混沌時間序列中最大Lyapunov指數(shù)與關(guān)聯(lián)維數(shù)的無標度區(qū)間自動識別研究
發(fā)布時間：2018-07-31 20:37
【摘要】：混沌與分形是非線性科學中一個重要的分支,其已經(jīng)滲透到天體物理、醫(yī)學、化學、計算機等各個學科,具有廣泛的應(yīng)用前景。隨著計算機的快速發(fā)展,數(shù)值方法刻畫混沌特征成為可能,其中從時間序列準確計算最大Lyapunov指數(shù)和關(guān)聯(lián)維數(shù)尤為重要。目前,計算最大Lyapuno v指數(shù)的小數(shù)據(jù)量方法和計算關(guān)聯(lián)維數(shù)的G-P方法是其中最主流的方法,但是在計算過程中人為選擇無標度區(qū)間,導(dǎo)致計算結(jié)果不準確,因此本文對無標度區(qū)間自動識別問題進行了研究,提出了新方法,并將該方法應(yīng)用到太陽長期活動規(guī)律的研究,取得了一些有意義的結(jié)果。本文的創(chuàng)新點如下：一、在小數(shù)據(jù)量方法計算最大Lyapunov指數(shù)的過程中,為了減少人為因素識別無標度區(qū)間帶來的誤差,提出一種基于模糊C均值聚類的方法,該方法根據(jù)平均發(fā)散程度指數(shù)曲線的變化特征,利用模糊C均值聚類進行識別。首先,利用小數(shù)據(jù)量方法對混沌時間序列進行計算得到平均發(fā)散程度指數(shù)集合。其次,利用模糊C均值聚類算法對平均發(fā)散程度指數(shù)集合進行分類,保留不飽和數(shù)據(jù)。然后,對不飽和的二階差分數(shù)據(jù)進行分類,得到零附近波動數(shù)據(jù)并剔除粗大誤差,再對保留的有效數(shù)據(jù)利用統(tǒng)計方法識別出無標度區(qū)間。最后,對無標度區(qū)間對應(yīng)的點進行最小二乘法擬合得到最大Lyapunov指數(shù)。通過實例仿真,新方法所得結(jié)果接近參考值。為了提高該方法的計算精度,利用模擬退火算法與遺傳算法對其優(yōu)化,優(yōu)化后的方法比已知方法,在大多數(shù)情況下,計算精度更高,但是在計算效率方面,基于模糊C均值聚類的新方法表現(xiàn)最好。二、將上述方法的思想應(yīng)用于計算混沌時間序列的關(guān)聯(lián)維數(shù),用來減少人為因素識別無標度區(qū)間帶來的誤差。該方法根據(jù)無標度區(qū)間對應(yīng)曲線的二階導(dǎo)數(shù)在零附近波動的變化特征,利用分類算法進行識別。首先計算雙對數(shù)關(guān)聯(lián)積分的二階差分,然后利用模擬退火遺傳模糊C均值聚類方法對該數(shù)據(jù)進行分類,選出在零附近波動的數(shù)據(jù),再剔除粗大誤差保留有效數(shù)據(jù),最后進行統(tǒng)計分析識別出線性度最好的作為無標度區(qū)間。應(yīng)用新方法對兩個著名的混沌系統(tǒng)Lorenz 和 Henon進行了仿真,計算結(jié)果與參考值非常吻合。實驗表明,所提出的新方法具有一定的抗噪性,與主觀識別、基于K-means方法和基于2-means方法比較,’可以更加精確的識別無標度區(qū)間。三、將新方法以及其它非線性分析技術(shù)應(yīng)用到太陽的長期活動規(guī)律研究中,研究結(jié)果表明：(1)太陽長期活動趨勢與過去的演化過程是密切相關(guān),具有長期記憶性；(2)太陽長期活動展示著低維混沌,由于其混沌的特征,只能進行中短期的太陽活動預(yù)報；(3)太陽黑子面積的變化規(guī)律比太陽黑子數(shù)更復(fù)雜,這與它們本身的物理意義相符合；(4)太陽黑子面積相對于太陽黑子數(shù)作為研究太陽活動的指標更有效。四、將新方法與其它方法結(jié)合對1952年2月至1998年6月的極區(qū)光斑和太陽黑子數(shù)的兩種太陽活動指標進行研究,通過實例仿真,得出了一些有意義的結(jié)果：(1)南北半球的太陽活動的混沌與分形特征存在統(tǒng)計上的差異性；(2)太陽高緯度活動現(xiàn)象比低緯度活動現(xiàn)象具有更強的混沌程度和更復(fù)雜的分形結(jié)構(gòu),其中北半球的高緯度太陽活動規(guī)律最復(fù)雜。
[Abstract]:Chaos and fractal are an important branch of nonlinear science. It has penetrated into various disciplines such as astrophysics, medicine, chemistry, computer and so on. It has extensive application prospects. With the rapid development of the computer, it is possible to describe the chaotic characteristics by numerical methods, in which the maximum Lyapunov index and the correlation dimension are accurately calculated from the time series. It is particularly important. At present, the calculation method of the maximum Lyapuno V index and the G-P method for calculating the correlation dimension are the most mainstream methods. However, in the calculation process, the artificial selection of the scale-free interval, which leads to the inaccuracy of the calculation results, is carried out in this paper. A new method is proposed and a new method is put forward. This method is applied to the study of the law of the long-term activity of the sun, and some meaningful results have been obtained. The innovation points of this paper are as follows: first, in the process of calculating the maximum Lyapunov exponent of small amount of data, a method based on fuzzy C mean clustering is proposed in order to reduce the error caused by the identification of the scale free interval of human factors. According to the variation characteristics of the average divergence index curve, the fuzzy C means clustering is used to identify. First, the mean divergence index set is obtained by using the small data quantity method to calculate the chaotic time series. Secondly, the fuzzy C mean clustering algorithm is used to classify the average divergence index set, and the unsaturated data is retained. After that, we classify the two order differential data of the unsaturated zone, get the wave data near zero and eliminate the rough error, and then identify the scale-free interval for the reserved effective data using statistical methods. Finally, the least square fitting is used to get the maximum Lyapunov exponent of the points corresponding to the scale-free interval. It is close to reference value. In order to improve the calculation accuracy of this method, the simulated annealing algorithm and genetic algorithm are used to optimize it. The optimization method is more accurate than the known method in most cases, but in the calculation efficiency, the new method based on fuzzy C means clustering is the best. Two, the idea of the above method is applied to the method. The correlation dimension of the chaotic time series is calculated to reduce the error caused by the artificial factor identification of the scale-free interval. The method is identified by the classification algorithm based on the variation characteristics of the fluctuation of the two order derivative of the scale-free interval corresponding to zero. First, the two order difference of the double logarithmic correlation integral is calculated, and then the simulated annealing remains are used. The fuzzy C means clustering method is used to classify the data, select the data that fluctuate near zero, and then remove the gross error and retain the effective data. Finally, the statistical analysis is used to identify the best line free degree as a scale-free interval. The new method is used to simulate the two famous chaotic systems, Lorenz and Henon, and the results and reference are calculated. The experimental results show that the proposed method has a certain noise resistance, and the subjective recognition, based on the K-means method and the 2-means based method, can more accurately identify the scale-free interval. Three, the new method and other nonlinear analysis techniques are applied to the study of the long-term activity law of the sun, and the results show that the results show that the new method and other nonlinear analysis techniques are applied to the study of the long-term activity of the sun. (1) the long-term activity trend of the sun is closely related to the evolution process of the past, and has long memory. (2) the long-term activity of the sun shows low dimensional chaos. Because of its chaotic characteristics, the solar activity can only be predicted in the middle and short term; (3) the variation of sunspot area is more complex than the number of sunspots, which is related to the physics of their own. The significance is conformed; (4) the area of sunspot area is more effective than the sunspot number. Four, the new method and other methods are combined to study two kinds of solar activity indexes of the polar spot and sunspot number from February 1952 to June 1998, and some meaningful results are obtained through the simulation of the case. (1) there is a statistical difference between the chaotic and fractal characteristics of the solar activity in the northern and southern hemispheres; (2) the high latitude activity of the sun is more chaotic and more complex than the low latitude activity, and the high latitude solar activity in the northern hemisphere is the most complex.
【學位授予單位】：中國科學院重慶綠色智能技術(shù)研究院
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O211.61
本文編號：2156761
[Abstract]:Chaos and fractal are an important branch of nonlinear science. It has penetrated into various disciplines such as astrophysics, medicine, chemistry, computer and so on. It has extensive application prospects. With the rapid development of the computer, it is possible to describe the chaotic characteristics by numerical methods, in which the maximum Lyapunov index and the correlation dimension are accurately calculated from the time series. It is particularly important. At present, the calculation method of the maximum Lyapuno V index and the G-P method for calculating the correlation dimension are the most mainstream methods. However, in the calculation process, the artificial selection of the scale-free interval, which leads to the inaccuracy of the calculation results, is carried out in this paper. A new method is proposed and a new method is put forward. This method is applied to the study of the law of the long-term activity of the sun, and some meaningful results have been obtained. The innovation points of this paper are as follows: first, in the process of calculating the maximum Lyapunov exponent of small amount of data, a method based on fuzzy C mean clustering is proposed in order to reduce the error caused by the identification of the scale free interval of human factors. According to the variation characteristics of the average divergence index curve, the fuzzy C means clustering is used to identify. First, the mean divergence index set is obtained by using the small data quantity method to calculate the chaotic time series. Secondly, the fuzzy C mean clustering algorithm is used to classify the average divergence index set, and the unsaturated data is retained. After that, we classify the two order differential data of the unsaturated zone, get the wave data near zero and eliminate the rough error, and then identify the scale-free interval for the reserved effective data using statistical methods. Finally, the least square fitting is used to get the maximum Lyapunov exponent of the points corresponding to the scale-free interval. It is close to reference value. In order to improve the calculation accuracy of this method, the simulated annealing algorithm and genetic algorithm are used to optimize it. The optimization method is more accurate than the known method in most cases, but in the calculation efficiency, the new method based on fuzzy C means clustering is the best. Two, the idea of the above method is applied to the method. The correlation dimension of the chaotic time series is calculated to reduce the error caused by the artificial factor identification of the scale-free interval. The method is identified by the classification algorithm based on the variation characteristics of the fluctuation of the two order derivative of the scale-free interval corresponding to zero. First, the two order difference of the double logarithmic correlation integral is calculated, and then the simulated annealing remains are used. The fuzzy C means clustering method is used to classify the data, select the data that fluctuate near zero, and then remove the gross error and retain the effective data. Finally, the statistical analysis is used to identify the best line free degree as a scale-free interval. The new method is used to simulate the two famous chaotic systems, Lorenz and Henon, and the results and reference are calculated. The experimental results show that the proposed method has a certain noise resistance, and the subjective recognition, based on the K-means method and the 2-means based method, can more accurately identify the scale-free interval. Three, the new method and other nonlinear analysis techniques are applied to the study of the long-term activity law of the sun, and the results show that the results show that the new method and other nonlinear analysis techniques are applied to the study of the long-term activity of the sun. (1) the long-term activity trend of the sun is closely related to the evolution process of the past, and has long memory. (2) the long-term activity of the sun shows low dimensional chaos. Because of its chaotic characteristics, the solar activity can only be predicted in the middle and short term; (3) the variation of sunspot area is more complex than the number of sunspots, which is related to the physics of their own. The significance is conformed; (4) the area of sunspot area is more effective than the sunspot number. Four, the new method and other methods are combined to study two kinds of solar activity indexes of the polar spot and sunspot number from February 1952 to June 1998, and some meaningful results are obtained through the simulation of the case. (1) there is a statistical difference between the chaotic and fractal characteristics of the solar activity in the northern and southern hemispheres; (2) the high latitude activity of the sun is more chaotic and more complex than the low latitude activity, and the high latitude solar activity in the northern hemisphere is the most complex.
【學位授予單位】：中國科學院重慶綠色智能技術(shù)研究院
【學位級別】：博士
【學位授予年份】：2016
【分類號】：O211.61
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