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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 dynamical decoupling 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 decoupling 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 dynamical decoupling 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 decoupling 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===
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, and obtain the following table:
{| class="wikitable"
|+Comparison of Different Quantitative Emergence Theories
|-
!Method!!Consider Causality?!!Involve Coarse-graining?!!Applicable Dynamical Systems!!Measurement Index
|-
|Hoel's causal emergence theory||Dynamic causality, the definition of EI introduces do-intervention||Requires specifying a coarse-graining method||Discrete Markov dynamics||Dynamic causality: effective information
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|Rosas's causal emergence theory||Approximation by correlation characterized by mutual information||When judged based on synergistic information, no coarse-graining is involved. When calculated based on redundant information, a coarse-graining method needs to be specified.||Arbitrary dynamics||Information decomposition: synergistic information or redundant information
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|Causal emergence theory based on reversibility||Dynamic causality, EI is equivalent to approximate dynamical reversibility||Does not depend on a specific coarse-graining strategy||Discrete Markov dynamics||Approximate dynamical reversibility: <math>\Gamma_{\alpha}</math>
|-
|Dynamic independence||Granger causality||Requires specifying a coarse-graining method||Arbitrary dynamics||Dynamic independence: transfer entropy
|}