R: Compute DPCA filter coefficients

dpca.filters {freqdom}

R Documentation

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)

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).

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.

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.

Shumway, R.H., and Stoffer, D.S. Time Series Analysis and Its Applications (2006), Springer, New York.