Extremal Geometry: Signal-Induced Supports for Time Series Representation

Local patterns in time series can occur over different durations, yet many representations use fixed windows, scales, or resolutions. We study extremal geometry, which represents each local maximum or minimum by a signed magnitude and a two-sided temporal support determined by the surrounding signal. Spikelet is a concrete realization developed for variable-duration time-series analysis. We characterize the prior one-pass Spikelet construction and give a linear-time completion that recovers exact signal-induced supports. The completed supports are noncrossing separately for maxima and minima, inducing an inclusion hierarchy across temporal extents. Magnitude pruning preserves the observed value at every retained apex and creates no new strict extrema. For two-sided interior events, the absolute magnitude also equals the corresponding zero-dimensional persistence lifetime, providing a scalar connection to persistent homology. We further compare extremal geometry with fixed-scale Gaussian smoothing and fixed-resolution Fourier reconstruction: smoothing changes global extremal values, whereas reconstruction can create false extrema even with lower $L^2$ error. Controlled experiments audit these structural claims, and a real-data audit shows that magnitude-selected events retain widely varying signal-induced temporal supports.