Factor-augmented Smoothing Model for Functional Data

TL;DR

This paper introduces a factor-augmented smoothing model for functional data that improves curve recovery and data variation capture by incorporating high-dimensional factors, especially under measurement errors or complex data structures.

Contribution

The paper proposes a novel factor-augmented smoothing model with an iterative estimation approach, addressing limitations of traditional smoothing methods in functional data analysis.

Findings

Enhanced estimation accuracy demonstrated in simulations.

Improved modeling of weather and temperature data.

Asymptotic normality of smoothing coefficients projected onto the complement space of factor loadings.

Abstract

We propose modeling raw functional data as a mixture of a smooth function and a highdimensional factor component. The conventional approach to retrieving the smooth function from the raw data is through various smoothing techniques. However, the smoothing model is not adequate 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. To address these challenges, a factor-augmented smoothing model is proposed, 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 is 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 Canadian weather data and Australian temperature data.