$ X in terms of its principal components is providing the best linear K-dimensional summary of $$\mathbf X$$ 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 Brillinger in 1981 in the vector case and, in the functional case, their functional extension proposed by Hörmann, Kidziński and Hallin in 2015.

Keywords: Dimension reduction; Time series; Principal components; Functional principal components; Dynamic principal components; Karhunen–Loève expansion (search for similar items in EconPapers)
Date: 2018
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Working Paper: Optimal Dimension Reduction for High-dimensional and Functional Time Series (2017)
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