mcfda

Mean and covariance estimation for functional data analysis

Install

devtools::install_github("linulysses/mcfda")

Demonstration

library(mcfda)

set.seed(1)

set up mean and cov functions and synthesize data functional dataset

mu <- function(s) sin(2pis)

cov <- synfd::matern

D <- synfd::irreg.fd(mu=mu, X=synfd::gaussian.process(cov), n=100, m=5)

estimate mean by 'FOURIER' (local polynomial) method

mu.obj <- meanfunc(D$t,D$y,newt=NULL,method='FOURIER', tuning='cv',weig=NULL, domain=c(0,1))

plot the object

plot(mu.obj)

lines(regular.grid(),mu(regular.grid()))

estimate mean by 'PACE' (local polynomial) method

mu.obj <- meanfunc(D$t,D$y,newt=NULL,method='PACE', tuning='cv',weig=NULL,kernel='epanechnikov',deg=1)

plot(mu.obj)

lines(regular.grid(),mu(regular.grid()))

estimate covariance by 'PACE'

cov.obj <- covfunc(D$t,D$y,newt=NULL,mu=mu.obj,method='PACE', tuning='GCV',weig=NULL,kernel='epanechnikov',delta=Inf)

grid <- regular.grid()

evaluate the estimated cov at grid

cov.hat <- predict(cov.obj,grid)

the true covariance at grid

cov0 <- cov(grid)

relative error in MISE

mean((cov.hat-cov0)^2) / mean(cov0^2)

estimate covariance by 'FOURIER'

cov.obj <- covfunc(D$t,D$y,newt=NULL,mu=mu.obj,method='FOURIER', tuning='lle',weig=NULL)

evaluate the estimated cov at grid

cov.hat <- predict(cov.obj,grid)

relative error in MISE

mean((cov.hat-cov0)^2) / mean(cov0^2)

estimate covariance by 'SP'

cov.obj <- covfunc(D$t,D$y,newt=NULL,method='SP',weig=NULL)

evaluate the estimated cov at grid

cov.hat <- predict(cov.obj,grid)

relative error in MISE

mean((cov.hat-cov0)^2) / mean(cov0^2)

estimate variance of the measurement noise

sig2 <- sigma2(D$t,D$y)

sig2

estimation error for the measurement variance

abs(sig2 - attr(D,'sig')^2)

FPCA via fdapace package

Once the mean and covariance functions are obtained, the fdapace package can used with the options userCov, userMu and userSigma2 to perform FPCA:

error <- ifelse(sig2==0,yes=FALSE,no=TRUE)

grid <- regular.grid(h=0)

optns <- list(userCov=list(t=grid,cov=predict(cov.obj,grid)), userMu=list(t=grid,mu=predict(mu.obj,grid)), userSigma2 = sig2, error=error, verbose=T)

R <- fdapace::FPCA(D$y,D$t,optns)

plot the first FPC, etc

plot(R$phi[,1])

References

Lin, Z. and Wang, J.-L. (2020+). Mean and covariance estimation for functional snippets. Journal of the American Statistical Association. to appear.

Lin, Z., Wang, J.-L. and Zhong, Q. (2020+). Basis expansions for functional snippets. preprint.

Yao, F., Müller, H.-G. and Wang, J.-L. (2005). Functional Data Analysis for Sparse Longitudinal Data. Journal Journal of the American Statistical Association. 100(470): 577-590.