The study, by Rongzhen Dai, an independent researcher based in Yangzhou, China, draws on the IBTrACS v04r01 satellite-era archive of 6661 storms with valid coordinates, containing 406,059 six-hourly fixes. The event set consists of 25 storms whose tracks overlap ±15-day windows around three benchmark geomagnetic storms, one from each recent solar cycle: the Halloween storm of 2003, with a peak Dst of roughly −383 nT; the St Patrick’s Day storm of 2015, at about −234 nT; and the Gannon storm of 2024, provisionally at about −406 nT. Against these, the author assembled 1020 calendar-matched control storms from the same calendar windows in every other year between 1981 and 2024, excluding the event years and two years on either side.

Persistent homology works by growing a ball of radius r around every point in a cloud and recording the radius at which each hole in the resulting union opens and closes. The result is a list of birth–death pairs, and the study restricts attention to connected components (H0) and loops (H1). Two such lists are compared by matching their holes and summing the cost of the matching, and the null distribution is built by resampling control storms, with the storm rather than the individual fix as the unit of resampling. The author is explicit that neither channel has a literal physical counterpart: H1 features are not closed storm tracks, and H0 components are not storm clusters.

When the same 25-storm event set and 1020-storm control pool were passed through three principled longitude encodings — a linear modular normalisation that ignores periodicity, a unit-sphere embedding, and a cylinder embedding — the three gave three different verdicts. The one-sided H1 resampling p-values were 0.108 for the linear encoding on a single longitude convention, 0.021 for the unit sphere, and 0.156 for the cylinder. Under the archived longitude convention, the linear and sphere encodings crossed the conventional 0.05 threshold while the cylinder did not; under the single convention, only the sphere did. The relative ordering of the three was stable across six event-subsample seeds — the sphere always smallest, the cylinder always largest — but their positions relative to 0.05 were not reproducible.

The paper traces part of the problem to a data defect discovered after the analyses were run: the archived IBTrACS snapshot does not use a single longitude convention, so 2.19 percent of fixes are recorded east of +180° while others carry the negative representation of the same meridians. Reducing the field to a single convention moved the linear H1 p-value from 0.037 to 0.108, and rotating the branch cut by 0, 90 and 180 degrees gave 0.108, 0.081 and 0.022. The sphere, cylinder and great-circle representations pass longitude through a periodic function before any distance is taken and are unaffected by this defect.

The repair the paper proposes is to abandon the encodings altogether and compute a great-circle distance directly between fixes, combined with an intensity term, feeding that distance matrix straight into the persistence computation. Under this physically interpretable distance, the H1 resampling p-value stayed within a narrow range across control-draw orderings, and the placebo calibration — 49 fake event-date sets at random years — was nominal, with per-ordering false positives of 2 to 4 out of 49. On the axis on which the three encodings disagreed, the physical distance returned a single value. The event–control difference under this distance survived per-window, basin, era and joint matching under the summed matching cost, though the era-matched design failed two-sided under the more conservative doubling convention, and the difference vanished entirely under the true degree-2 Wasserstein distance and the exact bottleneck distance.

What the repair left untouched proved substantial. The exponent of the matching cost — whether mismatched holes are charged linearly or by their squares — moved the resampling p-value from 0.003 to 0.087. The choice of which 600 of the ensemble’s fixes enter the diagram moved it from 0.012 to 0.069, and the choice of which fifty control diagrams form the reference set moved it over 0.036 to 0.047 on the linear encoding alone. The paper locates the mechanism in the persistence diagram itself: the event’s excess sits at intermediate persistence, in a band corresponding to great-circle separations of 3.6 to 7.2 degrees, while the single most persistent feature is unremarkable, at the 82nd percentile of its null. Squaring the matching cost concentrates the statistic on the largest features, doubling the noise floor while moving the event-to-control ratio by only 2.5 percent.

Rather than selecting any single exponent, the paper reports a specification-averaged test over a grid of seven exponents between 1 and 2, with the statistic averaged over ten independent subsamples of the event ensemble and of every control draw alike. The specification-averaged p-value was 0.082 over exponents under the basin-matched design, and 0.122 and 0.134 over wider unmatched families. The event lay above the control median in all 42 exponent-by-representation cells, at a median 86th percentile, and the averaged separation reached 1.87 standard deviations against the 1.96 required at the 5 percent level. The paper’s own conclusion is a design- and specification-limited directional residual: no detection, and no evidence of absence.

The author also identifies a structural limit that no statistical refinement can overcome. The three events were not selected on geomagnetic magnitude alone; the inclusion criteria also required sufficient cross-basin tropical-cyclone activity in the window, one benchmark per recent solar cycle, and two-year independence. Because cyclone activity is part of the outcome under study, event eligibility depends in part on the very quantity the test examines, so a design of this kind cannot establish the direction of influence regardless of how the statistics come out. Any physical test of the solar–cyclone question, the paper argues, must define its exposure from geomagnetic and solar-wind indices alone before any cyclone data is consulted.

The paper closes with a five-point minimum protocol for future ensemble-topology studies: report any positive H1 claim under at least two principled representations, register the matching exponent or report a statistic that selects none, average over at least ten subsample realisations for the event and every control draw alike, register the reference set alongside the exponent and subsample, and fix the control-draw ordering by an explicit sort key while reporting the Monte-Carlo uncertainty of the resulting p-value in both homology channels. The author acknowledges that the present analysis does not fully meet its own protocol, with a placebo set of 49 against a recommended minimum of 200 and only 100 null draws per placebo. The broader lesson, the paper suggests, extends beyond the solar–cyclone question: a pipeline whose verdict depends on how trajectories are represented, without converging, is not yet ready to test the substantive hypothesis at all.

Subject of Research: Specification dependence in ensemble persistent homology tests of cyclone trajectories around geomagnetic storms

Article Title: A physically grounded input distance resolves longitude-encoding ambiguity but not specification dependence in ensemble persistent homology of cyclone trajectories

Article References: Dai, R. (2026). A physically grounded input distance resolves longitude-encoding ambiguity but not specification dependence in ensemble persistent homology of cyclone trajectories. Nonlinear Processes in Geophysics, 33(4), 521-537. https://doi.org/10.5194/npg-33-521-2026

Image Credits: AI Generated

DOI: 10.5194/npg-33-521-2026

Keywords: persistent homology, tropical cyclones, geomagnetic storms, topological data analysis, longitude encoding, great-circle distance, resampling test, specification curve analysis, IBTrACS, solar–cyclone question, Monte Carlo calibration, research methodology