dpca.var: Proportion of variance explained

In freqdom: Frequency Domain Based Analysis: Dynamic PCA

Proportion of variance explained

Description

Computes the proportion of variance explained by a given dynamic principal component.

Usage

dpca.var(F)

Arguments

F	(d\times d) spectral density matrix, provided as an object of class freqdom. To guarantee accuracy of numerical integration it is important that F\$freq is a dense grid of frequencies in [-π,π].

Details

Consider a spectral density matrix \mathcal{F}_ω and let λ_\ell(ω) by the \ell-th dynamic eigenvalue. The proportion of variance described by the \ell-th dynamic principal component is given as

v_\ell:=\int_{-π}^π λ_\ell(ω)dω/\int_{-π}^π \mathrm{tr}(\mathcal{F}_ω)dω.

This function numerically computes the vectors (v_\ell\colon 1≤q \ell≤q d).

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

A d-dimensional vector containing the v_\ell.

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.filters, dpca.KLexpansion, dpca.scores

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