refund (version 0.1-37)

fpca.sc: Functional principal components analysis by smoothed covariance

Description

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

Usage

fpca.sc(
  Y = NULL,
  ydata = 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"
)

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

pve

The percent variance explained by the returned number of PCs

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.

Arguments

Y, ydata

the user must supply either Y, a matrix of functions observed on a regular grid, or a data frame ydata representing irregularly observed functions. See Details.

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.

Author

Jeff Goldsmith , Sonja Greven , Lan Huo , Lei Huang , and Philip Reiss

Details

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.

FPCA via kernel smoothing of the covariance function, with the diagonal treated separately, was proposed in Staniswalis and Lee (1998) and much extended by Yao et al. (2005), who introduced the 'PACE' method. fpca.sc uses penalized splines to smooth the covariance function, as developed by Di et al. (2009) and Goldsmith et al. (2013).

The functional data must be supplied as either

• an \(n \times d\) matrix Y, each row of which is one functional observation, with missing values allowed; or

• a data frame ydata, with columns '.id' (which curve the point belongs to, say \(i\)), '.index' (function argument such as time point \(t\)), and '.value' (observed function value \(Y_i(t)\)).

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

if (FALSE) {
library(ggplot2)
library(reshape2)
data(cd4)

Fit.MM = fpca.sc(cd4, var = TRUE, simul = TRUE)

Fit.mu = data.frame(mu = Fit.MM$mu,
                    d = as.numeric(colnames(cd4)))
Fit.basis = data.frame(phi = Fit.MM$efunctions,
                       d = as.numeric(colnames(cd4)))

## for one subject, examine curve estimate, pointwise and simultaneous itervals
EX = 1
EX.MM = data.frame(fitted = Fit.MM$Yhat[EX,],
           ptwise.UB = Fit.MM$Yhat[EX,] + 1.96 * sqrt(Fit.MM$diag.var[EX,]),
           ptwise.LB = Fit.MM$Yhat[EX,] - 1.96 * sqrt(Fit.MM$diag.var[EX,]),
           simul.UB = Fit.MM$Yhat[EX,] + Fit.MM$crit.val[EX] * sqrt(Fit.MM$diag.var[EX,]),
           simul.LB = Fit.MM$Yhat[EX,] - Fit.MM$crit.val[EX] * sqrt(Fit.MM$diag.var[EX,]),
           d = as.numeric(colnames(cd4)))

## plot data for one subject, with curve and interval estimates
EX.MM.m = melt(EX.MM, id = 'd')
ggplot(EX.MM.m, aes(x = d, y = value, group = variable, color = variable, linetype = variable)) +
  geom_path() +
  scale_linetype_manual(values = c(fitted = 1, ptwise.UB = 2,
                        ptwise.LB = 2, simul.UB = 3, simul.LB = 3)) +
  scale_color_manual(values = c(fitted = 1, ptwise.UB = 2,
                     ptwise.LB = 2, simul.UB = 3, simul.LB = 3)) +
  labs(x = 'Months since seroconversion', y = 'Total CD4 Cell Count')

## plot estimated mean function
ggplot(Fit.mu, aes(x = d, y = mu)) + geom_path() +
  labs(x = 'Months since seroconversion', y = 'Total CD4 Cell Count')

## plot the first two estimated basis functions
Fit.basis.m = melt(Fit.basis, id = 'd')
ggplot(subset(Fit.basis.m, variable %in% c('phi.1', 'phi.2')), aes(x = d,
y = value, group = variable, color = variable)) + geom_path()

## input a dataframe instead of a matrix
nid <- 20
nobs <- sample(10:20, nid, rep=TRUE)
ydata <- data.frame(
    .id = rep(1:nid, nobs),
    .index = round(runif(sum(nobs), 0, 1), 3))
ydata$.value <- unlist(tapply(ydata$.index,
                              ydata$.id,
                              function(x)
                                  runif(1, -.5, .5) +
                                  dbeta(x, runif(1, 6, 8), runif(1, 3, 5))
                              )
                       )

Fit.MM = fpca.sc(ydata=ydata, var = TRUE, simul = FALSE)

}