  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.
