fitted.FPCA {fdapace}

Fitted functional data from FPCA object

Description

Combines the zero-meaned fitted values and the interpolated mean to get the fitted values for the trajectories or the derivatives of these trajectories. Estimates are given on the work-grid, not on the observation grid. Use ConvertSupport to map the estimates to your desired domain. 100*(1-alpha)-percentage coverage intervals, or bands, for trajectory estimates (not derivatives) are provided. For details consult the example.

Usage

## S3 method for class 'FPCA'
fitted(
  object,
  K = NULL,
  derOptns = list(p = 0),
  ciOptns = list(alpha = NULL, cvgMethod = NULL),
  ...
)

Arguments

object

A object of class FPCA returned by the function FPCA().

K

The integer number of the first K components used for the representation. (default: length(fpcaObj$lambda ))

derOptns

A list of options to control the derivation parameters specified by list(name=value). See ‘Details’. (default = NULL)

ciOptns

A list of options to control the confidence interval/band specified by list(name=value). See ‘Details’. (default = NULL)

...

Additional arguments

Details

Available derivation control options are

p

The order of the derivatives returned (default: 0, max: 2)

method

The method used to produce the sample of derivatives ('FPC' (default) or 'QUO'). See Liu and Müller (2009) for more details

bw

Bandwidth for smoothing the derivatives (default: p * 0.10 * S)

kernelType

Smoothing kernel choice; same available types are FPCA(). default('epan')

Available confidence interval/band control options are

alpha

Significant level for confidence interval/band for trajectory coverage. default=0.05 (currently only work when p=0)

cvgMethod

Option for trajectory coverage method between 'interval' (pointwise coverage) and 'band' (simultaneous coverage). default='band'

Value

If alpha is NULL, p>1 or functional observations are dense, an n by length(workGrid) matrix, each row of which contains a sample. Otherwise, it returns a list which consists of the following items:

workGrid

An evaluation grid for fitted values.

fitted

An n by length(workGrid) matrix, each row of which contains a sample.

cvgUpper

An n by length(workGrid) matrix, each row of which contains the upper alpha-coverage limit

cvgLower

An n by length(workGrid) matrix, each row of which contains the lower alpha-coverage limit

References

Yao, F., Müller, H.-G. and Wang, J.-L. "Functional data analysis for sparse longitudinal data", Journal of the American Statistical Association, vol.100, No. 470 (2005): 577-590.

Liu, Bitao, and Hans-Georg Müller. "Estimating derivatives for samples of sparsely observed functions, with application to online auction dynamics." Journal of the American Statistical Association 104, no. 486 (2009): 704-717. (Sparse data FPCA)

Examples

set.seed(1)
n <- 100
pts <- seq(0, 1, by=0.05)
sampWiener <- Wiener(n, pts)
sampWiener <- Sparsify(sampWiener, pts, 5:10)
res <- FPCA(sampWiener$Ly, sampWiener$Lt, 
            list(dataType='Sparse', error=FALSE, kernel='epan', verbose=TRUE))
fittedY <- fitted(res, ciOptns = list(alpha=0.05))

workGrid <- res$workGrid
cvgUpper <- fittedY$cvgUpper
cvgLower <- fittedY$cvgLower

op <- par(mfrow=c(2,3))
ind <- sample(1:n,6)
for (i in 1:6) {
 j <- ind[i]
 plot(workGrid,cvgUpper[j,],type='l',ylim=c(min(cvgLower[j,]),max(cvgUpper[j,])),col=4,lty=2,
   xlab='t', ylab='X(t)', main=paste(j,'-th subject',sep=''))
 points(workGrid,cvgLower[j,],type='l',col=4,lty=2)
 points(res$inputData$Lt[[j]],res$inputData$Ly[[j]])
}
par(op)