uni.fpca: Functional Principle Component Analysis with Corpula

In cSFM: Covariate-adjusted Skewed Functional Model (cSFM)

Description

uni.fpca is used to do regular FPCA for univariate data with given covariance matrix. The sample covariance matrix will be used if covariance matrix is not given.

Usage

uni.fpca(Y, Kendall = NULL, Y.pred = NULL, 
         nbasis.mean = 10, nbasis = 10, pve = 0.99, npc = NULL, 
         center = TRUE, gam.method = "REML")

Arguments

Y	matrix for the fully observed curves; each row corresponds to one subject
Kendall	given covariance matrix for Y (raw matrix up to smoothing); typically obtained using Kendall's Tau approach. See 'Details'
Y.pred	partically observed curves to be predicted; default is null indicating no predication on new data set
nbasis.mean	number of basis functions to smooth the mean
nbasis	number of basis functions for each direction to smooth the covariance matrix
pve	threshold of the percentage of variance explained to select number of principal components
npc	prespecified value for the number of principal components to be included in the expansion (if given, this overrides 'pve').
center	if TRUE, we center the curves; otherwise, do not center the curves and assume the curves have mean 0 already
gam.method	estimation method when using gam

Details

This function is build in the spirit of the function fpca.sc in the package refund, but here the corpula plays a role. The covariance matrix Kendall is typically estimated before the application of this function, for instance, via Kendall's Tau approach. It is highly recommended that the Knedall's Tau covariance matris should be provided here since it is more stable.

Value

A list of the following components:

Yhat	matrix whose rows are the estimates through Functional FPCA of the curves in Y
scores	matrix of estimated principal component scores
mu	estimated mean function
efunctions	matrix of estimated eigenfunctions of the functional covariance operator, i.e., the FPC basis functions
evalues	estimated eigenvalues of the covariance operator, i.e., variances of FPC scores
npc	number of FPCs: either the supplied npc, or the minimum number of basis functions needed to explain proportion pve of the variance in the observed curves
K.tilde	Smoothed covariance matrix of Y after removing white noises
K.hat	Smoothed covariance matrix of Y
sigma2	estimated measurement error variance

References

[1]. Yao, F., Mueller, H.-G., and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association, 100(470):577-590.

[2]. Goldsmith, J., Greven, S., and Crainiceanu, C. (2013). Corrected confidence bands for functional data using principal components. Biometrics, 69(1), 41-51.

Examples

data(data.simulation)
fpca <- uni.fpca(Y = DST$obs, Y.pred=DSV$obs)

# visualize the data and predicted surface
par(mfrow = c(1,2))
persp(DSV$cp, DSV$tp, DSV$obs, theta=60, phi=15,
      ticktype = "detailed", col="lightblue", 
      xlab = "covariate", ylab = "time",
      zlab="data", main="data surface (partically observed)")
persp(DSV$cp, DSV$tp, fpca$Yhat, theta=60, phi=15,
      ticktype = "detailed", col="lightblue", 
      xlab = "covariate", ylab = "time",
      zlab="data", main="predicted surface via univariate FPCA")

# predication error
mean(((fpca$Yhat - DSV$obs.full)[!is.na(DSV$obs)])^2)

cSFM documentation built on May 29, 2017, 6:10 p.m.