Optimal dimension reduction for high-dimensional and functional time series

  • Marc HallinEmail author
  • Siegfried Hörmann
  • Marco Lippi


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 \(K\le p\), 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 Brillinger in 1981 in the vector case and, in the functional case, their functional extension proposed by Hörmann, Kidziński and Hallin in 2015.


Dimension reduction Time series Principal components Functional principal components Dynamic principal components Karhunen–Loève expansion 


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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.ECARESUniversité libre de BruxellesBrusselsBelgium
  2. 2.Département de MathématiqueUniversité libre de BruxellesBrusselsBelgium
  3. 3.Institute for StatisticsGraz University of TechnologyGrazAustria
  4. 4.Einaudi Institute for Economics and FinanceRomeItaly

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