dpca.filters: Compute DPCA filter coefficients

In kidzik/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

kidzik/freqdom documentation built on April 20, 2022, 9:47 p.m.