refund (version 0.1-11)

ccb.fpc: Corrected confidence bands using functional principal components

Description

Uses iterated expectation and variances to obtain corrected estimates and inference for functional expansions.

Usage

ccb.fpc(Y, nbasis = 10, pve = .99, n.boot = 100, simul = FALSE, sim.alpha = .95)

Arguments

Y

matrix of observed functions for which estimates and covariance matrices are desired.

nbasis

number of splines used in the estimation of the mean function and the bivariate smoothing of the covariance matrix

pve

proportion of variance explained used to choose the number of principal components to be included in the expansion.

n.boot

number of bootstrap iterations used to estimate the distribution of FPC decomposition objects.

simul

TRUE or FALSE, indicating whether critical values for simultaneous confidence intervals should be estimated

sim.alpha

alpha level of the simultaneous intervals.

Value

• Yhat
  a matrix whose rows are the estimates of the curves in Y.
• Yhat.boot
  a list containing the estimated curves within each bootstrap iteration.
• 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.

Details

To obtain corrected curve estimates and variances, this function accounts for uncertainty in FPC decomposition objects. Observed curves are resampled, and a FPC decomposition for each sample is constructed. A mixed-model framework is used to estimate curves and variances conditional on each decomposition, and iterated expectation and variances combines both model-based and decomposition-based uncertainty.

References

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

Examples

data(cd4)

# obtain a subsample of the data with 25 subjects
set.seed(1236)
sample = sample(1:dim(cd4)[1], 25)
Y.sub = cd4[sample,]

# obtain a mixed-model based FPCA decomposition
Fit.MM = fpca.sc(Y.sub, var = TRUE, simul = TRUE)

# use iterated variance to obtain curve estimates and variances
Fit.IV = ccb.fpc(Y.sub, n.boot = 25, simul = TRUE)

# for one subject, examine curve estimates, pointwise and simultaneous itervals
EX = 2
EX.IV =  cbind(Fit.IV$Yhat[EX,], 
      Fit.IV$Yhat[EX,] + 1.96 * sqrt(Fit.IV$diag.var[EX,]), 
      Fit.IV$Yhat[EX,] - 1.96 * sqrt(Fit.IV$diag.var[EX,]),
      Fit.IV$Yhat[EX,] + Fit.IV$crit.val[EX] * sqrt(Fit.IV$diag.var[EX,]),
      Fit.IV$Yhat[EX,] - Fit.IV$crit.val[EX] * sqrt(Fit.IV$diag.var[EX,]))

EX.MM =  cbind(Fit.MM$Yhat[EX,], 
      Fit.MM$Yhat[EX,] + 1.96 * sqrt(Fit.MM$diag.var[EX,]), 
      Fit.MM$Yhat[EX,] - 1.96 * sqrt(Fit.MM$diag.var[EX,]),
      Fit.MM$Yhat[EX,] + Fit.MM$crit.val[EX] * sqrt(Fit.MM$diag.var[EX,]),
      Fit.MM$Yhat[EX,] - Fit.MM$crit.val[EX] * sqrt(Fit.MM$diag.var[EX,]))

# plot data for one subject, with curve and interval estimates
d = as.numeric(colnames(cd4))
plot(d[which(!is.na(Y.sub[EX,]))], Y.sub[EX,which(!is.na(Y.sub[EX,]))], type = 'o', 
  pch = 19, cex=.75, ylim = range(0, 3400), xlim = range(d), 
    xlab = "Months since seroconversion", lwd = 1.2, ylab = "Total CD4 Cell Count", 
      main = "Est. & CI - Sampled Data")

matpoints(d, EX.IV, col = 2, type = 'l', lwd = c(2, 1, 1, 1, 1), lty = c(1,1,1,2,2))
matpoints(d, EX.MM, col = 4, type = 'l', lwd = c(2, 1, 1, 1, 1), lty = c(1,1,1,2,2))

legend("topright", c("IV Est", "IV PW Int", "IV Simul Int",
    expression(paste("MM - ", hat(theta), " Est", sep = "")), 
    expression(paste("MM - ", hat(theta), " PW Int", sep = "")), 
    expression(paste("MM - ", hat(theta), " Simul Int", sep = ""))), 
    lty=c(1,1,2,1,1,2), lwd = c(2.5,.75,.75,2.5,.75,.75), 
    col = c("red","red","red","blue","blue","blue"))