Factor-augmented model for functional data

We propose modeling raw functional data as a mixture of a smooth function and a high-dimensional factor component. The conventional approach to retrieving a smooth function from raw data is to use a smoothing technique. However, the smoothing model is unable to recover the smooth curve or capture the data variation in some situations, for example, when there is a large measurement error, the smoothing basis functions are incorrectly identified, or the step jumps in the functional mean levels are neglected. We propose a factor-augmented smoothing model to address these challenges, and implement an iterative numerical estimation approach. Including the factor model component in the proposed method solves the aforementioned problems because a few common factors often drive the variation that cannot be captured by the smoothing model. We also establish asymptotic theorems 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. Of independent interest, we present an estimator for the population covariance matrix of the raw data, based on the proposed model. Extensive simulation studies show that these factor adjustments are essential to improving the estimation accuracy and avoiding the curse of dimensionality. Lastly, we demonstrate the performance of our model by applying it to Australian temperature data.