fpca2s {refund}

Functional principal component analysis by a two-stage method

Description

This function performs functional PCA by performing an ordinary singular value decomposition on the functional data matrix, then smoothing the right singular vectors by smoothing splines.

Usage

fpca2s(
  Y = NULL,
  ydata = NULL,
  argvals = NULL,
  npc = NA,
  center = TRUE,
  smooth = TRUE
)

Arguments

Y

data matrix (rows: observations; columns: grid of eval. points)

ydata

a data frame ydata representing irregularly observed functions. NOT IMPLEMENTED for this method.

argvals

the argument values of the function evaluations in Y, defaults to a equidistant grid from 0 to 1. See Details.

npc

how many smooth SVs to try to extract, if NA (the default) the hard thresholding rule of Donoho, Gavish (2013) is used (see Details, References).

center

center Y so that its column-means are 0? Defaults to TRUE

smooth

logical; defaults to TRUE, if NULL, no smoothing of eigenvectors.

Details

Note that fpca2s computes smoothed orthonormal eigenvectors of the supplied function evaluations (and associated scores), not (!) evaluations of the smoothed orthonormal eigenfunctions. The smoothed orthonormal eigenvectors are then rescaled by the length of the domain defined by argvals to have a quadratic integral approximately equal to one (instead of crossproduct equal to one), so they approximate the behavior of smooth eigenfunctions. If argvals is not equidistant, fpca2s will simply return the smoothed eigenvectors without rescaling, with a warning.

Value

an fpca object like that returned from fpca.sc, with entries Yhat, the smoothed trajectories, Y, the observed data, scores, the estimated FPC loadings, mu, the column means of Y (or a vector of zeroes if !center), efunctions, the estimated smooth FPCs (note that these are orthonormal vectors, not evaluations of orthonormal functions if argvals is not equidistant), evalues, their associated eigenvalues, and npc, the number of smooth components that were extracted.

Author(s)

Luo Xiao [e-mail], Fabian Scheipl

References

Xiao, L., Ruppert, D., Zipunnikov, V., and Crainiceanu, C., (2013), Fast covariance estimation for high-dimensional functional data. (submitted) https://arxiv.org/abs/1306.5718.

Gavish, M., and Donoho, D. L. (2014). The optimal hard threshold for singular values is 4/sqrt(3). IEEE Transactions on Information Theory, 60(8), 5040–5053.

See Also

fpca.sc and fpca.face for FPCA based on smoothing a covariance estimate; fpca.ssvd for another SVD-based approach.

Examples

  #### settings
  I <- 50 # number of subjects
  J <- 3000 # dimension of the data
  t <- (1:J)/J # a regular grid on [0,1]
  N <- 4 #number of eigenfunctions
  sigma <- 2 ##standard deviation of random noises
  lambdaTrue <- c(1,0.5,0.5^2,0.5^3) # True eigenvalues

  case = 1
  ### True Eigenfunctions

  if(case==1) phi <- sqrt(2)*cbind(sin(2*pi*t),cos(2*pi*t),
                                   sin(4*pi*t),cos(4*pi*t))
  if(case==2) phi <- cbind(rep(1,J),sqrt(3)*(2*t-1),
                           sqrt(5)*(6*t^2-6*t+1),
                           sqrt(7)*(20*t^3-30*t^2+12*t-1))

  ###################################################
  ########     Generate Data            #############
  ###################################################
  xi <- matrix(rnorm(I*N),I,N);
  xi <- xi%*%diag(sqrt(lambdaTrue))
  X <- xi%*%t(phi); # of size I by J
  Y <- X + sigma*matrix(rnorm(I*J),I,J)

  results <- fpca2s(Y,npc=4,argvals=t)
  ###################################################
  ####               SVDS               ########
  ###################################################
  Phi <- results$efunctions
  eigenvalues <- results$evalues

  for(k in 1:N){
    if(Phi[,k]%*%phi[,k]< 0)
      Phi[,k] <- - Phi[,k]
  }

 ### plot eigenfunctions
 par(mfrow=c(N/2,2))
 seq <- (1:(J/10))*10
 for(k in 1:N){
      plot(t[seq],Phi[seq,k]*sqrt(J),type='l',lwd = 3,
           ylim = c(-2,2),col = 'red',
           ylab = paste('Eigenfunction ',k,sep=''),
           xlab='t',main='SVDS')

      lines(t[seq],phi[seq,k],lwd = 2, col = 'black')
      }