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Dynamic mode decomposition is a method of reducing the dimension of variables, dynamics, and observation functions simultaneously by using linear transformation <ref>B. Brunton, L. Johnson, J. Ojemann and J. N. Kutz, Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition arXiv:1409.5496</ref>. This method resembles the coarse-graining strategy in causal emergence, prioritizing error minimization for optimization. Although both model reduction and dynamic mode decomposition are very close to coarse-graining method, they are not optimized based on EI. In essence, they both suffer to a certain degree of information loss and may not enhance causal effects. The authors of <ref>Liu K, Yuan B, Zhang J. An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems[J]. arXiv preprint arXiv:2405.09207, 2024.</ref> demonstrated that the error minimization solution set includes the optimal solution set for maximizing EI. Therefore, if we want to optimize causal emergence, we can first minimize the error and find the best coarse-graining strategy in the error minimization solution set.

Dynamic mode decomposition is a method of reducing the dimension of variables, dynamics, and observation functions simultaneously by using linear transformation <ref>B. Brunton, L. Johnson, J. Ojemann and J. N. Kutz, Extracting spatial-temporal coherent patterns in large-scale neural recordings using dynamic mode decomposition arXiv:1409.5496</ref>. This method resembles the coarse-graining strategy in causal emergence, prioritizing error minimization for optimization. Although both model reduction and dynamic mode decomposition are very close to coarse-graining method, they are not optimized based on EI. In essence, they both suffer to a certain degree of information loss and may not enhance causal effects. The authors of <ref>Liu K, Yuan B, Zhang J. An Exact Theory of Causal Emergence for Linear Stochastic Iteration Systems[J]. arXiv preprint arXiv:2405.09207, 2024.</ref> demonstrated that the error minimization solution set includes the optimal solution set for maximizing EI. Therefore, if we want to optimize causal emergence, we can first minimize the error and find the best coarse-graining strategy in the error minimization solution set.

===Model Reduction of Markov chains===

===Model reduction of Markov chains===

The [[Model Reduction of a Markov Chain]], also referred to as [[coarse-graining of Markov chains]], is closely related to causal emergence. The coarse-graining process in causal emergence is essentially a model reduction of a Markov chain. Coarse-graining Markov processes <ref>Zhang A, Wang M. Spectral state compression of markov processes[J]. IEEE transactions on information theory, 2019, 66(5): 3202-3231.</ref> is an important problem for modeling a state transition system. It reduces the complexity of Markov chains by merging multiple states into one state.

The [[Model Reduction of a Markov Chain]], also referred to as [[coarse-graining of Markov chains]], is closely related to causal emergence. The coarse-graining process in causal emergence is essentially a model reduction of a Markov chain. Coarse-graining Markov processes <ref>Zhang A, Wang M. Spectral state compression of markov processes[J]. IEEE transactions on information theory, 2019, 66(5): 3202-3231.</ref> is an important problem for modeling a state transition system. It reduces the complexity of Markov chains by merging multiple states into one state.