FSVD: Functional Singular Value Decomposition

In functionaldata/tPACE: Functional Data Analysis and Empirical Dynamics

Functional Singular Value Decomposition

Description

FSVD for a pair of dense or sparse functional data.

Usage

FSVD(
  Ly1,
  Lt1,
  Ly2,
  Lt2,
  FPCAoptns1 = NULL,
  FPCAoptns2 = NULL,
  SVDoptns = list()
)

Arguments

Ly1	A list of n vectors containing the observed values for each individual. Missing values specified by NAs are supported for dense case (dataType='dense').
Lt1	A list of n vectors containing the observation time points for each individual corresponding to y. Each vector should be sorted in ascending order.
Ly2	A list of n vectors containing the observed values for each individual. Missing values specified by NAs are supported for dense case (dataType='dense').
Lt2	A list of n vectors containing the observation time points for each individual corresponding to y. Each vector should be sorted in ascending order.
FPCAoptns1	A list of options control parameters specified by list(name=value) for the FPC analysis of sample 1. See ‘?FPCA’.
FPCAoptns2	A list of options control parameters specified by list(name=value) for the FPC analysis of sample 2. See ‘?FPCA’.
SVDoptns	A list of options control parameters specified by list(name=value) for the FSVD analysis of samples 1 & 2. See 'Details'.

Details

Available control options for SVDoptns are:

bw1

The bandwidth value for the smoothed cross-covariance function across the direction of sample 1; positive numeric - default: determine automatically based on 'methodBwCov'

bw2

The bandwidth value for the smoothed cross-covariance function across the direction of sample 2; positive numeric - default: determine automatically based on 'methodBwCov'

methodBwCov

The bandwidth choice method for the smoothed covariance function; 'GMeanAndGCV' (the geometric mean of the GCV bandwidth and the minimum bandwidth),'CV','GCV' - default: 10% of the support

userMu1

The user defined mean of sample 1 used to centre it prior to the cross-covariance estimation. - default: determine automatically based by the FPCA of sample 1

userMu2

The user defined mean of sample 2 used to centre it prior to the cross-covariance estimation. - default: determine automatically based by the FPCA of sample 2

maxK

The maximum number of singular components to consider; default: min(20, N-1), N:# of curves.

kernel

Smoothing kernel choice, common for mu and covariance; "rect", "gauss", "epan", "gausvar", "quar" - default: "gauss"; dense data are assumed noise-less so no smoothing is performed.

rmDiag

Logical describing if the routine should remove diagonal raw cov for cross cov estimation (default: FALSE)

noScores

Logical describing if the routine should return functional singular scores or not (default: TRUE)

regulRS

String describing if the regularisation of the composite cross-covariance matrix should be done using 'sigma1' or 'rho' (see ?FPCA for details) (default: 'sigma2')

bwRoutine

String specifying the routine used to find the optimal bandwidth 'grid-search', 'bobyqa', 'l-bfgs-b' (default: 'l-bfgs-b')

flip

Logical describing if the routine should flip the sign of the singular components functions or not after the SVD of the cross-covariance matrix. (default: FALSE)

useGAM

Indicator to use gam smoothing instead of local-linear smoothing (semi-parametric option) (default: FALSE)

dataType1

The type of design we have for sample 1 (usually distinguishing between sparse or dense functional data); 'Sparse', 'Dense', 'DenseWithMV' - default: determine automatically based on 'IsRegular'

dataType2

The type of design we have for sample 2 (usually distinguishing between sparse or dense functional data); 'Sparse', 'Dense', 'DenseWithMV' - default: determine automatically based on 'IsRegular'

Value

A list containing the following fields:

bw1	The selected (or user specified) bandwidth for smoothing the cross-covariance function across the support of sample 1.
bw2	The selected (or user specified) bandwidth for smoothing the cross-covariance function across the support of sample 2.
CrCov	The smoothed cross-covariance between samples 1 & 2.
sValues	A list of length nsvd, each entry containing the singular value estimates for the FSC estimates.
nsvd	The number of singular components used.
canCorr	The canonical correlations for each dimension.
FVE	A percentage indicating the total variance explained by chosen FSCs with corresponding 'FVEthreshold'.
sFun1	An nWorkGrid by K matrix containing the estimated singular functions for sample 1.
sFun2	An nWorkGrid by K matrix containing the estimated singular functions for sample 2.
grid1	A vector of length nWorkGrid1. The internal regular grid on which the singular analysis is carried on the support of sample 1.
grid2	A vector of length nWorkGrid2. The internal regular grid on which the singular analysis is carried on the support of sample 2.
sScores1	A n by K matrix containing the singular scores for sample 1.
sScores2	A n by K matrix containing the singular scores for sample 2.
optns	A list of options used by the SVD and the FPCA's procedures.

References

Yang, W., Müller, H.G., Stadtmüller, U. (2011). Functional singular component analysis. J. Royal Statistical Society B73, 303-324.

functionaldata/tPACE documentation built on July 4, 2024, 5:09 a.m.