Interval multiscale sample entropy: A novel tool for interval-valued time series complexity analysis

doi:10.1088/1674-1056/adea56

Abstract To analyze the complexity of interval-valued time series (ITSs), a novel interval multiscale sample entropy (IMSE) methodology is proposed in this paper. To validate the effectiveness and feasibility of IMSE in characterizing ITS complexity, the method is initially implemented on simulated time series. The experimental results demonstrate that IMSE not only successfully identifies series complexity and long-range autocorrelation patterns but also effectively captures the intrinsic relationships between interval boundaries. Furthermore, the test results show that IMSE can also be applied to measure the complexity of multivariate time series of equal length. Subsequently, IMSE is applied to investigate interval temperature series (2000-2023) from four Chinese cities: Shanghai, Kunming, Chongqing, and Nagqu. The results show that IMSE not only distinctly differentiates temperature patterns across cities but also effectively quantifies complexity and long-term autocorrelation in ITSs. All the results indicate that IMSE is an alternative and effective method for studying the complexity of ITSs.

Keywords: interval multiscale sample entropy interval-valued time series

Received: 17 February 2025
Revised: 18 May 2025
Accepted manuscript online: 01 July 2025

PACS:	05.45.Tp	(Time series analysis)
	02.70.Rr	(General statistical methods)
	02.50.Sk	(Multivariate analysis)
	05.45.-a	(Nonlinear dynamics and chaos)

Fund: Project supported by Hubei Provincial Department of Education Science and Technology Plan Project (Grant No. B2022165).

Corresponding Authors: Bao-Gen Li
E-mail: baogenli@hbnu.edu.cn

About author: 2025-120508-250222.pdf

Cite this article:

Ping Tang(唐萍), Bao-Gen Li(李宝根), and Yang Wang(王阳) Interval multiscale sample entropy: A novel tool for interval-valued time series complexity analysis 2025 Chin. Phys. B 34 120508

[1] Shannon C E 1948 Bell Syst. Tech. J. 27 379
[2] Tang J and Liu X Q 2019 Acta Phys. Sin. 68 149801 (in Chinese)
[3] Zhang J C, Ren W K and Jin N D 2020 Chin. Phys. Lett. 37 090501
[4] Wang S Y, Shi C F, Qian G B and Wang W L 2018 Acta Phys. Sin. 67 018401 (in Chinese)
[5] Silva L E V and Murta L O J 2012 Chaos 22 043105
[6] Silva L E V, Cabella B C T, Neves U P C and Murta L O J 2015 Phys. A 422 143
[7] XuMJ, Shang P J and Huang J J 2016 Commun. Nonlinear Sci. Numer. Simul. 35 17
[8] Yao W P 2025 Acta Phys. Sin. 74 040502 (in Chinese)
[9] Ponta L, Murialdo P and Carbone A 2021 Phys. A 570 125777
[10] Pincus S M 1991 Proc. Natl. Acad. Sci. USA 88 2297
[11] Pincus S M 1995 Quantitative Neuroendocrinology (New York: Academic Press) p. 336
[12] Richman J S and Moorman J R 2000 Am. J. Physiol. Heart Circ. Physiol. 278 H2039
[13] Ramdani S, Bouchara F and Lagarde J 2009 Chaos 19 013123
[14] Costa M, Goldberger A L and Peng C K 2002 Phys. Rev. Lett. 89 068102
[15] Costa M, Goldberger A L and Peng C K 2005 Phys. Rev. E 71 021906
[16] Mitchell M 2009 Complexity: A Guided Tour (New York: Oxford University Press) pp. 94-114
[17] Zhang Y C 1991 J. Phys. I 1 971
[18] Wang J, Shang P J, Zhao X J and Xia J N 2013 Int. J. Mod. Phys. C 24 1350006
[19] Wang J, Shang P J, Xia J N and Shi W B 2015 Phys. A 421 583
[20] Ahmed M U, Li L, Cao J T and Mandic D P 2011 2011 Annual International Conference of the IEEE Engineering in Medicine and Biology Society, August 30-September 03, 2011, Boston, USA, p. 810
[21] Ahmed M U, and Mandic D P 2011 Phys. Rev. E 84 061918
[22] Li W J, Shen X H, Li Y A and Chen Z 2023 Chaos 33 063125
[23] Niu Y, Ding R N, Zhou M N, Sun J, Dou M L, Wen X, Cui X H, Yao R, Wei J and Xiang J 2025 Nonlinear Dyn. 113 8443
[24] Li W J, Shen X H, Li Y A, Chen Z and Shen Y P 2024 Chaos 34 093143
[25] Xiong T, Li C G and Bao Y K 2017 Econ. Model 60 11
[26] Jiang M R, Chen W, Xu H L and Liu Y X 2024 Pattern Recognit. 145 109920
[27] Zhang T, Zhou Y Y, Zhao K G, Zhu Z Y, Chen G, Hu J and Wang L 2022 Earth Syst. Sci. Data 14 5637
[28] Lu Y, Linderman G C, Mahajan S, Liu Y T, Huang C X, Khera R, Mortazavi B J, Spatz E S and Krumholz H M 2023 Circ. Cardiovasc. Qual. Outcomes 16 e009258
[29] Billard L and Diday E 2003 J. Am. Stat. Assoc. 98 470
[30] Yang D C, Guo J E, Sun S L, Han J and Wang S Y 2022 Appl. Energy 306 117992
[31] Xiong T, Bao Y K, Hu Z Y and Chiong R 2015 Inf. Sci. 305 77
[32] Zhou Y T, Fan Y, Chen Z Y and Sun J C 2017 Chin. Phys. Lett. 34 050502
[33] Li Q Q, Zheng R Z, Ji A B and Ma H Y 2025 Spat. Stat. 65 100875
[34] Yang K, Cheng Z S, LiMC,Wang S Y andWei Y J 2024 Appl. Energy 353 122102
[35] Tao Z F, NiW Q andWang P 2024 Eng. Appl. Artif. Intell. 133 108007
[36] Liu S H, Xie G, Wang Z Z and Wang S Y 2024 Appl. Energy 359 122613
[37] Maharaj E A, Teles P and Brito P 2019 Stat. Comput. 29 1011
[38] Sun L R, Zhu L J, Li W C, Zhang C H and Balezentis T 2022 Inf. Sci. 606 910
[39] Sun L R, Mao H F, Zheng C, Zhang C H and Balezentis T 2023 Appl. Math. Model. 123 627
[40] Hu C Y and Hu Z H 2020 Information Processing and Management of Uncertainty in Knowledge-Based Systems, June 15-19, 2020, Lisbon, Portugal, p. 407
[41] Hu C Y and Hu Z H 2020 Information Processing and Management of Uncertainty in Knowledge-Based Systems, June 15-19, 2020, Lisbon, Portugal, p. 422
[42] Takens F 1981 Dynamical Systems and Turbulence, Warwick 1980, 1979/80, Coventry, UK, p. 366
[43] Wang P, Gurmani S H, Tao Z F, Liu J P and Chen H Y 2024 J. Forecast. 43 249
[44] Moore R E, Kearfott R B and Cloud M J 2009 Introduction to Interval Analysis (Philadelphia: Society for Industrial and Applied Mathematics) pp. 51-83
[45] Wang X, Yu F S, PedryczWand Yu L 2019 Expert Syst. Appl. 125 293
[46] Golub G H and Van Loan C F 2013 Matrix Computations - 4th Edition, (Baltimore: Johns Hopkins University Press) pp. 71-73
[47] Omidvarnia A, Zalesky A, Mansour L S, Van De Ville D, Jackson G D and Pedersen M 2021 Neuroimage 230 117760

[1]	Analysis of spatiotemporal dynamic patterns of gene expression during mouse embryonic development based on Moran's I and spatial transcriptomics Qi-Chao Li(李啟超), Hai Lin(林海), Peng Wang(王鹏), Qiutong Dong(董秋彤), Kun Wang(王坤), Jian-Wei Shuai(帅建伟), and Fang-Fu Ye(叶方富). Chin. Phys. B, 2025, 34(8): 088703.
[2]	A novel baseline perspective visibility graph for time series analysis Huang-Jing Ni(倪黄晶), Zi-Jie Song(宋紫婕), Jiao-Long Qin(秦姣龙), Ye Wu(吴烨), Shi-Le Qi(戚世乐), and Ming Song(宋明). Chin. Phys. B, 2025, 34(8): 080504.
[3]	Associated network family of the unified piecewise linear chaotic family and their relevance Haoying Niu(牛浩瀛) and Jie Liu(刘杰). Chin. Phys. B, 2025, 34(4): 040503.
[4]	Model-free prediction of chaotic dynamics with parameter-aware reservoir computing Jianmin Guo(郭建敏), Yao Du(杜瑶), Haibo Luo(罗海波), Xuan Wang(王晅), Yizhen Yu(于一真), and Xingang Wang(王新刚). Chin. Phys. B, 2025, 34(4): 040505.
[5]	WT-FCTGN: A wavelet-enhanced fully connected time-gated neural network for complex noisy traffic flow modeling Zhifang Liao(廖志芳), Ke Sun(孙轲), Wenlong Liu(刘文龙), Zhiwu Yu(余志武), Chengguang Liu(刘承光), and Yucheng Song(宋禹成). Chin. Phys. B, 2024, 33(7): 078901.
[6]	A novel variable-order fractional chaotic map and its dynamics Zhouqing Tang(唐周青), Shaobo He(贺少波), Huihai Wang(王会海), Kehui Sun(孙克辉), Zhao Yao(姚昭), and Xianming Wu(吴先明). Chin. Phys. B, 2024, 33(3): 030503.
[7]	Visibility graph approach to extreme event series Jing Zhang(张晶), Xiaolu Chen(陈晓露), Haiying Wang(王海英), Changgui Gu(顾长贵), and Huijie Yang(杨会杰). Chin. Phys. B, 2023, 32(10): 100505.
[8]	Information flow between stock markets: A Koopman decomposition approach Semba Sherehe, Huiyun Wan(万慧云), Changgui Gu(顾长贵), and Huijie Yang(杨会杰). Chin. Phys. B, 2022, 31(1): 018902.
[9]	Sensitivity to external optical feedback of circular-side hexagonal resonator microcavity laser Tong Zhao(赵彤), Zhi-Ru Shen(申志儒), Wen-Li Xie(谢文丽), Yan-Qiang Guo(郭龑强), An-Bang Wang(王安帮), and Yun-Cai Wang(王云才). Chin. Phys. B, 2021, 30(12): 120513.
[10]	Synchronization mechanism of clapping rhythms in mutual interacting individuals Shi-Lan Su(苏世兰), Jing-Hua Xiao(肖井华), Wei-Qing Liu(刘维清), and Ye Wu(吴晔). Chin. Phys. B, 2021, 30(1): 010505.
[11]	Modeling and analysis of the ocean dynamic with Gaussian complex network Xin Sun(孙鑫), Yongbo Yu(于勇波), Yuting Yang(杨玉婷), Junyu Dong(董军宇)†, Christian B\"ohm, and Xueen Chen(陈学恩). Chin. Phys. B, 2020, 29(10): 108901.
[12]	Patterns of cross-correlation in time series: A case study of gait trails Jia Song(宋佳), Tong-Feng Weng(翁同峰), Chang-Gui Gu(顾长贵), Hui-Jie Yang(杨会杰). Chin. Phys. B, 2020, 29(8): 080501.
[13]	Inverse Ising techniques to infer underlying mechanisms from data Hong-Li Zeng(曾红丽), Erik Aurell. Chin. Phys. B, 2020, 29(8): 080201.
[14]	Reconstruction of dynamic structures of experimental setups based on measurable experimental data only Tian-Yu Chen(陈天宇), Yang Chen(陈阳), Hu-Jiang Yang(杨胡江), Jing-Hua Xiao(肖井华), Gang Hu(胡岗). Chin. Phys. B, 2018, 27(3): 030503.
[15]	Extracting hidden weak sinusoidal signal with short duration from noisy data:Analytical theory and computational realization Ying Zhang(张英), Zhaoyang Zhang(张朝阳), Hong Qian(钱弘), Gang Hu(胡岗). Chin. Phys. B, 2017, 26(10): 100508.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Interval multiscale sample entropy: A novel tool for interval-valued time series complexity analysis

Cite this article:

Online attention