更改

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.

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 effective information. 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 effective information. 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===