dpca.filters: Compute DPCA filter coefficients

In freqdom: Frequency Domain Based Analysis: Dynamic PCA

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,…, 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 q will be computed.

Details

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).

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 φ_{\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.

See Also

dpca.var, dpca.scores, dpca.KLexpansion

freqdom documentation built on Oct. 4, 2022, 5:05 p.m.