Computes the proportion of variance explained by a given dynamic principal component.
dpca.var(F)A \(d\)-dimensional vector containing the \(v_\ell\).
\((d\times d)\) spectral density matrix, provided as an object of class freqdom. To guarantee accuracy of numerical integration it is important that F\(\$\)freq is a dense grid of frequencies in \([-\pi,\pi]\).
Consider a spectral density matrix \(\mathcal{F}_\omega\) and let \(\lambda_\ell(\omega)\) by the \(\ell\)-th dynamic eigenvalue. The proportion of variance described by the \(\ell\)-th dynamic principal component is given as $$v_\ell:=\int_{-\pi}^\pi \lambda_\ell(\omega)d\omega/\int_{-\pi}^\pi \mathrm{tr}(\mathcal{F}_\omega)d\omega.$$ This function numerically computes the vectors \((v_\ell\colon 1\leq \ell\leq d)\).
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, dpca.KLexpansion, dpca.scores