  We propose modeling raw functional data as a mixture of a smooth function and
a high-dimensional factor component. The conventional approach to retrieving
the smooth function from the raw data is through various smoothing techniques.
However, the smoothing model is inadequate to recover the smooth curve or
capture the data variation in some situations. These include cases where there
is a large amount of measurement error, the smoothing basis functions are
incorrectly identified, or the step jumps in the functional mean levels are
neglected. A factor-augmented smoothing model is proposed to address these
challenges, and an iterative numerical estimation approach is implemented in
practice. Including the factor model component in the proposed method solves
the aforementioned problems since a few common factors often drive the
variation that cannot be captured by the smoothing model. Asymptotic theorems
are also established to demonstrate the effects of including factor structures
on the smoothing results. Specifically, we show that the smoothing coefficients
projected on the complement space of the factor loading matrix are
asymptotically normal. As a byproduct of independent interest, an estimator for
the population covariance matrix of the raw data is presented based on the
proposed model. Extensive simulation studies illustrate that these factor
adjustments are essential in improving estimation accuracy and avoiding the
curse of dimensionality. The superiority of our model is also shown in modeling
Australian temperature data.
