观测数据的降噪及动态建模技术的研究
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摘要
本文介绍了一种新型利用相空间重构(reconstructed phase space)和递归图法(recurrence plots)的非线性降噪方法:搜索平均法(searching average method,SAM)。应用该方法时必须确定一个局部邻域尺度,对局部邻域尺度的估计实质上就是要对含有噪声的混沌信号的噪声水平进行正确地估计。结合递归图法估计了最佳的局部邻域尺度,显著提高了降噪效果。同其他非线性降噪方法相比,SAM具有节省机时、精度较高等优点。将该方法应用于被高斯白噪声所污染的Henon映射时间序列,通过对比证实该方法能够取得比较好的降噪效果。利用气象局公布的呼玛站点1960年-2000年的逐日气温观测资料,并利用非线性预报方法衡量降噪的效果。比较降噪前后正规化预报误差的变化发现,在相同的预报步数条件下,降噪后数据的正规化预报误差的值比降噪前小许多,实际温度资料的可预报性有所提高。最后将该方法应用于中国气象局公布全国435站1960年-2000年的逐日气温观测时间序列发现,当预报步数相同时,所有站点资料的正规化预报误差均有明显减小,可预报性有了较大的提高。当预报步数为1-2步时,可预报性的增加由南向北呈递增趋势。
考虑到气候系统的层次性和奇异性特征,针对非线性时间序列信息的分离与提取这一关键的科学问题,引入了一种基于数据的机理模式(Date- based Mechanism Model),即先寻找和因变量关系密切的自变量,分析它们之间可能存在的函数关系,然后通过反演,建立一个以观测数据为基础的机理模式。先利用自记忆的思想设定一个可能的非线性常微分方程,然后运用双向差分方法对其进行反导得到方程的具体形式,称为反导模式。随后再利用自忆性原理推导一个差分-积分方程,运用最小二乘法求得该方程离散化形式中的记忆系数,可由此建立模式并进行预报试验。以长江三角洲地区1951-1998年的夏季降水为观测数据,分析了降水概率随时间的演化的规律,并对其成因进行了分析,并以此为例进行建模。将试报结果与实测时序作比较,表明经过自记忆处理之后的模式,其预报效果有较大改进。在此基础上,利用530年旱、涝等级序列,将干旱、洪涝纳入小概率事件的统一框架,从全局性观点的出发,利用降水概率随时间的分布特征探索小概率事件的时空演化规律。
In this paper we introduce a new method of noise reduction based on reconstructed phase space and recurrence plots: searching average method (SAM). In the application of this method we should determine a localized scale, and the essence of localized scale determination is the appropriate evaluation of noise level of chaotic signal. In this contribution we evaluate the optimal localized scale through recurrence plots; by doing so, we enhance the effectiveness of noise reduction significantly. SAM is more time saving and accurate compare with other nonlinear denoise method. We apply SAM in the processing of Henon mapping time series shuffled by white gaussian noise and through comparison we verified the effectiveness of this method in noise reduction. As an illustration of the preceding considerations we study here time series data of daily temperatures from 1960 to 2000 of Huma distributed by CMA, and evaluate the effect of denoise through nonlinear prediction methods. Through comparison of normalized prediction error before and after noise reduction we find that with the same number of predictive steps, the errors of normalized prediction is significantly reduced which means through noise reduction by SAM we can enhance the predictability of temperature observational data. Using SAM we also making noise reduction on daily temperature series of 435 stations in China from 1960 to 2000 distributed by CMA, obtained the change of normalized predictive error distribution before and after noise reduction, which indicates the effectiveness of denoise in different regions. Results indicate that with the same number of predictive steps, normalized predictive errors of all stations are significantly reduced, e.g. an obvious increasing of predictability. With numbers of predictive steps less than three the predictability shows a tendency of increasing from south to north China.
The extraction of information from nonlinear time series became crucial because of the hierarchy and nonstationarity of climate system. The recently raised Data-based Mechanism Model is to find those variables which strongly correlated with dependent variables and analyzed its possible correlations, then construct a model based on observational data through inversion. In this paper first we construct a possible nonlinear ordinary differential equation based on regression, and then through bi-directional difference we obtain the concrete form of equation, e.g. inversion model. Based on principle of regression we constructed a difference-integration equation, and obtain the memory index of discrete form of equation through least square method. Finally we develop a model for predictive test. In this paper we investigate the law of probability variation of precipitation using observational data of summer precipitation from 1951 to 1998 in the Yangtze delta, analyzed its caused and construct a model for prediction. Through comparison we find the ratio of correct predictions are greatly enhanced. And base on this, we study the character of temporal distributions of precipitation probability from the angle of small probability events using drought indices of 530 years, and try to induce law of spatiotemporal variation of small probability events.
考虑到气候系统的层次性和奇异性特征,针对非线性时间序列信息的分离与提取这一关键的科学问题,引入了一种基于数据的机理模式(Date- based Mechanism Model),即先寻找和因变量关系密切的自变量,分析它们之间可能存在的函数关系,然后通过反演,建立一个以观测数据为基础的机理模式。先利用自记忆的思想设定一个可能的非线性常微分方程,然后运用双向差分方法对其进行反导得到方程的具体形式,称为反导模式。随后再利用自忆性原理推导一个差分-积分方程,运用最小二乘法求得该方程离散化形式中的记忆系数,可由此建立模式并进行预报试验。以长江三角洲地区1951-1998年的夏季降水为观测数据,分析了降水概率随时间的演化的规律,并对其成因进行了分析,并以此为例进行建模。将试报结果与实测时序作比较,表明经过自记忆处理之后的模式,其预报效果有较大改进。在此基础上,利用530年旱、涝等级序列,将干旱、洪涝纳入小概率事件的统一框架,从全局性观点的出发,利用降水概率随时间的分布特征探索小概率事件的时空演化规律。
In this paper we introduce a new method of noise reduction based on reconstructed phase space and recurrence plots: searching average method (SAM). In the application of this method we should determine a localized scale, and the essence of localized scale determination is the appropriate evaluation of noise level of chaotic signal. In this contribution we evaluate the optimal localized scale through recurrence plots; by doing so, we enhance the effectiveness of noise reduction significantly. SAM is more time saving and accurate compare with other nonlinear denoise method. We apply SAM in the processing of Henon mapping time series shuffled by white gaussian noise and through comparison we verified the effectiveness of this method in noise reduction. As an illustration of the preceding considerations we study here time series data of daily temperatures from 1960 to 2000 of Huma distributed by CMA, and evaluate the effect of denoise through nonlinear prediction methods. Through comparison of normalized prediction error before and after noise reduction we find that with the same number of predictive steps, the errors of normalized prediction is significantly reduced which means through noise reduction by SAM we can enhance the predictability of temperature observational data. Using SAM we also making noise reduction on daily temperature series of 435 stations in China from 1960 to 2000 distributed by CMA, obtained the change of normalized predictive error distribution before and after noise reduction, which indicates the effectiveness of denoise in different regions. Results indicate that with the same number of predictive steps, normalized predictive errors of all stations are significantly reduced, e.g. an obvious increasing of predictability. With numbers of predictive steps less than three the predictability shows a tendency of increasing from south to north China.
The extraction of information from nonlinear time series became crucial because of the hierarchy and nonstationarity of climate system. The recently raised Data-based Mechanism Model is to find those variables which strongly correlated with dependent variables and analyzed its possible correlations, then construct a model based on observational data through inversion. In this paper first we construct a possible nonlinear ordinary differential equation based on regression, and then through bi-directional difference we obtain the concrete form of equation, e.g. inversion model. Based on principle of regression we constructed a difference-integration equation, and obtain the memory index of discrete form of equation through least square method. Finally we develop a model for predictive test. In this paper we investigate the law of probability variation of precipitation using observational data of summer precipitation from 1951 to 1998 in the Yangtze delta, analyzed its caused and construct a model for prediction. Through comparison we find the ratio of correct predictions are greatly enhanced. And base on this, we study the character of temporal distributions of precipitation probability from the angle of small probability events using drought indices of 530 years, and try to induce law of spatiotemporal variation of small probability events.
引文
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[2] David P.Feldman, James P. Crutchfield, Measures of statistical complexity: Why? Physics Letters A.,1998, Vo1238:244-252[3]
[3] M. E. Tomes, L. G. Gamero, Relative complexity changes in time series using information measures, Physica A.,2000, Vo1286:457-473
[4] J. J. Zebrowski,W. Poplawska, et. al.,Symbolic dynamics and complexity in a physiological time series, Chaos Solitons and Fractals, 2000, Volll: 1061一1075
[5] Steven M. Manson, Simplifying complexity: a review of complexity theory, Geoforum, 2001,Vol (32):405-414
[6] G.Boffetta, M.Cencini, M.Falcioni, A.Vulpiani,Predictability: a way to characterize complexity, Physics Reports, 2002,Vo1356:367-474
[7] How chaotic is the Weather and Where? Holland, PNAS: 1999,Vol(25):14210-14215
[8]苗东升,论复杂性,自然辩证法通讯,vo1(6) : 87-92, 2000
[9] M. Wordlop著,陈玲译,复杂一诞生于秩序与混沌边缘的科学,生活。读书。新知三联出版社,1997
[10] Nicolis G, Prigogine I.,Exploring Complexity, New York: W H Freeman&Co, 1986;中译本:尼科里斯,普里高津,探索复杂性,四川教育出版社,1989
[11]钱学森,于景元,戴汝为,一个科学新领域一开放的复杂巨系统及其方法论,自然杂志,1990, Vo113 (1)
[12] D. Rind,Complexity and Climate,Science,1999, Vo1284 (2):105-107
[13] Cheng SiWei,Complexity Science&Management, CAS BULLETIN,1999,Vo113 (3):176-183
[14] Kantz H,Shreiber T. Nolinear Time Series Analysis. Cambridge. Cambridge University Press. 2004.
[15] Tanaka N、Okamoto H and Natio M,Estimating the amplitude of measurement noise present in chaotic time series,Chaos,1999,9(2): 436-444
[16] Kantz H,A robust method to estimate the maximal Lyapunov exponent of a time series, Phys.Lett.A,1994,185(1):77-87
[17]杨绍清,章新华,赵长安,一种最大李雅普诺夫指数估计的稳健算法,物理学报,2000,49(4): 636-640
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