{"id":"c3d712c3-6df2-49dd-afde-bebb572970f2","arxiv_id":"2502.17453","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"Permutation extropy is introduced as a time series complexity measure, defined as the extropy of the permutation pattern distribution, and its validation is based on chaotic maps and financial/health data.","lead":"Researchers propose a new way to measure the complexity of a time series, called permutation extropy, which is the extropy of the ordinal patterns of a series. They claim it sometimes distinguishes chaotic from periodic behavior more accurately than the popular permutation entropy measure.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central comparison is under-specified: no embedding dimension k (or series length) is stated for any simulation, so the claimed superiority of permutation extropy over permutation entropy cannot be reproduced or tested.","rationale":"The strongest claim is that permutation extropy is a valid and better complexity measure. The only evidence for 'better' is a small number of visual comparisons; the values underlying those comparisons cannot be reproduced because k is missing. This is the single most load-bearing gap: if a standard k reverses the claimed orderings, the central claim is unsupported; if the orderings hold robustly across k, the paper's main empirical point survives, though the range error still needs correction. The reader identified the same missing-k problem as the weakest assumption, and I agree that it is decisive. The verdict should remain unchanged: reject in current form, with an opportunity to resubmit after specifying the protocol and correcting the range.","tokens_in":10124,"tokens_out":8863,"duration_ms":84468,"concrete_test":"Re-run the logistic-map comparison (Section 3.1) with an explicit protocol: x0=0.1, discard 10^3 transient iterations, keep m=10^5, embedding k in {3,4,5,6}, and standard ordinal-pattern assignment with deterministic tie breaking. For each k, compute J(k) from Eq. (3) and permutation entropy at r=3.6 and r=3.7. If the claimed ordering J(3.7)<J(3.6) fails for any k in this range, the central example supporting superiority is not robust. Also verify the Section 2 range statement by evaluating Eq. (3) on the uniform distribution over k! patterns.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Sections 3.1-3.3, where the authors conclude from visual alignment with bifurcation diagrams that permutation extropy 'more precisely captures' chaotic behavior than permutation entropy (Figs. 2, 6, 8-10). The inference depends entirely on the computed values, but no embedding dimension k is ever stated for Eq. (3), and no time-series length m or transient-removal procedure is given for the maps. Permutation measures are highly sensitive to k: for the logistic map at r=3.6 vs. 3.7, the relative order of permutation entropy can change with k, and the same applies to the proposed extropy. Without k, Fig. 2(b) is an unconstrained visual claim rather than a reproducible result. An independent red flag is the stated range in Section 2: [0,(k!-1)(log k! - log(k-1)!)] is not the range of Eq. (3). At the uniform pattern distribution, Eq. (3) attains (k!-1) log(k!/(k!-1)), not (k!-1) log k; for k=3 these are 0.91 and 5.49, respectively. This suggests the measure's own properties were not verified, reinforcing the need for a stated protocol before any superiority claim is accepted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":10416,"tokens_out":8945,"duration_ms":83778,"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":[{"comment":"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.","section":"Section 2"},{"comment":"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.","section":"Sections 3.1-3.3"},{"comment":"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.","section":"Section 3.1"},{"comment":"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.","section":"Sections 4 and 5"},{"comment":"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.","section":"Section 3.2"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Section 3.3"},{"comment":"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.","section":"Section 2"},{"comment":"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.","section":"Section 2"},{"comment":"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.","section":"Section 6"}],"recommendation":"major_revision","confidential_remarks":"The mathematical error in the range and the missing embedding dimension are the most serious issues and must be addressed before the paper can be considered further. If the authors can supply a corrected range and a complete experimental protocol, the central definition may still be publishable as a modest methodological contribution. However, I would also ask the editor to consider whether the paper's level of rigor meets the standards of nlin.CD; the qualitative visual comparison with bifurcation diagrams, without any statistical or quantitative validation, is far below what is typical for this venue."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe take-home: permutation extropy as defined here is a legitimate but extremely simple variant of permutation entropy—replace Shannon entropy by extropy on the ordinal pattern distribution. The paper's central claim that it outperforms permutation entropy is not supported by the evidence as presented.\n\nWhat is genuinely new: Eq. (3) is a clean explicit definition, and applying extropy to Bandt-Pompe patterns is a natural thing to try. The paper does that, walks through three chaotic maps, and gives financial and WHO applications. The prose is readable and the idea is easy to grasp.\n\nThe problems are real. No embedding dimension k is stated for any simulation, and no time-series length or transient-removal protocol is given. Permutation measures are highly sensitive to k; without it, Figures 2, 6, 8-10 are not reproducible and the visual comparisons with bifurcation diagrams are unconstrained. The stated range in Section 2 is mathematically wrong: the uniform distribution gives (k!-1) log(k!/(k!-1)), not (k!-1) log k. For k=3 that's 0.91 vs 5.49, so this is not a typo—it changes the qualitative picture. The superiority claims rely on cherry-picked parameter values (r=3.7, b=0.15) and visual alignment, not on any quantitative comparison. The applications are narratives with no statistical test or error bar.\n\nThe paper does deserve a referee. The idea is coherent, the definition is correct, and the flaws are fixable. A careful revision with stated experimental parameters, a corrected range, and a quantitative comparison (e.g., permutation entropy vs extropy across many parameter values) could make this a modest but useful contribution.\n\nMy call: send to peer review, but with the expectation of major revision. I wouldn't cite it as it stands.","headline":"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.","tokens_in":10929,"tokens_out":2308,"would_cite":false,"duration_ms":41913,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["permutation extropy","time series complexity","permutation entropy","chaotic maps","logistic map","Henon map","Burger map","bifurcation diagram"],"falsifier":"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.","tokens_in":9931,"feed_emoji":"🌀","tokens_out":7824,"duration_ms":73922,"temperature":0.7,"pith_summary":"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.","feed_headline":"New complexity measure outdoes entropy on chaotic maps","feed_subtitle":"Combines permutation entropy with extropy to flag dips in chaos that the older measure misses.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"supplies the ordinal-pattern embedding and the permutation entropy baseline that the new measure modifies.","marker":"[4]"},{"why":"defines extropy as the complementary dual of entropy, the formula the paper adapts to pattern probabilities.","marker":"[9]"},{"why":"provides the bifurcation analysis of the Burger map used as one of the validation systems.","marker":"[10]"},{"why":"defines the Henon map used as a second validation system.","marker":"[11]"},{"why":"supplies the NIFTY 50 closing-price series used in the financial application.","marker":"[12]"},{"why":"supplies the daily death-count series used in the COVID-19 application.","marker":"[13]"}],"fun_headline_variants":["New complexity measure beats permutation entropy on chaos","Permutation extropy: complexity measure that sees chaos dips","Extropy-based complexity measure outperforms on chaotic series","Permutation extropy: better chaos complexity than entropy"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["New complexity measure beats permutation entropy on chaos","Permutation extropy: complexity measure that sees chaos dips","Extropy-based complexity measure outperforms on chaotic series","Permutation extropy: better chaos complexity than entropy"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000525,"raw_usage":{"total_tokens":2565,"prompt_tokens":1002,"completion_tokens":1563,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":618,"completion_tokens_details":{"reasoning_tokens":1501}},"tokens_in":618,"tokens_out":1563,"duration_ms":11541,"temperature":1.0,"reasoning_tokens":1501,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T18:58:54.401138+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Permutation entropy: a natural complexity measure for time series","cited_arxiv_id":null,"evidence_quote":"supplies the ordinal-pattern embedding and the permutation entropy baseline that the new measure modifies."},{"cited_title":"Extropy: Complementary dual of entropy","cited_arxiv_id":null,"evidence_quote":"defines extropy as the complementary dual of entropy, the formula the paper adapts to pattern probabilities."},{"cited_title":"M., et al","cited_arxiv_id":null,"evidence_quote":"provides the bifurcation analysis of the Burger map used as one of the validation systems."},{"cited_title":"A two-dimensional mapping with a strange attractor","cited_arxiv_id":null,"evidence_quote":"defines the Henon map used as a second validation system."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the NIFTY 50 closing-price series used in the financial application."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the daily death-count series used in the COVID-19 application."}],"review_version":1}