IntLevPy: A Python library to classify and model intermittent and L\'evy processes

Pedro G. Lind; Pedro Lencastre; Sergiy Denysov; Shailendra Bhandari; Yurii Bystryk

REVIEW 4 major objections 5 minor 17 references

IntLevPy: A Python library to classify and model intermittent and L\'evy processes

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read IntLevPy classifies intermittent and Lévy processes from a single trajectory using moment fitting and a score function.

desk verdict IntLevPy is a usable, well-structured software package, but the paper needs a benchmark and error metrics before the classification claim is credible. read the letter →

arxiv 2506.03729 v1 pith:UW4K4B6F submitted 2025-06-04 cs.NE cs.MS

Shailendra Bhandari , Pedro Lencastre , Sergiy Denysov , Yurii Bystryk , Pedro G. Lind This is my paper

classification cs.NEcs.MS

keywords intermittentsearchLévywalkanomalousdiffusionmomentfittingprocessclassificationparameterestimationeye-trackingPythonlibrary

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

This paper presents IntLevPy, a Python library for simulating, fitting, and classifying two families of random movement: intermittent search and Lévy walks. Its claim is that the two can be told apart from one trajectory by comparing second and fourth velocity moments with analytical model moments, optimizing parameters, and reading a single score, $\Gamma = \bar{R}^2_{\mathrm{IS}} - \bar{R}^2_{\mathrm{LW}}$. On synthetic data the package recovered known simulation parameters, and on a real eye-tracking trajectory it returned $\Gamma > 0$, classifying the movement as intermittent search. If the claim holds, researchers in ecology, neuroscience, and finance gain a ready-made way to decide which movement strategy produced their data.

What carries the argument

The load-bearing mechanism is the moment-distance optimization loop. For time scales $t_s$, the empirical $k$-th order velocity moments $m^{\mathrm{exp}}_k(t_s)$ are computed; for each model family, parameters are adjusted to minimize $d_k(t_s) = [\log(m^{\mathrm{exp}}_k / m^{\mathrm{model}}_k)]^2$, and the quality of the optimized fits is summarized by adjusted $R^2$ for IS and LW. The classification score is $\Gamma = \bar{R}^2_{\mathrm{IS}} - \bar{R}^2_{\mathrm{LW}}$. The analytical moment expressions for the intermittent model and the approximate moment expressions for the Lévy walk come from the companion derivation the paper cites as Ref. [14].

What would settle it

Generate synthetic Lévy walks and intermittent trajectories with known parameters, then run IntLevPy's fitter and classifier: if Lévy walks with $\nu \in (1,2)$ frequently get $\Gamma > 0$, or if the recovered parameters deviate systematically from the generating values, the central claim fails.

Watch

Extended reading notes

Core claim

The central discovery is that a movement trajectory can be classified between these two process families from its velocity moments alone. The package estimates the empirical second and fourth moments $m_2(t_s)$ and $m_4(t_s)$, fits each candidate model by minimizing a logarithmic distance between empirical and model moments, and then compares the fit quality through adjusted $R^2$ measures. The score $\Gamma = \bar{R}^2_{\mathrm{IS}} - \bar{R}^2_{\mathrm{LW}}$ decides: positive means intermittent search, non-positive means Lévy walk. The authors report successful parameter recovery on synthetic trajectories and a positive $\Gamma$ for the analyzed eye-tracking data, which they read as evidence that the library can separate the two classes in practice.

Load-bearing premise

The whole pipeline stands on the assumption that the built-in moment formulas for intermittent and Lévy processes actually match those processes; if Ref. [14]'s derivations are inaccurate, every fitted parameter and the $\Gamma$ score inherit that error.

Editorial extensions

If this is right

Researchers can classify real trajectories without reimplementing simulation, moment formulas, or optimizer loops.
The same moments that drive classification also yield parameter estimates, so fitting and testing hypotheses about movement parameters becomes one workflow.
Applications that depend on knowing whether movement is intermittent or Lévy, such as eye-tracking, animal foraging, human travel, and financial time series, have a single entry-point tool.
Because the package provides unit tests and example scripts, published analyses can be reproduced and inspected.

Reading between the lines

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

The paper reports one real-data classification; a natural extension would be to compute confidence intervals for $\Gamma$ by bootstrap-resampling trajectories, which would show whether the classification is stable.
Since only second and fourth moments are used, the method could confuse two process classes that happen to share these moments; a stress test with deliberately matched moments would map the limits of the score.
The same comparison could be extended to three-dimensional trajectories or to mixed strategies where a searcher switches between intermittent and Lévy phases, neither of which the current package claims to cover.

Share X Bluesky LinkedIn Reddit HN

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

Colleague,

You should know this is a reasonably useful software paper with a validation hole at the center. The library itself looks well-structured: four clean modules (simulation, moments, optimization, classification), code on GitHub, a PyPI release, and a clear workflow. The Gamma = R2_IS - R2_LW score is a simple, interpretable heuristic for choosing between two process classes, and the package genuinely lowers the barrier for someone wanting to test IS versus LW on their own trajectories.

The problem is the paper never shows the core claim working. We are told the package "successfully recovered the known parameters" for synthetic data, but there is no table, no error bar, no convergence statistic. The real-data example compares fitted curves visually in Figure 2; no adjusted R2 values, no Gamma value, no uncertainty. Since the entire method reduces to minimizing d2 and d4 and then comparing R2s, the absence of any measured classification accuracy is a real gap, not a cosmetic one.

The load-bearing input is the moment expressions from Ref [14], a companion preprint by largely the same authors. They are not re-derived or verified here, and the paper doesn't even clearly state which parts of Gamma come from Ref [14] and which are new to IntLevPy. That is a circularity burden, though not a fatal one: the authors have shared code, so a referee can in principle check the moments against simulation. But the paper as written does not do that work.

One smaller thing: "adjusted R2" appears without a formula. Given that the number of parameters differs between the two models (4 vs 3), the degrees-of-freedom correction matters for the Gamma comparison. And the eye-tracking dataset is not released, so the illustrative example cannot be independently re-run.

The citation pattern is fine; the field's standard references are there, and Ref [14] is properly cited as the source of the moments. The prose in Sections 4 and 5 is a bit boosterish, but that's normal for software papers.

Who is this for? Researchers in movement ecology, anomalous diffusion, and eye-tracking who want a ready-made tool and are willing to trust Ref [14]. It's a legitimate niche. I'd send it to peer review, but with the clear expectation that the authors add a synthetic benchmark with error metrics, publish the data, and define adjusted R2. Without those, the classification claim is unsubstantiated.

Referee Report

4 major / 5 minor

Summary. The paper describes IntLevPy, a Python library for simulating, analyzing, and classifying intermittent search (IS) and Lévy walk (LW) processes. The workflow computes empirical second and fourth moments from a trajectory, fits the parameters of both model classes with an iterative optimizer, evaluates each fit with an adjusted-R2-type measure, and classifies the process through a score Gamma = R^2_IS - R^2_LW. The manuscript includes a flowchart, code metadata, module descriptions, and an illustrative application to a real eye-tracking trajectory that is classified as intermittent. For synthetic data, the paper states that known parameters were successfully recovered, but provides no quantitative results, tables, or error metrics.

Significance. If properly validated, a user-friendly library that correctly fits and distinguishes IS and LW processes would be a useful resource for anomaly-diffusion research in ecology, neuroscience, and related fields. The paper's strengths are its open-source availability, modular architecture, unit tests, documentation link, and a clear conceptual flowchart of the classification procedure. However, the central claim of reliable parameter estimation and classification is currently unsupported: no benchmark on labeled synthetic data is reported, no uncertainty quantification is given for the real-data classification, and the theoretical moment expressions that drive the entire pipeline are imported from a companion preprint (Ref. [14]) without independent verification in this manuscript. The contribution as presented is therefore more of a user guide than a validated software paper.

major comments (4)

[Section 3 and Section 4] The statement in Section 4 that "the package successfully recovered the known parameters of the simulated processes" is not supported by any reported results. Section 3 shows only a visual comparison of synthetic and model moments in Figure 2, with no parameter values, errors, or convergence statistics. Please add a quantitative benchmark on labeled synthetic data, including tables of true versus estimated parameters, mean errors or relative errors, and, where relevant, standard deviations across independent realizations.
[Section 3, Eq. (6) and Figure 2] The classification of the real eye-tracking trajectory rests on the visual claim that the IS model "aligns closely" with the empirical moments. The paper never reports the adjusted-R2 values for the two models, the value of Gamma, confidence intervals, or any statistical test of classification confidence. Please report these quantities, and ideally add a bootstrap or synthetic-label validation that quantifies classification accuracy.
[Section 2.2 and Section 3, Eqs. (5) and (6)] The parameter fitting and the classification score Gamma depend entirely on the analytical moments of IS and the approximate moments of LW, which are taken from Ref. [14] (a companion preprint by overlapping authors). The present paper neither reproduces nor independently verifies those moment expressions. If they have systematic errors in any parameter regime, both parameter recovery and Gamma would be biased. Please state explicitly whether Gamma is defined here or in Ref. [14], provide the functional form of the moment expressions used, and include a Monte Carlo or other numerical check of the moments over the relevant parameter ranges.
[Section 3, Eq. (5) and Eq. (6)] The text refers to "adjusted-R2-type measures" but does not give an exact definition. Since the classification decision is based on a difference of such values, the formula matters: how are the residuals d2 and d4 combined, and how many degrees of freedom are counted? Please define the exact adjusted-R2 calculation used by the library, including how the number of fitted parameters enters.

minor comments (5)

[Section 1] The text says "an LW with nu=1 (Cauchy walk)" but the exponent in Eq. (2) is denoted by the Greek letter nu; please use consistent notation and clarify whether this is nu (ν) or gamma (γ), since elsewhere gamma is the Lévy exponent.
[Abstract and Section 2] The phrase "a noble performance measure" appears to be a typo for "a novel performance measure"; please correct it.
[References] References [12] and [15] are the same paper (Bartumeus et al., Optimizing the encounter rate in biological interactions). Please remove the duplicate.
[References] Reference [14] is listed as "Submitted (2025)" with an arXiv identifier; please clarify its current status (under review, accepted, or preprint) and cite the arXiv version consistently.
[Table 1 and Section 2.1] The code metadata lists pomegranate as a dependency, but the paper does not describe its role in the package; please either mention its use or remove it from the dependency list.

Circularity Check

1 steps flagged · score 4.0 of 10

Central fitting and classification accuracy rests on moment expressions imported from a companion preprint by overlapping authors, without in-paper derivation or independent validation.

self citation load bearing [Section 1 (Motivation and significance), paragraph after Eq. (3); Ref. [14]]
"Recently, we have derived analytical expressions for the second and fourth-order moments of IS processes, as well as an accurate approximation of the moments of the LW class of processes. Moreover, we showed how these two statistical moments can be fitted with an iterative optimization scheme to estimate all parameters defining the underlying process. Details are given in Ref. [14]."

The package's moment calculation, parameter optimization (minimizing d2 and d4 in Eq. 5), and classification score Gamma (Eq. 6) all depend on the IS analytical moments and LW moment approximations from Ref. [14], a companion preprint by overlapping authors. The paper does not reproduce those derivations, state their assumptions, or validate them against independent benchmarks. Consequently, the central claim that IntLevPy 'enables to distinguish accurately between both classes of processes' is load-bearing on a self-citation whose correctness is not established within this manuscript. This is not a direct equation-level circularity, but the theoretical content of the classification reduces to an unverified result supplied by the same research group.

full rationale

The paper's own fitting and classification logic—Eq. (5) minimizing logarithmic moment discrepancies and Eq. (6) comparing adjusted-R2 values—is not circular by construction; these are standard model-comparison procedures. However, the moment functions being fitted and compared are not derived or stated in this paper; they are taken wholesale from Ref. [14], a companion preprint by overlapping authors. That self-citation is load-bearing because the claimed ability to 'distinguish accurately' between intermittent and Lévy processes depends entirely on the correctness of those moment expressions. The paper provides no quantitative recovery errors, no classification accuracy on labeled data, and no independent verification of the moment formulas, so the central empirical claim is not self-contained. This warrants a moderate circularity score, but not a higher one, because the software implementation, simulation routines, and model-comparison framework do have independent content and are not themselves defined in terms of the classification outcome.

Assumptions & free parameters 7 free parameters · 3 assumptions · 0 invented entities

The free parameters are the model parameters estimated from data; they are not reported in the paper. The axioms capture the paper's reliance on the companion preprint's moment formulas, on the sufficiency of the two moments, and on adjusted R2 as a model-selection criterion.

free parameters (7)

D (diffusion coefficient)
Model parameter for intermittent search, fitted to empirical moments; no fitted value reported.
V_B (ballistic velocity)
Model parameter for intermittent search, fitted to empirical moments; no fitted value reported.
λ_BD (switching rate ballistic to diffusive)
Model parameter for intermittent search, fitted to empirical moments; no fitted value reported.
λ_DB (switching rate diffusive to ballistic)
Model parameter for intermittent search, fitted to empirical moments; no fitted value reported.
τ0 (scale parameter for Lévy walk)
Model parameter for Lévy walk, fitted to empirical moments; no fitted value reported.
γ (Lévy exponent)
Model parameter for Lévy walk, fitted to empirical moments; no fitted value reported.
v (constant velocity for Lévy walk)
Model parameter for Lévy walk, fitted to empirical moments; no fitted value reported.

assumptions (3)

domain assumption The analytical moment expressions for IS and the approximate moment expressions for LW from Ref [14] are correct and applicable.
Section 1 states these were derived in Ref [14], and the paper relies on them without proof or independent verification.
domain assumption The empirical second and fourth moments computed from a trajectory are sufficient statistics to distinguish IS from LW.
The entire classification workflow is based solely on m2 and m4 as defined in Eq. (4), with no other features.
standard math Adjusted R-squared comparison is a valid model selection criterion for these fitted moment curves.
The paper assumes that adjusted R2 computed for each model and their difference Gamma provides a reliable decision rule, but the exact definition of adjusted R2 used is not given.

how reviews work

0 comments

Cite this review

Pith. "Pith review of IntLevPy: A Python library to classify and model intermittent and L\'evy processes." pith.science (2026). https://pith.science/paper/UW4K4B6F

@misc{pith2026250603729,
  author       = {Pith},
  title        = {Pith review of: IntLevPy: A Python library to classify and model intermittent and L\'evy processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UW4K4B6F}},
  note         = {Machine review of arXiv:2506.03729}
}

abstract

IntLevPy provides a comprehensive description of the IntLevPy Package, a Python library designed for simulating and analyzing intermittent and L\'evy processes. The package includes functionalities for process simulation, including full parameter estimation and fitting optimization for both families of processes, moment calculation, and classification methods. The classification methodology utilizes adjusted-$R^2$ and a noble performance measure {\Gamma}, enabling the distinction between intermittent and L\'evy processes. IntLevPy integrates iterative parameter optimization with simulation-based validation. This paper provides an in-depth user guide covering IntLevPy software architecture, installation, validation workflows, and usage examples. In this way, IntLevPy facilitates systematic exploration of these two broad classes of stochastic processes, bridging theoretical models and practical applications.

Figures

Figures reproduced from arXiv: 2506.03729 by the authors.

**Figure 2.** Comparison of synthetic, model-derived, and empirical moments for Lévy and intermittent processes. The top row shows the second [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

17 extracted references · 9 canonical work pages

[14]

Lencastre, Y ., Bystryk, S

P. Lencastre, Y ., Bystryk, S. Denysov, A. Yazidi, P. Lind, Foraging for visual information: Dis- tinguishing between lévy and intermittent strate- gies, Submitted (2025). URL:https://arxiv. org/abs/2505.00864

arXiv 2025
[1]

Patlak, Random walk with persistence and ex- ternal bias, Bulletin of Mathematical Biophysics 15 (1953) 311–338

C. Patlak, Random walk with persistence and ex- ternal bias, Bulletin of Mathematical Biophysics 15 (1953) 311–338. doi:10.1007/BF02476407

doi:10.1007/bf02476407 1953
[2]

E. A. Codling, M. J. Plank, S. Benhamou, Random walk models in biology, Journal of the Royal So- ciety Interface 5 (2008) 813–834. doi:10.1098/ rsif.2008.0014

arXiv 2008
[3]

G. H. Pyke, Understanding movements of or- ganisms: It’s time to abandon the lévy for- aging hypothesis, Methods in Ecology and Evolution 6 (2014) 1–16. URL:https:// doi.org/10.1111/2041-210X.12298. doi:10. 1111/2041-210X.12298. 5

work page doi:10.1111/2041-210x.12298 2014
[4]

Bénichou, C

O. Bénichou, C. Loverdo, M. Moreau, R. V oi- turiez, Intermittent search strategies, Rev. Mod. Phys. 83 (2011) 81–129. URL:https://link. aps.org/doi/10.1103/RevModPhys.83.81. doi:10.1103/RevModPhys.83.81

doi:10.1103/revmodphys.83.81 2011
[5]

Bénichou, M

O. Bénichou, M. Coppey, M. Moreau, R. V oi- turiez, Intermittent search strategies: When los- ing time becomes efficient, Europhysics Letters 75 (2006) 349. URL:https://dx.doi.org/10. 1209/epl/i2006-10100-3. doi:10.1209/epl/ i2006-10100-3

doi:10.1209/epl/ 2006
[6]

Viswanathan, E

G. Viswanathan, E. Raposo, M. da Luz, Lévy flights and superdiffusion in the context of biological encounters and random searches, Physics of Life Reviews 5 (2008) 133–150. URL:https://www.sciencedirect.com/ science/article/pii/S1571064508000146. doi:https://doi.org/10.1016/j.plrev. 2008.03.002

doi:10.1016/j.plrev 2008
[7]

G. M. Viswanathan, M. G. E. da Luz, E. P. Raposo, H. E. Stanley, Superdiffusive random searches, Cambridge University Press, 2011, p. 111–115. doi:10.1017/CBO9780511902680.013

work page doi:10.1017/cbo9780511902680.013 2011

Show all 17 references

[8]

Zaburdaev, S

V . Zaburdaev, S. Denisov, J. Klafter, Lévy walks, Rev. Mod. Phys. 87 (2015) 483–530. URL:https://link.aps. org/doi/10.1103/RevModPhys.87.483. doi:10.1103/RevModPhys.87.483

2015 doi
[9]

M. F. Shlesinger, J. Klafter, Lévy Walks Versus Lévy Flights, Springer Netherlands, Dordrecht, 1986, pp. 279–283. doi:10.1007/ 978-94-009-5165-5_29

1986
[10]

Viswanathan, et al., Optimizing the success of random searches, Nature 401 (1999) 911–914

G. Viswanathan, et al., Optimizing the success of random searches, Nature 401 (1999) 911–914

1999
[11]

Rebenshtok, S

A. Rebenshtok, S. Denisov, P. Hänggi, E. Barkai, Non-normalizable densities in strong anomalous diffusion: Beyond the central limit theorem, Phys- ical Review Letters 117 (2016) 270601. doi:10. 1103/PhysRevLett.117.270601

2016
[12]

Bartumeus, J

F. Bartumeus, J. Catalan, U. L. Fulco, M. L. Lyra, G. M. Viswanathan, Optimizing the en- counter rate in biological interactions: Lévy ver- sus brownian strategies, Physical Review Letters 88 (2002) 097901. doi:10.1103/PhysRevLett. 88.097901

2002 doi
[13]

Brockmann, L

D. Brockmann, L. Hufnagel, T. Geisel, The scal- ing laws of human travel, Nature 439 (2006) 462– 465

2006
[15]

Bartumeus, et al., Optimizing the encounter rate in biological interactions: Lévy versus brow- nian strategies, Physical Review Letters 88 (2002) 097901

F. Bartumeus, et al., Optimizing the encounter rate in biological interactions: Lévy versus brow- nian strategies, Physical Review Letters 88 (2002) 097901

2002
[16]

A. M. Reynolds, Current status and future di- rections of lévy walk research, Biology Open 7 (2018) bio030106

2018
[17]

R. N. Mantegna, H. E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance, Cambridge University Press, 2000. 6

2000

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [14]

Lencastre, Y ., Bystryk, S

P. Lencastre, Y ., Bystryk, S. Denysov, A. Yazidi, P. Lind, Foraging for visual information: Dis- tinguishing between lévy and intermittent strate- gies, Submitted (2025). URL:https://arxiv. org/abs/2505.00864

arXiv 2025

[2] [1]

Patlak, Random walk with persistence and ex- ternal bias, Bulletin of Mathematical Biophysics 15 (1953) 311–338

C. Patlak, Random walk with persistence and ex- ternal bias, Bulletin of Mathematical Biophysics 15 (1953) 311–338. doi:10.1007/BF02476407

doi:10.1007/bf02476407 1953

[3] [2]

E. A. Codling, M. J. Plank, S. Benhamou, Random walk models in biology, Journal of the Royal So- ciety Interface 5 (2008) 813–834. doi:10.1098/ rsif.2008.0014

arXiv 2008

[4] [3]

G. H. Pyke, Understanding movements of or- ganisms: It’s time to abandon the lévy for- aging hypothesis, Methods in Ecology and Evolution 6 (2014) 1–16. URL:https:// doi.org/10.1111/2041-210X.12298. doi:10. 1111/2041-210X.12298. 5

work page doi:10.1111/2041-210x.12298 2014

[5] [4]

Bénichou, C

O. Bénichou, C. Loverdo, M. Moreau, R. V oi- turiez, Intermittent search strategies, Rev. Mod. Phys. 83 (2011) 81–129. URL:https://link. aps.org/doi/10.1103/RevModPhys.83.81. doi:10.1103/RevModPhys.83.81

doi:10.1103/revmodphys.83.81 2011

[6] [5]

Bénichou, M

O. Bénichou, M. Coppey, M. Moreau, R. V oi- turiez, Intermittent search strategies: When los- ing time becomes efficient, Europhysics Letters 75 (2006) 349. URL:https://dx.doi.org/10. 1209/epl/i2006-10100-3. doi:10.1209/epl/ i2006-10100-3

doi:10.1209/epl/ 2006

[7] [6]

Viswanathan, E

G. Viswanathan, E. Raposo, M. da Luz, Lévy flights and superdiffusion in the context of biological encounters and random searches, Physics of Life Reviews 5 (2008) 133–150. URL:https://www.sciencedirect.com/ science/article/pii/S1571064508000146. doi:https://doi.org/10.1016/j.plrev. 2008.03.002

doi:10.1016/j.plrev 2008

[8] [7]

G. M. Viswanathan, M. G. E. da Luz, E. P. Raposo, H. E. Stanley, Superdiffusive random searches, Cambridge University Press, 2011, p. 111–115. doi:10.1017/CBO9780511902680.013

work page doi:10.1017/cbo9780511902680.013 2011