dpca.filters: Compute DPCA filter coefficients

Description

For a given spectral density matrix dynamic principal component filter sequences are computed.

Usage

dpca.filters(F, Ndpc = dim(F$operators)[1], q = 30)

Value

An object of class timedom. The list has the following components:

operators $\quad$ an array. Each matrix in this array has dimension Ndpc $\times d$ and is assigned to a certain lag. For a given lag $k$, the rows of the matrix correpsond to $\phi_{\ell k}$.
lags $\quad$ a vector with the lags of the filter coefficients.

Arguments

F: $(d\times d)$ spectral density matrix, provided as an object of class freqdom.
Ndpc: an integer $\in\{1,\ldots, d\}$. It is the number of dynamic principal components to be computed. By default it is set equal to $d$.
q: a non-negative integer. DPCA filter coefficients at lags $|h|\leq$ q will be computed.

Details

Dynamic principal components are linear filters $(\phi_{\ell k}\colon k\in \mathbf{Z})$, $1 \leq \ell \leq d$. They are defined as the Fourier coefficients of the dynamic eigenvector $\varphi_\ell(\omega)$ of a spectral density matrix $\mathcal{F}_\omega$: $$ \phi_{\ell k}:=\frac{1}{2\pi}\int_{-\pi}^\pi \varphi_\ell(\omega) \exp(-ik\omega) d\omega. $$ The index $\ell$ is referring to the $\ell$-th #'largest dynamic eigenvalue. Since the $\phi_{\ell k}$ are real, we have $$ \phi_{\ell k}^\prime=\phi_{\ell k}^*=\frac{1}{2\pi}\int_{-\pi}^\pi \varphi_\ell^* \exp(ik\omega)d\omega. $$ For a given spectral density (provided as on object of class freqdom) the function dpca.filters() computes $(\phi_{\ell k})$ for $|k| \leq$ q and $1 \leq \ell \leq$ Ndpc.

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).

References

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.