A seismograph records underground tremors: the smooth line suddenly jerks and changes its character. Similarly, functional data, such as ocean temperature maps or brain signals, may shift their shape at some point. The new algorithm FSBS catches such breakpoints, even when each curve is measured at only one point and noise drowns the signal.

Previously, detecting such changes required complete, error-free curves or one-dimensional domains. FSBS is the first algorithm to work on multidimensional domains with sparse, noisy observations. It first reconstructs the mean function at each time point using kernel smoothing, then compares the averages to the left and right of every candidate change point. To find all breakpoints, the algorithm scans a set of intervals of varying lengths and splits them in half if it finds a suspicious jump.

Theoretical analysis reveals a phase transition: when the number of curves is sufficiently large, the localization error of a change point does not depend on that number. If there are fewer curves, the error grows according to a power law. In numerical experiments, the algorithm outperformed three existing methods in all five scenarios, including the extremely sparse case with one point per curve.

On real sea surface temperature data from the southern Pacific Ocean, the algorithm detected two change points: 1981 and 1996. These coincide with extreme El Niño and Indian Ocean Dipole events. The seismograph went off: the tremors of climate shifts became visible in the flow of functional data.