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===Reduction of dynamical models===

===Reduction of dynamical models===

An important indicator of causal emergence is the selection of coarse-graining strategies. When the microscopic model is known, coarse-graining the microscopic state is equivalent to performing '''model reduction''' on the microscopic model. Model reduction is an important subfield in control theory. Antoulas once wrote a related review article <ref name=":15">Antoulas A C. An overview of approximation methods for large-scale dynamical systems[J]. Annual reviews in Control, 2005, 29(2): 181-190.</ref>.

An important indicator of causal emergence is the selection of coarse-graining strategies. When the microscopic model is known, coarse-graining the microscopic state is equivalent to performing a '''model reduction''' on the microscopic model. Model reduction is an important subfield in control theory. Antoulas once wrote a related review article <ref name=":15">Antoulas A C. An overview of approximation methods for large-scale dynamical systems[J]. Annual reviews in Control, 2005, 29(2): 181-190.</ref>.

In general, the error loss function of the output function before and after model reduction can be used to judge the coarse-graining parameters. This process assumes that system reduction inherently results in information loss. Therefore, minimizing the error is the only way to judge the effectiveness of the reduction method. However, from the perspective of causal emergence, [[effective information]] will increase due to dimensionality reduction. This highlights the primary difference between coarse-graining strategies in causal emergence research and model reduction in control theory. For stochastic systems <ref name=":17" />, directly calculating the loss function can be unstable due to randomness, making reduction effectiveness harder to measure. The effective information and causal emergence index based on stochastic dynamical systems can increase the effectiveness of evaluation indicators to a certain extent. Additionally, they make the control research of stochastic dynamical systems more rigorous.

In general, the error loss function of the output function before and after model reduction can be used to judge the coarse-graining parameters. This process assumes that system reduction inherently results in information loss. Therefore, minimizing the error is the only way to judge the effectiveness of the reduction method. However, from the perspective of causal emergence, [[effective information]] will increase due to dimensionality reduction. This highlights the primary difference between coarse-graining strategies in causal emergence research and model reduction in control theory. For stochastic systems <ref name=":17" />, directly calculating the loss function can be unstable due to randomness, making reduction effectiveness harder to measure. The effective information and causal emergence index based on stochastic dynamical systems can increase the effectiveness of evaluation indicators to a certain extent. Additionally, they make the control research of stochastic dynamical systems more rigorous.

===Dynamic mode decomposition===

===Dynamic mode decomposition===