fdapace (version 0.6.0)

FPCAder: Obtain the derivatives of eigenfunctions/ eigenfunctions of derivatives (note: these two are not the same)

Description

Obtain the derivatives of eigenfunctions/ eigenfunctions of derivatives (note: these two are not the same)

Usage

FPCAder(fpcaObj, derOptns = list(p = 1))

Arguments

fpcaObj

A object of class FPCA returned by the function FPCA().

derOptns

A list of options to control the derivation parameters specified by list(name=value). See `Details'. (default = NULL)

Details

Available derivative options are

method

The method used for obtaining the derivatives -- default is 'FPC', which is the derivatives of eigenfunctions; 'DPC': eigenfunctions of derivatives, with G^(1,1) estimated by an initial kernel local smoothing step for G^(1,0), then applying a 1D smoother in the second direction; 'FPC': functional principal component, based on smoothing the eigenfunctions; 'FPC1': functional principal component, based on smoothing G^(1,0). The latter may produce better estimates than 'FPC' but is slower.

p

The order of the derivatives returned (default: 1, max: 2).

bw

Bandwidth for the 1D and the 2D smoothers (default: p * 0.1 * S, where S is the length of the domain).

kernelType

Smoothing kernel choice; same available types are FPCA(). default('epan')

References

Dai, X., Tao, W., Müller, H.G. (2018). Derivative principal components for representing the time dynamics of longitudinal and functional data. Statistica Sinica 28, 1583--1609. (DPC) Liu, Bitao, and Hans-Georg Müller. "Estimating derivatives for samples of sparsely observed functions, with application to online auction dynamics." Journal of the American Statistical Association 104, no. 486 (2009): 704-717. (FPC)

Examples

bw <- 0.2
kern <- 'epan'
set.seed(1)
n <- 50 
M <- 30
pts <- seq(0, 1, length.out=M)
lambdaTrue <- c(1, 0.8, 0.1)^2
sigma2 <- 0.1

samp2 <- MakeGPFunctionalData(n, M, pts, K=length(lambdaTrue), 
                              lambda=lambdaTrue, sigma=sqrt(sigma2), basisType='legendre01')
samp2 <- c(samp2, MakeFPCAInputs(tVec=pts, yVec=samp2$Yn))
fpcaObj <- FPCA(samp2$Ly, samp2$Lt, list(methodMuCovEst='smooth',
                userBwCov=bw, userBwMu=bw, kernel=kern, error=TRUE)) 
CreatePathPlot(fpcaObj, showObs=FALSE)

FPCoptn <- list(bw=bw, kernelType=kern, method='FPC')
DPCoptn <- list(bw=bw, kernelType=kern, method='DPC')
FPC <- FPCAder(fpcaObj, FPCoptn)
DPC <- FPCAder(fpcaObj, DPCoptn)

CreatePathPlot(FPC, ylim=c(-5, 10))
CreatePathPlot(DPC, ylim=c(-5, 10))

# Get the true derivatives
phi <-  CreateBasis(K=3, type='legendre01', pts=pts)
basisDerMat <- apply(phi, 2, function(x) 
                       ConvertSupport(seq(0, 1, length.out=M - 1), pts, diff(x) * (M - 1)))
trueDer <- matrix(1, n, M, byrow=TRUE) + tcrossprod(samp2$xi, basisDerMat)
matplot(t(trueDer), type='l', ylim=c(-5, 10))

# DPC is slightly better in terms of RMSE
mean((fitted(FPC) - trueDer)^2)
mean((fitted(DPC) - trueDer)^2)