Is Persistent Entropy Simply a Volatility Proxy? Evidence from the WIG20 Index
DOI:
https://doi.org/10.15611/eada.2026.1.01Keywords:
topological data analysis, persistence homology, WIG20, market indices, volatilityAbstract
Aim: This article investigates whether persistent homology and persistence entropy capture structural properties of financial time series beyond variance-based risk measures. Using data from the WIG20 index (2019–2024), the study examines whether topological descriptors reflect intrinsic geometric and temporal organization rather than merely volatility intensity.
Methodology: Logarithmic returns are embedded using sliding-window delay coordinates and analysed with Vietoris–Rips persistent homology. Betti numbers, persistence diagrams and rolling 𝐻₁ persistence entropy are computed. Relationships with classical risk diagnostics are evaluated using linear correlations, nonlinear dependence measures, regime comparisons and shuffle-based tests.
Results: Persistence entropy shows weak association with volatility and second-moment risk. Stronger relationships appear with higher-order distributional characteristics such as skewness and kurtosis. Volatility-based regimes do not significantly separate entropy, whereas a structural split around the 2022 geopolitical shock reveals a significant increase, indicating a shift in return geometry. Shuffle experiments confirm dependence on temporal ordering.
Implications: Persistence entropy captures structural and temporal organization of financial returns and may complement classical econometric risk measures.
Originality/value: The study shows that persistent homology reflects structural organization of return dynamics rather than acting as a volatility proxy.
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References
Aguilar, A. & Ensor, K. (2020). Topology Data Analysis Using Mean Persistence Landscapes in Financial Crashes. Journal of Mathematical Finance, 10(4), 648-678. https://doi.org/10.4236/jmf.2020.104038
Akingbade, S. W., Gidea, M., Manzi, M., & Nateghi, V. (2024). Why topological data analysis detects financial bubbles? Communications in Nonlinear Science and Numerical Simulation, 128, 107665. https://doi.org/10.1016/j.cnsns.2023.107665
Barański, K., Gutman, Y., & Śpiewak, A. (2024). Prediction of dynamical systems from time-delayed measurements with self-intersections. Journal de Mathématiques Pures et Appliquées, 186, 103-149. https://doi.org/10.1016/j.matpur.2024.04.001
Bois, A. (2024). Topological data analysis for time series (Doctoral dissertation, Université Paris-Saclay). HAL Thèses. https://theses.hal.science/tel-05005755
Carlsson, G. (2009). Topology and data. Bulletin of the American Mathematical Society 46(2), 255–308. http://dx.doi.org/10.1090/S0273-0979-09-01249-X
Chaudhari, S., & Singh, S. K. (2024). A Review on Topological Data Analysis in Time Series. In Mahajan, V., Chowdhury, A., Singh, S.N., Shahidehpour, M. (eds) Emerging Technologies in Electrical Engineering for Reliable Green Intelligence. ICSTACE 2023. Lecture Notes in Electrical Engineering, vol 1117. Springer, Singapore. https://doi.org/10.1007/978-981-99-9235-5_36
Chazal, F., de Silva, V., Glisse, M., & Oudot, S. (2016). The Structure and Stability of Persistence Modules. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-42545-0
de Jesus, L. C., Fernández-Navarro, F., & Carbonero-Ruz, M. (2025). Enhancing financial time series forecasting through topological data analysis. Neural Computing & Applications, 37, 6527-6545. https://doi.org/10.1007/s00521-024-10787-x
El-Yaagoubi, A. B., Chung, M. K., & Ombao, H. (2023). Topological Data Analysis for Multivariate Time Series Data. Entropy, 25(11), 1509. https://doi.org/10.3390/e25111509
El-Yaagoubi, A., Freyermuth, J.-M., & Ombao, H. (2025). A Robust Topological Framework for Detecting Regime Changes in Multi-Trial Experiments With Application to Predictive Maintenance. Journal of Time Series Analysis, Early View, 1–19. https://doi.org/10.1111/jtsa.70032.
Emrani, S., Gentimis, T., & Krim, H. (2014). Persistent-homology of delay embeddings and its application to wheeze detection. IEEE Signal Processing Letters, 21(4), 459-463. https://doi.org/10.1109/LSP.2014.2305700
Ghrist, R. (2008). Barcodes: The persistent topology of data. Bulletin of the American Mathematical Society, 45(1), 61–75. http://dx.doi.org/10.1090/S0273-0979-07-01191-3
Gidea, M. (2017). Topological data analysis of critical transitions in financial networks. In E. Shmueli, B. Barzel, & R. Puzis (Eds.) 3rd International Winter School and Conference on Network Science (NetSci-X 2017). Springer Proceedings in Complexity. Springer. https://doi.org/10.1007/978-3-319-55471-6_5
Gidea, M., Goldsmith, D., Katz, Y., Roldan, P., & Shmalo, Y. (2020). Topological recognition of critical transitions in time series of cryptocurrencies. Physica A: Statistical Mechanics and its Applications 548, 123843. https://doi.org/10.1016/j.physa.2019.123843
Gidea, M., & Katz, Y. (2018). Topological data analysis of financial time series: Landscapes of crashes. Physica A: Statistical Mechanics and its Applications 491, 820–834. https://doi.org/10.1016/j.physa.2017.09.028
Girish, N., & Shahabudheen, A. (2024). Analysing Indian Stock Market Crashes Through a Topological Lens. In B. B. Mallik, K. Deyasi, S. Das, S. Ghosh, & S. Jana(Eds.) Proceedings of the Fifth International Conference on Emerging Trends in Mathematical Sciences & Computing (IEMSC-24). Information Systems Engineering and Management, vol 10. Springer. https://doi.org/10.1007/978-3-031-71125-1_33
Hedges, L. V. (2024). Interpretation of the Standardized Mean Difference Effect Size When Distributions Are Not Normal or Homoscedastic. Educational and Psychological Measurement, 85(2), 245-257. https://doi.org/10.1177/00131644241278928
Ichinomiya, T. (2023). Time series analysis using persistent homology of distance matrix. Nonlinear Theory and Its Applications, IEICE 14(2), 79–91. https://doi.org/10.1587/nolta.14.79
Islambekov, U., Pathirana, H., Khormali, O., Akçora, C., & Smirnova, E. (2024). A topological approach for capturing high-order interactions in graph data with applications to anomaly detection in time-varying cryptocurrency transaction graphs. Foundations of Data Science 6(4), 492–513. https://doi.org/10.3934/fods.2024024
Koltai, P., & Kunde, P. (2024). A Koopman–Takens theorem: Linear least squares prediction of nonlinear time series. Communications in Mathematical Physics, 405(5), 120. https://doi.org/10.1007/s00220-024-05004-8
Leaverton, L. A. (2020). Analysis of Financial Time Series using TDA: Theoretical and Empirical Results [Thesis, Universitat de Barcelona]. https://diposit.ub.edu/items/5f76bb5f-e836-4120-95cc-ae96201e3d3d
Li, Y., Li, X., Zhang, Y., & Cao, Y. (2023). A topological based feature extraction method for the stock market. Data Science in Finance and Economics 3(3), 208–229. https://doi.org/10.3934/DSFE.2023013
Manziy, O., Senyk, A., Sheremeta, D., & Senyk, Y. (2024). Information aspects of the manifestation of irrational influence of financial bubbles on financial markets. Galician Economic Journal, 91, 20-28. http://dx.doi.org/10.33108/galicianvisnyk_tntu2024.06.020
Noakes, L. (1991). The Takens Embedding Theorem. International Journal of Bifurcation and Chaos, 1(4), 867-872. https://doi.org/10.1142/S0218127491000634
Perea, J.A., Harer, J. (2015) Sliding Windows and Persistence: An Application of Topological Methods to Signal Analysis. Foundations of Computational Mathematics, 15, 799-838. https://doi.org/10.1007/s10208-014-9206-z
Rai, A., Nath Sharma, B., Rabindrajit Luwang, S., Nurujjaman, M., & Majhi, S. (2024). Identifying extreme events in the stock market: A topological data analysis. Chaos: An Interdisciplinary Journal of Nonlinear Science 34(10), 103106. https://doi.org/10.1063/5.0220424
Ravishanker, N., & Chen, R. (2021). An introduction to persistent homology for time series. WIREs Computational Statistics. 13, e1548. https://doi.org/10.1002/wics.1548
Robinson, J. C. (2010). The Takens Time-Delay Embedding Theorem.Dimensions, Embeddings, and Attractors. Cambridge University Press. https://doi.org/10.1017/CBO9780511933912
Rudkin, S., Rudkin, W., & Dłotko, P. (2023). On the topology of cryptocurrency markets. International Review of Financial Analysis, 89, 102759. https://doi.org/10.1016/j.irfa.2023.102759
Seversky, L. M., Davis, C., & Berger, M. (2016). On time-series topological data analysis: New data and opportunities. Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR) Workshops, 59–67. https://doi.org/10.1109/CVPRW.2016.131
Song, S., & Li, H. (2025). Can topological transitions in cryptocurrency systems serve as early warning signals for extreme fluctuations in traditional markets? Physica A: Statistical Mechanics and its Applications, 657, 130194. https://doi.org/10.1016/j.physa.2024.130194
Sugathadasa, H., Erfani, S.M., Leckie, C. (2025). S-CPD: Topological Smoothing-Based Change Point Detection. In: Wu, X., et al. Data Science: Foundations and Applications. PAKDD 2025. Lecture Notes in Computer Science(), vol 15875. Springer, Singapore. https://doi.org/10.1007/978-981-96-8295-9_10
Takens, F. (1981). Detecting strange attractors in turbulence. In D. Rand, & L. S. Young (Eds.), Dynamical Systems and Turbulence, Warwick 1980. Lecture Notes in Mathematics, Vol. 898, pp. 366-381. Springer. https://doi.org/10.1007/BFb0091924
Tauzin, G., Lupo, U., Tunstall, L., Pasrez, J. B., Caorsi, M., Medina-Mardones, A. M., Dassatti, A., & Hess, K. (2021). giotto-tda: A topological data analysis toolkit for machine learning and data exploration. Journal of Machine Learning Research, 22(39), 1-6. http://jmlr.org/papers/v22/20-325.html
West, R. M. (2021). Best practice in statistics: Use the Welch t-test when testing the difference between two groups. Annals of Clinical Biochemistry: International Journal of Laboratory Medicine, 58(4), 267-269. https://doi.org/10.1177/0004563221992088
Yao, J., Li, J., Wu, J., Yang, M., & Wang, X. (2025). Change Point Detection in Financial Market Using Topological Data Analysis. Systems, 13(10), 875. https://doi.org/10.3390/systems13100875
Yen, P. T.-W., & Cheong, S. A. (2021). Using topological data analysis (TDA) and persistent homology to analyze the stock markets in Singapore and Taiwan. Frontiers in Physics, 9. https://doi.org/10.3389/fphy.2021.572216
Zabaleta-Ortega, A., Masoller, C., & Guzmán-Vargas, L. (2023). Topological data analysis of the synchronization of a network of Rössler chaotic electronic oscillators. Chaos: An Interdisciplinary Journal of Nonlinear Science 3(11), 113110. https://doi.org/10.1063/5.0167523
Zheng, J., Feng, Z., & Ekstrom, A. D. (2024). Towards analysis of multivariate time series using topological data analysis. Mathematics 12(11). https://doi.org/10.3390/math12111727
Zheng, X., Mak, S., Xie, L., & Xie, Y. (2023). PERCEPT: A New Online Change-Point Detection Method using Topological Data Analysis. Technometrics, 65(2), 162–178. https://doi.org/10.1080/00401706.2022.2124312
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Copyright (c) 2026 Stanisław Halkiewicz
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Accepted 2026-03-02
Published 2026-04-01
