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The development of multi-channel data acquisition techniques has provided richer prior information for studying the nonlinear dynamic characteristics of complex systems. However, conventional nonlinear feature extraction algorithms prove unsuitable in the context of multi-channel data. Previously, the multivariate multiscale sample entropy (MMSE) algorithm was introduced for multi-channel data analysis. Although the MMSE algorithm generalized the multiscale sample entropy algorithm, presenting a novel method for multidimensional data analysis, it remains deficient in theoretical underpinning and suffers from shortcomings, such as missing cross-channel correlation information and having biased estimation results. In this paper, unbiased multivariate multiscale sample entropy algorithm (UMMSE) is proposed. UMMSE increases the embedding dimension from M to M + p. This increasing strategy facilitates the reconstruction of a deterministic phase space. By virtue of theoretical scrutiny grounded in probability theory and subsequent experimental validation, this paper illustrates the algorithm's effectiveness in extracting inter-channel correlation information through the integration of cross-channel conditional probabilities. The computation of similarities between sample points across different channels is recognized as a potential source of bias and instability in algorithms.Through simulation experiments, this study delineates the parameter selection range for the UMMSE algorithm. Subsequently, diverse simulation signals are employed to showcase the UMMSE algorithm’s efficacy in extracting both within-channel and cross-channel correlation information. Ultimately, this paper demonstrates that the new algorithm has the lowest computational cost compared with traditional MMSE algorithms.
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Google Scholar
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Liu B Z, Peng J H 2004 Nonlinear Dynamics (Beijing: Higher Education Press) pp301–466
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
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Google Scholar
[21] Costa M, Goldberger A L, Peng C K 2005 Phys. Rev. E 71 021906
Google Scholar
[22] Ahmed M U, Mandic D P 2011 Phys. Rev. E 84 061918
Google Scholar
[23] Li Y, Tang B, Jiao S, Zhou Y 2024 Chaos, Solitons Fractals 179 114436
Google Scholar
[24] Zhao C, Sun J, Lin S, Peng Y 2022 Measurement 195 111190
Google Scholar
[25] Cao L Y, Mees A, Judd K 1998 Physica D 121 75
Google Scholar
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Google Scholar
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图 1 无偏差的多通道多尺度样本熵算法参数选择 (a)数据长度; (b)嵌入维数; (c)阈值系数
Figure 1. The parameter selection of the unbiased multivariate multiscale sample entropy algorithm: (a) Data length; (b) embedding dimension; (c) coefficient of the threshold.
图 2 多通道高斯白噪声和$1/f$噪声的无偏差的多通道多尺度样本熵结果
Figure 2. Unbiased multivariate multiscale sample entropy results for multivariate white noise and $1/f$ noise.
图 3 相关高斯白噪声与不相关高斯白噪声的多通道多尺度样本熵结果 (a)朴素算法; (b)严格算法; (c)无偏差算法
Figure 3. The entropy results of different MMSE algorithms processing correlated and uncorrelated white noise: (a) Naive MMSE result; (b) rigorous MMSE result; (c) unbiased multivariate multiscale sample entropy results.
图 4 严格算法与无偏差的多通道多尺度样本熵算法稳定性分析 (a)通道间方差相同时的严格算法结果; (b)通道间方差不同时的严格算法结果; (c)通道间方差相同时的无偏差算法结果; (d)通道间方差不同时的无偏差算法结果
Figure 4. The robustness analysis for the different multivariate entropy algorithms: (a) Rigorous MMSE result for the same variance of each channel; (b) rigorous MMSE result for the different variance of each channel; (c) unbiased multivariate multiscale sample entropy results for the same variance of each channel; (d) unbiased multivariate multiscale sample entropy result for the different variance of each channel.
图 5 不同多通道熵算法的计算效率对比
Figure 5. Comparison of computational cost of different multi-channel entropy algorithms.
表 1 熵算法中涉及的事件及对应的符号
Table 1. Events involved in the entropy algorithm
符号 事件 A 通道1中长度为$m_{1}$的子序列相似 B 通道2中长度为$m_{2}$的子序列相似 C 通道1中长度为$m_{1}$的子序列在
第$m_{1}+1$个样本点依然相似D 通道2中长度为$m_{2}$的子序列在
第$m_{2}+1$个样本点依然相似
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表 2 不同的嵌入维数所对应的事件
Table 2. Events corresponding to different embedding dimensions
嵌入维数 可能的结果 $B^{M+i}(r)$ $m_1$ $m_1$ $P({\rm{A}})$ $m_1+1$ $m_1+1$ $P({\rm{AC}})$ M $[m_{1}, m_{2}]$ $P({\rm{AB}})$ $M+1$ $[m_{1}+1, m_{2}]$ $P({\rm{ABC}})$ $[m_{1}, m_{2}+1]$ $P({\rm{ABD}})$ $M+2$ $[m_{1}+1, m_{2}+1]$ $P({\rm{ABCD}})$
DownLoad: CSV
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[1] Fowler A C, Gibbon J D, McGuinness M J 1982 Physica D 4 139
Google Scholar
[2] Gao Z, Jin N 2009 Chaos 19 033137
Google Scholar
[3] López C, Naranjo á, Lu S, Moore K 2022 J. Sound Vib. 528 116890
Google Scholar
[4] Wang Q B, Yang Y J, Zhang X 2020 Chaos, Solitons Fractals 137 109832
Google Scholar
[5] Kunze M 2007 Non-Smooth Dynamical Systems (Heidelberg: Springer Berlin) pp63–140
[6] Zarei A, Asl B M 2018 IEEE J. Biomed. Health 23 1011
Google Scholar
[7] He H, Tan Y 2017 Appl. Soft Comput. 55 238
Google Scholar
[8] 刘秉正, 彭建华 2004 非线性动力学(北京: 高等教育出版社) 第301—466页
Liu B Z, Peng J H 2004 Nonlinear Dynamics (Beijing: Higher Education Press) pp301–466
[9] Cranch G A, Nash P J, Kirkendall C K 2003 IEEE Sens. J. 3 19
Google Scholar
[10] Zhu X, Murch R D 2002 IEEE Trans. Commun. 50 187
Google Scholar
[11] Van Trees H L 2002 Optimum Array Processing: Part IV of Detection, Estimation, and Modulation Theory (New York: John Wiley and Sons)pp17-230
[12] Eren F, Pe'eri S, Thein M W, Rzhanov Y, Celikkol B, Swift M R 2017 Sensors-Basel 17 1741
Google Scholar
[13] Shi Q Q, Li W, Tao R, Sun X, Gao L R 2019 Remote Sens-Basel 11 419
Google Scholar
[14] Xing X W, Ji K F, Zou H X, Chen W T, Sun J X 2013 IEEE Geosci. Remote Sens. Lett. 10 1562
Google Scholar
[15] Wang Z Y, Yao L G, Cai Y W 2020 Measurement 156 107574
Google Scholar
[16] Thuraisingham R A, Gottwald G A 2006 Physica A 366 323
Google Scholar
[17] Li W J, Shen X H, Li Y A 2019 Entropy-Switz 21 793
Google Scholar
[18] Pincus S M 1991 P. Natl. Acad. Sci. USA 88 2297
Google Scholar
[19] Richman J S, Moorman J R 2000 Am. J. Physiol-Heart. C 27 H2039
[20] Bandt C, Pompe B 2002 Phys. Rev. Lett 88 174102
Google Scholar
[21] Costa M, Goldberger A L, Peng C K 2005 Phys. Rev. E 71 021906
Google Scholar
[22] Ahmed M U, Mandic D P 2011 Phys. Rev. E 84 061918
Google Scholar
[23] Li Y, Tang B, Jiao S, Zhou Y 2024 Chaos, Solitons Fractals 179 114436
Google Scholar
[24] Zhao C, Sun J, Lin S, Peng Y 2022 Measurement 195 111190
Google Scholar
[25] Cao L Y, Mees A, Judd K 1998 Physica D 121 75
Google Scholar
[26] Zhang Y C 1991 J. Phys. I France 1 971
Google Scholar
[27] Takens F 1980 Dynamical Systems and Turbulence (Heidelberg: Springer Berlin) p366
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