dpca.KLexpansion {freqdom}

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\leq L\leq d. It is defined as

\sum_{\ell=1}^L\sum_{k\in\mathbf{Z}} Y_{\ell, t+k} \phi_{\ell k},

where \phi_{\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 \phi_{\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