-
Phase space reconstruction plays a pivotal role in calculating features of nonlinear systems. By mapping one-dimensional time series onto a high-dimensional phase space using phase space reconstruction techniques, the dynamical characteristics of nonlinear systems can be revealed. However, existing nonlinear analysis methods are primarily based on phase space reconstruction of single-channel data and cannot directly utilize the rich information contained in multi-channel array data. The reconstructed data matrix shows the structural similarities with multi-channel array data. The relationship between phase space reconstruction and array data structure, as well as the gain in nonlinear features brought by array data, has not been sufficiently studied. In this paper, two classical nonlinear features: multiscale sample entropy and multiscale permutation entropy are adopted. The array multi-channel data are used to replace the phase space reconstruction step in algorithms so as to enhance the algorithmic performance. Initially, the relationship between phase space reconstruction parameters and actual array structures is analyzed, and conversion relationships are established. Then, multiple sets of simulated and real-world array data are used to evaluate the performances of the two entropy algorithms. The results show that substituting array data for phase space reconstruction effectively improves the performances of both entropy algorithms. Specifically, the multiscale sample entropy algorithm, when applied to array data, allows for distinguishing between noisy target signals from background noise at low signal-to-noise ratios. At the same time, the multiscale permutation entropy algorithm using array data reveals the complex structure of signals on different time scales more accurately.
-
Keywords:
- nonlinear dynamics /
- array data analysis /
- multiscale sample entropy /
- multiscale permutation entropy
[1] Chen H T, Li L L, Shang C, Huang B 2022 IEEE T. Cybernetics 53 4259
Google Scholar
[2] Richard B, Kerry A E, Zhang F 2019 Science 363 342
Google Scholar
[3] Hsieh D A 1991 J. Financ. 46 1839
Google Scholar
[4] Xu F, Zou Z J, Yin J C, Cao J 2013 Ocean Eng. 67 68
Google Scholar
[5] Takens F 1980 Dynamical Systems and Turbulence (Heidelberg: Springer Berlin) p366
[6] 刘秉正, 彭建华 2004 非线性动力学 (北京: 高等教育出版社) 第389—403页
Liu B Z, Peng J H 2004 Nonlinear Dynamics pp389–403
[7] Zhao J Y, Jin N D 2012 Acta Phys. Sin. 61 094701 [赵俊英, 金宁德 2012 61 094701]
Google Scholar
Zhao J Y, Jin N D 2012 Acta Phys. Sin. 61 094701
Google Scholar
[8] 黄泽徽, 李亚安, 陈哲, 刘恋 2020 69 160501
Google Scholar
Huang Z H, Li Y A, Chen Z, Liu L 2020 Acta Phys. Sin. 69 160501
Google Scholar
[9] Grigoryeva L, Hart A, Ortega J P 2021 Phys. Rev. E 103 062204
Google Scholar
[10] Kennel M B, Brown R, Abarbanel H 1992 Phys. Rev. A 45 3403
Google Scholar
[11] 张淑清, 贾健, 高敏, 韩叙 2010 59 1576
Google Scholar
Zhang S Q, Jia J, Gao M, Han X 2010 Acta Phys. Sin 59 1576
Google Scholar
[12] Richman J S, Moorman J R 2000 Am. J. Physiol-Heart. C 278 H2039
Google Scholar
[13] Bandt C, Pompe B 2002 Phys. Rev. Lett 88 174102
Google Scholar
[14] Bryant P, Brown R, Abarbanel H 1990 Phys. Rev. Lett. 65 1523
Google Scholar
[15] Ferrara E, Parks T 1983 IEEE T. Antenn. Propag. 31 231
Google Scholar
[16] He J, Liu Z 2009 Signal Process. 89 1715
Google Scholar
[17] Costa M, Goldberger A L, Peng C K 2005 Phys. Rev. E 71 021906
Google Scholar
[18] Liu T B, Yao W P, Wu M, Shi Z R, Wang J, Ning X B 2017 Physica A 471 492
Google Scholar
[19] Humeau-Heurtier A, Wu C W, Wu S D 2015 IEEE Signal Proc. Let. 22 2364
Google Scholar
[20] Amigó J M, Dale R, Tempesta P 2021 Chaos 31 013115
Google Scholar
[21] Li W J, Shen X H, Li Y A 2019 Entropy-Switz 21 793
Google Scholar
[22] Li Y B, Xu M Q, Wang R X, Huang W H 2016 J. Sound Vib. 360 277
Google Scholar
[23] Gao S Z, Wang Q, Zhang Y M 2021 IEEE T. Instrum. Meas. 70 3514908
Google Scholar
[24] 何亮, 杜磊, 庄奕琪, 李伟华, 陈建平 2008 57 6545
Google Scholar
He L, Du L, Zhuang Y Q, Li W H, Chen J P 2008 Acta Phys. Sin. 57 6545
Google Scholar
[25] Zhang Y 1991 J. Phys. I 1 971
[26] Fogedby H C 1992 J. Stat. Phys. 69 411
Google Scholar
-
图 1 $ m=3$时6种不同的序数模式示意图, 黑色圆点代表数据点, 虚线连接的数据点之间的高低关系代表数据点之间的大小关系
Figure 1. All possible ordinal patterns when $ m = 3 $, the black dots represent data points, and the dashed lines connecting the data points indicate the relative magnitude between them.
图 2 远场条件下的均匀线列阵结构示意图
Figure 2. Schematic diagram of a uniform linear array structure under far-field conditions.
图 3 使用单通道数据的多尺度样本熵与多尺度排列熵算法结果 (a) 多尺度样本熵; (b) 多尺度排列熵
Figure 3. The entropy results of multiscale sample entropy and multiscale permutation entropy using signal-channel data: (a) Multiscale sample entropy result; (b) multiscale permutation entropy result.
图 4 使用按照相空间重构逻辑构造的阵列数据的多尺度样本熵与多尺度排列熵算法结果 (a) 多尺度样本熵; (b) 多尺度排列熵
Figure 4. The multiscale sample entropy and multiscale permutation entropy results using array data constructed according to phase space reconstruction technique: (a) Multiscale sample entropy result; (b) multiscale permutation entropy result.
图 5 仿真阵列数据的多尺度样本熵与多尺度排列熵算法结果 (a) 多尺度样本熵; (b) 多尺度排列熵
Figure 5. The multiscale sample entropy and multiscale permutation entropy results for simulated array data: (a) Multiscale sample entropy result; (b) multiscale permutation entropy result.
图 6 多尺度排列熵算法结果 (a) 基于相空间重构构造的阵列数据的多尺度排列熵算法结果; (b) 仿真阵列数据的多尺度排列熵算法结果
Figure 6. Multiscale permutation entropy results: (a) Array data constructed according to phase space reconstruction technique; (b) simulated array data.
图 7 使用一个通道数据计算实际数据的多尺度样本熵与多尺度排列熵结果 (a) 多尺度样本熵结果; (b) 多尺度排列熵结果
Figure 7. The multiscale sample entropy and multiscale permutation entropy results of the actual data calculated using only one channel data: (a) Multiscale sample entropy results; (b) multiscale permutation entropy results.
图 8 使用阵列数据代替相空间重构计算的多尺度样本熵与多尺度排列熵结果 (a) 多尺度样本熵结果; (b) 多尺度排列熵结果
Figure 8. The results of multiscale sample entropy and multiscale permutation entropy calculated by array data: (a) Multiscale sample entropy results; (b) multiscale permutation entropy results.
表 1 舰船辐射噪声相邻阵元的时间延迟
Table 1. Time delay of adjacent array elements of ship-radiated noise
舰船
类型通道1-
通道2通道2-
通道3通道3-
通道4通道4-
通道5平均时间
延迟舰船1 22 23 22 18 21.25 舰船2 20 21 21 17 19.75 
DownLoad: CSV
-
[1] Chen H T, Li L L, Shang C, Huang B 2022 IEEE T. Cybernetics 53 4259
Google Scholar
[2] Richard B, Kerry A E, Zhang F 2019 Science 363 342
Google Scholar
[3] Hsieh D A 1991 J. Financ. 46 1839
Google Scholar
[4] Xu F, Zou Z J, Yin J C, Cao J 2013 Ocean Eng. 67 68
Google Scholar
[5] Takens F 1980 Dynamical Systems and Turbulence (Heidelberg: Springer Berlin) p366
[6] 刘秉正, 彭建华 2004 非线性动力学 (北京: 高等教育出版社) 第389—403页
Liu B Z, Peng J H 2004 Nonlinear Dynamics pp389–403
[7] Zhao J Y, Jin N D 2012 Acta Phys. Sin. 61 094701 [赵俊英, 金宁德 2012 61 094701]
Google Scholar
Zhao J Y, Jin N D 2012 Acta Phys. Sin. 61 094701
Google Scholar
[8] 黄泽徽, 李亚安, 陈哲, 刘恋 2020 69 160501
Google Scholar
Huang Z H, Li Y A, Chen Z, Liu L 2020 Acta Phys. Sin. 69 160501
Google Scholar
[9] Grigoryeva L, Hart A, Ortega J P 2021 Phys. Rev. E 103 062204
Google Scholar
[10] Kennel M B, Brown R, Abarbanel H 1992 Phys. Rev. A 45 3403
Google Scholar
[11] 张淑清, 贾健, 高敏, 韩叙 2010 59 1576
Google Scholar
Zhang S Q, Jia J, Gao M, Han X 2010 Acta Phys. Sin 59 1576
Google Scholar
[12] Richman J S, Moorman J R 2000 Am. J. Physiol-Heart. C 278 H2039
Google Scholar
[13] Bandt C, Pompe B 2002 Phys. Rev. Lett 88 174102
Google Scholar
[14] Bryant P, Brown R, Abarbanel H 1990 Phys. Rev. Lett. 65 1523
Google Scholar
[15] Ferrara E, Parks T 1983 IEEE T. Antenn. Propag. 31 231
Google Scholar
[16] He J, Liu Z 2009 Signal Process. 89 1715
Google Scholar
[17] Costa M, Goldberger A L, Peng C K 2005 Phys. Rev. E 71 021906
Google Scholar
[18] Liu T B, Yao W P, Wu M, Shi Z R, Wang J, Ning X B 2017 Physica A 471 492
Google Scholar
[19] Humeau-Heurtier A, Wu C W, Wu S D 2015 IEEE Signal Proc. Let. 22 2364
Google Scholar
[20] Amigó J M, Dale R, Tempesta P 2021 Chaos 31 013115
Google Scholar
[21] Li W J, Shen X H, Li Y A 2019 Entropy-Switz 21 793
Google Scholar
[22] Li Y B, Xu M Q, Wang R X, Huang W H 2016 J. Sound Vib. 360 277
Google Scholar
[23] Gao S Z, Wang Q, Zhang Y M 2021 IEEE T. Instrum. Meas. 70 3514908
Google Scholar
[24] 何亮, 杜磊, 庄奕琪, 李伟华, 陈建平 2008 57 6545
Google Scholar
He L, Du L, Zhuang Y Q, Li W H, Chen J P 2008 Acta Phys. Sin. 57 6545
Google Scholar
[25] Zhang Y 1991 J. Phys. I 1 971
[26] Fogedby H C 1992 J. Stat. Phys. 69 411
Google Scholar
DownLoad:
Catalog
Metrics
- Abstract views: 599
- PDF Downloads: 10
- Cited By: 0
