foreca: Forecastable Component Analysis
In ForeCA: Forecastable Component Analysis

Description Usage Arguments Value Warning References Examples

foreca performs Forecastable Component Analysis (ForeCA) on \mathbf{X}_t – a K-dimensional time series with T observations. Users should only call foreca, rather than foreca.one_weightvector or foreca.multiple_weightvectors.

foreca.one_weightvector is a wrapper around several algorithms that solve the ForeCA optimization problem for a single weightvector \mathbf{w}_i and whitened time series \mathbf{U}_t.

foreca.multiple_weightvectors applies foreca.one_weightvector iteratively to \mathbf{U}_t in order to obtain multiple weightvectors that yield most forecastable, uncorrelated signals.

foreca(series, n.comp = 2, algorithm.control = list(type = "EM"), ...)

foreca.one_weightvector(
  U,
  f.U = NULL,
  spectrum.control = list(),
  entropy.control = list(),
  algorithm.control = list(),
  keep.all.optima = FALSE,
  dewhitening = NULL,
  ...
)

foreca.multiple_weightvectors(
  U,
  spectrum.control = list(),
  entropy.control = list(),
  algorithm.control = list(),
  n.comp = 2,
  plot = FALSE,
  dewhitening = NULL,
  ...
)

`series`	a T \times K array with `T` observations from the K-dimensional time series \mathbf{X}_t. Can be a `matrix`, `data.frame`, or a multivariate `ts` object.
`n.comp`	positive integer; number of components to be extracted. Default: `2`.
`algorithm.control`	list; control settings for any iterative ForeCA algorithm. See `complete_algorithm_control` for details.
`...`	additional arguments passed to available ForeCA algorithms.
`U`	a T \times K array with `T` observations from the K-dimensional whitened (`whiten`) time series \mathbf{U}_t. Can be a `matrix`, `data.frame`, or a multivariate `ts` object.
`f.U`	multivariate spectrum of class `'mvspectrum'` with `normalize = TRUE`.
`spectrum.control`	list; control settings for spectrum estimation. See `complete_spectrum_control` for details.
`entropy.control`	list; control settings for entropy estimation. See `complete_entropy_control` for details.
`keep.all.optima`	logical; if `TRUE`, it keeps the optimal solutions of each random start. Default: `FALSE` (only returns the best solution).
`dewhitening`	optional; if provided (returned by `whiten`) then it uses the dewhitening transformation to obtain the original series \mathbf{X}_t and it uses that vector (normalized) as the initial weightvector which corresponds to the series \mathbf{X}_{t,i} with larges `Omega`.
`plot`	logical; if `TRUE` a plot of the current optimal solution \mathbf{w}_i^* will be shown and updated for each iteration i = 1, ..., `n.comp` of any iterative algorithm. Default: `FALSE`.

An object of class foreca, which is similar to the output from princomp, with the following components (amongst others):

center: sample mean \widehat{μ}_X of each series,
whitening: whitening matrix of size K \times K from whiten: \mathbf{U}_t = (\mathbf{X}_t - \widehat{μ}_X) \cdot whitening; note that \mathbf{X}_t is centered prior to the whitening transformation,
weightvectors: orthonormal matrix of size K \times n.comp, which converts whitened data to n.comp forecastable components (ForeCs) \mathbf{F}_t = \mathbf{U}_t \cdot weightvectors,
loadings: combination of whitening \times weightvectors to obtain the final loadings for the original data: \mathbf{F}_t = (\mathbf{X}_t - \widehat{μ}_X) \cdot whitening \cdot weightvectors; again, it centers \mathbf{X}_t first,
loadings.normalized: normalized loadings (unit norm). Note though that if you use these normalized loadings the resulting signals do not have variance 1 anymore.
scores: n.comp forecastable components \mathbf{F}_t. They have mean 0, variance 1, and are uncorrelated.
Omega: forecastability score of each ForeC of \mathbf{F}_t.

ForeCs are ordered from most to least forecastable (according to Omega).

Estimating Omega directly from the ForeCs \mathbf{F}_t can be different to the reported $Omega estimates from foreca. Here is why:

In theory f_y(λ) of a linear combination y_t = \mathbf{X}_t \mathbf{w} can be analytically computed from the multivariate spectrum f_{\mathbf{X}}(λ) by the quadratic form f_y(λ) = \mathbf{w}' f_{\mathbf{X}}(λ) \mathbf{w} for all λ (see spectrum_of_linear_combination).

In practice, however, this identity does not hold always exactly since (often data-driven) control setting for spectrum estimation are not identical for the high-dimensional, noisy \mathbf{X}_t and the combined univariate time series y_t (which is usually more smooth, less variable). Thus estimating \widehat{f}_y directly from y_t can give slightly different estimates to computing it as \mathbf{w}'\widehat{f}_{\mathbf{X}}\mathbf{w}. Consequently also Omega estimates can be different.

In general, these differences are small and have no relevant implications for estimating ForeCs. However, in rare occasions the obtained ForeCs can have smaller Omega than the maximum Omega across all original series. In such a case users should not re-estimate Ω from the resulting ForeCs \mathbf{F}_t, but access them via $Omega provided by 'foreca' output (the univariate estimates are stored in $Omega.univ).

Goerg, G. M. (2013). “Forecastable Component Analysis”. Journal of Machine Learning Research (JMLR) W&CP 28 (2): 64-72, 2013. Available at http://jmlr.org/proceedings/papers/v28/goerg13.html.

XX <- diff(log(EuStockMarkets)) * 100
plot(ts(XX))
## Not run: 
ff <- foreca(XX[,1:4], n.comp = 4, plot = TRUE, spectrum.control=list(method="pspectrum"))
ff
summary(ff)
plot(ff)

## End(Not run)


## Not run: 
PW <- whiten(XX)
one.weight.em <- foreca.one_weightvector(U = PW$U,
                                        dewhitening = PW$dewhitening,
                                        algorithm.control =
                                          list(num.starts = 2,
                                               type = "EM"),
                                        spectrum.control =
                                          list(method = "mvspec"))
plot(one.weight.em)

## End(Not run)
## Not run: 

PW <- whiten(XX)
ff <- foreca.multiple_weightvectors(PW$U, n.comp = 2,
                                    dewhitening = PW$dewhitening)
ff
plot(ff$scores)

## End(Not run)