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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, EI 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, EI 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 EI 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===