dpca.KLexpansion: Dynamic KL expansion

In freqdom: Frequency Domain Based Analysis: Dynamic PCA

Dynamic KL expansion

Description

Computes the dynamic Karhunen-Loeve expansion of a vector time series up to a given order.

Usage

dpca.KLexpansion(X, dpcs)

Arguments

X	a vector time series given as a (T\times d)-matix. Each row corresponds to a timepoint.
dpcs	an object of class timedom, representing the dpca filters obtained from the sample X. If dpsc = NULL, then dpcs = dpca.filter(spectral.density(X)) is used.

Details

We obtain the dynamic Karhnunen-Loeve expansion of order L, 1≤q L≤q d. It is defined as

∑_{\ell=1}^L∑_{k\in\mathbf{Z}} Y_{\ell, t+k} φ_{\ell k},

where φ_{\ell k} are the dynamic PC filters as explained in dpca.filters and Y_{\ell k} are dynamic scores as explained in dpca.scores. For the sample version the sum in k extends over the range of lags for which the φ_{\ell k} are defined.

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 (T\times d)-matix. The \ell-th column contains the \ell-th data point.

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, filter.process, dpca.scores

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