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研究生:陳學金
研究生(外文):Hock-King Ting
論文名稱:Convergent Cross Mapping (CCM) 方法對預測因果關係的評估
論文名稱(外文):Assessment of Convergent Cross Mapping (CCM) Method by Using time-series data of Known Causality
指導教授:黎璧賢黎璧賢引用關係
指導教授(外文):Pik-Yin Lai
學位類別:碩士
校院名稱:國立中央大學
系所名稱:物理學系
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2016
畢業學年度:104
語文別:英文
論文頁數:132
中文關鍵詞:因果關係
外文關鍵詞:Convergent Cross MappingCCMcausality
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在許多不同的研究領域,揭開兩個物理量X和Y之間的因果關係
是一項重要的工作。X→Y表示X影響Y。Convergent Cross
Mapping (CCM) 是一個檢測因果關係的方法。這論文在介紹了
CCM 的運作原理之後,我們將使用兩個已知因果關係的時間序列
X(t)和Y(t) 去對CCM 進行評估,且 X(t)和Y(t) 具有相同或不同
(即混沌、週期振盪以及穩定點) 的動力學行為。當X(t)和Y(t) 同
步的時候,CCM 無法區分 X和Y 的因果關係。最後,這論文將討
論透過三個節點和一百個節點的環網絡對CCM 所進行的評估結果。
In many different areas of research, it is important to uncover the interaction or causal relation between two dynamical quantities (such as X(t) and Y (t)). If X is the cause and Y is the effect, then X will influence/drive Y . Such a causality or directed interaction can be denoted as: X drives Y . Convergent Cross Mapping (CCM) is a method used to detect the causality between X and Y .
In this thesis, the working principles behind CCM (i.e. state space reconstruction and cross estimation) will first be introduced. Then, the CCM method is assessed by using it to detect the causality of X and Y , in which they were generated by solving a system of coupled dynamical equations numerically. Since X(t) and Y(t) are two time series of known causality (e.g. X drives Y), the accuracy of the CCM method could be assessed by plotting sigma(X drives Y) versus gXY and sigma(Y drives X) versus gXY on the same graph, where sigma is a CCM accuracy indicator and gXY is the
coupling strength of X drives Y. For further assessment, CCM method is applied to detect the causality of X and Y of different combinations of dynamical behaviours (i.e. chaos, periodic oscillations and stable fixed point). It was found that when X(t) and Y(t) synchronize, CCM is unable to distinguish the true causality from the non-existing causality.
From the 3-node (X drives Y drives Z) motifs analysis, it was found that CCM method would misinterpret the existence of X drives Z, when both X and Y synchronize. Apart from that, the existence of the conflicting information would also affect the sigma value computed by the CCM method.
Finally, the sigma versus g curves for 2-node and 100-node (ring network) of different connectivities and ranges of g are plotted on the same graph. This is to investigate the universality of the sigma versus g relation. It was found that the true causality curve of 2-node unidirectional case fits the Power Law: sigma proportional to g^(-0.6). There
are a number of cases of points which fall on or close to the true causality curve of 2-node unidirectional case. The deviations of the rests of the points from the true causality curve are either due to the A drives B drives C effect or the conflicting information problem or both.
1 Introduction 1
1.1 Overview . . . . . . . . . . . . . . . . . . . . ..1
1.2 Introduction to Convergent Cross Mapping (CCM) Method . . . . . 6
1.3 Terminology in Nonlinear Dynamics . . . . . . . . . . . . . . . . . . 10
1.3.1 Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3.2 Limit Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3.3 Chaotic/Strange Attractor . . . . . . . . . . . . . . . . . . . 14
1.3.4 Synchronization . . . . . . . . . . . . . . . . . . . . . . . . . 16

2 Methodology 21
2.1 Procedures in CCM Performance Analysis . . . . . . . . . . . . . . 21
2.1.1 Data Management Before CCM Analysis . . . . . . . . . . . 21
2.1.2 Steps in Performing the CCM Analysis . . . . . . . . . . . . 22
2.2 Further Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.1 Investigation of the Existence of a Universal Empirical Relation Between sigma and Coupling Strength g . . . . . . . . . . 32
2.2.2 A Study of 3-node Configuration . . . . . . . . . . . . . . . 32
2.2.3 Performing a Network (100 Nodes) Study . . . . . . . . . . . 33

3 Results &; Discussions 35
3.1 Testing the accuracy of the CCM Method (Using 2-node motifs) . . 35
3.1.1 2 Independent (Random) Time Series . . . . . . . . . . . . . 35
3.1.2 2 Identical (Random) Time Series . . . . . . . . . . . . . . . 38
3.1.3 2 Random Time Series With a Phase Difference . . . . . . . 40
3.1.4 2 Chaotic Lorenz Systems . . . . . . . . . . . . . . . . . . . 42
3.1.5 2 Glycolytic Oscillators (Slow Oscillator Driving the Fast Oscillator) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.6 2 Glycolytic Oscillators (Fast Oscillator Driving the Slow Oscillator) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.7 Chaotic Lorenz Drives Lorenz Oscillator . . . . . . . . . . . 53
3.1.8 Lorenz Oscillator Drives Chaotic Lorenz . . . . . . . . . . . 56
3.1.9 Chaotic Lorenz Drives Stable Fixed Point Dynamics . . . . . 61
3.1.10 Coupled Logistic Map of 2 Nodes . . . . . . . . . . . . . . . 63
3.2 3-node motifs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
3.3 Network Study (1D ring with 100 nodes) . . . . . . . . . . . . . . . 79
3.4 Testing the accuracy of the CCM Method (2-node motifs with noise) 92
3.4.1 2 coupled map lattice (CML) of Logistic Maps with noise (chaotic when uncoupled) . . . . . . . . . . . . . . . . . . . 92
3.4.2 2 CML of Logistic maps with noise (periodic when uncoupled) 99

4 Conclusion, Future Work &; Recommendation 105
4.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
4.2 Future Work &; Recommendation . . . . . . . . . . . . . . . . . . . 109

Bibliography 111
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