We introduce Adaptive Functional Principal Component Analysis, a novel method to capture directions of variation in functional data that exhibit sharp changes in smoothness. We first propose a new adaptive scatterplot smoothing technique that is fast and scalable, and then integrate this technique into a probabilistic FPCA framework to adaptively smooth functional principal components. Our simulation results show that our approach is better able to model functional data with sharp changes in smoothness compared to standard approaches. We are motivated by the need to identify coordinated patterns of brain activity across multiple neurons during reaching movements prompted by an auditory cue, which enables understanding of the dynamics in the brain during dexterous movement. Our proposed method captures the underlying biological mechanisms that arise in data obtained from a mouse experiment focused on voluntary reaching movements, offering more interpretable activation patterns that reflect sharp changes in neural activity following the cue. We develop accompanying publicly available software for our proposed methodology, along with implementations to reproduce our results.

翻译：我们提出自适应函数型主成分分析，这是一种捕捉具有平滑度剧变的函数型数据变异方向的新方法。我们首先提出一种快速且可扩展的新型自适应散点图平滑技术，然后将该技术整合到概率函数型主成分分析框架中，对函数型主成分进行自适应平滑。模拟结果表明，与标准方法相比，我们的方法能更好地对具有平滑度剧变的函数型数据进行建模。我们的研究动机源于在听觉提示触发的伸手运动过程中识别多个神经元协调脑活动模式的需求，这有助于理解灵巧运动期间的脑动力学。我们提出的方法能够捕捉来自小鼠自愿伸手运动实验数据中潜在的生物学机制，并提供更可解释的激活模式，这些模式反映了提示后神经活动的剧变。我们开发了配套的公开软件，用于实现我们的方法论，并提供可复现结果的相关实现。