  We study the problem of change-point detection and localisation for
functional data sequentially observed on a general d-dimensional space, where
we allow the functional curves to be either sparsely or densely sampled. Data
of this form naturally arise in a wide range of applications such as biology,
neuroscience, climatology, and finance. To achieve such a task, we propose a
kernel-based algorithm named functional seeded binary segmentation (FSBS). FSBS
is computationally efficient, can handle discretely observed functional data,
and is theoretically sound for heavy-tailed and temporally-dependent
observations. Moreover, FSBS works for a general d-dimensional domain, which is
the first in the literature of change-point estimation for functional data. We
show the consistency of FSBS for multiple change-point estimations and further
provide a sharp localisation error rate, which reveals an interesting phase
transition phenomenon depending on the number of functional curves observed and
the sampling frequency for each curve. Extensive numerical experiments
illustrate the effectiveness of FSBS and its advantage over existing methods in
the literature under various settings. A real data application is further
conducted, where FSBS localises change-points of sea surface temperature
patterns in the south Pacific attributed to El Nino.
