dpca.filters | R Documentation |
For a given spectral density matrix dynamic principal component filter sequences are computed.
dpca.filters(F, Ndpc = dim(F$operators)[1], q = 30)
F |
(d\times d) spectral density matrix, provided as an object of class |
Ndpc |
an integer \in\{1,…, 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|≤q |
Dynamic principal components are linear filters (φ_{\ell k}\colon k\in \mathbf{Z}), 1 ≤q \ell ≤q d. They are defined as the Fourier coefficients of the dynamic eigenvector \varphi_\ell(ω) of a spectral density matrix \mathcal{F}_ω:
φ_{\ell k}:=\frac{1}{2π}\int_{-π}^π \varphi_\ell(ω) \exp(-ikω) dω.
The index \ell is referring to the \ell-th #'largest dynamic eigenvalue. Since the φ_{\ell k} are real, we have
φ_{\ell k}^\prime=φ_{\ell k}^*=\frac{1}{2π}\int_{-π}^π \varphi_\ell^* \exp(ikω)dω.
For a given
spectral density (provided as on object of class freqdom
) the function
dpca.filters()
computes (φ_{\ell k}) for |k| ≤q q
and 1 ≤q \ell ≤q 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).
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
φ_{\ell k}.
lags
\quad a vector with the lags of the filter coefficients.
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.var
, dpca.scores
, dpca.KLexpansion
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