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Chaos is a seemingly random and irregular movement, happening in a deterministic system without random factors. Chaotic theory has promising applications in various areas (e.g., communication, image encryption, geophysics, weak signal detection). However, observed chaotic signals are often contaminated by noise. The presence of noise hinders the chaos theory from being applied to related fields. Therefore, it is important to develop a new method of suppressing the noise of the chaotic signals. Recently, the denoising algorithm for chaotic signals based on collaborative filtering was proposed. Its denoising performance is better than those of the existing denoising algorithms for chaotic signals. The denoising algorithm for chaotic signals based on collaborative filtering makes full use of the self-similar structural feature of chaotic signals. However, in the parameter optimization issue of the denoising algorithm, the selection of the filter parameters is affected by signal characteristic, sampling frequency and noise level. In order to improve the adaptivity of the denoising algorithm, a criterion for selecting the optimal filter parameters is proposed based on permutation entropy in this paper. The permutation entropy can effectively measure the complexity of time series. It has been widely applied to physical, medical, engineering, and economic sciences. According to the difference among the permutation entropies of chaotic signals at different noise levels, first, different filter parameters are used for denoising noisy chaotic signals. Then, the permutation entropy of the reconstructed chaotic signal corresponding to each of filter parameters is computed. Finally, the permutation entropies of the reconstructed chaotic signals are compared with each other, and the filter parameter corresponding to the minimum permutation entropy is selected as an optimal filter parameter. The selections of the filter parameters are analyzed in the cases of different signal characteristics, different sampling frequencies and different noise levels. Simulation results show that this criterion can automatically optimize the filter parameter efficiently in different conditions, which improves the adaptivity of the denoising algorithm for chaotic signals based on collaborative filtering.
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Keywords:
- chaos /
- denoising /
- collaborative filtering /
- adaptive filtering
[1] L J H, Lu J A, Chen S H 2002 The Analysis and Applications of Chaotic Time Series (Wuhan:Wuhan University Press) pp1-8 (in Chinese) [吕金虎, 陆君安, 陈士华 2002 混沌时间序列分析及其应用(武汉:武汉大学出版社)第1–8页]
[2] Han M, Xu M L 2013 Acta Phys. Sin. 62 120510 (in Chinese) [韩敏, 许美玲 2013 62 120510]
[3] Sun J W, Shen Y, Yin Q, Xu C J 2013 Chaos 23 013140
[4] Li G Z, Zhang B 2017 IEEE Trans. Ind. Electron. 64 2255
[5] Peng G Y, Min F H 2017 Nonlinear Dynam. 90 1607
[6] Urbanowicz K, Hołyst J A 2003 Phys. Rev. E 67 046218
[7] Feng J C 2012 Chaotic Signals and Information Processing (Beijing:Tsinghua University Press) pp32-35 (in Chinese) [冯久超 2012 混沌信号与信息处理(北京:清华大学出版社)第32–35页]
[8] Badii R, Broggi G, Derighetti B, Ravani M 1988 Phys. Rev. Lett. 60 979
[9] Cawley R, Hsu G H 1992 Phys. Rev. A 46 3057
[10] Schreiber T, Richter M 1999 Int. J. Bifurcat. Chaos 9 2039
[11] Donoho D L 1995 IEEE Trans. Inf. Theory 41 613
[12] Han M, Liu Y H, Xi J H, Guo W 2007 IEEE Signal Process. Lett. 14 62
[13] Kopsinis Y, McLaughlin S 2009 IEEE Trans. Signal Process. 57 1351
[14] Wang W B, Zhang X D, Wang X L 2013 Acta Phys. Sin. 62 050201 (in Chinese) [王文波, 张晓东, 汪祥莉 2013 62 050201]
[15] Tung W W, Gao J B, Hu J, Yang L 2011 Phys. Rev. E 83 046210
[16] Gao J B, Sultan H, Hu J, Tung W W 2010 IEEE Signal Process. Lett. 17 237
[17] Chen Y, Liu X Y, Wu Z T, Fan Y, Ren Z L, Feng J C 2017 Acta Phys. Sin. 66 210501 (in Chinese) [陈越, 刘雄英, 吴中堂, 范艺, 任子良, 冯久超 2017 66 210501]
[18] Dabov K, Foi A, Katkovnik V, Egiazarian K 2007 IEEE Trans. Image Process. 16 2080
[19] Yadav S K, Sinha R, Bora P K 2015 IET Signal Process. 9 88
[20] Hou W, Feng G L, Dong W J, Li J P 2006 Acta Phys. Sin. 55 2663 (in Chinese) [侯威, 封国林, 董文杰, 李建平 2006 55 2663]
[21] Sun K H, He S B, Yin L Z, A D L·Duo L K 2012 Acta Phys. Sin. 61 130507 (in Chinese) [孙克辉, 贺少波, 尹林子, 阿地力·多力坤 2012 61 130507]
[22] Yu M Y, Sun K H, Liu W H, He S B 2018 Chaos Solitons Fractals 106 107
[23] Donoho D L, Johnstone I M 1994 Biometrika 81 425
[24] He S B, Sun K H, Wang H H 2016 Physical A 461 812
[25] Bandt C, Pompe B 2002 Phys. Rev. Lett. 88 174102
[26] Lorenz E N 1963 J. Atmos. Sci. 20 130
[27] Chen G R, Ueta T 1999 Int. J. Bifurcat. Chaos 9 1465
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[1] L J H, Lu J A, Chen S H 2002 The Analysis and Applications of Chaotic Time Series (Wuhan:Wuhan University Press) pp1-8 (in Chinese) [吕金虎, 陆君安, 陈士华 2002 混沌时间序列分析及其应用(武汉:武汉大学出版社)第1–8页]
[2] Han M, Xu M L 2013 Acta Phys. Sin. 62 120510 (in Chinese) [韩敏, 许美玲 2013 62 120510]
[3] Sun J W, Shen Y, Yin Q, Xu C J 2013 Chaos 23 013140
[4] Li G Z, Zhang B 2017 IEEE Trans. Ind. Electron. 64 2255
[5] Peng G Y, Min F H 2017 Nonlinear Dynam. 90 1607
[6] Urbanowicz K, Hołyst J A 2003 Phys. Rev. E 67 046218
[7] Feng J C 2012 Chaotic Signals and Information Processing (Beijing:Tsinghua University Press) pp32-35 (in Chinese) [冯久超 2012 混沌信号与信息处理(北京:清华大学出版社)第32–35页]
[8] Badii R, Broggi G, Derighetti B, Ravani M 1988 Phys. Rev. Lett. 60 979
[9] Cawley R, Hsu G H 1992 Phys. Rev. A 46 3057
[10] Schreiber T, Richter M 1999 Int. J. Bifurcat. Chaos 9 2039
[11] Donoho D L 1995 IEEE Trans. Inf. Theory 41 613
[12] Han M, Liu Y H, Xi J H, Guo W 2007 IEEE Signal Process. Lett. 14 62
[13] Kopsinis Y, McLaughlin S 2009 IEEE Trans. Signal Process. 57 1351
[14] Wang W B, Zhang X D, Wang X L 2013 Acta Phys. Sin. 62 050201 (in Chinese) [王文波, 张晓东, 汪祥莉 2013 62 050201]
[15] Tung W W, Gao J B, Hu J, Yang L 2011 Phys. Rev. E 83 046210
[16] Gao J B, Sultan H, Hu J, Tung W W 2010 IEEE Signal Process. Lett. 17 237
[17] Chen Y, Liu X Y, Wu Z T, Fan Y, Ren Z L, Feng J C 2017 Acta Phys. Sin. 66 210501 (in Chinese) [陈越, 刘雄英, 吴中堂, 范艺, 任子良, 冯久超 2017 66 210501]
[18] Dabov K, Foi A, Katkovnik V, Egiazarian K 2007 IEEE Trans. Image Process. 16 2080
[19] Yadav S K, Sinha R, Bora P K 2015 IET Signal Process. 9 88
[20] Hou W, Feng G L, Dong W J, Li J P 2006 Acta Phys. Sin. 55 2663 (in Chinese) [侯威, 封国林, 董文杰, 李建平 2006 55 2663]
[21] Sun K H, He S B, Yin L Z, A D L·Duo L K 2012 Acta Phys. Sin. 61 130507 (in Chinese) [孙克辉, 贺少波, 尹林子, 阿地力·多力坤 2012 61 130507]
[22] Yu M Y, Sun K H, Liu W H, He S B 2018 Chaos Solitons Fractals 106 107
[23] Donoho D L, Johnstone I M 1994 Biometrika 81 425
[24] He S B, Sun K H, Wang H H 2016 Physical A 461 812
[25] Bandt C, Pompe B 2002 Phys. Rev. Lett. 88 174102
[26] Lorenz E N 1963 J. Atmos. Sci. 20 130
[27] Chen G R, Ueta T 1999 Int. J. Bifurcat. Chaos 9 1465
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