A novel method for noise reduction in the setting of curve time series with error contamination is proposed, based on extending the framework of functional principal component analysis (FPCA). We employ the underlying, finite-dimensional dynamics of the functional time series to separate the serially dependent dynamical part of the observed curves from the noise. Upon identifying the subspaces of the signal and idiosyncratic components, we construct a projection of the observed curve time series along the noise subspace, resulting in an estimate of the underlying denoised curves. This projection is optimal in the sense that it minimizes the mean integrated squared error. By applying our method to similated and real data, we show the denoising estimator is consistent and outperforms existing denoising techniques. Furthermore, we show it can be used as a pre-processing step to improve forecasting.

翻译：针对含误差污染的曲线时间序列，本文提出了一种基于函数型主成分分析框架扩展的新型降噪方法。我们利用函数型时间序列潜在的有限维动力学特性，将观测曲线的序列相关动态部分与噪声分离。在识别信号子空间与异质成分子空间后，沿噪声子空间对观测曲线时间序列进行投影，从而得到潜在降噪曲线的估计。该投影在最小化均方积分误差意义下具有最优性。通过模拟数据与真实数据的应用，我们证明该降噪估计量具有一致性且优于现有降噪技术。此外，该方法可作为预处理步骤用于提升预测性能。