Implements the algorithm of Huang, Shen, Buja (2008) for finding smooth right
singular vectors of a matrix X containing (contaminated) evaluations of
functional random variables on a regular, equidistant grid. If the number of
smooth SVs to extract is not specified, the function hazards a guess for the
appropriate number based on the asymptotically optimal truncation threshold
under the assumption of a low rank matrix contaminated with i.i.d. Gaussian
noise with unknown variance derived in Donoho, Gavish (2013). Please note that
Donoho, Gavish (2013) should be regarded as experimental for functional PCA,
and will typically not work well if you have more observations than grid
points.
fpca.ssvd(
Y = NULL,
ydata = NULL,
argvals = NULL,
npc = NA,
center = TRUE,
maxiter = 15,
tol = 1e-04,
diffpen = 3,
gridsearch = TRUE,
alphagrid = 1.5^(-20:40),
lower.alpha = 1e-05,
upper.alpha = 1e+07,
verbose = FALSE,
integration = "trapezoidal"
)data matrix (rows: observations; columns: grid of eval. points)
a data frame ydata representing
irregularly observed functions. NOT IMPLEMENTED for this method.
the argument values of the function evaluations in Y,
defaults to a equidistant grid from 0 to 1. See Details.
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 Y so that its column-means are 0? Defaults to
TRUE
how many iterations of the power algorithm to perform at most (defaults to 15)
convergence tolerance for power algorithm (defaults to 1e-4)
difference penalty order controlling the desired smoothness of the right singular vectors, defaults to 3 (i.e., deviations from local quadratic polynomials).
use optimize or a grid search to find
GCV-optimal smoothing parameters? defaults to TRUE.
grid of smoothing parameter values for grid search
lower limit for for smoothing parameter if
!gridsearch
upper limit for smoothing parameter if !gridsearch
generate graphical summary of progress and diagnostic messages?
defaults to FALSE
ignored, see Details.
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.
Note that fpca.ssvd 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,
fpca.ssvd will simply return the smoothed eigenvectors without rescaling,
with a warning.
Huang, J. Z., Shen, H., and Buja, A. (2008). Functional principal components analysis via penalized rank one approximation. Electronic Journal of Statistics, 2, 678-695
Donoho, D.L., and Gavish, M. (2013). The Optimal Hard Threshold for Singular Values is 4/sqrt(3). eprint arXiv:1305.5870. Available from https://arxiv.org/abs/1305.5870.
fpca.sc and fpca.face for FPCA based on
smoothing a covariance estimate; fpca2s for a faster SVD-based
approach.
# NOT RUN {
## as in Sec. 6.2 of Huang, Shen, Buja (2008):
set.seed(2678695)
n <- 101
m <- 101
s1 <- 20
s2 <- 10
s <- 4
t <- seq(-1, 1, l=m)
v1 <- t + sin(pi*t)
v2 <- cos(3*pi*t)
V <- cbind(v1/sqrt(sum(v1^2)), v2/sqrt(sum(v2^2)))
U <- matrix(rnorm(n*2), n, 2)
D <- diag(c(s1^2, s2^2))
eps <- matrix(rnorm(m*n, sd=s), n, m)
Y <- U%*%D%*%t(V) + eps
smoothSV <- fpca.ssvd(Y, verbose=TRUE)
layout(t(matrix(1:4, nr=2)))
clrs <- sapply(rainbow(n), function(c)
do.call(rgb, as.list(c(col2rgb(c)/255, .1))))
matplot(V, type="l", lty=1, col=1:2, xlab="",
main="FPCs: true", bty="n")
matplot(smoothSV$efunctions, type="l", lty=1, col=1:5, xlab="",
main="FPCs: estimate", bty="n")
matplot(1:m, t(U%*%D%*%t(V)), type="l", lty=1, col=clrs, xlab="", ylab="",
main="true smooth Y", bty="n")
matplot(1:m, t(smoothSV$Yhat), xlab="", ylab="",
type="l", lty=1,col=clrs, main="estimated smooth Y", bty="n")
# }