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Extracting the signals from non-stationary time series is a difficult task in many fields such as physics, economics, and atmospheric sciences. The theory of hierarchy suggests that varying driving force leads to the non-stationary behavior, so extracting and analyzing the slowly varying features can help to study non-stationary dynamical system, which has become a compelling question recently. Slow feature analysis (SFA) is an effective technique for extracting slowly varying driving forces from quickly varying non-stationary time series. The basic idea of SFA is to nonlinearly extend the reconstructive signal into a combination form with one or higher order polynomials, and to apply the principal component analysis to this extended signal and its time derivatives. The algorithm is guaranteed to seek an optimal solution from a group of functions directly and can extract a lot of uncorrelated features that are ordered by slowness. A series of studies has shown its superiority in extracting the driving force of non-stationary time series. The extracted signal is found to be highly correlated with the real driving force. Results based on ideal models show that either the slow driving force itself or a slower subcomponent can be detected by SFA. Yet despite all that, the further investigating of SFA is still needed to reduce its uncertainty. In this study, we create two types of non-stationary models by the logistic map with time-varying parameters: one includes two varying driving forces with different time periods constraining the evolution of time series in a non-stationary way; and the other is a three-layer structure encompassing two superimposed signals in which the slower signal of driving force is modulated by the lowest one. According to the ideal model and SFA, we conduct the numerical experiments to develop corresponding analysis method and discuss its application prospect in extracting driving force signals. We find that for the system of first kind, either the slowest signal or the combination of two driving forces constructed by SFA contains some uncertain information. However, we can detect the two independent driving forces from the constructed signal by wavelet analysis. For the three-hierarchy system that includes two superimposed signals of driving force, successive applications through SFA on the original time series and the constructed SFA signal will in turn detect the slower varying driving force signal and the slowest varying driving forces signal. The successful application of SFA shows its promising prospect in analyzing the external driving forces in non-stationary system and understanding relevant dynamic mechanism.
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Keywords:
- non-stationary system /
- nonlinear system /
- driving force /
- slow feature analysis
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[18] Wiskott L 2003 Neural Comput. 15 2147
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[21] Robert M 1976 Nature 261 459
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[1] Packard N H, Crutchfield J P, Farmer J D, Shaw R S 1980 Phys. Rev. Lett. 45 712
[2] David R, Takens F 1971 Commun. Math. Phys. 20 167
[3] Bian J C, Yang P C 2003 Plateau Meteor. 22 315 (in Chinese) [卞建春, 杨培才 2003 高原气象 22 315]
[4] Tsonis A A 1996 Nature 382 700
[5] Wang S W, Zhu J H 2000 Quart. J. Appl. Meteor. 11 1 (in Chinese) [王绍武, 朱锦红 2000 应用气象学报 11 1]
[6] Wang H J, Zhou G Q, Lin Z H 2002 Climatic Environ. Res. 7 220 (in Chinese) [王会军, 周广庆, 林朝晖 2002 气候与环境研究 7 220]
[7] Ding Y H, Ren G Y, Zhao Z C, Xu Y, Luo Y, Li Q P, Zhang J 2007 Desert Oasis Meteor. 1 1 (in Chinese) [丁一汇, 任国玉, 赵宗慈, 徐影, 罗勇, 李巧萍, 张锦 2007 沙漠与绿洲气象 1 1]
[8] Wang S W 1998 Adv. Earth Sci. 13 8 (in Chinese) [王绍武 1998 地球科学进展 13 8]
[9] Chen B M, Ji L R, Yang P C, Zhang D M, Wang G L 2003 Chin. Sci. Bull. 48 513 (in Chinese) [陈伯民, 纪立人, 杨培才, 张道民, 王革丽 2003 科学通报 48 513]
[10] Wan S Q, Feng G L, Dong W J, Li J P 2005 Acta. Phys. Sin. 54 5487 (in Chinese) [万仕全, 封国林, 董文杰, 李建平 2005 物理学报 54 5487]
[11] Wang G L, Yang P C, Zhou X J 2016 Theor. Appl. Climatol. 124 985
[12] Chen X X, Wang G L, Jin L J 2015 China Environ. Sci. 35 694 (in Chinese) [陈潇潇, 王革丽, 金莲姬 2015 中国环境科学 35 694]
[13] Krner O 2002 J. Geophs. Res. 107 ACL1-1
[14] Davis A, Marshak A, Wiscombe W, Cahalan R 1996 J. Atmos. Sci. 53 1538
[15] Yang P C, Bian J C, Wang G L, Zhou X J 2003 Chin. Sci. Bull. 48 1470 (in Chinese) [杨培才, 卞建春, 王革丽, 周秀骥 2003 科学通报 48 1470]
[16] Yang P C, Zhou X J 2005 Acta. Meteor. Sinica 63 556 (in Chinese) [杨培才, 周秀骥 2005 气象学报 63 556]
[17] Verdes P F, Granitto P M, Navone H D, Ceccatto H A 2001 Phys. Rev. Lett. 87 124101
[18] Wiskott L 2003 Neural Comput. 15 2147
[19] Wiskott L, Sejnowski T J 2002 Neural Comput. 14 715
[20] Pan X N, Wang G L, Wang P F, Zhu K Y 2016 Meteor. Environ. Sci. 39 96 (in Chinese) [潘昕浓, 王革丽, 王鹏飞, 朱克云 2016 气象与环境科学 39 96]
[21] Robert M 1976 Nature 261 459
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