| Title: | Independent Components Analysis Techniques for Functional Data |
|---|---|
| Description: | This package includes a set of tools to perform smoothed (and non-smoothed) principal/independent components analysis of functional data. Various functional pre-whitening approaches are implemented as discussed in Vidal and Aguilera (2022) “Novel whitening approaches in functional settings", <doi:10.1002/sta4.516>. Further whitening representations of functional data can be derived in terms of a few principal components, providing a powerful avenue to explore hidden structures in low dimensional settings: see Vidal, Rosso and Aguilera (2021) “Bi-smoothed functional independent component analysis for EEG artifact removal”, <doi:10.3390/math9111243>. |
| Authors: | Marc Vidal [aut, cre], Ana Mª Aguilera [aut, ths] |
| Maintainer: | Marc Vidal <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.1.3 |
| Built: | 2025-02-25 04:31:25 UTC |
| Source: | https://github.com/marcvidalbadia/pfica |
This function computes the ordinary ICA procedure from a sample represented by basis functions (Fourier, B-splines...). The estimation method is based on the use of fourth moments (kurtosis), in which it is assumed that the independent components have different kurtosis values. The proposed algorithm is an implementation of the kurtosis operator introduced in Peña et. al (2014).
ffobi(fdx, ncomp = fdx$basis$nbasis, eigenfPar = fdPar(fdx), w = c("PCA", "PCA-cor","ZCA", "ZCA-cor", "Varimax", "Varimax-cor", "Cholesky"), pr = c("fdx", "wfdx"), center = TRUE)ffobi(fdx, ncomp = fdx$basis$nbasis, eigenfPar = fdPar(fdx), w = c("PCA", "PCA-cor","ZCA", "ZCA-cor", "Varimax", "Varimax-cor", "Cholesky"), pr = c("fdx", "wfdx"), center = TRUE)
fdx |
a functional data object obtained from the fda package. |
ncomp |
number of independent components to compute. |
eigenfPar |
a functional parameter object obtained from the fda package. |
w |
the whitening procedure. By default |
pr |
the functional data object to project on to the space spanned by the eigenfunctions of the kurtosis kernel function. To compute the independent components, the usual procedure is to use |
center |
a logical value indicating whether the mean function has to be subtracted from each functional observation. |
This functional ICA consists of performing the multivariate ICA of a transformation of the coordinate vectors associated with a basis of functions. The algorithm also incorporates a continuous penalty in the orthonormality constraint of the kurtosis eigenfunctions.
a list with the following named entries:
ICA.eigv |
a numeric vector giving the eigenvalues of the kurtosis kernel function. |
ICA.basis |
a functional data object for the kurtosis kernel eigenfunctions. |
ICA.scores |
a matrix whose column vectors are the projection coefficients for |
Marc Vidal, Ana Mª Aguilera
Peña, C., J. Prieto and C. Rendón (2014). Independent components techniques based on kurtosis for functional data analysis. Working paper 14–10 Statistics and Econometric Series (06); Universidad Carlos III de Madrid, Madrid, 2014.
## Canadian Weather data library(fda) arg <- 1:365 Temp <- CanadianWeather$dailyAv[,,1] q <- 14 B <- create.bspline.basis(rangeval=c(min(arg),max(arg)), nbasis=q) x <- Data2fd(Temp, argvals = arg, B) #plot(x) #plot data Lfdobj <- int2Lfd(max(0, norder(B)-2)) penf <- fdPar(B, Lfdobj, lambda=4200) #penalty parameter (lambda) ica.fd <- ffobi(x, eigenfPar = penf, w = "Cholesky") ## Plot by minimmum smoothed kurtosis sc <- ica.fd$ICA.scores plot(sc[,q], sc[,(q-1)], pch = 20, col = factor(c(rep("East",15),rep("West",20))), ylab = "IC q-1", xlab = "IC q")## Canadian Weather data library(fda) arg <- 1:365 Temp <- CanadianWeather$dailyAv[,,1] q <- 14 B <- create.bspline.basis(rangeval=c(min(arg),max(arg)), nbasis=q) x <- Data2fd(Temp, argvals = arg, B) #plot(x) #plot data Lfdobj <- int2Lfd(max(0, norder(B)-2)) penf <- fdPar(B, Lfdobj, lambda=4200) #penalty parameter (lambda) ica.fd <- ffobi(x, eigenfPar = penf, w = "Cholesky") ## Plot by minimmum smoothed kurtosis sc <- ica.fd$ICA.scores plot(sc[,q], sc[,(q-1)], pch = 20, col = factor(c(rep("East",15),rep("West",20))), ylab = "IC q-1", xlab = "IC q")
This function computes the ordinary ICA procedure on a smoothed principal component expansion (also known as Karhunen-Loeve expansion) whose eigenbasis is expressed in terms of basis functions (Fourier, B-splines...). The estimation method is based on the use of fourth moments (kurtosis), in which it is assumed that the independent components have different kurtosis values. The function further provides smoothed (and non-smoothed) functional PCA estimates based on Ocaña et al. (1999).
kffobi(fdx, ncomp = fdx$basis$nbasis, eigenfPar = fdPar(fdx), w = c("PCA", "PCA-cor","ZCA", "ZCA-cor","Cholesky"), pr = c("fdx", "wfdx", "KL", "wKL"), center = TRUE)kffobi(fdx, ncomp = fdx$basis$nbasis, eigenfPar = fdPar(fdx), w = c("PCA", "PCA-cor","ZCA", "ZCA-cor","Cholesky"), pr = c("fdx", "wfdx", "KL", "wKL"), center = TRUE)
fdx |
a functional data object obtained from the fda package. |
ncomp |
number of independent components to compute (must be > 1). |
eigenfPar |
a functional parameter object obtained from the fda package. |
w |
the whitening procedure. By default |
pr |
the functional data object to project on to the space spanned by the eigenfunctions of the kurtosis kernel function. To compute the independent components, the usual procedure is to use |
center |
a logical value indicating whether the mean function has to be subtracted from each functional observation. |
This functional ICA consists of performing the multivariate ICA of the PC coordinate vectors in terms of the principal component weight functions.
a list with the following named entries:
PCA.eigv |
a numeric vector giving the eigenvalues of the covariance kernel function. |
PCA.basis |
a functional data object for the eigenfunctions of the covariance kernel function. |
PCA.scores |
a matrix whose column vectors are the principal components. |
ICA.eigv |
a numeric vector giving the eigenvalues of the kurtosis kernel function. |
ICA.eigv |
a numeric vector giving the eigenvalues of the kurtosis kernel function. |
ICA.scores |
a matrix whose column vectors are the projection coefficients for |
wKL |
the whitened principal components expansion whith coefficients in terms of basis functions. |
Marc Vidal, Ana Mª Aguilera
Li, B., G. Van Bever, H. Oja, R. Sabolova and F. Critchley (2019). Functional independent component analysis: an extension of the fourth-orderblind identification. Technical Report, Univ. Namur.
Ocaña, F.A., A.M. Aguilera and M.J. Valderrama (1999). Functional Principal Components Analysis by Choice of Norm. Journal of Multivariate Analysis, 71(2), <doi:10.1006/jmva.1999.1844>.
ffobi, pspline.kffobi, whiten.fd
## Canadian Weather data library(fda) arg <- 1:365 Temp <- CanadianWeather$dailyAv[,,1] B <- create.bspline.basis(rangeval=c(min(arg),max(arg)), nbasis=14) x <- Data2fd(Temp, argvals = arg, B) #plot(x) #plot data Lfdobj <- int2Lfd(max(0, norder(B)-2)) penf <- fdPar(B, Lfdobj, lambda=10^9) #penalty parameter (lambda) ica.fd <- kffobi(x, 2, eigenfPar = penf, w="ZCA-cor") ## Whitened data using the two first smoothed principal components wKL <- ica.fd$wKL ## Plot by region sc <- ica.fd$ICA.scores plot(sc[,1], sc[,2], pch = 20, col = factor(CanadianWeather$region), ylab = "IC 1", xlab = "IC 2")## Canadian Weather data library(fda) arg <- 1:365 Temp <- CanadianWeather$dailyAv[,,1] B <- create.bspline.basis(rangeval=c(min(arg),max(arg)), nbasis=14) x <- Data2fd(Temp, argvals = arg, B) #plot(x) #plot data Lfdobj <- int2Lfd(max(0, norder(B)-2)) penf <- fdPar(B, Lfdobj, lambda=10^9) #penalty parameter (lambda) ica.fd <- kffobi(x, 2, eigenfPar = penf, w="ZCA-cor") ## Whitened data using the two first smoothed principal components wKL <- ica.fd$wKL ## Plot by region sc <- ica.fd$ICA.scores plot(sc[,1], sc[,2], pch = 20, col = factor(CanadianWeather$region), ylab = "IC 1", xlab = "IC 2")
This function is an alternative way of computing the smoothed functional ICA in terms of principal components (see kffobi). The function further provides smoothed (and non-smoothed) functional PCA estimates. A discrete penalty that measures the roughness of principal factors by summing squared r-order differences between adjacent B-spline coefficients (P-spline penalty) is used; see Aguilera and Aguilera-Morillo (2013) and Eliers and Marx (2021).
pspline.kffobi(fdx, ncomp = fdx$basis$nbasis, pp = 0, r = 2, w = c("PCA", "PCA-cor","ZCA", "ZCA-cor","Cholesky"), pr = c("fdx", "wfdx", "KL", "wKL"), center = TRUE)pspline.kffobi(fdx, ncomp = fdx$basis$nbasis, pp = 0, r = 2, w = c("PCA", "PCA-cor","ZCA", "ZCA-cor","Cholesky"), pr = c("fdx", "wfdx", "KL", "wKL"), center = TRUE)
fdx |
a functional data object obtained from the fda package. |
ncomp |
number of independent components to compute (must be > 1). |
pp |
the penalty parameter. It can be estimated using leave-one-out cross-validation. |
r |
a number indicating the order of the penalty. |
w |
the whitening procedure. By default |
pr |
the functional data object to project on to the space spanned by the eigenfunctions of the kurtosis kernel function. To compute the independent components, the usual procedure is to use |
center |
a logical value indicating whether the mean function has to be subtracted from each functional observation. |
a list with the following named entries:
PCA.eigv |
a numeric vector giving the eigenvalues of the covariance kernel function. |
PCA.basis |
a functional data object for the eigenfunctions of the covariance kernel function. |
PCA.scores |
a matrix whose column vectors are the principal components. |
ICA.eigv |
a numeric vector giving the eigenvalues of the kurtosis kernel function. |
ICA.eigv |
a numeric vector giving the eigenvalues of the kurtosis kernel function. |
ICA.scores |
a matrix whose column vectors are the projection coefficients for |
wKL |
the whitened principal components expansion whith coefficients in terms of basis functions. |
Marc Vidal, Ana Mª Aguilera
Aguilera, A.M. and M.C. Aguilera-Morillo (2013). Penalized PCA approaches for B-spline expansions of smooth functional data. Applied Mathematics and Computation 219(14), 7805–7819, <doi:10.1016/j.amc.2013.02.009>.
Eilers, P. and B. Marx (2021). Practical Smoothing: The Joys of P-splines. Cambridge: Cambridge University Press, <doi:10.1017/9781108610247>.
Vidal, M., M. Rosso and A.M. Aguilera. (2021). Bi-Smoothed Functional Independent Component Analysis for EEG Artifact Removal. Mathematics 9(11), 1243, <doi:10.3390/math9111243>.
## Canadian Weather data library(fda) arg <- 1:365 Temp <- CanadianWeather$dailyAv[,,1] B <- create.bspline.basis(rangeval=c(min(arg),max(arg)), nbasis=14) x <- Data2fd(Temp, argvals = arg, B) #plot(x) #plot data ica.fd <- pspline.kffobi(x, 2, pp = 10^4, w="ZCA-cor") ## Whitened data using the two first smoothed principal components wKL <- ica.fd$wKL ## Plot by region sc <- ica.fd$ICA.scores plot(sc[,2], sc[,1], pch = 20, col = factor(CanadianWeather$region), ylab = "IC 1", xlab = "IC 2")## Canadian Weather data library(fda) arg <- 1:365 Temp <- CanadianWeather$dailyAv[,,1] B <- create.bspline.basis(rangeval=c(min(arg),max(arg)), nbasis=14) x <- Data2fd(Temp, argvals = arg, B) #plot(x) #plot data ica.fd <- pspline.kffobi(x, 2, pp = 10^4, w="ZCA-cor") ## Whitened data using the two first smoothed principal components wKL <- ica.fd$wKL ## Plot by region sc <- ica.fd$ICA.scores plot(sc[,2], sc[,1], pch = 20, col = factor(CanadianWeather$region), ylab = "IC 1", xlab = "IC 2")
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).
whiten.fd(fdx, w = c("PCA", "PCA-cor","ZCA", "ZCA-cor", "Varimax", "Varimax-cor", "Cholesky"), pre.center = TRUE, post.center = FALSE)whiten.fd(fdx, w = c("PCA", "PCA-cor","ZCA", "ZCA-cor", "Varimax", "Varimax-cor", "Cholesky"), pre.center = TRUE, post.center = FALSE)
fdx |
a functional data object obtained from the fda package. |
w |
the whitening procedure. By default |
pre.center |
centering the functional observations before performing the transformation. |
post.center |
centering the functional observations after performing the transformation. |
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.
wfdx |
a functional data object with whitened coefficients. |
Marc Vidal, Ana Mª Aguilera
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>.
## Canadian Weather data library(fda) arg <- 1:365 Temp <- CanadianWeather$dailyAv[,,1] B <- create.bspline.basis(rangeval=c(min(arg),max(arg)), nbasis=16) x <- Data2fd(Temp, argvals = arg, B) wx <- whiten.fd(x)## Canadian Weather data library(fda) arg <- 1:365 Temp <- CanadianWeather$dailyAv[,,1] B <- create.bspline.basis(rangeval=c(min(arg),max(arg)), nbasis=16) x <- Data2fd(Temp, argvals = arg, B) wx <- whiten.fd(x)