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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 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.

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===

===Dynamical mode decomposition===

In addition to the reduction of dynamical models, dynamic mode decomposition is also closely related to coarse-graining. The basic idea of the dynamic mode decomposition (DMD) <ref>Schmid P J. Dynamic mode decomposition and its variants[J]. Annual Review of Fluid Mechanics, 2022, 54(1): 225-254.</ref><ref>J. Proctor, S. Brunton and J. N. Kutz, Dynamic mode decomposition with control, arXiv:1409.6358</ref> model is to directly obtain the dynamic information of the flow in the flow field from the data. It then finds the data mapping based on the flow field changes at different frequencies. This method transforms nonlinear infinite-dimensional dynamics into finite-dimensional linear dynamics, using Arnoldi method and singular value decomposition for dimensionality reduction. It draws on many key features of time series such as ARIMA, SARIMA and seasonal models, and is widely used in fields such as mathematics, physics, and finance <ref>J. Grosek and J. N. Kutz, Dynamic mode decomposition for real-time background/foreground separation in video, arXiv:1404.7592.</ref>. DMD sorts the system according to frequency, extracting the eigen-frequency to analyze how flow structures at different frequencies contribute to the flow field. At the same time, the dynamic mode decomposition method can decompose the system with eigenvectors of different modes and then to predict the flow field. While being continuously applied, the dynamic mode decomposition algorithm has also been improved on its original foundation, such as combining it with SPA testing to verify the strong effectiveness of stock price prediction compared to benchmark points, and by linking the dynamic mode decomposition algorithm with spectral research to simulate the vibration patterns of the stock market in a circular economy. These applications can effectively collect and analyze data, ultimately yielding results.

In addition to the reduction of dynamical models, dynamic mode decomposition is also closely related to coarse-graining. The basic idea of the dynamic mode decomposition (DMD) <ref>Schmid P J. Dynamic mode decomposition and its variants[J]. Annual Review of Fluid Mechanics, 2022, 54(1): 225-254.</ref><ref>J. Proctor, S. Brunton and J. N. Kutz, Dynamic mode decomposition with control, arXiv:1409.6358</ref> model is to directly obtain the dynamic information of the flow in the flow field from the data. It then finds the data mapping based on the flow field changes at different frequencies. This method transforms nonlinear infinite-dimensional dynamics into finite-dimensional linear dynamics, using Arnoldi method and singular value decomposition for dimensionality reduction. It draws on many key features of time series such as ARIMA, SARIMA and seasonal models, and is widely used in fields such as mathematics, physics, and finance <ref>J. Grosek and J. N. Kutz, Dynamic mode decomposition for real-time background/foreground separation in video, arXiv:1404.7592.</ref>. DMD sorts the system according to frequency, extracting the eigen-frequency to analyze how flow structures at different frequencies contribute to the flow field. At the same time, the dynamic mode decomposition method can decompose the system with eigenvectors of different modes and then to predict the flow field. While being continuously applied, the dynamic mode decomposition algorithm has also been improved on its original foundation, such as combining it with SPA testing to verify the strong effectiveness of stock price prediction compared to benchmark points, and by linking the dynamic mode decomposition algorithm with spectral research to simulate the vibration patterns of the stock market in a circular economy. These applications can effectively collect and analyze data, ultimately yielding results.