fpca.tfd: Functional principal components analysis by smoothed...

In gekim0519/tidyfunfun:

Description

Decomposes functional observations using functional principal components analysis. A mixed model framework is used to estimate scores and obtain variance estimates.

Usage

fpca.tfd(data = NULL, col = NULL, Y.pred = NULL, argvals = NULL,
  random.int = FALSE, nbasis = 10, pve = 0.99, npc = NULL,
  var = FALSE, simul = FALSE, sim.alpha = 0.95, useSymm = FALSE,
  makePD = FALSE, center = TRUE, cov.est.method = 2,
  integration = "trapezoidal")

Arguments

data	the user must supply data, a dataframe containing a tfd column, col
col	tfd-vector of regular data with a common grid and irregular or sparse data
Y.pred	if desired, a matrix of functions to be approximated using the FPC decomposition.
argvals	the argument values of the function evaluations in Y, defaults to a equidistant grid from 0 to 1.
random.int	If TRUE, the mean is estimated by gamm4 with random intercepts. If FALSE (the default), the mean is estimated by gam treating all the data as independent.
nbasis	number of B-spline basis functions used for estimation of the mean function and bivariate smoothing of the covariance surface.
pve	proportion of variance explained: used to choose the number of principal components.
npc	prespecified value for the number of principal components (if given, this overrides pve).
var	TRUE or FALSE indicating whether model-based estimates for the variance of FPCA expansions should be computed.
simul	logical: should critical values be estimated for simultaneous confidence intervals?
sim.alpha	1 - coverage probability of the simultaneous intervals.
useSymm	logical, indicating whether to smooth only the upper triangular part of the naive covariance (when cov.est.method==2). This can save computation time for large data sets, and allows for covariance surfaces that are very peaked on the diagonal.
makePD	logical: should positive definiteness be enforced for the covariance surface estimate?
center	logical: should an estimated mean function be subtracted from Y? Set to FALSE if you have already demeaned the data using your favorite mean function estimate.
cov.est.method	covariance estimation method. If set to 1, a one-step method that applies a bivariate smooth to the y(s_1)y(s_2) values. This can be very slow. If set to 2 (the default), a two-step method that obtains a naive covariance estimate which is then smoothed.
integration	quadrature method for numerical integration; only 'trapezoidal' is currently supported.

Details

Altered from 'refund::fpca.sc', this function allows users to apply the function directly on dataframes with a tfd column, specified in the col argument.

This function computes a FPC decomposition for a set of observed curves, which may be sparsely observed and/or measured with error. A mixed model framework is used to estimate curve-specific scores and variances.

Value

An object of class fpca containing:

Yhat	FPC approximation (projection onto leading components) of Y.pred if specified, or else of Y.
Y	the observed data
scores	n \times npc matrix of estimated FPC scores.
mu	estimated mean function (or a vector of zeroes if center==FALSE).
efunctions	d \times npc matrix of estimated eigenfunctions of the functional covariance, i.e., the FPC basis functions.
evalues	estimated eigenvalues of the covariance operator, i.e., variances of FPC scores.
npc	number of FPCs: either the supplied npc, or the minimum number of basis functions needed to explain proportion pve of the variance in the observed curves.
argvals	argument values of eigenfunction evaluations
sigma2	estimated measurement error variance.
diag.var	diagonal elements of the covariance matrices for each estimated curve.
VarMats	a list containing the estimated covariance matrices for each curve in Y.
crit.val	estimated critical values for constructing simultaneous confidence intervals.

Author(s)

Gaeun Kim [e-mail] and Jeff Goldsmith [e-mail]

References

Di, C., Crainiceanu, C., Caffo, B., and Punjabi, N. (2009). Multilevel functional principal component analysis. Annals of Applied Statistics, 3, 458–488.

Goldsmith, J., Greven, S., and Crainiceanu, C. (2013). Corrected confidence bands for functional data using principal components. Biometrics, 69(1), 41–51.

Staniswalis, J. G., and Lee, J. J. (1998). Nonparametric regression analysis of longitudinal data. Journal of the American Statistical Association, 93, 1403–1418.

Yao, F., Mueller, H.-G., and Wang, J.-L. (2005). Functional data analysis for sparse longitudinal data. Journal of the American Statistical Association, 100, 577–590.

Examples

## Not run: 
library(ggplot2)
library(reshape2)
data(dti)

fit.cca = fpca.tfd(data = dti, col = cca)
fit.mu = data.frame(mu = fit.cca$mu,
                   n = 1:ncol(fit.cca$Yhat))
fit.basis = data.frame(phi = fit.cca$efunctions, #the FPC basis functions.
                      n = 1:ncol(fit.cca$Yhat))

## plot estimated mean function
ggplot(fit.mu, aes(x = n, y = mu)) + geom_path() +
      theme_bw() + ggtitle("Estimated mean function of corpus callosum")

## plot the first two estimated basis functions
fit.basis.m = melt(fit.basis, id = 'n')
ggplot(subset(fit.basis.m, variable %in% c('phi.1', 'phi.2')),
       aes(x = n, y = value, group = variable, color = variable)) +
       geom_path() + theme_bw() + ggtitle("First two estimated FPC basis functions")


## End(Not run)

gekim0519/tidyfunfun documentation built on Aug. 2, 2019, 5:18 a.m.