Sliced average variance estimation for multivariate time series

Abstract

Supervised dimension reduction for time series is challenging as there may be temporal dependence between the response y and the predictors x. Recently a time series version of sliced inverse regression, TSIR, was suggested, which applies approximate joint diagonalization of several supervised lagged covariance matrices to consider the temporal nature of the data. In this paper we develop this concept further and propose a time series version of sliced average variance estimation, TSAVE. As both TSIR and TSAVE have their own advantages and disadvantages, we consider furthermore a hybrid version of TSIR and TSAVE.Based on examples and simulations we demonstrate and evaluate the differences between the three methods and show also that they are superior to apply their iid counterparts to when also using lagged values of the explaining variables as predictors.

Dimension Reduction for Time Series in a Blind Source Separation Context Using R

“…Supervised dimension reduction in case of time series data is however much more difficult as the dependence between y t and x t may also lag in time. To address this problem Matilainen, Croux, Nordhausen, and Oja (2017a) and Matilainen, Croux, Nordhausen, and Oja (2019) suggested time series versions of SIR and SAVE as well as a weighted linear combination of the two methods. The three approaches are denoted TSIR, TSAVE and TSSH (time series SIR SAVE hybrid), respectively, and recalled in the following.…”

“…TSIR and TSAVE use similar approximate joint diagonalization as the unsupervised methods considered in previous sections. For TSIR Matilainen et al (2017a) suggest to use matrices G 0,τ (z t , y t ) = Cov(E(z t |y t+τ )), and for TSAVE the matrices G 1,τ (z t , y t ) = E((I p − Cov(z t |y t+τ )) 2 ), are suggested in Matilainen et al (2019). The solutions are then given as W = U Cov(x t ) −1/2 , where the orthogonal matrix U ∈ R p×q maximizes…”

“…Also the important lags regarding the sources need to be found. So far no statistical tests for these are available, however Matilainen et al (2017aMatilainen et al ( , 2019 used the pseudo-eigenvalues λ ij = u i G s,τ j (x st t , y t )u i for determining the value of q as follows. Notice that now T = {τ 1 , .…”

“…≥ λ q• and q = p. Then Matilainen et al (2017a) suggested four different strategies for finding the appropriate amount of sources and the lags corresponding to the sources by, similarly to the principal component analysis (PCA), trying to explain 100 × P % of the dependence between the latent sources and the response series. The recommended strategy according to Matilainen et al (2019) is the so called "biggest value" strategy which finds the smallest number r of elements (i 1 , j 1 ), . .…”

“…For details about the other strategies see Matilainen et al (2017a). For the number of slices Matilainen et al (2019) recommend to use H = 10 for TSIR and H = 5 for TSAVE.…”