Online Estimation for Functional Data

Functional data analysis has attracted considerable interest and is facing new challenges, one of which is the increasingly available data in a streaming manner. In this article we develop an online nonparametric method to dynamically update the estimates of mean and covariance functions for functional data. The kernel-type estimates can be decomposed into two sufficient statistics depending on the data-driven bandwidths. We propose to approximate the future optimal bandwidths by a sequence of dynamically changing candidates and combine the corresponding statistics across blocks to form the updated estimation. The proposed online method is easy to compute based on the stored sufficient statistics and the current data block. We derive the asymptotic normality and, more importantly, the relative efficiency lower bounds of the online estimates of mean and covariance functions. This provides insight into the relationship between estimation accuracy and computational cost driven by the length of candidate bandwidth sequence. Simulations and real data examples are provided to support such findings. Supplementary materials for this article are available online.

函数数据分析（Functional Data Analysis）已受到学界广泛关注，同时亦面临诸多新挑战，其中一项便是流式获取的数据愈发普遍。本文提出一种在线非参数方法，可对函数数据的均值函数与协方差函数的估计值进行动态更新。核型估计可分解为两个充分统计量，二者均依赖于数据驱动的带宽参数。我们提出通过一组动态变化的候选带宽序列来近似未来的最优带宽，并跨数据区块合并对应统计量以生成更新后的估计结果。所提在线方法仅需依托已存储的充分统计量与当前数据块即可完成计算，实现便捷。本文推导了均值与协方差函数在线估计的渐近正态性，更重要的是，还推导了其相对效率下界。该结论揭示了估计精度与由候选带宽序列长度决定的计算成本之间的内在关联。本文通过模拟实验与真实数据案例验证了上述结论。本文的补充材料可在线获取。