whiten.fd {pfica}

Whitening (or sphering) functional data

Description

This function whitens functional data in terms of basis functions expansions.

Functional whitening procedures available:

• PCA: uses a projection that combines the covariance function eigenfactors and an arbitrary orthonormal basis

• PCA-cor: PCA whitening based on the correlation function

• ZCA: or Mahalanobis whitening, is a symmetric solution based on the spectral decomposition of the inverse square root of the covariance function

• ZCA-cor: Mahalanobis whitening based on the correlation function

• Varimax: combines the covariance function eigenfactors and its varimax rotation

• Varimax-cor: Varimax whitening based on the correlation function

• Cholesky: uses the Cholesky factorization of the inverse covariance function.

Which functional whitening procedure to use? PCA and PCA-cor allow for maximum compression of the functional observations and therefore the new data representations might be rather noisy. ZCA and ZCA-cor maximize the similarity with the original data, providing most robust whitening. The Varimax/Varimax-cor procedures, which are analogous to ZCA/ZCA-cor, can be used when the first component of the functional PCA accounts for a high percentage of the total variance. Cholesky takes an intermediate position in scoring for maximum compression and similarity indices. To explore low dimensional structures in the data, kffobi and pspline.kffobi allows to compute these functional whitening representations using a few principal components.

For further details, see Vidal and Aguilera (2022).

Usage

whiten.fd(fdx, w = c("PCA", "PCA-cor","ZCA", "ZCA-cor",
                     "Varimax", "Varimax-cor", "Cholesky"),
          pre.center = TRUE, post.center = FALSE)

Arguments

fdx

a functional data object obtained from the fda package.

w

the whitening procedure. By default ZCA (Mahalanobis whitening) is used.

pre.center

centering the functional observations before performing the transformation.

post.center

centering the functional observations after performing the transformation.

Details

As in the whitening package (see Kessy et al. (2018)), to solve the sign ambiguity in PCA, PCA-cor we use eigenvector matrices with a positive diagonal. The pre.centrer/post.centrer options can be used interchangeably.

Value

wfdx

a functional data object with whitened coefficients.

Author(s)

Marc Vidal, Ana Mª Aguilera

References

Acal, C., A.M. Aguilera and M. Escabias (2020). New Modeling Approaches Based on Varimax Rotation of Functional Principal Components. Mathematics, 8(11), 1-15. <doi:10.3390/math8112085>

Kessy, A., A. Lewin and K. Strimmer (2018). Optimal Whitening and Decorrelation. The American Statistician, 72(4), 309-314, <doi:10.1080/00031305.2016.1277159>.

Vidal, M. and and A.M. Aguilera (2022). Novel whitening approaches in functional settings. Stat, e516, <doi:10.1002/sta4.516>.

See Also

ffobi

Examples

## Canadian Weather data
library(fda)
arg <- 1:365
Temp <- CanadianWeather$dailyAv[,,1]
B <- create.bspline.basis(rangeval=c(min(arg),max(arg)), nbasis=16)
x <- Data2fd(Temp, argvals = arg, B)
wx <- whiten.fd(x)