# Obtain the derivatives of eigenfunctions/ eigenfunctions of derivatives (note: these two are not the same) ```r 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') ## Examples ```r 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) ``` ## References Dai, X., Tao, W., Müller, H.G. (2018). Derivative principal componentsfor representing the time dynamics of longitudinal and functional data.Statistica Sinica 28, 1583-1609. (DPC) Liu, Bitao, and Hans-Georg Müller. "Estimating derivatives for samplesof sparsely observed functions, with application to online auctiondynamics." Journal of the American Statistical Association 104, no. 486(2009): 704-717. (FPC)

FPCAder function