Cited by 249Complex Systems and Time Series AnalysisFinancial Markets and Investment StrategiesStock Market Forecasting Methods

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Forbidden patterns, permutation entropy and stock market inefficiency

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Zunino, Luciano, Zanin, Massimiliano, Tabak, Benjamin M., Pérez, Darío G., Rosso, Osvaldo A.·2009·Physica A Statistical Mechanics and its Applications

DOI 10.1016/j.physa.2009.03.042

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EFFICIENT CAPITAL MARKETS: A REVIEW OF THEORY AND EMPIRICAL WORK*
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Permutation Entropy: A Natural Complexity Measure for Time Series
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Distinguishing Noise from Chaos
Rosso, O. A., Larrondo, H. A., Martín, M. T., Plastino, A., & Fuentes, M. (2007) · Physical Review Letters · doi:10.1103/physrevlett.99.154102 · Cited by 716
A review of symbolic analysis of experimental data
Daw, C. S., Finney, C. E. A., & Tracy, E. R. (2003) · Review of Scientific Instruments · doi:10.1063/1.1531823 · Cited by 595
Detecting dynamical changes in time series using the permutation entropy
Cao, Y., Tung, W., Gao, J., Protopopescu, V., & Hively, L. M. (2004) · Physical Review E · doi:10.1103/physreve.70.046217 · Cited by 545
Long-term memories of developed and emerging markets: Using the scaling analysis to characterize their stage of development
Matteo, T. D., Aste, T., & Dacorogna, M. M. (2004) · Journal of Banking & Finance · doi:10.1016/j.jbankfin.2004.08.004 · Cited by 433
The Hurst exponent over time: testing the assertion that emerging markets are becoming more efficient
Cajueiro, D. O., & Tabak, B. M. (2004) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2003.12.031 · Cited by 411
A multifractal approach for stock market inefficiency
Zunino, L., Tabak, B. M., Figliola, A., Pérez, D. G., Garavaglia, M., & Rosso, O. A. (2008) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2008.08.028 · Cited by 305

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Refined Composite Multi-Scale Reverse Weighted Permutation Entropy and Its Applications in Ship-Radiated Noise
Li, Y., Geng, B., & Jiao, S. (2021) · Entropy · doi:10.3390/e23040476 · Cited by 18
“Its applications can also be found in the fields of economics [19,20], mechanical engineering [21,22], and underwater acoustics [23,24].”
Background
Long-term recurrence patterns in the late 2000 economic crisis: Evidences from entropy analysis of the Dow Jones index
Álvarez‐Ramírez, J., & Rodríguez, E. (2011) · Technological Forecasting and Social Change · doi:10.1016/j.techfore.2011.03.003 · Cited by 17
Quantifying stochasticity in the dynamics of delay-coupled semiconductor lasers via forbidden patterns
Tiana‐Alsina, J., Buldú, J. M., Torrent, M. C., & García‐Ojalvo, J. (2009) · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences · doi:10.1098/rsta.2009.0241 · Cited by 17
“In its original implementation this technique used ordinal patterns in order to convert a continuous into a discrete time series (Zunino et al., 2009).”
Method
Multivariate multiscale fractional order weighted permutation entropy of nonlinear time series
Chen, S., Shang, P., & Wu, Y. (2018) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2018.09.165 · Cited by 16
Is human atrial fibrillation stochastic or deterministic?—Insights from missing ordinal patterns and causal entropy-complexity plane analysis
Aronis, K. N., Berger, R. D., Calkins, H. G., Chrispin, J., Marine, J. E., SPRAGG, D. D., Tao, S., Tandri, H. S., & Ashikaga, H. (2018) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.5023588 · Cited by 16
Analysis of crude oil markets with improved multiscale weighted permutation entropy
Niu, H., Wang, J., & Liu, C. (2017) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2017.12.049 · Cited by 16
Spillover effects of Great Recession on Hong-Kong’s Real Estate Market: An analysis based on Causality Plane and Tsallis Curves of Complexity–Entropy
Argyroudis, G. S., & Siokis, F. (2019) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2019.04.052 · Cited by 16
“[13] are its robustness, simplicity, its invariance considering nonlinear monotonous transformations, applicability to any type of time series, the fact that it does not require a large sample size like for instance in fractal analysis, and lastly that it provides a fast computational algorithm.”
Background
Permutation entropy and its variants for measuring temporal dependence
Huang, X., Shang, H. L., & Pitt, D. (2022) · Australian & New Zealand Journal of Statistics · doi:10.1111/anzs.12376 · Cited by 15
Permutation Entropy as a Measure of Information Gain/Loss in the Different Symbolic Descriptions of Financial Data
Kozák, J. T., Kania, K., & Juszczuk, P. (2020) · Entropy · doi:10.3390/e22030330 · Cited by 15
“This dependency was also used in [21] for comparison of emerging and mature stock markets.”
Method
Revisiting the European sovereign bonds with a permutation-information-theory approach
Bariviera, A. F., Zunino, L., Guercio, M., Martínez, L. B., & Rosso, O. A. (2013) · The European Physical Journal B · doi:10.1140/epjb/e2013-40660-7 · Cited by 15
Entropy as a Tool for the Analysis of Stock Market Efficiency During Periods of Crisis
Papla, D., & Siedlecki, R. (2024) · Entropy · doi:10.3390/e26121079 · Cited by 14
Information-theoretic characterization of eye-tracking signals with relation to cognitive tasks
Iaconis, F. R., Gandica, A. A. J., Punta, J. A. D., Delrieux, C. A., & Gasaneo, G. (2021) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0042104 · Cited by 14
Generalized weighted permutation entropy
Stošić, D., Stošić, D., Stošić, T., & Stosic, B. D. (2022) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0107427 · Cited by 14
Decision trees unearth return sign predictability in the S&P 500
Fiévet, L., & Sornette, D. (2018) · Quantitative Finance · doi:10.1080/14697688.2018.1441535 · Cited by 14
“…anomalies in stock market returns reported by Niederhoffer and Osborne (1966), Zhang (1999), Leung et al. (2000), Christoffersen and Diebold (2003), Andersen and Sornette (2005), Christoffersen et al. (2006), Zunino et al. (2009), Satinover and Sornette (2012a, 2012b) and James et al. (2014).”
Background
Forecasting Events in the Complex Dynamics of a Semiconductor Laser with Optical Feedback
Colet, M., & Aragoneses, A. (2018) · Scientific Reports · doi:10.1038/s41598-018-29110-5 · Cited by 14
Using the Information Provided by Forbidden Ordinal Patterns in Permutation Entropy to Reinforce Time Series Discrimination Capabilities
Cuesta–Frau, D. (2020) · Entropy · doi:10.3390/e22050494 · Cited by 14
“Actually, the numbers of admissible/forbidden patterns have been demonstrated to provide information with system distinguishing power [21], and a few studies have exploited this new feature independently of PE for signal classification [22].”
BackgroundResultInfluential
The Impact of Linear Filter Preprocessing in the Interpretation of Permutation Entropy
Dávalos, A., Jabloun, M., Ravier, P., & Buttelli, O. (2021) · Entropy · doi:10.3390/e23070787 · Cited by 14
“There are still several challenges in the understanding of the filters’ effects on ordinal information, in particular the characterization of forbidden patterns [24] of filtered signals.”
Background
Slope Entropy Characterisation: The Role of the δ Parameter
Kouka, M., & Cuesta–Frau, D. (2022) · Entropy · doi:10.3390/e24101456 · Cited by 14
“A lot of scientific fields have benefited from the high segmentation power of these methods [9,10], which have been used as features to distinguish among time series types in classification or clustering algorithms [11,12].”
Method
Generalized permutation entropy analysis based on the two-index entropic form Sq,δ
Xu, M., & Shang, P. (2015) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/1.4921552 · Cited by 14
Testing implications of the Adaptive Market Hypothesis via computational intelligence
Butler, M., & Kazakov, D. (2012) · IEEE Conference on Computational Intelligence for Financial Engineering & Economics · doi:10.1109/cifer.2012.6327799 · Cited by 14
“The results from each of the studies [20] [13] revealed that emerging markets tended to be less efficient than mature markets.”
BackgroundResult
Permutation Entropy: Enhancing Discriminating Power by Using Relative Frequencies Vector of Ordinal Patterns Instead of Their Shannon Entropy
Cuesta–Frau, D., Molina, Á., Vargas, B., & González, P. (2019) · Entropy · doi:10.3390/e21101013 · Cited by 14
“At the opposite end of the spectrum of randomness, Figure 1b shows the histogram of a sinusoidal time series, with a more polarised distribution of motifs, some of them forbidden (zero probability) [22].”
Background
Modeling the predictive power of the singular value decomposition-based entropy. Empirical evidence from the Dow Jones Global Titans 50 Index
Bușu, C., & Bușu, M. (2019) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2019.04.055 · Cited by 13
Fault diagnosis of rolling bearing using a refined composite multiscale weighted permutation entropy
Li, Y., Gao, Q., Li, P., Liu, J., & Zhu, Y. (2021) · Journal of Mechanical Science and Technology · doi:10.1007/s12206-021-0408-4 · Cited by 13
Global Collective Dynamics of Financial Market Efficiency Using Attention Entropy with Hierarchical Clustering
Cho, P., & Kim, K. (2022) · Fractal and Fractional · doi:10.3390/fractalfract6100562 · Cited by 13
Variance of permutation entropy and the influence of ordinal pattern selection
Little, D. J., & Kane, D. M. (2017) · Physical review. E · doi:10.1103/physreve.95.052126 · Cited by 13
Multiscale dynamics under the lens of permutation entropy
Olivares, F., & Zunino, L. (2020) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2020.125081 · Cited by 13
Characterizing unstructured data with the nearest neighbor permutation entropy
Voltarelli, L. G. J. M., Pessa, A. A. B., Zunino, L., Zola, R. S., Lenzi, E. K., Perc, M., & Ribeiro, H. V. (2024) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0209206 · Cited by 12
Multivariate generalized information entropy of financial time series
Zhang, Y., Shang, P., & Xiong, H. (2019) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2019.04.029 · Cited by 12
Permutation Entropy-Based Analysis of Temperature Complexity Spatial-Temporal Variation and Its Driving Factors in China
Zhang, T., Cheng, C., & Gao, P. (2019) · Entropy · doi:10.3390/e21101001 · Cited by 12
Ranking efficiency for twenty-six emerging stock markets and financial crisis: Evidence from the shannon entropy approach
Mensi, W. (2012) · International Journal of Management Science and Engineering Management · doi:10.1080/17509653.2012.10671207 · Cited by 12
“(2009) [40] used forbidden patterns and permutation entropy concepts to quantify stock market inefficiency.”
BackgroundMethod
Randomness in denoised stock returns: The case of Moroccan family business companies
Lahmiri, S. (2017) · Physics Letters A · doi:10.1016/j.physleta.2017.12.020 · Cited by 12
Nonlinear Multiscale Entropy and Recurrence Quantification Analysis of Foreign Exchange Markets Efficiency
Niu, H., & Zhang, L. (2017) · Entropy · doi:10.3390/e20010017 · Cited by 11
“It shows advantages of simplicity, fast calculation, robustness for analyzing different kinds of time series, including random, chaotic and real-world time series, which has been employed in the context of neural [25], physiological signals [26–28], climate system [29] and financial market time series [30].”
Background
Ensemble Improved Permutation Entropy: A New Approach for Time Series Analysis
Chen, Z., Ma, X., Fu, J., & Li, Y. (2023) · Entropy · doi:10.3390/e25081175 · Cited by 11
“The development of new entropy metrics has shed new light on a wide range of engineering problems, including fault diagnosis [18,19], underwater target recognition [1,7,20–22], stock market analysis [23,24], and biomedical signal processing [8,9,15,25], among others.”
Background
Generalized Statistical Complexity: A New Tool for Dynamical Systems
(2013) · BENTHAM SCIENCE PUBLISHERS eBooks · doi:10.2174/9781608057603113010013 · Cited by 11
Transportation Mode Detection Based on Permutation Entropy and Extreme Learning Machine
Zhang, L., Liu, L., Bao, S., Qiang, M., & Zou, X. (2015) · Mathematical Problems in Engineering · doi:10.1155/2015/940624 · Cited by 11
“PE has been applied in many applications [15], such as neural applications [16], electroencephalography (EEG) signal analysis [17], electrocardiograph (ECG) [18], and stock market analysis [19].”
Method
Permutation entropy with vector embedding delays
Little, D. J., & Kane, D. M. (2017) · Physical review. E · doi:10.1103/physreve.96.062205 · Cited by 11
A universal algorithm to generate pseudo-random numbers based on uniform mapping as homeomorphism
Fu-Lai, W. (2010) · Chinese Physics B · doi:10.1088/1674-1056/19/9/090505 · Cited by 10
A novel method of visualizing q-complexity-entropy curve in the multiscale fashion
Yeh, C.-H., Fang, Y., Shi, W., & Hong, Y. E. (2019) · Nonlinear Dynamics · doi:10.1007/s11071-019-05165-5 · Cited by 10
Comparative study of feature extraction and classification based on dispersion Lempel‐Ziv complexity
Jiao, S., Geng, B., Li, Y., Zhang, Q., Wang, Q., Li, Y., & Wang, W. (2021) · International Journal of Numerical Modelling Electronic Networks Devices and Fields · doi:10.1002/jnm.2949 · Cited by 10
Complexity of Magnetic-field Turbulence at Reconnection Exhausts in the Solar Wind at 1 au
Miranda, R. A., Valdivia, J. A., Chian, A. C. ‐., & Muñoz, P. R. (2021) · The Astrophysical Journal · doi:10.3847/1538-4357/ac2dfe · Cited by 10
“It has been successfully applied to data from experiments with electronic oscillators (Soriano et al. 2011), stock market data (Zunino et al. 2009), the Southern Oscillation index (Bandt 2005), and heart rate variability (Bian et al. 2012), among others.”
Method
Reverse Weighted-Permutation Entropy: A Novel Complexity Metric Incorporating Distance and Amplitude Information
Li, Y. (2019) · International Electronic Conference on Entropy and Its Applications (ECEA) · doi:10.3390/ecea-5-06688 · Cited by 10
“In addition, more applications can also be found in the fields of economics [4,5], mechanical engineering [6,7] and underwater acoustics [8,9].”
Background
Random Walk Null Models for Time Series Data
DeFord, D., & Moore, K. (2017) · Entropy · doi:10.3390/e19110615 · Cited by 9
“For instance, permutation entropy and the number of forbidden patterns for both closing values and returns were suggested as methods for distinguishing developed and emerging markets with the aim of using these measures to quantify stock market ineﬃciency [41].”
BackgroundMethodInfluential
Unsupervised selection of the singular spectrum components based on information theory for bioacoustic signal filtering
Colonna, J. G., & Nakamura, E. F. (2018) · Digital Signal Processing · doi:10.1016/j.dsp.2018.07.009 · Cited by 9
Efficiency of Chinese Real Estate Market Based on Complexity-Entropy Binary Causal Plane Method
Chen, Y., Cai, Y., & Zheng, C. (2020) · Complexity · doi:10.1155/2020/2791352 · Cited by 9
“[14, 15] pointed out that it can effectively detect the structural information hidden in system noise, even if the system is at the edge of chaos.”
Background
Research on Dynamic Characteristics of Stock Market Based on Big Data Analysis
Ping, Y., & Hou, X. (2022) · Discrete Dynamics in Nature and Society · doi:10.1155/2022/8758976 · Cited by 8
“Zunino et al. [10] believed that entropy can be used to quantify the degree of randomness of time series, so entropy can be used to quantify the degree of market effectiveness, and further use entropy to study the degree of market eﬀectiveness of American market and Asian market during Black Monday…”
Method
Detrended fluctuation analysis and entropy-complexity causality analysis of temperatures in an urbanized mountain stream
Thaxton, C. S., Anderson, W. P., Gu, C., Stosic, B. D., & Stošić, T. (2017) · Stochastic Environmental Research and Risk Assessment · doi:10.1007/s00477-017-1434-8 · Cited by 8
Distinguishing Stock Indices and Detecting Economic Crises Based on Weighted Symbolic Permutation Entropy
Zhang, N., Sun, Y., Zhang, Y., Yang, P., Lin, A., & Shang, P. (2019) · Fluctuation and Noise Letters · doi:10.1142/s0219477519500263 · Cited by 8
Analysis of financial time series through forbidden patterns
Ji, A., & Shang, P. (2019) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2019.122038 · Cited by 8
Multiscale permutation mutual information quantify the information interaction for traffic time series
Yin, Y., Wang, X., Li, Q., Shang, P., Gao, H., & Ma, Y. (2020) · Nonlinear Dynamics · doi:10.1007/s11071-020-05981-0 · Cited by 8
Stock index dynamics worldwide: a comparative analysis
Cortines, A., Anteneodo, C., & Riera, R. (2008) · The European Physical Journal B · doi:10.1140/epjb/e2008-00347-x · Cited by 8
A permutation entropy analysis of Bitcoin volatility
Obanya, P. O., Seitshiro, M. B., Olivier, C. P., & Verster, T. (2024) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2024.129609 · Cited by 7
Variable contribution analysis in multivariate process monitoring using permutation entropy
Obanya, P. O., Coetzer, R. L. J., Olivier, C. P., & Verster, T. (2024) · Computers & Industrial Engineering · doi:10.1016/j.cie.2024.110064 · Cited by 7
Permutation Entropy and Its Niche in Hydrology: A Review
Mihailović, D. T. (2025) · Entropy · doi:10.3390/e27060598 · Cited by 7
“This information measure has quickly found applications across a wide range of scientific fields: biomedical sciences [16–19], econophysics [20,21], signal processing [22,23], climate research [24,25], laser physics and geophysics [22,26], machine learning [23] as well as in hydrology ([27–31],…”
Unraveling the decay of the number of unobserved ordinal patterns in noisy chaotic dynamics
Olivares, F., Zunino, L., Soriano, M. C., & Pérez, D. G. (2019) · Physical review. E · doi:10.1103/physreve.100.042215 · Cited by 7
The Method for Determining Optimal Analysis Length of Vibration Data Based on Improved Multiscale Permutation Entropy
Zhang, J., Li, Z., Peng, Y., Li, Y., & Huang, J. (2021) · Shock and Vibration · doi:10.1155/2021/6654089 · Cited by 7
Study of Speculative Bubbles: The Contribution of Approximate Entropy
Mahmoud, I., Naoui, K., & Jemmali, H. (2013) · International Journal of Economics and Financial Issues · Cited by 7
Algorithmic Complexity Theory Detects Decreases in the Relative Efficiency of Stock Markets in the Aftermath of the 2008 Financial Crisis
Taufemback, C. G., Giglio, R., & Silva, S. D. (2011) · RePEc: Research Papers in Economics · Cited by 7
Application of permutation entropy method in the analysis of chaotic, noisy, and chaotic noisy series
Makarkin, S. A., Starodubov, A. V., & Kalinin, Y. A. (2017) · Technical Physics · doi:10.1134/s1063784217110202 · Cited by 7
Revisiting the decay of missing ordinal patterns in long-term correlated time series
Olivares, F., Zunino, L., & Pérez, D. G. (2019) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2019.122100 · Cited by 7
Information measures through velocity time series in a seepage affected alluvial sinuous channel
Taye, J., Lade, A. D., Mihailović, A., Mihailović, D. T., & Kumar, B. (2020) · Stochastic Environmental Research and Risk Assessment · doi:10.1007/s00477-020-01849-2 · Cited by 7
Crude oil market efficiency: An empirical investigation via the Shannon entropy
Mensi, W., Aloui, C., Hamdi, M., & Nguyen, D. K. (2012) · Économie internationale · doi:10.3917/ecoi.129.0119 · Cited by 7
Evaluating Methods for Detrending Time Series Using Ordinal Patterns, with an Application to Air Transport Delays
Olivares, F., Marín-Rodríguez, F. J., Acharya, K., & Zanin, M. (2025) · Entropy · doi:10.3390/e27030230 · Cited by 6
“The existence of forbidden [28,34] and missing ordinal patterns [35,36] have been applied to different areas from finance [36,37], semiconductor lasers [38], atmospheric turbulence [36], chaotic optoelectronic systems [39], and hydrology [36].”
Pipelined reconfigurable accelerator for ordinal pattern encoding
Guo, C., Luk, W. W., & Weston, S. (2014) · IEEE International Conference on Application-specific Systems, Architectures and Processors (ASAP) · doi:10.1109/asap.2014.6868662 · Cited by 6
“[23] propose to use this measure to describe the degree of stock market inefficiency.”
Multiscale weighted Rényi entropy causality plane for financial time series
Gao, J. W., & Shang, P. (2019) · International Journal of Modern Physics C · doi:10.1142/s0129183119500372 · Cited by 6
Multiscale adaptive multifractal analysis and its applications
Han, G., Zhou, F.-X., & Jiang, H. (2021) · Chaos An Interdisciplinary Journal of Nonlinear Science · doi:10.1063/5.0028215 · Cited by 6
A new comprehensive automatic fault detection method for rotating machinery using HmvAAPE and VNWOA-KELM
Gong, J., Yang, X., Han, J., Shen, J., Zhou, F., & Liu, W. (2022) · Applied Intelligence · doi:10.1007/s10489-022-03505-4 · Cited by 6
Statistical properties of one-dimensional binary sequences with power-law power spectrum
Gong, L., Zhou, Z., Tong, P., & Zhao, S. (2011) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2011.04.010 · Cited by 6
Chaotic edge density fluctuations in the Alcator C-Mod tokamakPreprint
Zhu, Z., Carter, T. A., White, A. E., Baek, S. G., & Terry, J. L. (2016) · DSpace@MIT (Massachusetts Institute of Technology) · Cited by 6
Predictability of monthly streamflow by considering complexity measures
Mihailović, D. T., Milićević, S. M., Frau, F. J., Singh, V. P., & Han, J. (2024) · Journal of Hydrology · doi:10.1016/j.jhydrol.2024.131103 · Cited by 5
Efficient Bayesian estimation of permutation entropy with Dirichlet priors
Little, D. J., Toomey, J. P., & Kane, D. M. (2021) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2021.106216 · Cited by 5
The frequency of pattern occurrence in random walks
Elizalde, S., & Martinez, M. A. (2015) · Discrete Mathematics & Theoretical Computer Science · doi:10.46298/dmtcs.2476 · Cited by 5
Complex and composite entropy fluctuation behaviors of statistical physics interacting financial model
Wang, G., Zheng, S., & Wang, J. (2018) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2018.11.014 · Cited by 5
Slope Entropy Normalisation by Means of Analytical and Heuristic Reference Values
Cuesta–Frau, D., Kouka, M., Silvestre-Blanes, J., & Sempere-Payá, V. (2022) · Entropy · doi:10.3390/e25010066 · Cited by 5
“This can be due to two reasons: the length of the time series is not enough to allocate all the pattern variations, especially when some of them have a low appearance probability, or due to the concept of forbidden patterns [25,31,42].”
Background
Characterization of autoregressive processes using entropic quantifiers
Traversaro, F., & Redelico, F. O. (2017) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2017.07.025 · Cited by 5
Share Price Reaction to Earnings Announcement on the JSE-ALTXPreprint
Kruger, J. W., Mlonzi, V., & Nthoesane, M. G. (2011) · SSRN Electronic Journal · doi:10.2139/ssrn.1935198 · Cited by 5
Terrestrial and Martian space weather: A complex systems approach
Chian, A. C. ‐., Miranda, R. A., Bertucci, C. L., Blanco-Cano, X., Borovsky, J., Dasso, S., ECHER, E., Franco, A. M. D. S., Girgis, K. M., González-Esparza, A., et al. (2024) · Journal of Atmospheric and Solar-Terrestrial Physics · doi:10.1016/j.jastp.2024.106253 · Cited by 4
Novel Order Patterns Recurrence Plot-Based Quantification Measures to Unveil Deterministic Dynamics from Stochastic Processes
Lu, S., Oberst, S. M., Zhang, G., & Luo, Z. (2019) · Contributions to statistics · doi:10.1007/978-3-030-26036-1_5 · Cited by 4
Una medida de eficiencia de mercado. Un enfoque de teoría de la información
Ruíz, R. S. G., Aké, S. C., & Venegas-Martı́nez, F. (2014) · Contaduría y Administración · doi:10.1016/s0186-1042(14)70158-5 · Cited by 4
Using Rényi parameter to improve the predictive power of singular value decomposition entropy on stock market
Jiang, J., & Gu, R. (2016) · Physica A Statistical Mechanics and its Applications · doi:10.1016/j.physa.2015.12.070 · Cited by 4
Nonlinear Dynamic Characteristics of Oil-in-Water Emulsions
Yin, Z., Han, Y., Ren, Y., Yang, Q., & Jin, N. (2016) · Zeitschrift für Naturforschung A · doi:10.1515/zna-2016-0157 · Cited by 4
The application of multiscale joint permutation entropy on multichannel sleep electroencephalography
Yin, Y., Peng, C., Hou, F., Gao, H., Shang, P., Li, Q., & Ma, Y. (2019) · AIP Advances · doi:10.1063/1.5124366 · Cited by 4
A model sufficiency test using permutation entropy
Huang, X., Shang, H. L., & Pitt, D. (2022) · Journal of Forecasting · doi:10.1002/for.2849 · Cited by 4
“In empirical applications, PE has been shown to be a powerful tool in capturing the characteristics of complex systems, such as brain (electrical) activity (Costa et al., 2002), heart rate rhythms (Rubinov and Sporns, 2010) and ﬁnancial markets (Zunino et al., 2009).”
Method
The Measurement and Elimination of Mode Splitting: From the Perspective of the Partly Ensemble Empirical Mode Decomposition
Liu, B., Zheng, P. J., Dai, Q., & Zhou, Z. (2018) · Complexity · doi:10.1155/2018/4230649 · Cited by 4
“financial dynamics [17, 18], and breakage detection in mechanical systems [19].”
Background
Nonlinear Complexity and Chaotic Behaviors on Finite-Range Stochastic Epidemic Financial Dynamics
Wang, G., Zheng, S., & Wang, J. (2019) · International Journal of Bifurcation and Chaos · doi:10.1142/s0218127419500834 · Cited by 4
Modeling Expected Shortfall Using Tail Entropy
Pele, D. T., Lazar, E., & Mazurencu-Marinescu-Pele, M. (2019) · Entropy · doi:10.3390/e21121204 · Cited by 4
“[39], while Risso [40] links it to stock market crashes, arguing that the probability of having a crash increases as entropy decreases.”
On the permutation entropy Bayesian estimation
Pose, F., Bautista, L., Gianmuso, F., & Redelico, F. O. (2021) · Communications in Nonlinear Science and Numerical Simulation · doi:10.1016/j.cnsns.2021.105779 · Cited by 4
Multiscale permutation entropy analysis of oil-in-water type two-phase flow pattern
Du, M., Jin, N., Gao, Z., Zhu, L., & Zhen-Ya, W. (2012) · Acta Physica Sinica · doi:10.7498/aps.61.230507 · Cited by 4
Multivariate permutation entropy, a Cartesian graph product approach
Fabila-Carrasco, J. S., Tan, C., & Escudero, J. (2022) · European Signal Processing Conference (EUSIPCO) · doi:10.23919/eusipco55093.2022.9909930 · Cited by 4
Is Human Atrial Fibrillation Stochastic or Deterministic?Preprint
Aronis, K. N., Berger, R. D., Calkins, H. G., Chrispin, J., Marine, J. E., SPRAGG, D. D., Tao, S., Tandri, H. S., & Ashikaga, H. (2018) · arXiv (Cornell University) · doi:10.48550/arxiv.1802.01496 · Cited by 4
Experimental study of feedback-induced dynamics in semiconductor lasers : from symbolic analysis to subwavelength position sensing
Aguado, A. A. (2014) · doi:10.5821/dissertation-2117-95425 · Cited by 4
Rényi Entropy and Surrogate Data Analysis for Stock Markets
Sun, Y., Shang, P., He, J., & Xu, M. (2018) · Fluctuation and Noise Letters · doi:10.1142/s0219477518500359 · Cited by 3
KHẢO SÁT MỐI LIÊN HỆ GIỮA CÁC THỊ TRƯỜNG CHỨNG KHOÁN ĐÔNG NAM Á: TIẾP CẬN BẰNG KIỂM ĐỊNH NHÂN QUẢ GRANGER TÍNH HIỆU QUẢ THÔNG TIN GIỮA CÁC THỊ TRƯỜNG
Anh, T. T. T. (2020) · Dalat University Journal of Science · doi:10.37569/dalatuniversity.10.4.614(2020 · Cited by 3
Robust Detection of Epilepsy Using Weighted-Permutation Entropy: Methods and Analysis
Fadlallah, B. H., Fadlallah, A., Razafsha, M., Karnib, N., Wang, K., & Kobeissy, F. H. (2018) · Elsevier eBooks · doi:10.1016/b978-0-12-809556-0.00006-x · Cited by 3
Information-Theoretic ESG Index Direction Forecasting: A Complexity-Aware Framework
Öztürk, K. N., & Yiğit, Ö. E. (2025) · Entropy · doi:10.3390/e27111164 · Cited by 2
“Implemented in a rolling, time-respecting manner, these descriptors explicitly capture regime dependence and distributional shifts in the return-generating process, complementing conventional fundamental and technical indicators [21,22].”
Determining Stock Market Changes using Permutation Entropy
Obanya, P. O., Olivier, C. P., & Verster, T. (2025) · Computational Economics · doi:10.1007/s10614-025-11198-3 · Cited by 2
“These patterns are referred to as forbidden patterns (Zunino et al., 2009 ; Obanya et al., 2024 ).”
Entropy-Augmented Forecasting and Portfolio Construction at the Industry-Group Level: A Causal Machine-Learning Approach Using Gradient-Boosted Decision Trees
Cohen, G., Aiche, A., & Eichel, R. (2026) · Entropy · doi:10.3390/e28010108 · Cited by 2
“Zunino et al. [17] demonstrated that permutation entropy provides a robust framework for identifying forbidden patterns and structural inefficiencies in asset returns, capturing dynamical properties that are not accessible through second-moment statistics or linear models.”
A chaotic signal detection method based on the component permutation of the incomplete two-dimensional phase-space
Sheng-Li, Z., & Gan, L. (2016) · Acta Physica Sinica · doi:10.7498/aps.65.070502 · Cited by 2
A robust complexity measure for noisy EEG time series under dynamic transitions during anesthesia
Joo, P., Kim, S., Noh, G., & Choi, B. (2021) · Journal of the Korean Physical Society · doi:10.1007/s40042-021-00332-7 · Cited by 2
Ranking Stock Markets Informational (In)Efficiency During the COVID-19 Pandemic
Olbryś, J., & Majewska, E. (2023) · Springer proceedings in business and economics · doi:10.1007/978-3-031-22749-3_29 · Cited by 2
Graph-Based Permutation Patterns for the Analysis of Task-Related FMRI Signals on DTI Networks in Mild Cognitive Impairment
Fabila-Carrasco, J. S., Campbell-Cousins, A., Parra, M. A., & Escudero, J. (2024) · IEEE International Conference on Acoustics Speech and Signal Processing · doi:10.1109/icassp48485.2024.10447332 · Cited by 2
“Advanced quantifiers further refine time series analysis, like ‘forbidden patterns’ in finance [32] and weighted differences of pattern likelihoods for univariate brain signals [3].”
Background

Authors: Zunino, L., Zanin, M., Tabak, B. M., Pérez, D. G., & Rosso, O. A.
Affiliations: Consejo Nacional de Investigaciones Científicas y Técnicas (AR)
Universidad Autónoma de Madrid (ES)
Central Bank of Brazil (BR)
Universidad de Buenos Aires (AR)
Corresponding author: Zunino, Luciano
ORCID: Zunino, Luciano
Zanin, Massimiliano
Tabak, Benjamin M.
Rosso, Osvaldo A.
Venue: Physica A Statistical Mechanics and its Applications
Publisher: Elsevier BV
Published: 2009-07
Volume: 388
Issue: 14
Pages: 2854-2864
ISSN: 0378-4371 (ISSN-L)
DOI: 10.1016/j.physa.2009.03.042
OpenAlex: W2018876117
Cited by: 249
References: 55
Citations by year: 2026: 6 · 2025: 10 · 2024: 11 · 2023: 5 · 2022: 19 · 2021: 17 · 2020: 21 · 2019: 36 · 2018: 18 · 2017: 18 · 2016: 17 · 2015: 10 · 2014: 10 · 2013: 11 · 2012: 16
FWCI: 26.94
Type: journal article
Language: en
Open access: green
www.rcaap.pt
Licence: other-oa
www.elsevier.com/tdm/userlicense/1.0/ (tdm)
Locations: Physica A Statistical Mechanics and its Applications (publishedVersion)
Portuguese National Funding Agency for Science, Research and Technology (RCAAP Project by FCT) (publishedVersion, other-oa)
LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) (submittedVersion, cc-by-nc-nd)
RePEc: Research Papers in Economics (submittedVersion)
Americanae (AECID Library) (submittedVersion, other-oa)
LA Referencia (Red Federada de Repositorios Institucionales de Publicaciones Científicas) (submittedVersion, other-oa)
Funders: Consejo Nacional de Investigaciones Científicas y Técnicas
Conselho Nacional de Desenvolvimento Científico e Tecnológico
Pontificia Universidad Católica de Valparaíso
Comisión Nacional de Investigación Científica y Tecnológica
Topics: Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Complex Systems and Time Series Analysis
Social Sciences › Economics, Econometrics and Finance › Finance › Financial Markets and Investment Strategies
Social Sciences › Decision Sciences › Management Science and Operations Research › Stock Market Forecasting Methods
Keywords: forbidden patterns, permutation entropy, time series complexity

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Forbidden patterns, permutation entropy and stock market inefficiency

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