Pith. sign in

REVIEW 3 major objections 7 minor 48 references

Extropy Rate: Properties and Application in Feature Selection

T0 review · 3 major / 7 minor · reviewed 2026-08-06 · deepseek-v4-flash

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

Pith's one-line read The paper defines an extropy rate for discrete stochastic processes and claims it equals the entropy rate almost surely for stationary ergodic processes, with applications in time-series complexity, chaos detection, and feature selection.

desk verdict The extropy rate collapses to support growth and Theorem 4 is false; a few salvageable conditional-extropy properties do not save it. read the letter →

arxiv 2507.11242 v1 pith:YW236MVB submitted 2025-07-15 cs.IT math.ITmath.PR

Naveen Kumar , Vivek Vijay This is my paper
classification cs.ITmath.ITmath.PR MSC 94A1762B1060G1062H3062P30
keywords extropyrateconditionalentropyShannon–McMillan–BreimantheoremfeatureselectionstochasticprocessesSimpson'sdiversityindextime-seriescomplexity
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

Extropy, the complementary dual of Shannon entropy, is built from the probabilities that events do not occur, and it lacks the additive structure that makes entropy rates easy to define. This paper introduces an extropy rate for discrete-time, discrete-state stochastic processes by pairing the logarithm of the support size of the n-dimensional joint distribution with its joint extropy, and it claims that for stationary ergodic processes with positive entropy rate, the extropy rate equals the entropy rate almost surely. The paper also establishes bounds, subadditivity, and Lipschitz continuity for joint and conditional extropy, and it gives numerical evidence that the estimated extropy rate tracks time-series complexity, signals bifurcations in logistic and Hénon maps, and aligns with Simpson's diversity index. A feature-selection method that ranks features by their estimated extropy rate is then tested on six public datasets, where the paper reports accuracy and F1 scores at least matching mutual information, chi-square, and F-score methods. A sympathetic reader would care because, if the equivalence claim holds, extropy rate becomes a legitimate alternative measure of average uncertainty per variable with its own emphasis on frequent rather than rare events.

What carries the argument

The defining mechanism is the rescaled identity $H(\{q_i\})=\log(m-1)+J(\{p_i\})/(m-1)$ with $q_i=(1-p_i)/(m-1)$, which expresses the entropy of a rescaled distribution in terms of extropy; removing the rescaling yields the paper's extropy-rate definition. The load-bearing analytic tool in the equivalence claim is the Shannon–McMillan–Breiman theorem, used to show that with high probability the per-symbol log-likelihood converges to the entropy rate, so the log-support term $\frac{1}{n}\log(S_n-1)$ can be sandwiched between $H-\epsilon$ and $H+\epsilon$. For finite samples, the estimator $J^F_{\mathrm{rate}}$ is backed by the Lipschitz-continuity result for the extropy functional under the $\ell^1$ metric, which the paper proves on the space of distributions with probabilities bounded below $r$.

What would settle it

Take a stationary ergodic Markov chain over a countably infinite state space where the stationary distribution has infinite support and the transition graph is irreducible, so that the number of n-step paths grows like $e^{cn}$ with $c>H_{\mathrm{rate}}$. If $\frac{1}{n}\log(S_n-1)$ converges to a value strictly larger than the entropy rate for this process, then $J_{\mathrm{rate}}=H_{\mathrm{rate}}$ fails. The paper's equations (54)-(57) require the non-typical set to be exponentially negligible in cardinality, so a direct computation of $S_n$ for such a chain would settle the claim.

Watch

Extended reading notes

Core claim

For a discrete stochastic process $X=(X_1,X_2,\ldots)$, the paper defines the extropy rate as $J_{\mathrm{rate}}(X)=\lim_{n\to\infty}\frac{1}{n}\left(\log(S_n-1)+\frac{J(X_1,\ldots,X_n)}{S_n-1}\right)$, where $S_n$ is the support size of the joint distribution; since the joint extropy term divided by $n$ vanishes, the rate reduces to $\lim_n \frac{1}{n}\log(S_n-1)$, which the authors interpret as an average zeroth-order Rényi entropy. The central theoretical claim is Theorem 4: for a stationary ergodic process with entropy rate $H_{\mathrm{rate}}(X)>0$, $J_{\mathrm{rate}}(X)=H_{\mathrm{rate}}(X)$ almost surely, derived from the Shannon–McMillan–Breiman theorem by bounding the support size with the size of the $\epsilon$-typical set. The paper further shows that for an IID sequence on $k$ symbols the extropy rate is approximately $\log k$, that joint extropy is strictly subadditive for independent variables, and that the empirical version $J^F_{\mathrm{rate}}$ can be computed for finite samples. On the applied side, the paper claims that features with higher estimated extropy rates carry more information, and reports that this selection rule outperforms or matches mutual information, chi-square, and F-score on six datasets.

Load-bearing premise

The proof that extropy rate equals entropy rate assumes that for every stationary ergodic process, the support size of the n-dimensional distribution stays within the size of the epsilon-typical set plus a small number of low-probability sequences outside it; many stationary ergodic processes have far more low-probability sequences than that bound allows.

Editorial extensions

If this is right

  • If $J_{\mathrm{rate}}=H_{\mathrm{rate}}$ almost surely, then extropy-rate estimators become alternative estimators of entropy rate for stationary ergodic data, with the practical difference that they are computed from non-occurrence probabilities.
  • For IID sequences with support size $k$, the extropy rate is approximately $\log k$, so the measure doubles as a support-size and diversity index for discrete data.
  • The strict subadditivity $J(X,Y)<J(X)+J(Y)$ for independent variables implies that joint extropy grows sublinearly, which is why a plain time-average of joint extropy is degenerate and why the support-size correction is necessary.
  • The numerical demonstrations indicate that estimated extropy rate can rank time series on a complexity axis from constant to random walk and can flag bifurcation parameters in the logistic and Hénon maps.
  • A feature-selection rule based on highest estimated extropy rate is claimed to reach accuracy, F1-score, and true-positive rates at least as high as mutual information, chi-square, and F-score across the six tested datasets.

Reading between the lines

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

  • The equivalence $J_{\mathrm{rate}}=H_{\mathrm{rate}}$ is only as robust as the claim that the support size grows like the typical set; for stationary ergodic processes over countably infinite alphabets with many low-probability sequences, $\frac{1}{n}\log S_n$ could exceed the entropy rate, so the paper's own definition may yield a different limit outside the finite-alphabet regime.
  • Because the infinite-process rate reduces to the growth rate of $\log S_n$, it is essentially the zeroth-order Rényi entropy rate; a natural testable extension is to compare finite-sample extropy-rate estimates with direct Rényi entropy rate estimators on heavy-tailed data.
  • The observed alignment with Simpson's diversity index suggests an ecological reading of extropy rate as a temporal diversity index, but the paper proves no quantitative relation between the two, so a formal comparison would be a useful next step.
  • The feature-selection method ranks features by an estimated marginal or accumulated extropy rate; an unstated risk is that high-extropy features may be redundant, so a natural extension is to combine the criterion with a redundancy-penalizing step and test it on high-dimensional data.
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.

Referee Report

3 major / 7 minor

Summary. The paper defines conditional extropy and an extropy rate for discrete stochastic processes via Eq. (16), proves several properties of joint and conditional extropy, and claims that for stationary ergodic processes the extropy rate equals the entropy rate almost surely (Theorem 4). It then applies an estimated extropy rate to time-series complexity, chaotic map characterization, and feature selection on six datasets.

Significance. If the central Theorem 4 were correct, the extropy rate would be a genuine probability-sensitive alternative to the entropy rate for stochastic processes. The paper also contains a substantial numerical and experimental component. However, the central theorem is false, and the definition itself reduces to a support-growth rate, so the theoretical contribution collapses. The feature-selection experiments may still be of empirical interest, but they are not supported by the claimed theory.

major comments (3)
  1. [Definition 2 and Remark 2] Equation (16) defines Jrate(X) = lim (1/n)[log(S_n-1) + J(X_1,...,X_n)/(S_n-1)]. Remark 2 correctly observes that J(X_1,...,X_n)/n tends to 0, so the definition reduces to Jrate(X) = lim (1/n) log(S_n-1). Thus the proposed extropy rate is a pure support-growth rate, analogous to topological entropy, and it carries no information about the actual probabilities. This already contradicts the paper's motivation: the extropy term is asymptotically irrelevant. Consequently, Theorem 1's result for IID sequences, Jrate = log k, follows directly from S_n = k^n and is a consequence of the definition rather than a property of extropy.
  2. [Theorem 4] Theorem 4 claims that for a stationary ergodic process X with entropy rate Hrate(X)>0, Jrate(X)=Hrate(X) almost surely. This is false. For an IID Bernoulli(0.9) process on {0,1}, every binary n-tuple has positive probability, so S_n = 2^n and Jrate = log 2 ≈ 0.693 nats, while Hrate = H(0.9) ≈ 0.325 nats. Hence Jrate does not equal Hrate. The same failure occurs for any irreducible finite Markov chain with full support: S_n = k^n, so Jrate = log k, while Hrate can be any value less than log k. The claimed equivalence is therefore not a meaningful property of the extropy rate.
  3. [Theorem 4 proof, Eq. (54)] The proof bounds the non-typical set by E_2^n ≤ ε e^{n(H(P)-ε)}. The Shannon-McMillan-Breiman theorem controls only the total probability of the non-typical set, not its cardinality. In the Bernoulli(0.9) example, the non-typical set contains roughly 2^n - 2^{nH} strings, all of which belong to the support S_n, so the bound S_n ≤ e^{n(H+ε)} fails. This invalidates the chain of inequalities leading to Eq. (57) and is the exact technical gap behind the false Theorem 4.
minor comments (7)
  1. [Proposition 4] The displayed identity has a sign error: summing (p_j - p_{ij}) ln p_j over i gives (m_X-1)∑_j p_j ln p_j = -(m_X-1)H(Y), so the correct formula is J(X|Y) = {p_j}J(X|Y) - (m_X-1)H(Y). As stated, the proposition is false.
  2. [Proposition 2 proof] The first line of the proof writes J(X|Y) = -∑_j p_j J(X|Y=j), but Definition 1 already includes the minus sign; the subsequent expansion is consistent with the definition, so this is a sign typo.
  3. [Theorem 3 proof, Eq. (51)] The inequality (1-z)ln(1-z) ≥ -z implies -p_j(1-p_i)ln(1-p_i) ≤ p_j p_i, not ≥. The displayed lower bound therefore does not follow. The theorem may be true, but this proof is invalid.
  4. [Section 4.2, Eq. (58)] The estimator appears to contain a single term -(1-P(X_1,...,X_n))ln(1-P(X_1,...,X_n)) rather than a sum over all m^n states. Under the independence approximation, the joint extropy should be -∑_{state}(1-p(state))ln(1-p(state)). As written, Eq. (58) does not compute the quantity defined in Eq. (16).
  5. [Section 4.4, Eq. (63)] This equation defines the empirical extropy rate as J(X_1,...,X_i)/i, but Remark 2 and Proposition 1 show this quantity tends to 0 as i grows. Figure 6 therefore appears to plot a different quantity from the extropy rate of Definition 2.
  6. [Section 5.4, Table 2] The perfect scores (accuracy and F1 of 1.0000) on Diabetes and Boston Housing require details on train/test splitting, cross-validation, and random seeds to be reproducible. The comparison should also state whether the same classifier configuration was used for all methods.
  7. [Throughout] There are typographical issues, including 'preposition' for 'proposition' in the proof of Theorem 1, 'defination' in Remark 2, and missing accents in 'Henon' and 'Renyi'.

Circularity Check

2 steps flagged · score 6.0 of 10

The infinite-process extropy rate is defined into the support-growth (Rényi-0/topological) entropy rate, so Theorem 1 is an algebraic restatement rather than a derived extropy result; Theorem 4's proof additionally rests on a false support-size bound.

  1. renaming known result [Section 3, Definition 2 (Eq. 16), Remark 2 (Eq. 18), Remark 3]
    "Note that the extropy of any discrete distribution is bounded above by 1. Therefore, we have limn→∞ J(X1, X2, ..., Xn)/n = 0. Further, we can rewrite the defination of extropy rate for a infinite stochastic process X = {Xi}i as Jrate(X) = limn→∞ log(Sn − 1)/n. This expression approximately equal to the average of zeroth-order Rényi entropy."

    The paper itself reduces the infinite-process extropy rate to the normalized logarithm of support size. That object is the Rényi-0 entropy rate / topological-entropy growth rate, not a functional of extropy: the J-term in Eq. (16) is discarded by the boundedness observation in Remark 2, so no extropy content remains in the asymptotic quantity. Presenting this support-growth rate as a new extropy-based uncertainty measure is a relabeling of a known quantity rather than a derivation from extropy.

  2. self definitional [Section 3, Theorem 1 (Eq. 22)]
    "From preposition (1), we know that limn→∞ J(X1, X2, ..., Xn)/n = 0 and the random variables in the process are IID, this implies that Sn = kn. Thus from definition of extropy rate, we have ... = limn→∞ log(kn − 1)/n ≈ log(k)."

    Theorem 1 is not an independent prediction about extropy: once Definition 2 is reduced to Jrate = lim (1/n) log(Sn − 1) in Remark 2, the IID result is simply the evaluation of that definition at the deterministic value Sn = k^n. No property of the probability distribution or of the extropy functional enters the limit except the support size. The claimed theorem is therefore forced by the definition, not derived from extropy as a substantive measure.

full rationale

The main circularity is definitional. Definition 2 includes a J-term, but Remark 2 immediately shows that the term vanishes asymptotically and rewrites the extropy rate as Jrate(X) = lim (1/n) log(Sn − 1). This is the support-growth rate, equivalently the Rényi-0 entropy rate or topological entropy rate, as the paper itself notes in Remarks 2 and 3. Consequently, Theorem 1's IID result Jrate ≈ log k is just the algebraic evaluation of the definition at Sn = k^n; it carries no independent extropy content. I do not count the few self-citations here as load-bearing: the authors' own review [24] is an incidental background citation, and no uniqueness theorem is imported from the authors' prior work. The numerical feature-selection comparisons are external benchmarks, but they do not cure the definitional collapse of the theoretical quantity. Theorem 4 is not itself circular, but it is mathematically invalid for a separate reason: the proof's bound E2 ≤ ε e^{n(H−ε)} at Eq. (54) is false because the Shannon–McMillan–Breiman theorem controls the probability of atypical sequences, not their cardinality. For a Bernoulli(0.9) source every binary string is in the support, so Jrate = log 2 while Hrate ≈ 0.325, contradicting Theorem 4. That is a correctness failure, not a reduction-by-construction, and I accordingly do not list it as a circular step. Overall, the central theoretical quantity is defined into a known support-growth rate and the headline IID result reduces to that definition, so the circularity score is 6.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central theoretical claim depends on standard results from information theory and real analysis, but the proofs contain errors. The only free parameters appear in the application sections, where the number of selected features and the implicit discretization of continuous maps are chosen by hand. No new physical entities are introduced.

free parameters (2)
  • Number of selected features k = 3 (diabetes, blood, EEG, electricity), 4 (Boston), 6 (forest fires)
    The number of features chosen per dataset in Section 5.4 is hand-selected to demonstrate the method's superiority; the reported accuracy depends on this choice and no principled rule is given.
  • Implicit discretization for logistic and Henon maps = not specified
    Continuous states are used with the empirical estimator in Eq 58 without specifying a binning or discretization scheme; the resulting extropy rate values depend on this implicit parameter.
assumptions (4)
  • standard math Shannon-McMillan-Breiman theorem
    Used in Theorem 4 to bound typical set sizes, but applied incorrectly; the theorem itself is standard.
  • standard math Calculus and real analysis results, including Lipschitz continuity and series expansions
    Used in Theorem 2 and Proposition 3; the proofs have errors in the series manipulation and inequality directions.
  • domain assumption Finite state space for the stochastic process
    Definition 2 restricts to finite support sizes; the theoretical results do not cover countable or continuous alphabets.
  • domain assumption Independence assumption in the empirical estimator
    Section 4.2 approximates the joint pmf as a product of marginal probabilities, which is not valid for dependent time series; this assumption underlies all numerical experiments.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Extropy Rate: Properties and Application in Feature Selection." pith.science (2026). https://pith.science/paper/YW236MVB

@misc{pith2026250711242,
  author       = {Pith},
  title        = {Pith review of: Extropy Rate: Properties and Application in Feature Selection},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YW236MVB}},
  note         = {Machine review of arXiv:2507.11242}
}
read the original abstract

Extropy, a complementary dual of entropy, (proposed by Lad et al. \cite{lad2015extropy} in 2015) has attracted considerable interest from the research community. In this study, we focus on discrete random variables and define conditional extropy, establishing key properties of joint and conditional extropy such as bounds, uncertainty reduction due to additional information, and Lipschitz continuity. We further introduce the concept of extropy rate for a stochastic process of discrete random variables as a measure of the average uncertainty per random variable within the process. It is observed that for infinite stationary and ergodic stochastic processes, as well as for identically and independently distributed sequences, the extropy rate exhibits asymptotic equivalence. We explore the extropy rate for finite stochastic processes and numerically illustrate its effectiveness in capturing the underlying information across various distributions, quantifying complexity in time series data, and characterizing chaotic dynamics in dynamical systems. The behaviour of estimated extropy rate is observed to be closely aligned with Simpson's diversity index. The real-life applicability of the extropy rate is presented through a novel feature selection method based on the fact that features with higher extropy rates contain greater inherent information. Using six publicly available datasets, we show the superiority of the proposed feature selection method over some other existing popular approaches.

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

48 extracted references · 46 canonical work pages

  1. [1]

    Topological entropy

    RL Adler, AG Konheim, and MH McAndrew. Topological entropy. Transactions of the American Mathematical Society, 114(2):309–319, 1965

    work page 1965

  2. [2]

    A sandwich proof of the shannon–mcmillan–breiman theorem

    PH Algoet and TM Cover. A sandwich proof of the shannon–mcmillan–breiman theorem. The annals of probability, pages 899–909, 1988

    work page 1988

  3. [3]

    On tsallis extropy with an application to pattern recognition

    N Balakrishnan, F Buono, and M Longobardi. On tsallis extropy with an application to pattern recognition. Statistics & Probability Letters, 180:109241, 2022. 25

    work page 2022

  4. [4]

    On weighted extropies

    Narayanaswamy Balakrishnan, Francesco Buono, and Maria Longobardi. On weighted extropies. Communications in Statistics-Theory and Methods , 51(18):6250–6267, 2022

    work page 2022

  5. [5]

    Permutation entropy: a natural complexity measure for time series

    C Bandt and B Pompe. Permutation entropy: a natural complexity measure for time series. Physical review letters, 88(17):174102, 2002

    work page 2002

  6. [6]

    The unified extropy and its versions in classical and dempster–shafer theories

    F Buono, Y Deng, and M Longobardi. The unified extropy and its versions in classical and dempster–shafer theories. Journal of Applied Probability, 61(2):685– 696, 2024

    work page 2024

  7. [7]

    A dual measure of uncertainty: The deng extropy

    F Buono and M Longobardi. A dual measure of uncertainty: The deng extropy. Entropy, 22(5):582, 2020

    work page 2020

  8. [8]

    On weighted cumulative tsallis residual and past entropy measures

    S Chakraborty and B Pradhan. On weighted cumulative tsallis residual and past entropy measures. Communications in Statistics-Simulation and Computation, 52(5):2058–2072, 2023

    work page 2023

Show all 48 references
  1. [9]

    A data mining approach to predict forest fires using meteorological data

    P Cortez and AJR Morais. A data mining approach to predict forest fires using meteorological data. 2007

    2007

  2. [10]

    Elements of information theory

    TM Cover. Elements of information theory. John Wiley & Sons, 1999

    1999

  3. [11]

    Approximate entropy and sample entropy: A comprehensive tutorial

    A Delgado-Bonal and A Marshak. Approximate entropy and sample entropy: A comprehensive tutorial. Entropy, 21(6):541, 2019

    2019

  4. [12]

    Plausibility extropy: The complemen- tary dual of plausibility entropy

    X Deng, S Xue, W Jiang, and X Zhang. Plausibility extropy: The complemen- tary dual of plausibility entropy. IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2024

    2024

  5. [13]

    Least angle regression

    B Efron, T Hastie, I Johnstone, and R Tibshirani. Least angle regression. 2004

    2004

  6. [14]

    Discrete chaos: with applications in science and engineering

    SN Elaydi. Discrete chaos: with applications in science and engineering . Chapman and Hall/CRC, 2007

    2007

  7. [15]

    Permutation extropy: a time series complexity measure

    RR Giri and S Kayal. Permutation extropy: a time series complexity measure. arXiv preprint arXiv:2502.17453, 2025

    2025 arXiv

  8. [16]

    Entropy rate of diffusion processes on complex networks

    J G´ omez-Gardenes and V Latora. Entropy rate of diffusion processes on complex networks. Physical Review E—Statistical, Nonlinear, and Soft Matter Physics, 78(6):065102, 2008

    2008

  9. [17]

    Some characterizations of continuous symmetric distributions based on extropy of record values

    N Gupta and SK Chaudhary. Some characterizations of continuous symmetric distributions based on extropy of record values. Statistical Papers, 65(1):291–308, 2024

    2024

  10. [18]

    Splice-2 comparative evaluation: Electricity pricing

    M Harries and NS Wales. Splice-2 comparative evaluation: Electricity pricing. 1999

    1999

  11. [19]

    Hedonic housing prices and the demand for clean air

    D Harrison Jr and DL Rubinfeld. Hedonic housing prices and the demand for clean air. Journal of environmental economics and management, 5(1):81–102, 1978

    1978

  12. [20]

    Chaos and nonlinear dynamics: an introduction for scientists and engineers

    RC Hilborn. Chaos and nonlinear dynamics: an introduction for scientists and engineers. Oxford university press, 2000

    2000

  13. [21]

    On cumulative residual extropy

    SMA Jahanshahi, H Zarei, and AH Khammar. On cumulative residual extropy. Probability in the Engineering and Informational Sciences, 34(4):605–625, 2020

    2020

  14. [22]

    Fractional deng entropy and extropy and some applications

    MR Kazemi, S Tahmasebi, F Buono, and M Longobardi. Fractional deng entropy and extropy and some applications. Entropy, 23(5):623, 2021

    2021

  15. [23]

    Estimating entropy rates with bayesian confidence intervals

    MB Kennel, J Shlens, HDI Abarbanel, and EJ Chichilnisky. Estimating entropy rates with bayesian confidence intervals. Neural Computation, 17(7):1531–1576, 2005. 26

    2005

  16. [24]

    Entropy measures and their applications: A comprehensive review

    N Kumar, A Dixit, and V Vijay. Entropy measures and their applications: A comprehensive review. arXiv preprint arXiv:2503.15660, 2025

    2025 arXiv

  17. [25]

    Extropy: Complementary dual of entropy

    F Lad, G Sanfilippo, and G Agro. Extropy: Complementary dual of entropy. 2015

    2015

  18. [26]

    Kolmogorov-sinai entropy rate versus physical entropy

    V Latora and M Baranger. Kolmogorov-sinai entropy rate versus physical entropy. Physical Review Letters, 82(3):520, 1999

    1999

  19. [27]

    The rate of entropy increase at the edge of chaos

    V Latora, M Baranger, A Rapisarda, and C Tsallis. The rate of entropy increase at the edge of chaos. Physics Letters A, 273(1-2):97–103, 2000

    2000

  20. [28]

    Renyi extropy

    J Liu and F Xiao. Renyi extropy. Communications in Statistics-Theory and Methods, 52(16):5836–5847, 2023

    2023

  21. [29]

    Entropy rate superpixel segmentation

    MY Liu, O Tuzel, S Ramalingam, and R Chellappa. Entropy rate superpixel segmentation. In CVPR 2011, pages 2097–2104. IEEE, 2011

    2011

  22. [30]

    How can i evaluate P∞ n=0(n + 1)xn? Mathematics Stack Exchange

    Micah. How can i evaluate P∞ n=0(n + 1)xn? Mathematics Stack Exchange. URL:https://math.stackexchange.com/q/223857 (version: 2012-10-29)

    2012

  23. [31]

    Choosing and using diversity indices: insights for ecological applications from the german biodiversity exploratories

    EK Morris, T Caruso, F Buscot, M Fischer, C Hancock, TS Maier, T Mein- ers, C M¨ uller, E Obermaier, D Prati, SA Socher, I Sonnermann, N Waschke, T Wubet, S Wurst, and MC Rillig. Choosing and using diversity indices: insights for ecological applications from the german biodive...

    2014

  24. [32]

    Coarse-grained entropy rates for characterization of complex time series

    M Paluˇ s. Coarse-grained entropy rates for characterization of complex time series. Physica D: Nonlinear Phenomena, 93(1-2):64–77, 1996

    1996

  25. [33]

    Approximate entropy as a measure of system complexity

    SM Pincus. Approximate entropy as a measure of system complexity. Proceedings of the national academy of sciences, 88(6):2297–2301, 1991

    1991

  26. [34]

    Entropy, entropy rate, and pattern classification as tools to typify complex- ity in short heart period variability series

    A Porta, S Guzzetti, N Montano, R Furlan, M Pagani, A Malliani, and S Cerutti. Entropy, entropy rate, and pattern classification as tools to typify complex- ity in short heart period variability series. IEEE Transactions on Biomedical Engineering, 48(11):1282–1291, 2002

    2002

  27. [35]

    Extropy estimators with applications in testing uniformity

    G Qiu and K Jia. Extropy estimators with applications in testing uniformity. Journal of Nonparametric Statistics, 30(1):182–196, 2018

    2018

  28. [36]

    On extropy properties of mixed systems.Probability in the Engineering and Informational Sciences, 33(3):471–486, 2019

    G Qiu, L Wang, and X Wang. On extropy properties of mixed systems.Probability in the Engineering and Informational Sciences, 33(3):471–486, 2019

    2019

  29. [37]

    On measures of entropy and information

    A R´ enyi. On measures of entropy and information. In Proceedings of the fourth Berkeley symposium on mathematical statistics and probability, volume 1: con- tributions to the theory of statistics, volume 4, pages 547–562. University of California Press, 1961

    1961

  30. [38]

    Sample entropy

    JS Richman, DE Lake, and JR Moorman. Sample entropy. In Methods in enzymology, volume 384, pages 172–184. Elsevier, 2004

    2004

  31. [39]

    EEG Eye State

    O Roesler. EEG Eye State. UCI Machine Learning Repository, 2013. DOI: https://doi.org/10.24432/C57G7J

    2013 doi

  32. [40]

    Copula-based extropy measures, properties, and dependence in bivariate distributions

    S Saha and S Kayal. Copula-based extropy measures, properties, and dependence in bivariate distributions. Communications in Statistics-Theory and Methods, pages 1–28, 2025

    2025

  33. [41]

    On relative cumulative extropy, its residual (past) measures and their applications in estimation and testing

    P Saranya and SM Sunoj. On relative cumulative extropy, its residual (past) measures and their applications in estimation and testing. Journal of the Indian Society for Probability and Statistics, 25(1):199–225, 2024. 27

    2024

  34. [42]

    On dynamic weighted extropy

    EIA Sathar and RD Nair. On dynamic weighted extropy. Journal of Computa- tional and Applied Mathematics, 393:113507, 2021

    2021

  35. [43]

    A mathematical theory of communication

    CE Shannon. A mathematical theory of communication. The Bell system technical journal, 27(3):379–423, 1948

    1948

  36. [44]

    On negative cumulative extropy with applications

    S Tahmasebi and A Toomaj. On negative cumulative extropy with applications. Communications in Statistics-Theory and Methods, 51(15):5025–5047, 2022

    2022

  37. [45]

    Extropy: Characterizations and dynamic versions

    A Toomaj, M Hashempour, and N Balakrishnan. Extropy: Characterizations and dynamic versions. Journal of Applied Probability, 60(4):1333–1351, 2023

    2023

  38. [46]

    Measuring visual saliency by site entropy rate

    W Wang, Y Wang, Q Huang, and W Gao. Measuring visual saliency by site entropy rate. In 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, pages 2368–2375. IEEE, 2010

    2010

  39. [47]

    A comparative study on feature selection in text categorization

    Y Yang and JO Pedersen. A comparative study on feature selection in text categorization. In ICML, volume 97, page 35. Citeseer, 1997

    1997

  40. [48]

    Knowledge discovery on rfm model using bernoulli sequence

    IC Yeh, KJ Yang, and TM Ting. Knowledge discovery on rfm model using bernoulli sequence. Expert Systems with applications, 36(3):5866–5871, 2009. 28

    2009

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.