Several research lines surround this work: methods for complete curves without measurement error, for one-dimensional domains, and for single change points. Our approach stands apart: it is the first to handle multidimensional domains, sparse observations, and multiple change points simultaneously. Neighboring papers in the archive use neural networks and gravitational lensing for cosmology or improve medical images, but here the task is different: to find the moment when the function's shape itself changes, not to reconstruct it or classify an object.
The FSBS algorithm works in two stages. First, for each time point, it estimates the mean function from discrete observations using kernel smoothing. Then it computes a cumulative sum (CUSUM) that compares the mean functions to the left and right of each candidate change point. To find all change points, it uses seeded binary segmentation: a set of intervals of varying lengths covers the time axis, and on each interval it searches for the point with the maximum CUSUM value. If that value exceeds a threshold, the interval is split in half and the procedure repeats. A key detail: to speed up computations, it checks only randomly selected points in the domain, not the entire grid.
Theorem 1 guarantees that, with probability tending to one, the algorithm finds the correct number of change points, and the localization error for each point is bounded by a term of order a logarithmic factor times $κ^{-2}$, where κ is the minimal jump size. In numerical experiments, FSBS outperforms three existing methods (BGHK, HK, SN) in terms of Hausdorff distance and accuracy of the number of change points across all five scenarios, including the extremely sparse case n=1 and a two-dimensional domain. On real sea surface temperature data from the southern Pacific Ocean, the algorithm detected two change points: 1981 and 1996, which align with extreme El Niño and Indian Ocean Dipole events.
This work differs from neighboring studies by being the first to provide a theoretically justified method for functional data on a multidimensional domain with discrete observations and dependent noise. Neighboring works either require complete curves without error, are limited to one-dimensional domains, or offer no guarantees for multiple change points. Here, we prove consistency and derive the exact localization rate, which reveals a phase transition: for n≥$T^{1/4}$, the error does not depend on n, while for smaller n it grows as $T^{1/5}$$n^{-4/5}$. This is a new contribution to change point detection theory for functional data.
The method can be extended to the case where the minimum distance between change points is much smaller than T, by modifying the set of seeded intervals and using the selection of the smallest interval above the threshold. One could also relax the assumption of equal grid sizes for each curve, allowing different $n_t$, and then the results would depend on the minimum and maximum number of points. Another direction is adaptive selection of the bandwidth h without cross-validation, to speed up computations on large data.
The algorithm is applicable in neuroscience for analyzing EEG or fMRI signals, in climatology for detecting changes in spatial temperature or precipitation fields, and in finance for identifying regime shifts in yield curves. It may also be useful in biology for analyzing gene expression time series measured across different tissue regions.
The next step is to test FSBS on more complex real data with strong dependence and heavy tails, such as financial time series. It would also be worthwhile to implement automatic selection of all three tuning parameters without human intervention.
In the field of change point detection for functional data, the question of optimal localization rate for arbitrary n and T remains open. Our work provides an upper bound, but the lower bound (minimax) for the multidimensional case is not proven. Also unresolved is the problem of varying smoothness of the functions or change points that affect only part of the domain.
The algorithm can detect a change point even if each curve is measured at just one point (n=1). This is equivalent to detecting a change in the mean over time for scalar observations, but with guarantees that account for the functional nature of the data.
In Isaac Asimov's novel 'Foundation', psychohistorians predict crises in the development of a galactic empire by analyzing vast arrays of social data. FSBS does something similar, but for continuous functions: it finds moments when a system's behavior shifts abruptly, which may herald a crisis.