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The paper gives an example of a linear dynamical system. Its dynamics is a vector autoregressive model. By using genetic algorithms to iteratively evolve different initial conditions, the degree of dynamic independence of the system can also gradually increase. At the same time, it is found that different coarse-graining scales will affect the degree of optimization to dynamic independence. The experiment finds that [[dynamic independence]] can only be achieved at certain scales, but not at other scales. Therefore, the choice of scale is also very important.

The paper gives an example of a linear dynamical system. Its dynamics is a vector autoregressive model. By using genetic algorithms to iteratively evolve different initial conditions, the degree of dynamic independence of the system can also gradually increase. At the same time, it is found that different coarse-graining scales will affect the degree of optimization to dynamic independence. The experiment finds that [[dynamic independence]] can only be achieved at certain scales, but not at other scales. Therefore, the choice of scale is also very important.

===Comparison of Several Causal Emergence Theories===

===Comparison of several causal emergence theories===

We can compare the above four different quantitative causal emergence theories from several different dimensions such as whether causality is considered, whether a coarse-graining function needs to be specified, the applicable dynamical systems, and quantitative indicators. The comparison is summarized in the following table:

We can compare the above four different quantitative causal emergence theories from several different dimensions such as whether causality is considered, whether a coarse-graining function needs to be specified, the applicable dynamical systems, and quantitative indicators. The comparison is summarized in the following table: