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REVIEW 2 major objections 5 minor 103 references

Unveiling Amplitude Distributions via the Ordinal Language of Random Walks

T0 review · 2 major / 5 minor · reviewed 2026-07-31 · grok-4.5

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Pith's one-line read Integrating a time series before ordinal analysis turns amplitude shape into pattern probabilities that can recover non-Gaussian laws.

desk verdict Solid analytical core: integration turns amplitude shape into ordinal geometry for i.i.d. walks, with clean symmetry formulas and a real centering warning; the finance cubic-law claim is only a consistency check inside one family. read the letter →

arxiv 2607.28266 v1 pith:RMJGFRQL submitted 2026-07-30 physics.data-an

Alberto Mateos Roig , Luciano Zunino , Felipe Olivares This is my paper · ORCID
classification physics.data-an
keywords ordinalpatternsrandomwalksnon-Gaussianfluctuationsq-GaussianpermutationJensen-Shannondistancefinancialreturnsinversecubiclawmediancentering
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

Ordinal patterns usually throw away how large the fluctuations are and only keep their relative order, so they cannot see whether a signal is Gaussian or heavy-tailed. This paper shows that first integrating the series into a random walk changes that: the walk's geometry forces some ordinal patterns to have fixed probabilities fixed by symmetry alone, while the remaining patterns' probabilities become sensitive to the shape of the increment distribution. The authors derive those probabilities analytically for short patterns, confirm them with q-Gaussian walks, and show that the centering used to build the walk matters: mean-centering can invent artificial trends in heavy-tailed or asymmetric data, whereas median-centering keeps the ordinal signature clean. Applied to shuffled financial log-returns, the same signature matches q-Gaussian walks near q = 5/3, recovering the inverse-cubic-law tails reported for markets. A sympathetic reader cares because a simple, robust symbolic tool can now read amplitude information that ordinal methods were thought to discard.

What carries the argument

The ordinal probability of a length-D pattern on the integrated walk, written as the (D-1)-fold integral of the product of increment densities over the region of jump values that produce that ordering. Symmetry of the density collapses some of those regions to universal fractions; the others retain explicit dependence on the density shape.

What would settle it

Compute the full D=4 ordinal histogram on median-centered walks of shuffled high-frequency log-returns and check whether its Jensen-Shannon distance to synthetic q-Gaussian walks still reaches the finite-sample floor near q in [1.4, 1.7]; a clear minimum elsewhere, or no minimum at all, would falsify the claimed recovery of cubic-law tails.

Watch

Extended reading notes

Core claim

For random walks built from i.i.d. continuous increments, the ordinal pattern probabilities of the integrated series are not uniform. Under symmetry, a subset of D=3 and D=4 pattern probabilities is completely fixed by symmetry (for D=3: the two monotone patterns each equal 1/4 and the four turning patterns each equal 1/8; for D=4 the six pairs listed in the paper), while the remaining patterns depend explicitly on the increment density and thereby encode non-Gaussian shape. With median centering this structure recovers the heavy-tailed features associated with the cubic law in financial log-return walks.

Load-bearing premise

After shuffling removes temporal structure, the empirical log-return series can be treated as independent draws from a stationary continuous amplitude law, and median-centering produces a walk whose ordinal histogram is free of residual bias from microstructure or non-stationarity.

Editorial extensions

If this is right

  • Ordinal analysis of integrated series becomes a practical probe of amplitude shape, not only of temporal order.
  • A short list of D=4 patterns whose probabilities are fixed by symmetry can serve as a built-in null check that the walk is consistent with continuous symmetric increments.
  • The remaining shape-sensitive patterns can be matched to candidate densities (q-Gaussian, Laplace, uniform, stable) without estimating high moments.
  • Median rather than mean centering is required when the same pipeline is applied to heavy-tailed or asymmetric increments, otherwise artificial drift contaminates the ordinal signature.
  • Shuffled financial returns whose integrated ordinal spectra peak near q=5/3 supply independent symbolic evidence for inverse-cubic tails.

Reading between the lines

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

  • The same integral geometry should let ordinal spectra of integrated series discriminate compact-support from power-law increments in turbulence or physiological data where large samples are hard to obtain.
  • Because only a few patterns carry the distributional information, one could design a low-dimensional summary statistic from those patterns alone and bypass full D!-dimensional histogram comparison.
  • If discrete or tied measurements are common, the continuous-density integrals will need an explicit tie-breaking or lattice correction before the method is applied to quantized sensor streams.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper argues that ordinal pattern probabilities computed on integrated (random-walk) series encode the amplitude law of i.i.d. increments, even though direct ordinal analysis of the increments is distribution-blind. For continuous symmetric f, a subset of D=3 and D=4 pattern probabilities is fixed by symmetry alone (e.g. P(012)=P(210)=1/4 and the four turning patterns =1/8 for D=3; the six pairs in Eq. (36) for D=4), while the remaining patterns are given by multidimensional integrals over f and thus depend on shape (heavy tails, compact support). These formulas are checked against Monte Carlo walks with q-Gaussian increments (Figs. 1–2, Appendix A). The authors further show that the centering used to build the walk (mean vs zero vs median) can strongly bias ordinal histograms for asymmetric or very heavy-tailed increments (Fig. 4), and recommend median centering. Empirically, shuffled median-centered log-return walks from S&P500 and NASDAQ yield PJSD minima versus q-Gaussian walks near q∈[1.4,1.7], consistent with the inverse-cubic / q≈5/3 literature.

Significance. If the analytical and numerical core holds—as it appears to—the work cleanly removes a standard limitation of Bandt–Pompe analysis: amplitude information becomes accessible via integration without abandoning the ordinal language. The symmetry-fixed probabilities are parameter-free and falsifiable; the f-dependent integrals and q-Gaussian checks are reproducible; and the centering analysis is a concrete methodological contribution for heavy-tailed data. The finance application is mainly corroborative rather than a new market discovery, but it shows the framework can recover a well-known non-Gaussian signature under controlled preprocessing. Overall this is a solid, usable addition to ordinal time-series methodology for non-Gaussian fluctuations.

major comments (2)
  1. [Section V, Fig. 5] Section V and Fig. 5: the claim that integrated ordinal distributions “capture non-Gaussian features consistent with a cubic law” rests on a PJSD minimum versus a one-parameter q-Gaussian grid that reaches the finite-sample floor near q∈[1.4,1.7]. That shows consistency with the q≈5/3 family after shuffling and median centering, not that the amplitude law is identified or that cubic tails are preferred over other heavy-tailed laws with similar shape. At minimum, report PJSD (or pattern-wise) comparisons against alternative families used in the finance literature (e.g. Student-t, stable/Lévy with comparable tail index, double exponential) on the same shuffled walks; if several families also hit the baseline, tone the cubic-law identification down to “compatible with q-Gaussian / inverse-cubic phenomenology.”
  2. [Section V] Section V construction (Eqs. (45), (51)): after random shuffling, the returns are treated as i.i.d. continuous draws whose ordinal walk histogram is driven only by the stationary amplitude law. The paper does not show controls that would support this load-bearing step—e.g. PJSD or pattern probabilities for non-shuffled median-centered walks, or a brief check that residual microstructure/discreteness (tick size, overnight gaps, heterogeneous trading intensity across the three sampling frequencies) does not move the minimum. A short control panel or appendix table would substantially strengthen the empirical claim; without it, the finance result remains a consistency check under aggressive preprocessing rather than a robust amplitude readout.
minor comments (5)
  1. [Section IV, Eqs. (6) and (45)] Eq. (6) vs Eq. (45): the main text introduces mean-centering first, then generalizes to arbitrary γ. State earlier (when Eq. (6) appears) that γ will be varied and that median is preferred for asymmetric/heavy-tailed cases, so readers do not carry mean-centering into Section V by default.
  2. [Fig. 1] Fig. 1 caption and axis labels: pattern labels are readable but the q color/legend encoding is not fully specified in the caption (only the q grid). Add an explicit colorbar or legend mapping q to symbol/color.
  3. [Section IV.A.3 and Section V] For q>5/3 the text notes divergent variance and slower convergence, yet Fig. 5 extends to q=2.5. Briefly state how synthetic walks were standardized (or not) in that regime so the PJSD comparison remains well-defined.
  4. [Title page / Introduction] Typos/notation: “V alencia” and “V alencia” spacing in affiliations; “man-made” vs “human-made” inconsistency between abstract block and Introduction; arXiv stamp date “31 July 2026” looks like a placeholder—verify before publication.
  5. [Appendix A] Appendix A is valuable; a one-row table summarizing which of the 24 patterns are symmetry-fixed vs f-dependent (with the closed values) would help readers implement the method without parsing all integrals.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: symmetry-fixed and f-dependent ordinal probabilities are derived from i.i.d. integrals; finance q≈5/3 is external consistency, not a forced fit.

full rationale

The paper’s load-bearing analytical chain is self-contained. For continuous i.i.d. symmetric increments, D=3 probabilities (P(012)=P(210)=1/4; four turning patterns =1/8) and the six D=4 pairs in Eq. (36) follow from symmetry of integration domains alone (Secs. IV.A.1–2, App. A); the remaining twelve D=4 patterns are written as explicit triple integrals over f (Eq. (40) and App. A) and evaluated numerically, not fitted to the target claim. q-Gaussian simulations (Figs. 1–2) validate those formulas rather than define them. Median vs mean centering (Sec. IV.B) is a methodological control on the walk construction (Eqs. (6)/(45)), not a circular redefinition of the amplitude law. The finance section shuffles log-returns, builds median-centered walks, and reports a PJSD minimum vs synthetic q-Gaussian walks near q∈[1.4,1.7] that reaches the finite-sample floor—presented as consistency with the externally cited inverse-cubic / Tsallis literature (Gopikrishnan et al.; Nayak et al.), not as a parameter fit that is then relabeled a prediction. PJSD (Zunino et al. 2022, overlapping authors) is only a dissimilarity metric; the ordinal probabilities and the location of the minimum do not reduce to that citation by construction. No self-definitional loop, no fitted-input-as-prediction, and no uniqueness theorem imported from the authors. Score 0.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central theory rests on standard probability (i.i.d. continuous increments, symmetry implying half-mass on each side) plus the Bandt–Pompe ordinal map and the definition of the centered cumulative sum. No new physical entities. Free choices are the embedding (D,τ), the q-Gaussian family used as a probe, the centering functional for asymmetric data, and the discrete q grid / series length in numerics and finance. Finance interpretation further assumes shuffled log-returns behave as i.i.d. draws from a stationary amplitude law comparable to q-Gaussians.

free parameters (4)
  • q (q-Gaussian shape) = empirical minimum near q∈[1.4,1.7], consistent with 5/3
    Scanned over a grid and selected by minimum PJSD to empirical ordinal histograms in the finance section; also the control parameter in synthetic validation. Not predicted a priori for markets.
  • embedding dimension D and delay τ = D=3 or 4, τ=1
    Chosen as D=3 for analytics/centering demos and D=4, τ=1 for main numerics and finance; admissible max D constrained by series length.
  • centering parameter γ = median(x_t) for asymmetric and empirical cases
    Mean, zero, or median; median is selected as the robust default for asymmetric/heavy-tailed increments after comparative numerics.
  • synthetic series length and ensemble size = N=1e5, 20 runs
    N=10^5 points, 20 realizations; affects convergence especially for q>5/3 where variance diverges.
assumptions (7)
  • domain assumption Increments are i.i.d. continuous random variables so P(ties)=0 and ordinal probabilities reduce to integrals of ∏f(x_k) over ordering regions S_π.
    Stated in Section IV and Eq. (8); foundation of all analytical probabilities.
  • domain assumption For the closed-form subset, f is symmetric about 0 (f(x)=f(-x)), hence E[x]=0 and positive-half mass equals 1/2.
    Section IV.A; used to obtain Eqs. (14), (23)–(26), (36) without knowing f further.
  • standard math Ordinal patterns are Bandt–Pompe rank permutations of delay vectors; probabilities are relative frequencies.
    Section II; standard definition from Bandt & Pompe 2002.
  • domain assumption The integrated process is the centered cumulative sum y_t = Σ(x_i − γ); ordinal geometry is read on y, not on x.
    Eqs. (6) and (45); the paper’s core modeling move.
  • standard math Time-reversal symmetry of symmetric-increment walks equates each pattern probability with its reverse.
    Used throughout IV.A and Appendix A to halve the pattern list.
  • domain assumption After random shuffling, financial log-returns may be treated as i.i.d. samples from the empirical amplitude law for comparison to synthetic walks.
    Section V; isolates amplitude from temporal dependence.
  • domain assumption q-Gaussian family (Tsallis) is an adequate one-parameter probe spanning compact-support to heavy-tailed regimes, recovering Gaussian as q→1.
    Section IV.A.3 and finance comparison; standard in nonextensive stats but still a model choice.

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Pith. "Pith review of Unveiling Amplitude Distributions via the Ordinal Language of Random Walks." pith.science (2026). https://pith.science/paper/RMJGFRQL

@misc{pith2026260728266,
  author       = {Pith},
  title        = {Pith review of: Unveiling Amplitude Distributions via the Ordinal Language of Random Walks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RMJGFRQL}},
  note         = {Machine review of arXiv:2607.28266}
}
abstract

Ordinal patterns are widely used to characterize temporal organization in time series, yet they are often considered insensitive to the amplitude distribution of the data. In this work, we show that this limitation can be overcome by considering the ordinal structure of integrated time series. We investigate random walks generated from independent non-Gaussian increments and derive analytical expressions for the ordinal pattern probabilities associated with them. We show that for symmetric distributions, the probabilities of some ordinal patterns are fully determined by symmetry arguments, while those for the remaining patterns depend explicitly on the shape of the increments distribution. Numerical simulations based on $q$-Gaussian increments validate the theoretical predictions. We further show that the construction of the random walk itself plays a fundamental role in the accurate characterization of non-Gaussian fluctuations, as different centering procedures may significantly affect the resulting ordinal statistics. Finally, we validate the proposed framework using financial time series, showing that the ordinal distributions of integrated logarithmic returns capture non-Gaussian features consistent with a cubic law.

Figures

Figures reproduced from arXiv: 2607.28266 by the authors.

Figure 1
Figure 1. FIG. 1. Ordinal pattern probabilities for the integrated process gener [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Ordinal pattern probabilities for the integrated process gen [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Graphical representation of the ordinal patterns (a) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Ordinal pattern probabilities ( [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Permutation Jensen-Shannon Distance (PJSD) between the [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

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Reference graph

Works this paper leans on

103 extracted references · 3 canonical work pages

  1. [1]

    SIAM Review , volume=

    Power-law distributions in empirical data , author=. SIAM Review , volume=. 2009 , publisher=

    2009

  2. [2]

    Physics Reports , volume=

    The random walk's guide to anomalous diffusion: a fractional dynamics approach , author=. Physics Reports , volume=. 2000 , publisher=

    2000

  3. [3]

    2009 , publisher=

    Introduction to Nonextensive Statistical Mechanics , author=. 2009 , publisher=

    2009

  4. [4]

    Velocity probability density functions of high

    Castaing, B and Gagne, Y and Hopfinger, E J , journal=. Velocity probability density functions of high. 1990 , publisher=

    1990

  5. [5]

    Physical Review Letters , volume=

    Dynamical foundations of nonextensive statistical mechanics , author=. Physical Review Letters , volume=. 2001 , publisher=

    2001

  6. [6]

    Long-range anticorrelations and non-

    Peng, C-K and Mietus, J M and Hausdorff, J M and Havlin, Shlomo and Stanley, H Eugene and Goldberger, Ary L , journal=. Long-range anticorrelations and non-. 1993 , publisher=

    1993

  7. [7]

    Scientific Reports , volume=

    Statistical characterization of airplane delays , author=. Scientific Reports , volume=. 2021 , publisher=

    2021

  8. [8]

    Quantifying deviations from

    Olivares, Felipe and Zanin, Massimiliano , journal=. Quantifying deviations from. 2025 , publisher=

    2025

Show all 103 references
  1. [9]

    Journal of Political Economy , volume=

    New methods in statistical economics , author=. Journal of Political Economy , volume=. 1963 , publisher=

    1963

  2. [10]

    The European Physical Journal B-Condensed Matter and Complex Systems , volume=

    Inverse cubic law for the distribution of stock price variations , author=. The European Physical Journal B-Condensed Matter and Complex Systems , volume=. 1998 , publisher=

    1998

  3. [11]

    Computational modeling of non-

    Nayak, Gangadhar and Singh, Amit Kumar and Senapati, Dilip , journal=. Computational modeling of non-. 2021 , publisher=

    2021

  4. [12]

    Nature , volume=

    A theory of power-law distributions in financial market fluctuations , author=. Nature , volume=. 2003 , publisher=

    2003

  5. [13]

    Chaos, Solitons & Fractals , volume=

    Complexity testing techniques for time series data: A comprehensive literature review , author=. Chaos, Solitons & Fractals , volume=. 2015 , publisher=

    2015

  6. [14]

    Physical Review Letters , volume=

    Permutation entropy: a natural complexity measure for time series , author=. Physical Review Letters , volume=. 2002 , publisher=

    2002

  7. [15]

    Physical Review Letters , volume=

    Distinguishing noise from chaos , author=. Physical Review Letters , volume=. 2007 , publisher=

    2007

  8. [16]

    Communications Physics , volume=

    Ordinal patterns-based methodologies for distinguishing chaos from noise in discrete time series , author=. Communications Physics , volume=. 2021 , publisher=

    2021

  9. [17]

    Physica A: Statistical Mechanics and its Applications , volume=

    Quantifying long-range correlations with a multiscale ordinal pattern approach , author=. Physica A: Statistical Mechanics and its Applications , volume=. 2016 , publisher=

    2016

  10. [18]

    Physica A: Statistical Mechanics and its Applications , volume=

    Multiscale dynamics under the lens of permutation entropy , author=. Physica A: Statistical Mechanics and its Applications , volume=. 2020 , publisher=

    2020

  11. [19]

    Permutation

    Zunino, Luciano and Olivares, Felipe and Ribeiro, Haroldo V and Rosso, Osvaldo A , journal=. Permutation. 2022 , publisher=

    2022

  12. [20]

    Physical Review E , volume=

    Permutation-information-theory approach to unveil delay dynamics from time-series analysis , author=. Physical Review E , volume=. 2010 , publisher=

    2010

  13. [21]

    Optics Letters , volume=

    Distinguishing fingerprints of hyperchaotic and stochastic dynamics in optical chaos from a delayed optoelectronic oscillator , author=. Optics Letters , volume=. 2011 , publisher=

    2011

  14. [22]

    Physical Review E , volume=

    Distinguishing chaotic and stochastic dynamics from time series by using a multiscale symbolic approach , author=. Physical Review E , volume=. 2012 , publisher=

    2012

  15. [23]

    Entropy , volume=

    Identifying ordinal similarities at different temporal scales , author=. Entropy , volume=. 2024 , publisher=

    2024

  16. [24]

    Entropy , volume=

    Assessing time series reversibility through permutation patterns , author=. Entropy , volume=. 2018 , publisher=

    2018

  17. [25]

    Symmetry , volume=

    Manipulating time series irreversibility through continuous ordinal patterns , author=. Symmetry , volume=. 2024 , publisher=

    2024

  18. [26]

    Mathematical Biosciences and Engineering , volume=

    Permutation entropy: Influence of amplitude information on time series classification performance , author=. Mathematical Biosciences and Engineering , volume=. 2019 , publisher=

    2019

  19. [27]

    Chinese Physics B , volume=

    Fine-grained permutation entropy as a measure of natural complexity for time series , author=. Chinese Physics B , volume=

  20. [28]

    Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=

    Weighted-permutation entropy: A complexity measure for time series incorporating amplitude information , author=. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics , volume=. 2013 , publisher=

    2013

  21. [29]

    Computer Methods and Programs in Biomedicine , volume=

    Amplitude-aware permutation entropy: Illustration in spike detection and signal segmentation , author=. Computer Methods and Programs in Biomedicine , volume=. 2016 , publisher=

    2016

  22. [30]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Continuous ordinal patterns: Creating a bridge between ordinal analysis and deep learning , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2023 , publisher=

    2023

  23. [31]

    Entropy , volume=

    Random walk null models for time series data , author=. Entropy , volume=. 2017 , publisher=

    2017

  24. [32]

    Complexity

    Patterns with equal values in permutation entropy: Do they really matter for biosignal classification? , author=. Complexity. 2018 , publisher=

    2018

  25. [33]

    Divergence measures based on the

    Lin, Jianhua , journal=. Divergence measures based on the. 1991 , publisher=

    1991

  26. [34]

    Possible generalization of

    Tsallis, Constantino , journal=. Possible generalization of. 1988 , publisher=

    1988

  27. [35]

    Generalized

    Thistleton, William J and Marsh, John A and Nelson, Kenric and Tsallis, Constantino , journal=. Generalized. 2007 , publisher=

    2007

  28. [36]

    2009 , publisher=

    Robust Statistics , author=. 2009 , publisher=

    2009

  29. [37]

    Journal of Time Series Analysis , volume=

    Order patterns in time series , author=. Journal of Time Series Analysis , volume=. 2007 , publisher=

    2007

  30. [38]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Statistics and contrasts of order patterns in univariate time series , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2023 , publisher=

    2023

  31. [39]

    Physica A: Statistical Mechanics and its Applications , volume=

    Stock market return distributions: From past to present , author=. Physica A: Statistical Mechanics and its Applications , volume=. 2007 , publisher=

    2007

  32. [40]

    Modeling and simulation of financial returns under non-

    De Domenico, Federica and Livan, Giacomo and Montagna, Guido and Nicrosini, Oreste , journal=. Modeling and simulation of financial returns under non-. 2023 , publisher=

    2023

  33. [41]

    Physical Review E , volume=

    Scaling of the distribution of fluctuations of financial market indices , author=. Physical Review E , volume=. 1999 , publisher=

    1999

  34. [42]

    Entropy , volume=

    Time-delay identification using multiscale ordinal quantifiers , author=. Entropy , volume=. 2021 , publisher=

    2021

  35. [43]

    Entropy , volume=

    Including the Magnitude Variability of a Signal in the Ordinal Pattern Analysis , author=. Entropy , volume=. 2025 , publisher=

    2025

  36. [44]

    , Title =

    Malkiel, Burton G. , Title =. Journal of Economic Perspectives , Volume =. 2003 , Month =. doi:10.1257/089533003321164958 , URL =

    2003 doi

  37. [45]

    The Variation of Certain Speculative Prices , urldate =

    Benoit Mandelbrot , journal =. The Variation of Certain Speculative Prices , urldate =

  38. [46]

    Zunino and B.M

    L. Zunino and B.M. Tabak and A. Figliola and D.G. Pérez and M. Garavaglia and O.A. Rosso , keywords =. A multifractal approach for stock market inefficiency , journal =. 2008 , issn =. doi:https://doi.org/10.1016/j.physa.2008.08.028 , url =

    2008 doi

  39. [47]

    The European Physical Journal B , volume=

    Inefficiency in Latin-American market indices , author=. The European Physical Journal B , volume=. 2007 , publisher=

    2007

  40. [48]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Contrasting chaotic with stochastic dynamics via ordinal transition networks , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2020 , publisher=

    2020

  41. [49]

    EPL (Europhysics Letters) , volume=

    Permutation min-entropy: An improved quantifier for unveiling subtle temporal correlations , author=. EPL (Europhysics Letters) , volume=. 2015 , publisher=

    2015

  42. [50]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Markov-modulated model for landing flow dynamics: An ordinal analysis validation , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2023 , publisher=

    2023

  43. [51]

    Language organization and temporal correlations in the spiking activity of an excitable laser: Experiments and model comparison , author =. Phys. Rev. E , volume =. 2011 , month =. doi:10.1103/PhysRevE.84.026202 , url =

    2011 doi

  44. [52]

    Physical Review E , volume=

    Using missing ordinal patterns to detect nonlinearity in time series data , author=. Physical Review E , volume=. 2017 , publisher=

    2017

  45. [53]

    Estimation of ordinal pattern probabilities in

    Sinn, Mathieu and Keller, Karsten , journal=. Estimation of ordinal pattern probabilities in. 2011 , publisher=

    2011

  46. [54]

    Characterization of chaotic maps using the permutation

    Rosso, Osvaldo A and Olivares, Felipe and Zunino, Luciano and De Micco, Luciana and Aquino, Andr. Characterization of chaotic maps using the permutation. The European Physical Journal B , volume=. 2013 , publisher=

    2013

  47. [55]

    Physics Letters A , volume=

    Order patterns and chaos , author=. Physics Letters A , volume=. 2006 , publisher=

    2006

  48. [56]

    EPL (Europhysics Letters) , volume=

    True and false forbidden patterns in deterministic and random dynamics , author=. EPL (Europhysics Letters) , volume=. 2007 , publisher=

    2007

  49. [57]

    EPL (Europhysics Letters) , volume=

    Combinatorial detection of determinism in noisy time series , author=. EPL (Europhysics Letters) , volume=. 2008 , publisher=

    2008

  50. [58]

    2017 AAAI Fall Symposium Series , year=

    Analysis of automatic dependent surveillance-broadcast data , author=. 2017 AAAI Fall Symposium Series , year=

    2017

  51. [59]

    2009 , publisher=

    Williams, Genevra , journal=. 2009 , publisher=

    2009

  52. [60]

    Bringing up

    Sch. Bringing up. IPSN-14 Proceedings of the 13th International Symposium on Information Processing in Sensor Networks , pages=. 2014 , organization=

    2014

  53. [61]

    Short-term correlation of solar activity and sunspot:

    Blanter, E M and Mou. Short-term correlation of solar activity and sunspot:. Solar Physics , volume=. 2006 , publisher=

    2006

  54. [62]

    Physical Review E , volume=

    Long-term persistence of the spatial organization of temperature fluctuation lifetime in turbulent air avalanches , author=. Physical Review E , volume=. 2006 , publisher=

    2006

  55. [63]

    Physical Review A , volume=

    Persistence and stochastic periodicity in the intensity dynamics of a fiber laser during the transition to optical turbulence , author=. Physical Review A , volume=. 2018 , publisher=

    2018

  56. [64]

    Physica A: Statistical Mechanics and its Applications , volume=

    Multifractal detrended fluctuation analysis on air traffic flow time series: A single airport case , author=. Physica A: Statistical Mechanics and its Applications , volume=. 2019 , publisher=

    2019

  57. [65]

    IEEE Access , volume=

    Exploring dynamic characteristics of multi-state air traffic flow: A time series approach , author=. IEEE Access , volume=. 2020 , publisher=

    2020

  58. [66]

    Physica A: Statistical Mechanics and its Applications , volume=

    Exploring dynamic evolution and fluctuation characteristics of air traffic flow volume time series: A single waypoint case , author=. Physica A: Statistical Mechanics and its Applications , volume=. 2018 , publisher=

    2018

  59. [67]

    IEEE Transactions on Control Systems Technology , volume=

    Adjoint-based control of a new Eulerian network model of air traffic flow , author=. IEEE Transactions on Control Systems Technology , volume=. 2006 , publisher=

    2006

  60. [68]

    Acta Aeronautica et Astronautica Sinica , volume=

    Analysis on generalized following behavior and complex phase-transition law of approaching traffic flow in terminal airspace , author=. Acta Aeronautica et Astronautica Sinica , volume=

  61. [69]

    Journal of Transportation Systems Engineering and Information Technology , volume=

    Impacts of parallel runway operation modes on air traffic flow characteristics in terminal areas , author=. Journal of Transportation Systems Engineering and Information Technology , volume=

  62. [70]

    Journal of Guidance, Control, and Dynamics , volume=

    Computer-aided Eulerian air traffic flow modeling and predictive control , author=. Journal of Guidance, Control, and Dynamics , volume=

  63. [71]

    Transportation Research Part B: Methodological , volume=

    Fundamental diagrams of airport surface traffic: Models and applications , author=. Transportation Research Part B: Methodological , volume=. 2017 , publisher=

    2017

  64. [72]

    Transportation Research Part B: Methodological , volume=

    Total unimodularity and decomposition method for large-scale air traffic cell transmission model , author=. Transportation Research Part B: Methodological , volume=. 2013 , publisher=

    2013

  65. [73]

    Physica A: Statistical Mechanics and its Applications , volume=

    Multiscale complexity analysis on airport air traffic flow volume time series , author=. Physica A: Statistical Mechanics and its Applications , volume=. 2020 , publisher=

    2020

  66. [74]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Assessing time series irreversibility through micro-scale trends , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2021 , publisher=

    2021

  67. [75]

    2021 , publisher=

    Deep Learning and Physics , author=. 2021 , publisher=

    2021

  68. [76]

    Journal of the Royal Society Interface , volume=

    Opportunities and obstacles for deep learning in biology and medicine , author=. Journal of the Royal Society Interface , volume=. 2018 , publisher=

    2018

  69. [77]

    Journal of Computational Chemistry , volume=

    Deep learning for computational chemistry , author=. Journal of Computational Chemistry , volume=. 2017 , publisher=

    2017

  70. [78]

    Knowledge and Information Systems , volume=

    Interpretable deep learning: Interpretation, interpretability, trustworthiness, and beyond , author=. Knowledge and Information Systems , volume=. 2022 , publisher=

    2022

  71. [79]

    The structure and function of

    Murschel, Andrea , journal=. The structure and function of. 1995 , publisher=

    1995

  72. [80]

    Journal of Air Transport Management , volume=

    Applying complexity science to air traffic management , author=. Journal of Air Transport Management , volume=. 2015 , publisher=

    2015

  73. [81]

    Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition , pages=

    Deep residual learning for image recognition , author=. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition , pages=

  74. [82]

    2017 International Joint Conference on Neural Networks (IJCNN) , pages=

    Time series classification from scratch with deep neural networks: A strong baseline , author=. 2017 International Joint Conference on Neural Networks (IJCNN) , pages=. 2017 , organization=

    2017

  75. [83]

    Zanin, Massimiliano , journal=. Can. 2022 , publisher=

    2022

  76. [84]

    IEEE Access , volume=

    Assessing Identifiability in Airport Delay Propagation Roles Through Deep Learning Classification , author=. IEEE Access , volume=. 2022 , publisher=

    2022

  77. [85]

    Nature , volume=

    Deep learning , author=. Nature , volume=. 2015 , publisher=

    2015

  78. [86]

    IEEE Transactions on Control Systems Technology , volume=

    Airport capacity: Representation, estimation, optimization , author=. IEEE Transactions on Control Systems Technology , volume=. 1993 , publisher=

    1993

  79. [87]

    Europhysics Letters , volume=

    20 years of ordinal patterns: Perspectives and challenges , author=. Europhysics Letters , volume=. 2022 , publisher=

    2022

  80. [88]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Forbidden patterns in financial time series , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2008 , publisher=

    2008

  81. [89]

    Counting forbidden patterns in irregularly sampled time series

    McCullough, Michael and Sakellariou, Konstantinos and Stemler, Thomas and Small, Michael , journal=. Counting forbidden patterns in irregularly sampled time series. 2016 , publisher=

    2016

  82. [90]

    Counting forbidden patterns in irregularly sampled time series

    Sakellariou, Konstantinos and McCullough, Michael and Stemler, Thomas and Small, Michael , journal=. Counting forbidden patterns in irregularly sampled time series. 2016 , publisher=

    2016

  83. [91]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    An analysis of high-frequency cryptocurrencies prices dynamics using permutation-information-theory quantifiers , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2018 , publisher=

    2018

  84. [92]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Discriminating chaotic and stochastic dynamics through the permutation spectrum test , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2014 , publisher=

    2014

  85. [93]

    Assessing serial dependence in ordinal patterns processes using chi-squared tests with application to

    Yamashita Rios de Sousa, Arthur Matsuo and Hlinka, Jaroslav , journal=. Assessing serial dependence in ordinal patterns processes using chi-squared tests with application to. 2022 , publisher=

    2022

  86. [94]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Differentiating resting brain states using ordinal symbolic analysis , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2018 , publisher=

    2018

  87. [95]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Detection of time reversibility in time series by ordinal patterns analysis , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2018 , publisher=

    2018

  88. [96]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Using forbidden ordinal patterns to detect determinism in irregularly sampled time series , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2016 , publisher=

    2016

  89. [97]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Inferring long memory processes in the climate network via ordinal pattern analysis , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2011 , publisher=

    2011

  90. [98]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Characterizing system dynamics with a weighted and directed network constructed from time series data , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2014 , publisher=

    2014

  91. [99]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Regenerating time series from ordinal networks , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2017 , publisher=

    2017

  92. [100]

    Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=

    Time lagged ordinal partition networks for capturing dynamics of continuous dynamical systems , author=. Chaos: An Interdisciplinary Journal of Nonlinear Science , volume=. 2015 , publisher=

    2015

  93. [101]

    Physica A: Statistical Mechanics and its Applications , volume=

    Revisiting the decay of missing ordinal patterns in long-term correlated time series , author=. Physica A: Statistical Mechanics and its Applications , volume=. 2019 , publisher=

    2019

  94. [102]

    Physica A: Statistical Mechanics and its Applications , volume=

    Forbidden patterns, permutation entropy and stock market inefficiency , author=. Physica A: Statistical Mechanics and its Applications , volume=. 2009 , publisher=

    2009

  95. [103]

    Journal of business , volume=

    The variation of certain speculative prices , author=. Journal of business , volume=. 1963 , publisher=

    1963

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Reviewed July 31, 2026 · model on record in the stance chip above.