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：本文研究在离散网格上记录的部分观测函数型数据的线性重建问题。我们提出了一种基于近似因子模型的新颖估计方法，该方法通过考虑潜在协变量信息并允许因子数量递增。与通常涉及初步平滑的替代重建程序不同，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，并提出了构建缺失轨迹同步预测带的新方法。模拟研究检验了所提方法在有限样本中的表现。最后，基于温度曲线的真实数据应用表明，我们的理论为恢复缺失片段提供了简洁有效的解决方案。