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.

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