Optimal dimension reduction for high-dimensional and functional time series

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 $$K\le p$$ , 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 Hormann, Kidzinski and Hallin in 2015.