Computes the dynamic Karhunen-Loeve expansion of a vector time series up to a given order.
dpca.KLexpansion(X, dpcs)A \((T\times d)\)-matix. The \(\ell\)-th column contains the \(\ell\)-th data point.
a vector time series given as a \((T\times d)\)-matix. Each row corresponds to a timepoint.
an object of class timedom, representing the dpca filters obtained from the sample X. If dpsc = NULL,
then dpcs = dpca.filter(spectral.density(X)) is used.
We obtain the dynamic Karhnunen-Loeve expansion of order \(L\), \(1\leq L\leq d\). It is defined as
$$
\sum_{\ell=1}^L\sum_{k\in\mathbf{Z}} Y_{\ell, t+k} \phi_{\ell k},
$$
where \(\phi_{\ell k}\) are the dynamic PC filters as explained in dpca.filters and \(Y_{\ell k}\) are dynamic scores as explained in dpca.scores. For the sample version the sum in \(k\) extends over the range of lags for which the \(\phi_{\ell k}\) are defined.
For more details we refer to Chapter 9 in Brillinger (2001), Chapter 7.8 in Shumway and Stoffer (2006) and to Hormann et al. (2015).
Hormann, S., Kidzinski, L., and Hallin, M. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 77.2 (2015): 319-348.
Brillinger, D. Time Series (2001), SIAM, San Francisco.
Shumway, R.H., and Stoffer, D.S. Time Series Analysis and Its Applications (2006), Springer, New York.
dpca.filters, filter.process, dpca.scores