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Permutation extropy: a time series complexity measure

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

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Pith's one-line read The paper proposes permutation extropy, a complexity measure built from ordinal patterns and complementary probabilities, and argues it tracks chaotic regimes more faithfully than permutation entropy.

desk verdict A simple, clearly defined extropy variant of permutation entropy whose claimed superiority is undercut by a wrong range formula, unstated embedding dimensions, and cherry-picked visual comparisons. read the letter →

arxiv 2502.17453 v1 pith:7ZPEZ3ZC submitted 2025-02-08 nlin.CD stat.AP

Ritik Roshan Giri , Suchandan Kayal This is my paper · ORCID
classification nlin.CDstat.AP
keywords permutationextropytimeseriescomplexityentropychaoticmapslogisticmapHenonBurgerbifurcationdiagram
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 introduces permutation extropy, a time series complexity measure built by applying the extropy formula $J(k)=-\sum_{i=1}^{k!}(1-p(\pi_i))\log(1-p(\pi_i))$ to the frequencies of ordinal patterns of length $k$. The authors argue that this measure is a valid indicator of uncertainty and that it tracks the bifurcation structure of chaotic maps more faithfully than permutation entropy in specific cases, most notably for the logistic map at $r=3.6$ versus $r=3.7$, where permutation entropy ranks the less chaotic state as more uncertain. They support the claim with visual comparisons against bifurcation diagrams for the logistic, Henon, and Burger maps, and with applications to NIFTY 50 closing prices and WHO daily death counts during the COVID-19 period. If the claim holds, the measure gives researchers a cheap, fast complexity score that is invariant under monotone transformations and can flag when a system is actually becoming more chaotic.

What carries the argument

The central object is the ordinal pattern, the relative ordering of $k$ consecutive values in a sliding window of the time series; there are at most $k!$ such patterns. Permutation extropy assigns each pattern the probability $p(\pi_i)$ of occurrence and then evaluates $J(k)=-\sum(1-p)\log(1-p)$, the extropy of that pattern distribution. Where permutation entropy measures how evenly the patterns are spread, permutation extropy measures the same spread through the complementary probabilities, so it gives different weight to patterns that are absent or rare. This is the mechanism the paper invokes, repeatedly comparing the parameter-by-$J$ curve against a bifurcation diagram to conclude that the new curve's rises and dips match the onset of periodicity and chaos.

What would settle it

Run both measures on a long logistic-map time series at $r=3.6$ and $r=3.7$ with the sliding-window length stated and varied ($k=3,4,5,6$), repeating over trajectory lengths and initial conditions; if permutation extropy does not consistently rank $r=3.6$ above $r=3.7$ wherever the bifurcation diagram shows more chaos, the paper's central superiority claim fails. The same test with a small amount of observational noise added would show whether the ordering is robust.

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Extended reading notes

Core claim

The paper's central claim is that replacing the factor $p\log p$ in permutation entropy with the complementary factor $(1-p)\log(1-p)$ yields a quantity that is still a legitimate complexity measure and agrees better with the bifurcation diagram in some regimes. Concretely, the paper defines permutation extropy as $J(k) = -\sum_{i=1}^{k!}(1-p(\pi_i))\log(1-p(\pi_i))$, where $p(\pi_i)$ is the relative frequency with which the $i$-th ordinal pattern of length $k$ appears among the embedded windows of a time series. The authors assert that this quantity is zero for monotone series, reaches its maximum under equiprobable patterns, is fast to compute, and is invariant under monotonically increasing transformations just as permutation entropy is. Their evidence is qualitative: on the logistic map the permutation extropy curve dips at $r=3.7$ in line with the bifurcation diagram, where the permutation entropy curve does not; on the Henon and Burger maps the two measures mostly agree; and on real data the measure assigns lower complexity to the COVID-years' NIFTY 50 series and to the early and late phases of the Indian COVID death series, which they read as matching the known course of the pandemic.

Load-bearing premise

The argument stands on visual agreement with bifurcation diagrams, and on an embedding dimension that the paper never reports; without those fixed, the claimed advantage cannot be independently checked.

Editorial extensions

If this is right

  • For the logistic map, the two measures disagree about which of the neighbouring regimes $r=3.6$ and $r=3.7$ is more complex; the paper's extropy curve follows the bifurcation diagram, so the measure would be preferable for locating chaos onset in one-dimensional maps.
  • Because permutation extropy inherits phase-space independence from ordinal patterns, it can be computed for any real-valued time series without knowing the underlying dynamical equations.
  • Applied to NIFTY 50 closing prices, the measure gives lower values in 2020 and 2021 than in 2019 and 2022, so it can serve as a market-stress diagnostic across calendar years.
  • Applied to WHO daily death counts, the measure rises, falls, and rises again across the 2020-2021 phases of COVID in India, providing a phase-tracking complexity score for epidemic curves.

Reading between the lines

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

  • Because extropy weights complementary probabilities, permutation extropy is plausibly more sensitive to rare or missing ordinal patterns; if that sensitivity is real, the measure may detect intermittency and laminar phases that permutation entropy averages away, but the paper does not test this on stochastic signals.
  • The paper never states the sliding-window length $k$ used for its figures, so its claimed advantage should be rechecked across $k=3,4,5,6$ and across series lengths; if the $r=3.6$ versus $3.7$ ordering flips for some $k$, the superiority claim narrows to a parameter-specific statement.
  • A natural next test would be to compute permutation extropy on sliding windows of a financial or epidemiological series and compare turning points with known events; the paper reports year and phase averages, not online monitoring.
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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

5 major / 5 minor

Summary. The paper proposes a time-series complexity measure called permutation extropy, defined as the extropy of the ordinal-pattern distribution of an embedded time series (Eq. 3). It claims that this measure is a valid uncertainty measure and that it 'more precisely captures' the behavior of chaotic maps than permutation entropy, supporting this with qualitative comparisons against bifurcation diagrams of the logistic, Hénon, and Burgers maps. The paper also applies the measure to NIFTY50 closing prices and WHO COVID-19 death data, claiming better complexity specification than permutation entropy for these real-world series.

Significance. If the claims were established, permutation extropy would be a simple and computationally cheap variant of permutation entropy, potentially useful in diverse applied fields. The paper does make one concrete contribution: it gives a clear formula for a new measure and illustrates its behavior on standard chaotic maps. However, the significance is severely limited by three problems: a mathematical error in the stated maximum of the measure, the complete omission of the embedding dimension and other experimental parameters needed for reproducibility, and the absence of any quantitative or statistical comparison against permutation entropy, especially in the real-data sections. The paper also provides no code or data, so the visual claims cannot be independently checked.

major comments (5)
  1. [Section 2] The stated range of the permutation extropy is incorrect. The paper claims the value lies in [0, (k!-1)(log k! - log(k-1)!)] and that the upper bound is attained when all k! ordinal patterns are equally likely. For the uniform distribution p(pi_i) = 1/k!, Eq. (3) gives J = - sum_{i=1}^{k!} (1 - 1/k!) log(1 - 1/k!) = (k!-1) log(k!/(k!-1)), not (k!-1) log k. For k=3 these values are approximately 0.91 and 5.49, so the discrepancy is large. The following sentence also contains a typo, writing '(m! - 1)(log k! - log(k-1)!)' instead of k!. Since the range is a fundamental property of the proposed measure, this error must be corrected and the maximum verified.
  2. [Sections 3.1-3.3] No experiment states the embedding dimension k used to compute Eq. (3), nor the time-series length m, the number of transient iterations discarded, or the rule for handling ties in the ordinal patterns. Permutation entropy and extropy are highly sensitive to k; the relative ordering of the two measures at a given parameter value can change with k. Consequently, the visual comparisons in Figures 1-14 and the repeated conclusion that permutation extropy 'more precisely captures' chaotic behavior are not reproducible or testable. The authors must specify k, m, transient removal, and tie-breaking for every simulation, and ideally show that the qualitative conclusions are stable over a range of k.
  3. [Section 3.1] The central comparative claim for the logistic map rests on a single arbitrary parameter choice: the paper states that at r=3.7 permutation entropy is higher than at r=3.6 while the bifurcation diagram suggests more complexity at r=3.6. No quantitative criterion is offered for judging which measure is 'better', and no evidence is given that this example is representative rather than cherry-picked. Additionally, the text refers to the fully chaotic case r=4, but the list of simulated r values is only 3, 3.1, ..., 3.8; if r=4 was used, it must be stated explicitly.
  4. [Sections 4 and 5] The abstract and introduction claim that the proposed measure gives 'a better complexity specification than permutation entropy' for financial time series and WHO data, but these sections compute only permutation extropy and never report permutation entropy values for the same series. Without a direct comparison, the real-data sections do not support the stated superiority claim. The authors should either add permutation entropy results for these datasets or revise the abstract and conclusions to remove the unsupported comparison.
  5. [Section 3.2] The description of the Hénon map experiment is internally inconsistent and does not specify which parameter is varied in each figure. The text says 'by taking b = 0.3, varying parameter a' and then discusses 'varying parameter b in place of parameter a which takes a value of 1.4'; Figure 3(a) is labeled 'when b=0.3' while Figure 5(a) is labeled 'when a=1.4'. The captions and text must be reconciled so that the reader can determine whether each permutation extropy and permutation entropy curve is a function of a or b. This ambiguity makes the claimed differences at b=0.15 versus b=0.10 impossible to verify.
minor comments (5)
  1. [Throughout] There are numerous typos, including 'Mathemtics' in the affiliation, 'Bifuraction' in several figure captions, 'Shanon' for Shannon, 'extopy' in the abstract, and 'Burger' for Burgers in the map name.
  2. [Section 3.3] The sentence 'When b = 0.4, the probability of occurrence of each permutation pattern is high in comparison to b = 0.2, which implies the permutation extropy value at b = 0.4 in comparison to b = 0.2' is incomplete; it should state that the value at b = 0.4 is higher than at b = 0.2. Similar incomplete comparative sentences appear nearby.
  3. [Section 2] The logarithm in Eqs. (1)-(3) is not specified; any base could be used, but the base should be stated and kept consistent so that reported values are meaningful.
  4. [Section 2] The definition of ordinal patterns assumes no equal values in the embedded vectors, but this is not discussed. Real-world data such as stock closing prices can contain ties; the paper should state whether it uses the standard Bandt-Pompe convention or a tie-breaking rule.
  5. [Section 6] The conclusion says the proposed measure is 'kind of robust', but robustness is not defined and no supporting experiments are provided. Either remove this claim or define and test it.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: permutation extropy is defined independently of the data used to illustrate it; the comparisons are external validations, not fits.

full rationale

The paper's central construction (Eq. 3) is a direct definition: J(k) = -sum (1-p(pi_i)) log(1-p(pi_i)) evaluated on the empirical ordinal-pattern distribution p(pi_i) from Eq. 4. No parameter of J(k) is fitted to the bifurcation diagrams, financial data, or WHO data. The claims that extropy 'is a valid measure of uncertainty' and 'more precisely captures' chaotic behavior (Sections 3.1-3.2, Figs. 1-6) are supported by visual comparison of computed curves with bifurcation diagrams; this is an external empirical validation, not an equivalence by construction. No step uses a definition that presupposes the target conclusion. There are no self-citations or imported uniqueness theorems; the only cited prior results (Bandt-Pompe, Lad et al.) are external and do not carry the paper's conclusion. The paper does contain serious correctness/reproducibility problems - the claimed range in Section 2 does not match Eq. (3) at uniform patterns, and no embedding dimension k or series length is stated - but these are errors of validation and reporting, not circularity. In particular, choosing parameter values (e.g., r=3.7, b=0.15) where permutation entropy misorders the expected complexity is post hoc selection, not a fitted parameter masquerading as a prediction. Therefore no circular step can be exhibited.

Assumptions & free parameters 1 free parameters · 2 assumptions · 0 invented entities

The paper introduces no new entities, only a new measure built from existing concepts. The main free parameter is the embedding dimension, which is left unspecified. The validity rests on the domain assumption that extropy is a meaningful complexity measure and that ordinal patterns capture relevant dynamics.

free parameters (1)
  • Embedding dimension k
    The number of ordinal patterns depends on k, and the permutation extropy values and all comparisons are sensitive to this choice. The paper never states the value of k used in the simulations or applications.
assumptions (2)
  • domain assumption Extropy is a valid complement to entropy for measuring uncertainty or complexity
    The paper inherits Lad et al.'s extropy as a complexity measure without providing independent justification for why extropy of the permutation distribution is meaningful for time series complexity.
  • standard math Ordinal patterns capture the dynamical complexity of a time series
    This is the standard assumption behind permutation entropy from Bandt and Pompe, which the paper accepts without further discussion.

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Cite this review

Pith. "Pith review of Permutation extropy: a time series complexity measure." pith.science (2026). https://pith.science/paper/7ZPEZ3ZC

@misc{pith2026250217453,
  author       = {Pith},
  title        = {Pith review of: Permutation extropy: a time series complexity measure},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7ZPEZ3ZC}},
  note         = {Machine review of arXiv:2502.17453}
}
read the original abstract

On account of a greater need for understanding the complexity of time series like physiological time series, financial time series, and many more that enter into picture for their inculpation with real-world problems, several complexity parameters have already been proposed in the literature. Permutation entropy, Lyapunov exponents are such complexity parameters out of many. In this article, we introduce a new time series complexity parameter, that is, the permutation extropy. The failure of permutation entropy in correctly specifying complexity of some chaotic time series motivates us to come up with a better complexity parameter, hence we propose this permutation extropy measure. We try to combine the ideas behind the permutation entropy and extopy to construct this measure. We also validate our proposed measure using several chaotic maps like logistic map, Henon map and Burger map. We apply the proposed complexity parameter to study the complexity of financial time series of the stock market and time series constructed using WHO data, finding a better complexity specification than permutation entropy. The proposed measure is kind of robust, fast calculation and invariant with respect to monotonous nonlinear transformation like permutation entropy, but it gives us a better result in specifying complexity in some cases.

Figures

Figures reproduced from arXiv: 2502.17453 by the authors.

Figure 1
Figure 1. Bifurcation diagram and permutation extropy of logistic map. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Permutation entropy and comparison between permutation entropy and extropy of the logistic map. [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Bifurcation diagram and permutation extropy of the Henon map. [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (15 more)
Figure 4
Figure 4. Figure 4: Permutation entropy and comparison between permutation entropy and extropy of the Henon map. [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: Bifurcation diagram and permutation extropy of the Henon map. [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]
Figure 6
Figure 6. Figure 6: Permutation entropy and comparison between permutation entropy and extropy of Henon map. [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]
Figure 7
Figure 7. Figure 7: Bifurcation diagram and permutation extropy of the Burger map. [PITH_FULL_IMAGE:figures/full_fig_p008_7.png]
Figure 8
Figure 8. Figure 8: Permutation entropy and comparison between permutation entropy and extropy of the Burger map. [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]
Figure 9
Figure 9. Figure 9: Bifurcation diagram and permutation extropy of the Burger map. [PITH_FULL_IMAGE:figures/full_fig_p010_9.png]
Figure 10
Figure 10. Figure 10: Permutation entropy and comparison between permutation entropy and extropy of the Burger map. [PITH_FULL_IMAGE:figures/full_fig_p010_10.png]
Figure 11
Figure 11. Figure 11: Bifurcation diagram of the Burger map. (a) Permutation entropy for the Burger map when b=0. (b) Permutation entropy for the Burger map [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: Permutation entropy and permutation extropy of the Burger map. [PITH_FULL_IMAGE:figures/full_fig_p011_12.png]
Figure 13
Figure 13. Figure 13: Bifurcation diagram of the Burger map [PITH_FULL_IMAGE:figures/full_fig_p012_13.png]
Figure 14
Figure 14. Figure 14: Comparison of permutation extropy taking different values of [PITH_FULL_IMAGE:figures/full_fig_p012_14.png]
Figure 15
Figure 15. Figure 15: Permutation extropy plot of different financial years. [PITH_FULL_IMAGE:figures/full_fig_p013_15.png]
Figure 16
Figure 16. Figure 16: Permutation extropy value of different months during a financial year (2019-2022). [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]
Figure 17
Figure 17. Figure 17: Permutation extropy value of different years and phases during a year (2020-2022). [PITH_FULL_IMAGE:figures/full_fig_p016_17.png]
Figure 18
Figure 18. Figure 18: Comparison of permutation extropy values between different years. [PITH_FULL_IMAGE:figures/full_fig_p017_18.png]

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Extropy Rate: Properties and Application in Feature Selection

    cs.IT 2025-07 reject novelty 3.0 of 10

    The proposed extropy rate reduces to the logarithmic growth rate of the support size, and the claimed equivalence with the entropy rate for stationary ergodic processes is incorrect.

Reference graph

Works this paper leans on

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Reviewed August 8, 2026 · model on record in the stance chip above.