Modality Analysis via Spacing with the Dimodal Software Libraries

Greg Kreider

REVIEW 4 major objections 5 minor 22 references

Modality Analysis via Spacing with the Dimodal Software Libraries

T0 review · 4 major / 5 minor · reviewed 2026-07-10 · grok-4.5

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Pith's one-line read Spacing between ordered data points both signals multi-modality and locates modes and anti-modes, and Dimodal turns those features into practical detectors and tests.

desk verdict Useful open toolkit that turns classical spacing into a multi-test modality pipeline with real software and a clean Kirkwood demo; significance models are empirical and domain-limited, not fatal. read the letter →

arxiv 2607.06722 v1 pith:NIXOKT4N submitted 2026-07-07 stat.ME astro-ph.EPastro-ph.IM

Greg Kreider This is my paper

classification stat.MEastro-ph.EPastro-ph.IM MSC 62G3062G1062-04

keywords multi-modalityspacingorderstatisticsDimodalpeakandflatdetectionbootstrapexcursiontestschangepointfusionKirkwoodgaps

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that the differences between consecutive ordered observations (spacing) carry two clear signatures of modality: stretches of nearly constant small spacing sit around modes, while local rises mark the transitions between them. Those signatures not only decide whether data are multi-modal; they also place the modes and anti-modes. Because raw spacing is noisy, the author smooths it with low-pass filters or with wider intervals, then screens for peaks and flats and evaluates them with a battery of parametric models, runs tests, permutations, bootstrap excursions, and a majority vote of changepoint detectors. The whole pipeline is packaged as the R library Dimodal (with C and Python ports) and is shown to recover the classical Kirkwood gaps and asteroid families from semi-major-axis data. A reader who cares about unimodal-versus-multimodal questions therefore gets both a conceptual tool and ready software that works on one-dimensional samples up to roughly a hundred thousand points.

What carries the argument

Spacing Di = Ti − Ti−1 (and its interval and low-pass versions), together with screened peaks and flats whose significance is assessed by parametric critical-value models, runs/permutation tests, bootstrap excursion tests, and majority-vote changepoint fusion.

What would settle it

On synthetic multi-modal draws of known size and separation, check whether the package’s accepted peaks and flats recover the true anti-mode and mode locations (and whether the same features appear under both low-pass and interval analyses) at the claimed false-positive rates.

Watch

Extended reading notes

Core claim

Spacing of ordered data exhibits two modality-reflecting features—stable plateaus around modes and local increases at anti-modes—that both detect multi-modality and locate the modes and anti-modes; Dimodal’s detectors and complementary significance tests make those features usable on real data.

Load-bearing premise

The critical-value models that decide whether a peak or flat is real are purely empirical regressions fitted to simulated unimodal draws; they have no theoretical derivation and lose accuracy for small samples or extreme filter sizes.

Editorial extensions

If this is right

One-dimensional multi-modality questions can be answered by inspecting spacing features rather than by fitting mixture models or running density-mode trees.
Modes can be located by the mid-points of accepted flats, and anti-modes by accepted peaks, with complementary checks from several independent tests.
The same pipeline recovers physically meaningful structure (Kirkwood gaps and asteroid families) from orbital data without prior dynamical modelling.
C and Python ports make the detectors usable outside R for samples up to a few hundred thousand points.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the method is strictly one-dimensional and ordering-based, it supplies a natural pre-filter before higher-dimensional clustering or mixture modelling.
The documented sensitivity of flat critical values to the choice of null base distribution suggests a diagnostic that compares results under logistic, Weibull and Gumbel bases before claiming a mode.
Interval-spacing runs tests, being free of parametric assumptions, may remain usable when the low-pass models break down at the extreme tails of the data.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The punchline is that this is a methods-and-software paper that actually ships: screened peaks and flats on low-pass and interval spacing, a battery of complementary tests (empirical height/length models, Markov runs, permutation, excursion bootstrap, majority-vote changepoints), R/C/Python implementations, and a Kirkwood-gap demonstration that recovers the known resonances at plausible locations.

What is new is the finished package, not the observation that spacing is small near modes and large at transitions (Pyke 1965 is cited correctly). The detectors are carefully specified, the tests are shown to have low mutual correlation, and the software is engineered for large n (segtrees for flats, PCG RNG, etc.). The asteroid example is honest: it reports when the flat-length model is already outside its fitted domain and falls back on the excursion test, and it recovers 3:1, 5:2, 7:3, 2:1, 7:4 without circular fitting.

The soft spot the stress-test flags is real but proportionate. The critical-value models are pure regressions on unimodal simulations (Wald for heights, polynomial-logistic for lengths) with no derivation from the spacing density; accuracy degrades for small n or extreme filter fractions, and the flat null base distribution matters. In the Kirkwood run the length model is already out of range, so those flats rest on the non-parametric tests. Comparative power against dip, excess mass, or mode-tree methods is only qualitative. None of that invents circularity or breaks the internal logic; it just means the p-values are provisional outside the simulation grid and should be treated as such.

Citation pattern is clean: classical spacing, filter, runs, and changepoint literature plus the author’s own technical reports that supply the pseudo-code. Free parameters exist (height fractions, ripple, windows, acceptance levels) but are exposed and documented.

This is for people who already look at histograms or order-statistic diagnostics and want a reproducible multi-test pipeline with location estimates. It deserves a serious referee, not a desk reject. I would engage with the software and cite the feature definitions; I would not treat the empirical critical values as theory. Worth a careful read and a conditional accept after the domain-of-validity and power comparisons are tightened.

Referee Report

4 major / 5 minor

Summary. The paper argues that spacing (differences of consecutive order statistics) exhibits two modality-reflecting features—stable plateaus around modes and local increases at anti-modes—and that these features both signal multi-modality and locate modes/anti-modes. It presents Dimodal (R, with C and Python ports) implementing screened peak and flat detectors on low-pass-filtered and interval spacing, parametric critical-value models and bootstrap/excursion tests for the filtered signal, runs/permutation tests for interval spacing, and majority-vote fusion of external changepoint libraries on raw spacing. The methods are illustrated by recovering known Kirkwood gaps and asteroid families from semi-major-axis data.

Significance. If the feature detectors and complementary tests are reliable, the work supplies a practical, multi-test toolkit for one-dimensional modality analysis that both detects multi-modality and estimates mode/anti-mode locations, with open implementations (R package, C CLI, Python wrapper), documented performance on large n, and a non-toy astronomical application that recovers established resonances (3:1, 5:2, 7:3, 2:1, 7:4). The low mutual correlation among tests and the explicit software engineering (segtrees for flats, PCG RNG, mid-quantile handling of ties) are genuine strengths. The contribution is primarily methodological/software rather than a new asymptotic theory of spacing under multi-modality.

major comments (4)

[Section 2.3] §2.3: The peak-height and flat-length critical-value models are purely empirical regressions (Wald/inverse-Gaussian location-scale plus height scaling for peaks; linear-in-flp / quadratic-in-n / quadratic-in-q plus logistic for lengths) fitted to unimodal simulations, with no derivation from the spacing density (1)–(2) or its moments. The manuscript itself states that height accuracy degrades for n≤70 or flp≤0.075 or flp>0.30, and that flat critical values change materially with the arbitrary base distribution (Weibull vs logistic vs Gumbel). Because these models supply the parametric significance numbers attached to located features, the central claim that features can be evaluated for significance rests on unvalidated extrapolations outside a narrow (n, flp) window.
[Section 5] §5 (Kirkwood, n=2093, flp=0.05): The flat-length model is already outside its stated domain—the 0.05 critical length exceeds available data—so acceptance of the Ceres/Themis flats relies solely on the excursion test. The paper should either restrict claims for flats to the validated regime, supply a recalibrated model for large n, or clearly demote the length-model p-values in the example and in the software defaults.
[Section 2.3] §2.3 and overall: Claims that the low-pass tests are “a little more sensitive than existing uni-modality checks” are not supported by a systematic power/size study against standard procedures (Hartigan dip, Silverman bandwidth test, excess mass, etc.) under controlled multi-modal alternatives with known mode locations. Without that comparison, the practical advantage of the spacing suite over established tools remains unquantified, and the significance numbers stay provisional.
[Section 2.4] §2.4 / package interface: Changepoint fusion is majority voting over heterogeneous external libraries with no per-point significance, and the CRAN version omits it entirely because of optional-dependency constraints. The manuscript should either provide a minimal, self-contained changepoint baseline with documented size/power, or clearly separate the fusion as an optional exploratory layer that does not itself evaluate feature significance (as already noted in the text).

minor comments (5)

[Section 5] Many free detector and acceptance parameters (f_ht, f_relht, f_ripple, L_min, f_minlen, window fractions, base distribution, p-cutoffs) are tuned by trial and error in the Kirkwood example; a short sensitivity or default-robustness note would help users.
[Section 5] Figure 3 (mode track) and Figure 4 would benefit from explicit resonance markers or vertical lines at the known Kirkwood locations so the reader can judge positional accuracy without computing (7).
[References] Several supporting details (full pseudo-code, Markov recursion, model coefficients) are deferred to technical reports (Kreider 2024, 2025b, 2025c) that are not yet archival; either archive them or move the essential equations into the main text/supplement.
[Section 3.3] Table 1 and Table 2 report median times but not variability across the 25 replicates; a brief note on run-to-run dispersion would strengthen the performance claims.
[Section 2.1] Minor notation: Di,w is introduced as interval spacing but the signed-difference runs tests also use “signed”; a single consistent symbol table would reduce friction.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: null critical-value models are calibrated on independent unimodal simulations, features are operationally defined from spacing theory, and Kirkwood gaps are external astronomical facts used only for illustration.

full rationale

The paper's derivation chain begins from the classical spacing density (Eq. 1) and its expectation (Eq. 2), shows the characteristic U-shape for unimodal logistic draws (Eq. 3), and numerically exhibits flats-plus-peaks for a known tri-modal mixture (Eq. 4 and Fig. 1). Feature detectors (peaks, flats, fuzzy runs) are then defined operationally by height/ripple/length thresholds; significance is assessed either by empirical quantile regressions fitted exclusively to simulated unimodal draws (Wald/inverse-Gaussian for heights; polynomial-plus-logistic for lengths) or by non-parametric runs/permutation/excursion procedures that sample from the observed spacing differences themselves. None of these steps reduces a claimed multi-modality conclusion to a quantity that was fitted on the same multi-modal data, nor does any uniqueness theorem or ansatz get imported solely via self-citation. Self-citations (Kreider 2023 for the logistic moment, 2025c for full pseudo-code) supply supporting formulae and implementation details but are not load-bearing: the qualitative spacing argument, the external references (Pyke, Harris, Kaplansky & Riordan, Venter, etc.), and the independent simulation calibration stand on their own. The Kirkwood application recovers known orbital resonances that pre-exist in the astronomical literature; they are not free parameters of the method. Consequently the central claim that spacing features both signal and locate modes remains non-circular, even while the empirical models themselves have acknowledged domain limitations (a correctness rather than circularity issue).

Assumptions & free parameters 6 free parameters · 5 assumptions · 3 invented entities

The central claim rests on classical spacing theory plus a large set of empirically chosen detector thresholds and simulation-calibrated quantile models. No new physical entities are postulated; the free parameters are the usual tuning knobs of a feature-detection pipeline. Domain assumptions (univariate ordered data, positive spacing, filterable noise) are standard for the setting.

free parameters (6)

peak height fractions (f_ht, f_relht)
Screening thresholds that decide which local maxima survive; set by hand (e.g., 0.015 on Kirkwood) and control false-positive rate.
flat ripple and min-length (f_ripple, L_min, f_minlen, n_outlier)
Define what counts as a stable region; trial-and-error values (0.0075 ripple on Kirkwood) directly affect which modes are reported.
low-pass / interval window fractions
Kernel or interval width (default 0.05, tracked 0.01–0.40); chosen by mode-tree stability and strongly influences feature locations and counts.
test acceptance levels (p-value cut-offs)
Separate thresholds for height model, excursion, runs-count, longest-run, and permutation tests; tightened for runs tests to control false positives.
null base distribution for flat-length model
Logistic default with Weibull/Gumbel/Gaussian alternatives; choice moves critical lengths across quantiles and is not theoretically fixed.
Wald location/scale and height-scaling regressions for peak model
Coefficients fitted to simulated unimodal draws; the model has no closed-form derivation and degrades outside the fitted (n, f_lp) grid.

assumptions (5)

standard math Spacing density and moments follow Pyke’s integral formula; closed forms exist only for a few distributions (uniform, exponential, logistic, Gumbel).
Section 2.1; used as background, not re-derived.
standard math Expected logistic spacing equals σ n / ((i-1)(n-i+1)); multi-modal spacing cannot be obtained in closed form.
Eqs. (3)–(4) and surrounding text; motivates the feature-based rather than fully parametric approach.
domain assumption Univariate ordered data; higher-dimensional data lack a natural total order so the method does not apply.
Explicit limitation in Section 2.5.
domain assumption Kaiser (or other low-pass) filtering and interval sums preserve the modality-relevant features while reducing variance enough for detection.
Section 2.1–2.2; supported by development experience and Harris (1978) but not proved optimal.
ad hoc to paper Empirical quantile models calibrated on unimodal simulations supply usable critical values for peaks and flats under the null.
Section 2.3: “The models do not have a theoretical basis and were picked for their fit to the collected data.”

invented entities (3)

Screened peak and flat feature detectors on filtered/interval spacing independent evidence
purpose: Operationalize the two modality signatures so they can be tested for significance.
Core algorithmic contribution; independent evidence is the recovery of known Kirkwood gaps and internal consistency across test families.
Wald/inverse-Gaussian peak-height quantile model and polynomial-logistic flat-length model
purpose: Supply parametric critical values without requiring a full multi-modal spacing density.
Fitted to simulations; no external theoretical derivation; accuracy claims are internal to the simulation grid.
Majority-vote fusion of heterogeneous changepoint libraries on raw spacing
purpose: Locate transitions between modes/anti-modes when individual detectors are noisy or inconsistent.
Engineering construct; no significance measure; recommended minimal library set is author-chosen.

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Cite this review

Pith. "Pith review of Modality Analysis via Spacing with the Dimodal Software Libraries." pith.science (2026). https://pith.science/paper/NIXOKT4N

@misc{pith2026260706722,
  author       = {Pith},
  title        = {Pith review of: Modality Analysis via Spacing with the Dimodal Software Libraries},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/NIXOKT4N}},
  note         = {Machine review of arXiv:2607.06722}
}

read the original abstract

Spacing, the difference between consecutive order statistics, has two features that reflect the modality of the data. Consistent, stable values occur around modes while local increases mark the transitions between them. These features not only signal multi-modality, they also locate modes and anti-modes. Dimodal is an R package for detecting and evaluating these situations. It includes parametric feature models and bootstrap tests for spacing smoothed by low-pass filtering, non-parametric runs and permutation tests for the interval spacing, and a fusion of changepoints in the raw spacing. We introduce the analysis, describe the package, its implementation and performance, and apply it to identifying Kirkwood gaps in the asteroid belt. We also present ports of the software, with DimodalCPy a command-line program written in C with a Python interface.

Figures

Figures reproduced from arXiv: 2607.06722 by the authors.

**Figure 1.** Example of spacing in a tri-modal setup. the second variate. The points in the right graph are the spacing of a sample draw from (4). Increases exist around the expected peaks and the second variate produces consistent smaller spacings, but there is a wide spread in values in the first and third modes. Dealing with this high variance is the challenge when working with the spacing. For exponential variates it equals … view at source ↗

**Figure 2.** Screening requirements for peaks (left) and flats (right). have been screened to eliminate small subsidiaries to the side of larger increases. The detector first compresses series of nearly equal values to a single point. This has the advantage of allowing some noise at the feature, which is useful when working with the interval spacing, and turns an ideal square wave into alternating high and low points. It then id… view at source ↗

**Figure 3.** Position and probability of low-pass spacing features as kernel size changes. Dashes in the left graph mark minima. This data set is large and even the interval spacing is smooth because the spacing is locally dense and consistent. The values are taken to six decimal places so ties are not a problem. The features are small, but the smooth filtered signal supports tightening the detector parameters, allowing smaller … view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Low-pass and interval spacing analysis of the kirkwood data. R> m <- Dimodal(kirkwood) R> plot(m) [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]

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Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

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Pith tools

Reviewed July 10, 2026 · model on record in the stance chip above.

[1] [1]

Probal Chaudhuri and J. S. Marron. Scale Space View of Curve Estimation. The Annals of Statistics

work page

[2] [2]

Gradie and Clark R

Jonathan C. Gradie and Clark R. Chapman and James G. Williams. Families of Minor Planets. Asteroids

work page

[3] [3]

Fredric J. Harris. On the Use of Windows for Harmonic Analysis with D iscrete F ourier T ransforms. Proceedings of the IEEE

work page

[4] [4]

Kaylea Haynes and Rebecca Killick

work page

[5] [5]

Multiple Matching and Runs By The Symbolic Method

Irving Kaplansky and John Riordan. Multiple Matching and Runs By The Symbolic Method. The Annals of Mathematical Statistics

work page

[6] [6]

Siegfried Koestlmeier

work page

[7] [7]

The Dynamical Evolution of the Hirayama Family

Yoshihide Kozai. The Dynamical Evolution of the Hirayama Family. Asteroids

work page

[8] [8]

Expected Spacing

Greg Kreider. Expected Spacing. Communications in Statistics - Theory and Methods

work page