dpca.var: Proportion of variance explained

In kidzik/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

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