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Neuro-Fuzzy Modeling Techniques in Economics

Neuro-Fuzzy Modeling Techniques in Economics

ISSN 2415-3516
Белінський Андрій  
Andrii Bielinskyi  
Соловйов Володимир Миколайович  
Vladimir Soloviev  
Семеріков Сергій Олексійович  
Serhiy Semerikov  
Соловйова Вікторія  
Viktoria Solovieva  

Identifying stock market crashes by fuzzy measures of complexity

DOI:

10.33111/nfmte.2021.003

Анотація:
Abstract: This study, for the first time, presents the possibility of using fuzzy set theory in combination with information theory and recurrent analysis to construct indicators (indicators-precursors) of crisis phenomena in complex nonlinear systems. In our study, we analyze the 4 most important crisis periods in the history of the stock market – 1929, 1987, 2008 and the COVID-19 pandemic in 2020. In particular, using the sliding window procedure, we analyze how the complexity of the studied crashes changes over time, and how it depends on events such as the global stock market crises. For comparative analysis, we take classical Shannon entropy, approximation and permutation entropy, recurrent diagrams, and their fuzzy alternatives. Each of the fuzzy modifications uses three membership functions: exponential, sigmoidal, and simple linear functions. Empirical results demonstrate the fact that the fuzzification of classical entropy and recurrence approaches opens up prospects for constructing effective and reliable indicators-precursors of critical events in the studied complex systems
Ключові слова:
Key words: crash, critical event, stock market, entropy, recurrence plot, fuzzy set theory, indicator-precursor of crisis phenomena, fuzzy measure of complexity
УДК:
UDC:

JEL: C22 C58 G01 G17

To cite paper
In APA style
Bielinskyi, A., Soloviev, V., Semerikov, S., & Solovieva, V. (2021). Identifying stock market crashes by fuzzy measures of complexity. Neuro-Fuzzy Modeling Techniques in Economics, 10, 3-45. http://doi.org/10.33111/nfmte.2021.003
In MON style
Белінський А., Соловйов В.М., Семеріков С.О., Соловйова В. Identifying stock market crashes by fuzzy measures of complexity. Нейро-нечіткі технології моделювання в економіці. 2021. № 10. С. 3-45. http://doi.org/10.33111/nfmte.2021.003 (дата звернення: 19.09.2025).
With transliteration
Bielinskyi, A., Soloviev, V., Semerikov, S., Solovieva, V. (2021) Identifying stock market crashes by fuzzy measures of complexity. Neuro-Fuzzy Modeling Techniques in Economics, no. 10. pp. 3-45. http://doi.org/10.33111/nfmte.2021.003 (accessed 19 Sep 2025).
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  1. Acharya, U.R., Sree, S.V., Chattopadhyay, S., Yu, W., & Ang, P.C.A. (2011). Application of recurrence quantification analysis for the automated identification of epileptic EEG signals. International journal of neural systems, 21(03), 199-211. https://doi.org/10.1142/s0129065711002808
  2. Aldalou, E., & Perçin, S. (2020). Application of integrated fuzzy MCDM approach for financial performance evaluation of Turkish technology sector. International Journal of Procurement Management, 13(1), 1-23. https://doi.org/10.1504/ijpm.2020.105198
  3. Al-Sharhan, S., Karray, F., Gueaieb, W., & Basir, O. (2001). Fuzzy entropy: a brief survey. In Proceedings of the 10th IEEE international conference on fuzzy systems (Cat. No. 01CH37297): Vol. 3 (pp. 1135-1139). IEEE. https://doi.org/10.1109/FUZZ.2001.1008855
  4. Alves, P. (2019). Chaos in historical prices and volatilities with five-dimensional Euclidean spaces. Chaos, Solitons & Fractals: X, 1, Article 100002. https://doi.org/10.1016/j.csfx.2019.100002
  5. Argyroudis, G. S., & Siokis, F. M. (2019). Spillover effects of Great Recession on Hong-Kong’s Real Estate Market: An analysis based on Causality Plane and Tsallis Curves of Complexity–Entropy. Physica A: Statistical Mechanics and its Applications, 524, 576–586. https://doi.org/10.1016/j.physa.2019.04.052
  6. Azami, H., Fernández, A., & Escudero, J. (2017). Refined multiscale fuzzy entropy based on standard deviation for biomedical signal analysis. Medical & Biological Engineering & Computing, 55(11), 2037–2052. https://doi.org/10.1007/s11517-017-1647-5
  7. Bandt, C., & Pompe, B. (2002). Permutation Entropy: A Natural Complexity Measure for Time Series. Physical Review Letters, 88(17), Article 174102. https://doi.org/10.1103/physrevlett.88.174102
  8. Bastos, J. A., & Caiado, J. (2011). Recurrence quantification analysis of global stock markets. Physica A: Statistical Mechanics and its Appli-cations, 390(7), 1315–1325. https://doi.org/10.1016/j.physa.2010.12.008
  9. Bielinskyi, A., Semerikov, S., Serdyuk, O., Solovieva, V., Soloviev, V., & Pichl, L. (2020). Econophysics of sustainability indices. CEUR Workshop Proceedings, 2713, 372-392. http://ceur-ws.org/Vol-2713/paper41.pdf
  10. Bielinskyi, A., & Soloviev, V. (2018). Complex network precursors of crashes and critical events in the cryptocurrency market. CEUR Workshop Proceedings, 2292, 37-45. http://ceur-ws.org/Vol-2292/paper02.pdf
  11. Bielinskyi, A., Soloviev, V., Semerikov, S., & Solovieva, V. (2019). Detecting stock crashes using Levy distribution. CEUR Workshop Proceedings, 2422, 420-433. http://ceur-ws.org/Vol-2422/paper34.pdf
  12. Boccara, N. (2010). Modeling complex systems. Springer Science & Business Media. https://doi.org/10.1007/978-1-4419-6562-2
  13. Castiglioni, P., & Di Rienzo, M. (2008). How the threshold “r” influences approximate entropy analysis of heart-rate variability. In Proceedings of the 2008 Computers in Cardiology (pp. 561-564). IEEE. https://doi.org/10.1109/CIC.2008.4749103
  14. Chen, W., Wang, Z., Xie, H., & Yu, W. (2007). Characterization of Surface EMG Signal Based on Fuzzy Entropy. IEEE Transactions on Neural Systems and Rehabilitation Engineering, 15(2), 266–272. https://doi.org/10.1109/tnsre.2007.897025
  15. Chen, W., Zhuang, J., Yu, W., & Wang, Z. (2009). Measuring complexity using FuzzyEn, ApEn, and SampEn. Medical Engineering & Physics, 31(1), 61–68. https://doi.org/10.1016/j.medengphy.2008.04.005
  16. Collet, P., Eckmann, J. P., & Koch, H. (1981). Period doubling bifurcations for families of maps on . Journal of Statistical Physics, 25(1), 1–14. https://doi.org/10.1007/bf01008475
  17. De Luca, A., & Termini, S. (1972). A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory. Information and Control, 20(4), 301–312. https://doi.org/10.1016/s0019-9958(72)90199-4
  18. Duran, N.D., Dale, R., Kello, C.T., Street, C.N.H., & Richardson, D.C. (2013). Exploring the movement dynamics of deception. Frontiers in Psychology, 4, Article 140. https://doi.org/10.3389/fpsyg.2013.00140
  19. Eckmann, J. P., & Ruelle, D. (1985). Ergodic theory of chaos and strange attractors. Reviews of Modern Physics, 57(3), 617–656. https://doi.org/10.1103/revmodphys.57.617
  20. Eckmann, J. P., Kamphorst, S. O., & Ruelle, D. (1987). Recurrence Plots of Dynamical Systems. Europhysics Letters, 4(9), 973–977. https://doi.org/10.1209/0295-5075/4/9/004
  21. Elias, J., & Narayanan Namboothiri, V. N. (2013). Cross-recurrence plot quantification analysis of input and output signals for the detection of chatter in turning. Nonlinear Dynamics, 76(1), 255–261. https://doi.org/10.1007/s11071-013-1124-0
  22. Eroglu, D., McRobie, F. H., Ozken, I., Stemler, T., Wyrwoll, K. H., Breitenbach, S. F. M., Marwan, N., & Kurths, J. (2016). See–saw rela-tionship of the Holocene East Asian–Australian summer monsoon. Nature Communications, 7(1), Article 12929. https://doi.org/10.1038/ncomms12929
  23. Farmer, J. D. (1982). Information Dimension and the Probabilistic Structure of Chaos. Zeitschrift Für Naturforschung A, 37(11), 1304–1326. https://doi.org/10.1515/zna-1982-1117
  24. García-Ochoa, E., González-Sánchez, J., Acuña, N., & Euan, J. (2008). Analysis of the dynamics of Intergranular corrosion process of sensitised 304 stainless steel using recurrence plots. Journal of Applied Electrochemistry, 39(5), 637–645. https://doi.org/10.1007/s10800-008-9702-4
  25. Gardini, L., Lupini, R., & Messia, M. G. (1989). Hopf bifurcation and transition to chaos in Lotka-Volterra equation. Journal of Mathematical Biology, 27(3), 259–272. https://doi.org/10.1007/bf00275811
  26. GitHub. (2021). Complex systems measures. https://github.com/Butman2099/Complex-systems-measures
  27. GitHub. (2021). EntropyHub: An open-source toolkit for entropic time series analysis. https://github.com/MattWillFlood/EntropyHub
  28. Graf, S. (1987). Statistically self-similar fractals. Probability Theory and Related Fields, 74(3), 357–392. https://doi.org/10.1007/bf00699096
  29. Grassberger, P., & Procaccia, I. (1983). Characterization of Strange Attractors. Physical Review Letters, 50(5), 346–349. https://doi.org/10.1103/physrevlett.50.346
  30. Grebogi, C., Ott, E., Pelikan, S., & Yorke, J. A. (1984). Strange attractors that are not chaotic. Physica D: Nonlinear Phenomena, 13(1-2), 261-268. https://doi.org/10.1016/0167-2789(84)90282-3
  31. Harris, P., Litak, G., Iwaniec, J., & Bowen, C. R. (2016). Recurrence Plot and Recurrence Quantification of the Dynamic Properties of Cross-Shaped Laminated Energy Harvester. Applied Mechanics and Materials, 849, 95–105. https://doi.org/10.4028/www.scientific.net/amm.849.95
  32. He, S., Sun, K., & Wang, R. (2018). Fractional fuzzy entropy algorithm and the complexity analysis for nonlinear time series. The European Physical Journal Special Topics, 227(7), 943-957. https://doi.org/10.1140/epjst/e2018-700098-x
  33. Hou, Y., Aldrich, C., Lepkova, K., Machuca, L., & Kinsella, B. (2016). Monitoring of carbon steel corrosion by use of electrochemical noise and recurrence quantification analysis. Corrosion Science, 112, 63–72. https://doi.org/10.1016/j.corsci.2016.07.009
  34. Humeau-Heurtier, A. (2015). The Multiscale Entropy Algorithm and Its Variants: A Review. Entropy, 17(5), 3110–3123. https://doi.org/10.3390/e17053110
  35. Hutchinson, J. E. (1981). Fractals and Self Similarity. Indiana University Mathematics Journal, 30(5), 713-747. https://www.jstor.org/stable/24893080
  36. Ishizaki, R., & Inoue, M. (2020). Analysis of local and global instability in foreign exchange rates using short-term information entropy. Physica A: Statistical Mechanics and its Applications, 555, Article 124595. https://doi.org/10.1016/j.physa.2020.124595
  37. Iwaniec, J., Uhl, T., Staszewski, W. J., & Klepka, A. (2012). Detection of changes in cracked aluminium plate determinism by recurrence analysis. Nonlinear Dynamics, 70(1), 125–140. https://doi.org/10.1007/s11071-012-0436-9
  38. Jahanshahi, H., Yousefpour, A., Wei, Z., Alcaraz, R., & Bekiros, S. (2019). A financial hyperchaotic system with coexisting attractors: Dynamic investigation, entropy analysis, control and synchronization. Chaos, Solitons & Fractals, 126, 66–77. https://doi.org/10.1016/j.chaos.2019.05.023
  39. Kantz, H. (1994). A robust method to estimate the maximal Lyapunov exponent of a time series. Physics Letters A, 185(1), 77–87. https://doi.org/10.1016/0375-9601(94)90991-1
  40. Konvalinka, I., Xygalatas, D., Bulbulia, J., Schjødt, U., Jegindø, E. M., Wallot, S., van Orden, G., & Roepstorff, A. (2011). Synchronized arousal between performers and related spectators in a fire-walking ritual. Proceedings of the National Academy of Sciences of the United States of America, 108(20), 8514–8519. https://doi.org/10.1073/pnas.1016955108
  41. Lahmiri, S., & Bekiros, S. (2017). Disturbances and complexity in volatility time series. Chaos, Solitons & Fractals, 105, 38–42. https://doi.org/10.1016/j.chaos.2017.10.006
  42. Lahmiri, S., & Bekiros, S. (2019). Nonlinear analysis of Casablanca Stock Exchange, Dow Jones and S&P500 industrial sectors with a comparison. Physica A: Statistical Mechanics and its Applications, 539, Article 122923. https://doi.org/10.1016/j.physa.2019.122923
  43. Lahmiri, S., & Bekiros, S. (2020). Randomness, Informational Entropy, and Volatility Interdependencies among the Major World Markets: The Role of the COVID-19 Pandemic. Entropy, 22(8), Article 833. https://doi.org/10.3390/e22080833
  44. Lahmiri, S., Bekiros, S., & Avdoulas, C. (2018). Time-dependent complexity measurement of causality in international equity markets: A spatial approach. Chaos, Solitons & Fractals, 116, 215–219. https://doi.org/10.1016/j.chaos.2018.09.030
  45. Lahmiri, S., Uddin, G. S., & Bekiros, S. (2017). Nonlinear dynamics of equity, currency and commodity markets in the aftermath of the global financial crisis. Chaos, Solitons & Fractals, 103, 342–346. https://doi.org/10.1016/j.chaos.2017.06.019
  46. Lam, W. S., Lam, W. H., Jaaman, S. H., & Liew, K. F. (2021). Performance Evaluation of Construction Companies Using Integrated Entropy–Fuzzy VIKOR Model. Entropy, 23(3), Article 320. https://doi.org/10.3390/e23030320
  47. Li, P., Liu, C., Li, K., Zheng, D., Liu, C., & Hou, Y. (2014). Assessing the complexity of short-term heartbeat interval series by distribution entropy. Medical & Biological Engineering & Computing, 53(1), 77–87. https://doi.org/10.1007/s11517-014-1216-0
  48. Li, S., Zhao, Z., Wang, Y., & Wang, Y. (2011). Identifying spatial patterns of synchronization between NDVI and climatic determinants using joint recurrence plots. Environmental Earth Sciences, 64(3), 851–859. https://doi.org/10.1007/s12665-011-0909-z
  49. List of stock market crashes and bear markets. (2021, August 1). In Wikipedia. https://en.wikipedia.org/w/index.php?title=List_of_stock_market_crashes_and_bear_markets
  50. Longwic, R., Litak, G., & Sen, A. K. (2009). Recurrence Plots for Diesel Engine Variability Tests. Zeitschrift Für Naturforschung A, 64(1–2), 96–102. https://doi.org/10.1515/zna-2009-1-214
  51. Mandelbrot, B. B. (1985). Self-Affine Fractals and Fractal Dimension. Physica Scripta, 32(4), 257–260. https://doi.org/10.1088/0031-8949/32/4/001
  52. Marwan, N., Trauth, M. H., Vuille, M., & Kurths, J. (2003). Comparing modern and Pleistocene ENSO-like influences in NW Argentina using nonlinear time series analysis methods. Climate Dynamics, 21(3–4), 317–326. https://doi.org/10.1007/s00382-003-0335-3
  53. Marwan, N., Wessel, N., Meyerfeldt, U., Schirdewan, A., & Kurths, J. (2002). Recurrence-plot-based measures of complexity and their application to heart-rate-variability data. Physical Review E, 66(2), Article 026702. https://doi.org/10.1103/physreve.66.026702
  54. Moore, J. M., Corrêa, D. C., & Small, M. (2018). Is Bach’s brain a Markov chain? Recurrence quantification to assess Markov order for short, symbolic, musical compositions. Chaos: An Interdisciplinary Journal of Nonlinear Science, 28(8), Article 085715. https://doi.org/10.1063/1.5024814
  55. Nair, V., & Sujith, R. I. (2013). Identifying homoclinic orbits in the dynamics of intermittent signals through recurrence quantification. Chaos: An Interdisciplinary Journal of Nonlinear Science, 23(3), Article 033136. https://doi.org/10.1063/1.4821475
  56. Nichols, J., Trickey, S., & Seaver, M. (2006). Damage detection using multivariate recurrence quantification analysis. Mechanical Systems and Signal Processing, 20(2), 421–437. https://doi.org/10.1016/j.ymssp.2004.08.007
  57. Palmieri, F., & Fiore, U. (2009). A nonlinear, recurrence-based approach to traffic classification. Computer Networks, 53(6), 761–773. https://doi.org/10.1016/j.comnet.2008.12.015
  58. Pham, T. D. (2016). Fuzzy recurrence plots. Europhysics Letters, 116(5), Article 50008. https://doi.org/10.1209/0295-5075/116/50008
  59. Pham, T. D. (2019). Fuzzy weighted recurrence networks of time series. Physica A: Statistical Mechanics and its Applications, 513, 409–417. https://doi.org/10.1016/j.physa.2018.09.035
  60. Pham, T. D. (2020). Fuzzy cross and fuzzy joint recurrence plots. Physica A: Statistical Mechanics and its Applications, 540, Article 123026. https://doi.org/10.1016/j.physa.2019.123026
  61. Pham, T. D. (2020). Fuzzy recurrence entropy. Europhysics Letters, 130(4), Article 40004. https://doi.org/10.1209/0295-5075/130/40004
  62. Pham, T. D. (2020). Fuzzy recurrence plots. In Fuzzy Recurrence Plots and Networks with Applications in Biomedicine (pp. 29-55). Springer. https://doi.org/10.1007/978-3-030-37530-0_4
  63. Pham, T. D., Wardell, K., Eklund, A., & Salerud, G. (2019). Classification of short time series in early Parkinsons disease with deep learning of fuzzy recurrence plots. IEEE/CAA Journal of Automatica Sinica, 6(6), 1306–1317. https://doi.org/10.1109/jas.2019.1911774
  64. Pincus, S. M. (1991). Approximate entropy as a measure of system complexity. Proceedings of the National Academy of Sciences of the United States of America, 88(6), 2297–2301. https://doi.org/10.1073/pnas.88.6.2297
  65. Pincus, S. M., & Goldberger, A. L. (1994). Physiological time-series analysis: what does regularity quantify? American Journal of Physiology-Heart and Circulatory Physiology, 266(4), H1643–H1656. https://doi.org/10.1152/ajpheart.1994.266.4.h1643
  66. Pincus, S. M., & Huang, W. M. (1992). Approximate entropy: Statistical properties and applications. Communications in Statistics – Theory and Methods, 21(11), 3061–3077. https://doi.org/10.1080/03610929208830963
  67. Qian, Y., Yan, R., & Hu, S. (2014). Bearing Degradation Evaluation Using Recurrence Quantification Analysis and Kalman Filter. IEEE Transactions on Instrumentation and Measurement, 63(11), 2599–2610. https://doi.org/10.1109/tim.2014.2313034
  68. Rand, R., & Holmes, P. (1980). Bifurcation of periodic motions in two weakly coupled van der Pol oscillators. International Journal of Non-Linear Mechanics, 15(4–5), 387–399. https://doi.org/10.1016/0020-7462(80)90024-4
  69. Reinertsen, E., Osipov, M., Liu, C., Kane, J. M., Petrides, G., & Clifford, G. D. (2017). Continuous assessment of schizophrenia using heart rate and accelerometer data. Physiological Measurement, 38(7), 1456–1471. https://doi.org/10.1088/1361-6579/aa724d
  70. Richardson, D. C., & Dale, R. (2005). Looking to Understand: The Coupling Between Speakers’ and Listeners’ Eye Movements and Its Relationship to Discourse Comprehension. Cognitive Science, 29(6), 1045–1060. https://doi.org/10.1207/s15516709cog0000_29
  71. Richman, J. S., & Moorman, J. R. (2000). Physiological time-series analysis using approximate entropy and sample entropy. American Journal of Physiology-Heart and Circulatory Physiology, 278(6), H2039–H2049. https://doi.org/10.1152/ajpheart.2000.278.6.h2039
  72. Roncagliolo Barrera, P., Rodríguez Gómez, F., & García Ochoa, E. (2019). Assessing of New Coatings for Iron Artifacts Conservation by Recurrence Plots Analysis. Coatings, 9(1), Article 12. https://doi.org/10.3390/coatings9010012
  73. Rostaghi, M., & Azami, H. (2016). Dispersion Entropy: A Measure for Time-Series Analysis. IEEE Signal Processing Letters, 23(5), 610–614. https://doi.org/10.1109/lsp.2016.2542881
  74. Ruelle, D. (1981). Small random perturbations of dynamical systems and the definition of attractors. Communications in Mathematical Physics, 82(1), 137–151. https://doi.org/10.1007/bf01206949
  75. Sanchez-Roger, M., Oliver-Alfonso, M. D., & Sanchís-Pedregosa, C. (2019). Fuzzy Logic and Its Uses in Finance: A Systematic Review Exploring Its Potential to Deal with Banking Crises. Mathematics, 7(11), Article 1091. https://doi.org/10.3390/math7111091
  76. Serrà, J., Serra, X., & Andrzejak, R. G. (2009). Cross recurrence quantification for cover song identification. New Journal of Physics, 11, Article 093017. https://doi.org/10.1088/1367-2630/11/9/093017
  77. Shannon, C. E. (1948). A Mathematical Theory of Communication. Bell System Technical Journal, 27(3), 379–423. https://doi.org/10.1002/j.1538-7305.1948.tb01338.x
  78. Shao, W., & Wang, J. (2020). Does the “ice-breaking” of South and North Korea affect the South Korean financial market? Chaos, Solitons & Fractals, 132, Article 109564. https://doi.org/10.1016/j.chaos.2019.109564
  79. Shaw, R. (1981). Strange Attractors, Chaotic Behavior, and Information Flow. Zeitschrift Für Naturforschung A, 36(1), 80–112. https://doi.org/10.1515/zna-1981-0115
  80. Shi, B., Wang, L., Yan, C., Chen, D., Liu, M., & Li, P. (2019). Nonlinear heart rate variability biomarkers for gastric cancer severity: A pilot study. Scientific Reports, 9(1), Article 13833. https://doi.org/10.1038/s41598-019-50358-y
  81. Silva, D. F., De Souza, V. M., & Batista, G. E. (2013). Time series classification using compression distance of recurrence plots. In Proceedings of 2013 IEEE 13th International Conference on Data Mining (pp. 687-696). IEEE. https://doi.org/10.1109/ICDM.2013.128
  82. Soloviev, V., & Belinskiy, A. (2018). Complex systems theory and crashes of cryptocurrency market. In V. Ermolayev, M. Suárez-Figueroa, V. Yakovyna, H. Mayr, M. Nikitchenko, & A. Spivakovsky (Eds.), Communications in Computer and Information Science: Vol. 1007. Information and Communication Technologies in Education, Research, and Industrial Applications (pp. 276-297). Springer. https://doi.org/10.1007/978-3-030-13929-2_14
  83. Soloviev, V., & Belinskij, A. (2018). Methods of nonlinear dynamics and the construction of cryptocurrency crisis phenomena precursors. CEUR Workshop Proceedings, 2104, 116-127. http://ceur-ws.org/Vol-2104/paper_175.pdf
  84. Soloviev, V., Bielinskyi, A., & Kharadzjan, N. (2020). Coverage of the coronavirus pandemic through entropy measures. CEUR Workshop Proceedings, 2832, 24-42. http://ceur-ws.org/Vol-2832/paper02.pdf
  85. Soloviev, V., Bielinskyi, A., Serdyuk, O., Solovieva, V., & Semerikov, S. (2020). Lyapunov exponents as indicators of the stock market crashes. CEUR Workshop Proceedings, 2732, 455-470. http://ceur-ws.org/Vol-2732/20200455.pdf
  86. Soloviev, V., Bielinskyi, A., & Solovieva, V. (2019). Entropy analysis of crisis phenomena for DJIA index. CEUR Workshop Proceedings, 2393, 434-449. http://ceur-ws.org/Vol-2393/paper_375.pdf
  87. Soloviev, V., Serdiuk, O., Semerikov, S., & Kohut-Ferens, O. (2019). Recurrence entropy and financial crashes. Advances in Economics, Business and Management Research, 99, 385-388. https://dx.doi.org/10.2991/mdsmes-19.2019.73
  88. Stangalini, M., Ermolli, I., Consolini, G., & Giorgi, F. (2017). Recurrence quantification analysis of two solar cycle indices. Journal of Space Weather and Space Climate, 7, Article A5. https://doi.org/10.1051/swsc/2017004
  89. Stender, M., Oberst, S., Tiedemann, M., & Hoffmann, N. (2019). Complex machine dynamics: systematic recurrence quantification analysis of disk brake vibration data. Nonlinear Dynamics, 97(4), 2483–2497. https://doi.org/10.1007/s11071-019-05143-x
  90. Strozzi, F., Zaldı́Var, J. M., & Zbilut, J. P. (2002). Application of nonlinear time series analysis techniques to high-frequency currency exchange data. Physica A: Statistical Mechanics and its Applications, 312(3–4), 520–538. https://doi.org/10.1016/s0378-4371(02)00846-4
  91. Takens, F. (1981). Detecting strange attractors in turbulence. In D. Rand, & L. Young (Eds.), Lecture Notes in Mathematics: Vol. 898. Dynamical systems and turbulence, Warwick 1980 (pp. 366-381). Springer-Verlag. https://doi.org/10.1007/BFb0091924
  92. Voss, A., Schroeder, R., Vallverdu, M., Cygankiewicz, I., Vazquez, R., de Luna, A. B., & Caminal, P. (2008). Linear and nonlinear heart rate variability risk stratification in heart failure patients. In Proceedings of the 2008 Computers in Cardiology (pp. 557-560). IEEE. https://doi.org/10.1109/CIC.2008.4749102
  93. Wang, G., & Wang, J. (2017). New approach of financial volatility duration dynamics by stochastic finite-range interacting voter system. Chaos: An Interdisciplinary Journal of Nonlinear Science, 27(1), Article 013117. https://doi.org/10.1063/1.4974216
  94. Wang, Y., Zheng, S., Zhang, W., Wang, G., & Wang, J. (2018). Fuzzy entropy complexity and multifractal behavior of statistical physics financial dynamics. Physica A: Statistical Mechanics and its Applications, 506, 486–498. https://doi.org/10.1016/j.physa.2018.04.086
  95. Webber, C. L., & Zbilut, J. P. (1994). Dynamical assessment of physiological systems and states using recurrence plot strategies. Journal of Applied Physiology, 76(2), 965–973. https://doi.org/10.1152/jappl.1994.76.2.965
  96. Xie, H. B., He, W. X., & Liu, H. (2008). Measuring time series regularity using nonlinear similarity-based sample entropy. Physics Letters A, 372(48), 7140–7146. https://doi.org/10.1016/j.physleta.2008.10.049
  97. Xie, H. B., Sivakumar, B., Boonstra, T. W., & Mengersen, K. (2018). Fuzzy Entropy and Its Application for Enhanced Subspace Filtering. IEEE Transactions on Fuzzy Systems, 26(4), 1970–1982. https://doi.org/10.1109/tfuzz.2017.2756829
  98. Yang, Y. G., Pan, Q. X., Sun, S. J., & Xu, P. (2015). Novel Image Encryption based on Quantum Walks. Scientific Reports, 5(1), Article 7784. https://doi.org/10.1038/srep07784
  99. Zadeh, L. A. (1965). Fuzzy sets. Information and Control, 8(3), 338–353. https://doi.org/10.1016/S0019-9958(65)90241-X
  100. Zaitouny, A., Walker, D. M., & Small, M. (2019). Quadrant scan for multi-scale transition detection. Chaos: An Interdisciplinary Journal of Nonlinear Science, 29(10), Article 103117. https://doi.org/10.1063/1.5109925
  101. Zbilut, J. P., & Marwan, N. (2008). The Wiener–Khinchin theorem and recurrence quantification. Physics Letters A, 372(44), 6622–6626. https://doi.org/10.1016/j.physleta.2008.09.027
  102. Zhang, Z., Xiang, Z., Chen, Y., & Xu, J. (2020). Fuzzy permutation entropy derived from a novel distance between segments of time series. AIMS Mathematics, 5(6), 6244–6260. https://doi.org/10.3934/math.2020402
  103. Zhao, H., Deng, W., Yao, R., Sun, M., Luo, Y., & Dong, C. (2017). Study on a novel fault diagnosis method based on integrating EMD, fuzzy entropy, improved PSO and SVM. Journal of Vibroengineering, 19(4), 2562–2577. https://doi.org/10.21595/jve.2017.18052
  104. Zhao, X., & Zhang, P. (2020). Multiscale horizontal visibility entropy: Measuring the temporal complexity of financial time series. Physica A: Statistical Mechanics and its Applications, 537, Article 122674. https://doi.org/10.1016/j.physa.2019.122674
  105. Zhao, Z. Q., Li, S. C., Gao, J. B., & Wang, Y. L. (2011). Identifying Spatial Patterns and Dynamics of Climate Change Using Recurrence Quantification Analysis: A Case Study of Qinghai–Tibet Plateau. International Journal of Bifurcation and Chaos, 21(04), 1127-1139. https://doi.org/10.1142/s0218127411028933
  106. Zhou, C., & Zhang, W. (2015). Recurrence Plot Based Damage Detection Method by Integrating Control Chart. Entropy, 17(5), 2624-2641. https://doi.org/10.3390/e17052624
  107. Zhou, Q., & Shang, P. (2020). Weighted multiscale cumulative residual Rényi permutation entropy of financial time series. Physica A: Statistical Mechanics and its Applications, 540, Article 123089. https://doi.org/10.1016/j.physa.2019.123089
  108. Zhou, R., Wang, X., Wan, J., & Xiong, N. (2021). EDM-Fuzzy: An Euclidean Distance Based Multiscale Fuzzy Entropy Technology for Diagnosing Faults of Industrial Systems. IEEE Transactions on Industrial Informatics, 17(6), 4046–4054. https://doi.org/10.1109/tii.2020.3009139
  109. Zolotova, N. V., & Ponyavin, D. I. (2006). Phase asynchrony of the north-south sunspot activity. Astronomy & Astrophysics, 449(1), L1–L4. https://doi.org/10.1051/0004-6361:200600013