This paper studies linear reconstruction of partially observed functional data which are recorded on a discrete grid. We propose a novel estimation approach based on approximate factor models with increasing rank taking into account potential covariate information. Whereas alternative reconstruction procedures commonly involve some preliminary smoothing, our method separates the signal from noise and reconstructs missing fragments at once. We establish uniform convergence rates of our estimator and introduce a new method for constructing simultaneous prediction bands for the missing trajectories. A simulation study examines the performance of the proposed methods in finite samples. Finally, a real data application of temperature curves demonstrates that our theory provides a simple and effective method to recover missing fragments.

翻译：本文研究离散网格上记录的部分观测函数数据的线性重建问题。我们提出一种基于近似因子模型的新型估计方法，该模型具有递增秩并考虑潜在协变量信息。与通常涉及预平滑处理的替代重建方法不同，我们的方法能同时实现信号与噪声分离及缺失片段重建。我们建立了估计量的一致收敛速率，并提出一种为缺失轨迹构建联合预测带的新方法。通过模拟研究检验了所提方法在有限样本下的表现。最后，温度曲线的实际数据应用表明，我们的理论为恢复缺失片段提供了一种简单有效的方法。