Usually, the complexity of systems, from satellite motions to heart rhythms, is assessed with averaged entropy (a measure of unpredictability, as understood by Boltzmann). But that's like evaluating the ocean only by wave height: the average figure hides the full richness of underwater currents.
The CWMMSE method dives deeper. It automatically groups similar trajectories, as if isolating distinct currents in the ocean, and computes the complexity for each one. Then the overall picture is assembled from these pieces, with the differences between currents being just as important, like warm and cold flows combining to create something greater than the sum of their parts.
This explains paradoxes: in heart disease, rhythm diversity drops even though averaged entropy stays unchanged. And an earthquake can be seen as a collapse of complexity — many small underground movements suddenly merging into one giant jolt.
Thus the method reminds us: in the ocean, organism, or society, the value lies not in average chaos, but in the hidden mosaic of patterns.
as written by the authors arXiv:2607.14738
Across the physical and life sciences, data increasingly appear as ensembles of trajectories, from chaotic flows and satellite constellations to clinical cohorts. Established sample-entropy measures characterize individual time series, while averaging across an ensemble discards population structure and cannot distinguish redundancy from diversity. We introduce clustering-weighted multivariate multiscale sample entropy (CWMMSE), which groups trajectories into behavioral patterns and weights each by its dynamical complexity. CWMMSE is a weighted entropy of the population's pattern distribution. Its empirical plug-in estimator is strongly consistent for a fixed finite partition, and it separates two components that can diverge in real data: individual complexity and population diversity. Both are essential. Averaging ignores diversity, whereas spread alone can mistake a varied but predictable population for a complex one. Across eleven physical, environmental, engineering, and biomedical systems, CWMMSE ranks a calm ocean region above an energetic but individually more complex one, identifies a major earthquake as a collapse in system complexity, and reverses the conclusion from averaging in cardiac cohorts, where disease reduces population diversity. Supported by an open, reproducible implementation, these results show that population complexity should be measured rather than averaged.
Entropy, originally devised for steam engines in the 19th century, now reveals the hidden rhythms of hearts.
Just as Asimov's psychohistory predicts the fate of civilizations by capturing the statistical patterns of crowds, CWMMSE sees order in the chaos of many trajectories.