||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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