Persistent homology of featured time series data and its applications

Eunwoo Heo; Jae-Hun Jung; Eunwoo Heo; Jae-Hun Jung

doi:10.3934/math.20241315

AIMS Mathematics

2024, Volume 9, Issue 10: 27028-27057. doi: 10.3934/math.20241315

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Research article

Persistent homology of featured time series data and its applications

Eunwoo Heo ,
Jae-Hun Jung ^,

Department of Mathematics, and Mathematical Institute for Data Science, Pohang University of Science and Technology, Pohang 37673, Korea

Received: 03 July 2024 Revised: 19 August 2024 Accepted: 09 September 2024 Published: 18 September 2024
MSC : 00A69, 37M10, 55N31, 91B84

Recent studies have actively employed persistent homology (PH), a topological data analysis technique, to analyze the topological information in time series data. Many successful studies have utilized graph representations of time series data for PH calculation. Given the diverse nature of time series data, it is crucial to have mechanisms that can adjust the PH calculations by incorporating domain-specific knowledge. In this context, we introduce a methodology that allows the adjustment of PH calculations by reflecting relevant domain knowledge in specific fields. We introduce the concept of featured time series, which is the pair of a time series augmented with specific features such as domain knowledge, and an influence vector that assigns a value to each feature to fine-tune the results of the PH. We then prove the stability theorem of the proposed method, which states that adjusting the influence vectors grants stability to the PH calculations. The proposed approach enables the tailored analysis of a time series based on the graph representation methodology, which makes it applicable to real-world domains. We consider two examples to verify the proposed method's advantages: anomaly detection of stock data and topological analysis of music data.
- topological data analysis,
- persistent homology,
- time series analysis,
- featured time series,
- graph representation,
- stability theorem
Citation: Eunwoo Heo, Jae-Hun Jung. Persistent homology of featured time series data and its applications[J]. AIMS Mathematics, 2024, 9(10): 27028-27057. doi: 10.3934/math.20241315

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Abstract

Recent studies have actively employed persistent homology (PH), a topological data analysis technique, to analyze the topological information in time series data. Many successful studies have utilized graph representations of time series data for PH calculation. Given the diverse nature of time series data, it is crucial to have mechanisms that can adjust the PH calculations by incorporating domain-specific knowledge. In this context, we introduce a methodology that allows the adjustment of PH calculations by reflecting relevant domain knowledge in specific fields. We introduce the concept of featured time series, which is the pair of a time series augmented with specific features such as domain knowledge, and an influence vector that assigns a value to each feature to fine-tune the results of the PH. We then prove the stability theorem of the proposed method, which states that adjusting the influence vectors grants stability to the PH calculations. The proposed approach enables the tailored analysis of a time series based on the graph representation methodology, which makes it applicable to real-world domains. We consider two examples to verify the proposed method's advantages: anomaly detection of stock data and topological analysis of music data.

References

[1]	M. Braei, S. Wagner, Anomaly detection in univariate time-series: A survey on the state-of-the-art, 2020. https://doi.org/10.48550/arXiv.2004.00433
[2]	A. Blázquez-García, A. Conde, U. Mori, J. A Lozano, A review on outlier/anomaly detection in time series data, ACM Comput. Surv., 54 (2021), 1–33. https://doi.org/10.1145/3444690 doi: 10.1145/3444690
[3]	B. Lim, S. Zohren, Time-series forecasting with deep learning: a survey, Philos. T. R. Soc. A, 379 (2021), 20200209. https://doi.org/10.1098/rsta.2020.0209 doi: 10.1098/rsta.2020.0209
[4]	O. B. Sezer, M. U. Gudelek, A. M. Ozbayoglu, Financial time series forecasting with deep learning: A systematic literature review: 2005–2019, Appl. soft comput., 90 (2020), 106181. https://doi.org/10.1016/j.asoc.2020.106181 doi: 10.1016/j.asoc.2020.106181
[5]	J. F. Torres, D. Hadjout, A. Sebaa, F. Martínez-Álvarez, A. Troncoso, Deep learning for time series forecasting: a survey, Big Data, 9 (2021), 3–21. https://doi.org/10.1089/big.2020.0159 doi: 10.1089/big.2020.0159
[6]	P. Lara-Benítez, M. Carranza-García, J. C. Riquelme, An experimental review on deep learning architectures for time series forecasting, Int. J. Neural Syst., 31 (2021), 2130001. https://doi.org/10.1142/S0129065721300011 doi: 10.1142/S0129065721300011
[7]	A. Bagnall, J. Lines, A. Bostrom, J. Large, E. Keogh, The great time series classification bake off: a review and experimental evaluation of recent algorithmic advances, Data Min. Knowl. Disc., 31 (2017), 606–660. https://doi.org/10.1007/s10618-016-0483-9 doi: 10.1007/s10618-016-0483-9
[8]	G. A. Susto, A. Cenedese, M. Terzi, Time-series classification methods: Review and applications to power systems data, Big Data Appl. Power Syst., 2018 (2018), 179–220. https://doi.org/10.1016/B978-0-12-811968-6.00009-7 doi: 10.1016/B978-0-12-811968-6.00009-7
[9]	H. Ismail Fawaz, G. Forestier, J. Weber, L. Idoumghar, P. A. Muller, Deep learning for time series classification: a review, Data Min. Knowl. Disc., 33 (2019), 917–963. https://doi.org/10.1007/s10618-019-00619-1 doi: 10.1007/s10618-019-00619-1
[10]	A. Abanda, U. Mori, J. A Lozano, A review on distance based time series classification, Data Min. Knowl. Dis., 33 (2019), 378–412. https://doi.org/10.1007/s10618-018-0596-4 doi: 10.1007/s10618-018-0596-4
[11]	S. Aghabozorgi, A. S. Shirkhorshidi, T. Y. Wah, Time-series clustering-a decade review, Inform. syst., 53 (2015), 16–38. https://doi.org/10.1016/j.is.2015.04.007 doi: 10.1016/j.is.2015.04.007
[12]	E. A. Maharaj, P. D'Urso, J. Caiado, Time series clustering and classification, Chapman and Hall/CRC, 2019. https://doi.org/10.1201/9780429058264
[13]	M. Ali, A. Alqahtani, M. W. Jones, X. Xie, Clustering and classification for time series data in visual analytics: A survey, IEEE Access, 7 (2019), 181314–181338. https://doi.org/10.1109/ACCESS.2019.2958551 doi: 10.1109/ACCESS.2019.2958551
[14]	N. H. Packard, J. P. Crutchfield, J. D. Farmer, R. S. Shaw, Geometry from a time series, Phys. Rev. Lett., 45 (1980), 712. https://doi.org/10.1103/PhysRevLett.45.712
[15]	F. Takens, Detecting strange attractors in turbulence, In: Dynamical Systems and Turbulence, Warwick 1980, Berlin: Springer, 1981. https://doi.org/10.1007/BFb0091924
[16]	P. Skraba, V. De Silva, M. Vejdemo-Johansson, Topological analysis of recurrent systems, In: NIPS 2012 Workshop on Algebraic Topology and Machine Learning, 2012, 1–5.
[17]	J. A. Perea, J. Harer, Sliding windows and persistence: An application of topological methods to signal analysis, Found. Comput. Math., 15 (2015), 799–838. https://doi.org/10.1007/s10208-014-9206-z doi: 10.1007/s10208-014-9206-z
[18]	J. A. Perea, Persistent homology of toroidal sliding window embeddings, In: 2016 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2016, 6435–6439. https://doi.org/10.1109/ICASSP.2016.7472916
[19]	J. Berwald, M. Gidea, Critical transitions in a model of a genetic regulatory system, 2013. https://doi.org/10.48550/arXiv.1309.7919
[20]	M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, et al., Early-warning signals for critical transitions, Nature, 461 (2009), 53–59. https://doi.org/10.1038/nature08227 doi: 10.1038/nature08227
[21]	F. A. Khasawneh, E. Munch, Stability determination in turning using persistent homology and time series analysis, In: Proceedings of the ASME 2014 International Mechanical Engineering Congress and Exposition, 2014, 14–20. https://doi.org/10.1115/IMECE2014-40221
[22]	C. M. M. Pereira, R. F. de Mello, Persistent homology for time series and spatial data clustering, Expert Syst. Appl., 42 (2015), 6026–6038. https://doi.org/10.1016/j.eswa.2015.04.010 doi: 10.1016/j.eswa.2015.04.010
[23]	Q. H. Tran, Y. Hasegawa, Topological time-series analysis with delay-variant embedding, Phys. Rev. E, 99 (2019), 032209. https://doi.org/10.1103/PhysRevE.99.032209 doi: 10.1103/PhysRevE.99.032209
[24]	C. Wu, C. A. Hargreaves, Topological machine learning for multivariate time series, J. Exp. Theor. Artif. In., 34 (2022), 311–326. https://doi.org/10.1080/0952813X.2021.1871971 doi: 10.1080/0952813X.2021.1871971
[25]	S. Emrani, T. Gentimis, H. Krim, Persistent homology of delay embeddings and its application to wheeze detection, IEEE Signal Proc. Let., 21 (2014), 459–463. https://doi.org/10.1109/LSP.2014.2305700 doi: 10.1109/LSP.2014.2305700
[26]	M. Gidea, Y. Katz, Topological data analysis of financial time series: Landscapes of crashes, Physica A, 491 (2018), 820–834. https://doi.org/10.1016/j.physa.2017.09.028 doi: 10.1016/j.physa.2017.09.028
[27]	M. S. Ismail, M. S. Md Noorani, M. Ismail, F. A. Razak, M. A. Alias, Early warning signals of financial crises using persistent homology, Physica A, 586 (2022), 126459. https://doi.org/10.1016/j.physa.2021.126459 doi: 10.1016/j.physa.2021.126459
[28]	V. Venkataraman, K. N. Ramamurthy, P. Turaga, Persistent homology of attractors for action recognition, In: 2016 IEEE international conference on image processing (ICIP), 2016, 4150–4154.. https://doi.org/10.1109/ICIP.2016.7533141
[29]	J. A Perea, A. Deckard, S. B. Haase, J. Harer, Sw1pers: Sliding windows and 1-persistence scoring; discovering periodicity in gene expression time series data, BMC Bioinformatics, 16 (2015), 257. https://doi.org/10.1186/s12859-015-0645-6 doi: 10.1186/s12859-015-0645-6
[30]	B. J. Stolz, H. A. Harrington, M. A. Porter, Persistent homology of time-dependent functional networks constructed from coupled time series, Chaos, 27 (2017), 047410. https://doi.org/10.1063/1.4978997 doi: 10.1063/1.4978997
[31]	M. Gidea, Topological data analysis of critical transitions in financial networks, In: 3rd International Winter School and Conference on Network Science, 2017, 47–59. https://doi.org/10.1007/978-3-319-55471-6_5
[32]	S. Gholizadeh, A. Seyeditabari, W. Zadrozny, Topological signature of 19th century novelists: Persistent homology in text mining, Big Data Cogn. Comput., 2 (2018), 33. https://doi.org/10.3390/bdcc2040033 doi: 10.3390/bdcc2040033
[33]	M. E. Aktas, E. Akbas, J. Papayik, Y. Kovankaya, Classification of turkish makam music: a topological approach, J. Math. Music, 13 (2019), 135–149. https://doi.org/10.1080/17459737.2019.1622810 doi: 10.1080/17459737.2019.1622810
[34]	M. L. Tran, C. Park, J. H. Jung, Topological data analysis of korean music in jeongganbo: a cycle structure, J. Math. Music, 17 (2023), 403–432. https://doi.org/10.1080/17459737.2022.2164626 doi: 10.1080/17459737.2022.2164626
[35]	M. L. Tran, D. Lee, J. H. Jung, Machine composition of korean music via topological data analysis and artificial neural network, J. Math. Music, 18 (2024), 20–41. https://doi.org/10.1080/17459737.2023.2197905 doi: 10.1080/17459737.2023.2197905
[36]	M. G Bergomi, A. Baratè, Homological persistence in time series: an application to music classification., J. Math. Music, 14 (2020), 204–221. https://doi.org/10.1080/17459737.2020.1786745 doi: 10.1080/17459737.2020.1786745
[37]	M. Mijangos, A. Bravetti, P. Padilla-Longoria, Musical stylistic analysis: a study of intervallic transition graphs via persistent homology, J. Math. Music, 18 (2024), 89–108. https://doi.org/10.1080/17459737.2023.2232811 doi: 10.1080/17459737.2023.2232811
[38]	D. Cao, Y. Wang, J. Duan, C. Zhang, X. Zhu, C. Huang, et al., Spectral temporal graph neural network for multivariate time-series forecasting, Adv. Neural Inform. Proc. Syst., 33 (2020), 17766–17778.
[39]	A. Deng, B. Hooi, Graph neural network-based anomaly detection in multivariate time series, In: Proceedings of the AAAI conference on artificial intelligence, 2021, 4027–4035. https://doi.org/10.1609/aaai.v35i5.16523
[40]	D. Zha, K. Lai, K. Zhou, X. Hu, Towards similarity-aware time-series classification, In: Proceedings of the 2022 SIAM International Conference on Data Mining (SDM), 2022,199–207. https://doi.org/10.1137/1.9781611977172.23
[41]	R. Gilmore, Topological analysis of chaotic dynamical systems, Rev. Mod. Phys., 70 (1998), 1455. https://doi.org/10.1103/RevModPhys.70.1455 doi: 10.1103/RevModPhys.70.1455
[42]	D. S. Broomhead, G. P. King, Extracting qualitative dynamics from experimental data, Physica D, 20 (1986), 217–236. https://doi.org/10.1016/0167-2789(86)90031-X doi: 10.1016/0167-2789(86)90031-X
[43]	E. Tan, S. Algar, D. Corrêa, M. Small, T. Stemler, D. Walker, Selecting embedding delays: An overview of embedding techniques and a new method using persistent homology, Chaos, 33 (2023), 032101. https://doi.org/10.1063/5.0137223 doi: 10.1063/5.0137223
[44]	L. Vietoris, Über den höheren zusammenhang kompakter räume und eine klasse von zusammenhangstreuen abbildungen, Math. Ann., 97 (1927), 454–472. https://doi.org/10.1007/BF01447877 doi: 10.1007/BF01447877
[45]	M. Gromov, Hyperbolic groups, In: Essays in group theory, 1987. https://doi.org/10.1007/978-1-4613-9586-7_3
[46]	Edelsbrunner, Letscher, Zomorodian, Topological persistence, simplification, Discrete Comput. Geom., 28 (2002), 511–533. https://doi.org/10.1007/s00454-002-2885-2
[47]	H. Adams, A. Tausz, M. Vejdemo-Johansson, Javaplex: A research software package for persistent (co) homology, In: Mathematical Software-ICMS 2014, 2014,129–136. https://doi.org/10.1007/978-3-662-44199-2_23
[48]	C. Maria, J. Boissonnat, M. Glisse, M. Yvinec, The gudhi library: Simplicial complexes and persistent homology, In: Mathematical Software-ICMS 2014, 2014,167–174. https://doi.org/10.1007/978-3-662-44199-2_28
[49]	U. Bauer, Ripser: efficient computation of vietoris-rips persistence barcodes, J. Appl. Comput. Topology, 5 (2021), 391–423. https://doi.org/10.1007/s41468-021-00071-5 doi: 10.1007/s41468-021-00071-5
[50]	D. Attali, A. Lieutier, D. Salinas, Vietoris-rips complexes also provide topologically correct reconstructions of sampled shapes, In: Proceedings of the twenty-seventh annual symposium on Computational geometry, 2011,491–500. https://doi.org/10.1145/1998196.1998276
[51]	T. K. Dey, Y. Wang, Computational topology for data analysis, Cambridge University Press, 2022. https://doi.org/10.1017/9781009099950
[52]	E. W. Dijkstra, A note on two problems in connexion with graphs, Numer. Math., 1 (1959), 269–271.
[53]	D. Cohen-Steiner, H. Edelsbrunner, J. Harer, Stability of persistence diagrams, In: Proceedings of the twenty-first annual symposium on Computational geometry, 2005,263–271. https://doi.org/10.1145/1064092.1064133
[54]	M. M. Ghazani, A. A. M. Malekshah, R. Khosravi, Analyzing time–frequency connectedness between cryptocurrencies, stock indices, and benchmark crude oils during the covid-19 pandemic, Financ. Innova., 10 (2024), 119. https://doi.org/10.1186/s40854-024-00645-z doi: 10.1186/s40854-024-00645-z
[55]	S. C. Nayak, S. Dehuri, S. B. Cho, Elitist-opposition-based artificial electric field algorithm for higher-order neural network optimization and financial time series forecasting, Financ. Innov., 10 (2024), 5. https://doi.org/10.1186/s40854-023-00534-x doi: 10.1186/s40854-023-00534-x
[56]	M. Anas, S. J. H. Shahzad, L. Yarovaya, The use of high-frequency data in cryptocurrency research: A meta-review of literature with bibliometric analysis, Financ. Innov., 10 (2024), 90. https://doi.org/10.1186/s40854-023-00595-y doi: 10.1186/s40854-023-00595-y
[57]	S. W. Akingbade, M. Gidea, M. Manzi, V. Nateghi, Why topological data analysis detects financial bubbles? Commun. Nonlinear Sci., 128 (2024), 107665. https://doi.org/10.1016/j.cnsns.2023.107665
[58]	P. Bubenik, Statistical topological data analysis using persistence landscapes, J. Mach. Learn. Res., 16 (2015), 77–102.

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