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A random fractal exhibits self-similarity over the scaling region, this is different from the regular fractal. The scaling region obtained by the proper method for the exact fractal dimension is very important. And the correlation dimension is one of the fractal dimensions which is used widely in many fields. Therefore, it is necessary and timely to identify the scaling region that plays a critical role in calculating the correlation dimension accurately in various chaotic systems. Visual identification is widely used to determine the scaling region as a quick and simple subjective method. However, this method may lead to an inaccurate result in Grassberger Procaccia algorithm. In order to reduce the error caused by human factors from computing the correlation dimension, a novel method of identifying the scaling region based on simulated annealing genetic fuzzy C-means clustering algorithm is proposed. This new method is based on the fluctuating characteristics that the second-order derivative of the curve within the scaling region is zero or nearly zero. Firstly, the second-order differential of the double logarithm correlation integral discrete data is calculated. Secondly, the simulated annealing genetic fuzzy C-means clustering method is used for dividing the data into three groups: positive fluctuation data, zero fluctuation data, and negative fluctuation data. The zero fluctuation data are selected to retain, the rest is excluded. Thirdly, the 3 σ criteria are used for excluding gross errors to retain those valid from the zero fluctuation data. Fourthly, the data of the consecutive nature point interval are chosen from the retained data. Finally, the regression analysis and statistical test are used to identify the scaling region from the data valid. In order to verify the effectiveness of the proposed method, it is used to simulate the Lorenz and Henon systems. The calculated results are in good agreement with the theoretical values. Experimental results show that the proposed new approach is easy to operate, more efficient and more accurate than the subjective recognition, K-means method, and 2-means method in identifying the scaling region.
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Keywords:
- correlation dimension /
- scaling region /
- fractal /
- fuzzy clustering
[1] Ostryakov V M, Usoskin I G 1990 Solar Physics 127 405
[2] Wang W J, Wu Z T 2000 Journal of Shanghai Jiaotong 34 1265 (in Chinese) [汪慰军, 吴昭同 2000 上海交通大学学报 34 1265]
[3] Casaleggio A, Corana A 2000 Chaos, Solitons & Fractals 11 2017
[4] Huang G R, Rui X F 2004 Advances in Water Science 15 255 (in Chinese) [黄国如, 芮孝芳 2004 水科学进展 15 255]
[5] Huang R S, Huang H 2007 Chaos and Its Applications (Second Edition) (Wuhan: Wuhan University Press) p217 (in Chinese) [黄润生, 黄浩 2007 混沌及其应用 (第二版) (武汉: 武汉大学出版社) 第217页]
[6] Sheng Y G, Xu Y, Li Z S, Wu D, Sun Y H, Wu Z H 2005 Acta Phys. Sin. 54 221 (in Chinese) [盛永刚, 徐耀, 李志宏, 吴东, 孙予罕, 吴中华 2005 物理学报 54 221]
[7] Yokoya N, Yamamoto K, Funakubo N 1989 Computer Vision, Graphics, and Image Processing 46 284
[8] Maragos P, Sun F K 1993 IEEE Transactions on signal Processing 41 108
[9] Kim H S, Eykholt R, Salas J D 1999 Physica D 127 48
[10] Harikrishnan K P, Misra R, Ambika G, Kembhavi A K 2006 Physica D 215 137
[11] Wang F Q, Luo C S, Cheng G X 1993 Chinese Journal of Computational Physics 10 345 (in Chinese) [汪富泉,罗朝盛,陈国先 1993 计算物理 10 345]
[12] Bolea J, Laguna P, Remartínez J M, Rovira E, Navarro A, Bailón R 2014 Computational and Mathematical Methods in Medicine 2014 1
[13] Judd K 1994 Physica D 71 421
[14] Xiong J, Chen S K, Wei W, Liu S, Guan W 2014 Acta Phys. Sin. 63 200504 (in Chinese) [熊杰, 陈绍宽, 韦伟, 刘爽, 关伟 2014 物理学报 63 200504]
[15] Dang J W, Shi Y, Huang J G 2003 Computer Engineering and Applications 23 35 (in Chinese) [党建武, 施怡, 黄建国 2003 计算机工程与应用 23 35]
[16] Wu Z C 2002 Acta Geodaetica et Cartographica Sinica 31 240 (in Chinese) [巫兆聪 2002 测绘学报 31 240]
[17] Du B Q, Jia Z W, Tang G J 2013 Journal of Vibration and Shock 32 40 (in Chinese) [杜必强, 贾子文, 唐贵基 2013 震动与冲击 32 40]
[18] Wang C D, Ling D, Miao Q 2012 Computer Engineering and Applications 48 9 (in Chinese) [王成栋, 凌丹, 苗强 2012 计算机工程与应用 48 9]
[19] Wu H S, Ni L P, Zhang F M, Zhou X, Du J Y 2014 Control and Decision 29 455 (in Chinese) [吴虎胜, 倪丽萍, 张凤鸣, 周漩, 杜继勇 2014 控制与决策 29 455]
[20] Yang H Y, Ye H, Wang G Z, Pan G D 2008 3rd IEEE Conference on Industrial Electronics and Applications Singapore, Jun 3-5, 2008 p1383
[21] Ji C C, Zhu H, Jiang W 2010 Chinese Science Bulletin 31 3069 (in Chinese) [姬翠翠, 朱华, 江炜 2010 科学通报 31 3069]
[22] Wang A L, Yang C X 2002 Acta Phys. Sin. 51 2719 (in Chinese) [王安良, 杨春信 2002 物理学报 51 2719]
[23] Grassberger P, Procaccia I 1983 Physical Review Letters 50 346
[24] Grassberger P, Procaccia I 1983 Physica D 9 189
[25] Kirkpatrick S, Gelatt J C D, Vecchi M P 1983 Science 220 671
[26] Holland J H 1975 Adaptation in Natural and Artifical Systems (Ann Arbor: The University of Michigan Press)
[27] Jain A K 2010 Pattern Recognition Letters 31 651
[28] Bai L Y, Hu S Y, Liu S H 2005 Computer Engineering and Applications 41 56 (in Chinese) [白莉媛, 胡声艳, 刘素华 2005 计算机工程与应用 41 56]
[29] Liang J W, Chen L C, He G 2001 Error Theory and Data Processing (Revised Edition) (Beijing: China Metrology Press) p57 (in Chinese) [梁晋文, 陈林才, 何贡 2001 误差理论与数据处理 (修订版) (北京: 中国计量出版社) 第57页]
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[1] Ostryakov V M, Usoskin I G 1990 Solar Physics 127 405
[2] Wang W J, Wu Z T 2000 Journal of Shanghai Jiaotong 34 1265 (in Chinese) [汪慰军, 吴昭同 2000 上海交通大学学报 34 1265]
[3] Casaleggio A, Corana A 2000 Chaos, Solitons & Fractals 11 2017
[4] Huang G R, Rui X F 2004 Advances in Water Science 15 255 (in Chinese) [黄国如, 芮孝芳 2004 水科学进展 15 255]
[5] Huang R S, Huang H 2007 Chaos and Its Applications (Second Edition) (Wuhan: Wuhan University Press) p217 (in Chinese) [黄润生, 黄浩 2007 混沌及其应用 (第二版) (武汉: 武汉大学出版社) 第217页]
[6] Sheng Y G, Xu Y, Li Z S, Wu D, Sun Y H, Wu Z H 2005 Acta Phys. Sin. 54 221 (in Chinese) [盛永刚, 徐耀, 李志宏, 吴东, 孙予罕, 吴中华 2005 物理学报 54 221]
[7] Yokoya N, Yamamoto K, Funakubo N 1989 Computer Vision, Graphics, and Image Processing 46 284
[8] Maragos P, Sun F K 1993 IEEE Transactions on signal Processing 41 108
[9] Kim H S, Eykholt R, Salas J D 1999 Physica D 127 48
[10] Harikrishnan K P, Misra R, Ambika G, Kembhavi A K 2006 Physica D 215 137
[11] Wang F Q, Luo C S, Cheng G X 1993 Chinese Journal of Computational Physics 10 345 (in Chinese) [汪富泉,罗朝盛,陈国先 1993 计算物理 10 345]
[12] Bolea J, Laguna P, Remartínez J M, Rovira E, Navarro A, Bailón R 2014 Computational and Mathematical Methods in Medicine 2014 1
[13] Judd K 1994 Physica D 71 421
[14] Xiong J, Chen S K, Wei W, Liu S, Guan W 2014 Acta Phys. Sin. 63 200504 (in Chinese) [熊杰, 陈绍宽, 韦伟, 刘爽, 关伟 2014 物理学报 63 200504]
[15] Dang J W, Shi Y, Huang J G 2003 Computer Engineering and Applications 23 35 (in Chinese) [党建武, 施怡, 黄建国 2003 计算机工程与应用 23 35]
[16] Wu Z C 2002 Acta Geodaetica et Cartographica Sinica 31 240 (in Chinese) [巫兆聪 2002 测绘学报 31 240]
[17] Du B Q, Jia Z W, Tang G J 2013 Journal of Vibration and Shock 32 40 (in Chinese) [杜必强, 贾子文, 唐贵基 2013 震动与冲击 32 40]
[18] Wang C D, Ling D, Miao Q 2012 Computer Engineering and Applications 48 9 (in Chinese) [王成栋, 凌丹, 苗强 2012 计算机工程与应用 48 9]
[19] Wu H S, Ni L P, Zhang F M, Zhou X, Du J Y 2014 Control and Decision 29 455 (in Chinese) [吴虎胜, 倪丽萍, 张凤鸣, 周漩, 杜继勇 2014 控制与决策 29 455]
[20] Yang H Y, Ye H, Wang G Z, Pan G D 2008 3rd IEEE Conference on Industrial Electronics and Applications Singapore, Jun 3-5, 2008 p1383
[21] Ji C C, Zhu H, Jiang W 2010 Chinese Science Bulletin 31 3069 (in Chinese) [姬翠翠, 朱华, 江炜 2010 科学通报 31 3069]
[22] Wang A L, Yang C X 2002 Acta Phys. Sin. 51 2719 (in Chinese) [王安良, 杨春信 2002 物理学报 51 2719]
[23] Grassberger P, Procaccia I 1983 Physical Review Letters 50 346
[24] Grassberger P, Procaccia I 1983 Physica D 9 189
[25] Kirkpatrick S, Gelatt J C D, Vecchi M P 1983 Science 220 671
[26] Holland J H 1975 Adaptation in Natural and Artifical Systems (Ann Arbor: The University of Michigan Press)
[27] Jain A K 2010 Pattern Recognition Letters 31 651
[28] Bai L Y, Hu S Y, Liu S H 2005 Computer Engineering and Applications 41 56 (in Chinese) [白莉媛, 胡声艳, 刘素华 2005 计算机工程与应用 41 56]
[29] Liang J W, Chen L C, He G 2001 Error Theory and Data Processing (Revised Edition) (Beijing: China Metrology Press) p57 (in Chinese) [梁晋文, 陈林才, 何贡 2001 误差理论与数据处理 (修订版) (北京: 中国计量出版社) 第57页]
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