  In this article, we consider the problem of recovering the underlying
trajectory when the longitudinal data are sparsely and irregularly observed and
noise-contaminated. Such data are popularly analyzed with functional principal
component analysis via the Principal Analysis by Conditional Estimation (PACE)
method. The PACE method may sometimes be numerically unstable because it
involves the inverse of the covariance matrix. We propose a sparse orthonormal
approximation (SOAP) method as an alternative. It estimates the optimal
empirical basis functions in the best approximation framework rather than
eigen-decomposing the covariance function. The SOAP method avoids estimating
the mean and covariance function, which is challenging when the assembled time
points with observations for all subjects are not sufficiently dense. The SOAP
method avoids the inverse of the covariance matrix, hence the computation is
more stable. It does not require the functional principal component scores to
follow the Gaussian distribution. We show that the SOAP estimate for the
optimal empirical basis function is asymptotically consistent. The finite
sample performance of the SOAP method is investigated in simulation studies in
comparison with the PACE method. Our method is demonstrated by recovering the
CD4 percentage curves from sparse and irregular data in the Multi-center AIDS
Cohort Study.
