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==Related research fields==

==Related research fields==

There are some related research fields that are closely related to causal emergence theory. Here we focus on introducing the differences and connections with three related fields: [[reduction of dynamical models]], [[dynamic mode decomposition]], and [[simplification of Markov chains]].

Several research fields are closely tied to causal emergence theory. This section explores the differences and connections with three related fields: [[reduction of dynamical models]], [[dynamic mode decomposition]], and [[simplification of Markov chains]].

Model reduction is to simplify and reduce the dimension of the high-dimensional complex system dynamics model and describe the evolution law of the original system with low-dimensional dynamics. This process is actually the coarse-graining process in the study of causal emergence. There are mainly two types of approximation methods for large-scale dynamical systems, namely approximation methods based on singular value decomposition <ref name=":15" /><ref>Gallivan K, Grimme E, Van Dooren P. Asymptotic waveform evaluation via a Lanczos method[J]. Applied Mathematics Letters, 1994, 7(5): 75-80.</ref> and approximation methods based on Krylov <ref name=":15" /><ref name=":17">CHRISTIAN DE VILLEMAGNE & ROBERT E. SKELTON (1987) Model reductions using a projection formulation, International Journal of Control, 46:6, 2141-2169, DOI: 10.1080/00207178708934040 </ref><ref>Boley D L. Krylov space methods on state-space control models[J]. Circuits, Systems and Signal Processing, 1994, 13: 733-758.</ref>. The former is based on singular value decomposition, and the latter is based on moment matching. Although the former has many ideal properties, including error bounds, it cannot be applied to systems with high complexity. On the other hand, the advantage of the latter is that it can be implemented iteratively and is therefore suitable for high-dimensional complex systems. Combining the advantages of these two methods gives rise to a third type of approximation method, namely the SVD/Krylov method <ref>Gugercin S. An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems[J]. Linear Algebra and its Applications, 2008, 428(8-9): 1964-1986.</ref><ref>Khatibi M, Zargarzadeh H, Barzegaran M. Power system dynamic model reduction by means of an iterative SVD-Krylov model reduction method[C]//2016 IEEE Power & Energy Society Innovative Smart Grid Technologies Conference (ISGT). IEEE, 2016: 1-6.</ref>. Both methods evaluate the model reduction effect based on the error loss function of the output function before and after coarse-graining. Therefore, the goal of model reduction is to find the reduced parameter matrix that minimizes the error.

Model reduction simplifies high-dimensional system dynamics, describing the evolution of the original system with low-dimensional dynamics. This process is actually the coarse-graining process in the study of causal emergence. There are mainly two types of approximation methods for large-scale dynamical systems, namely approximation methods based on singular value decomposition <ref name=":15" /><ref>Gallivan K, Grimme E, Van Dooren P. Asymptotic waveform evaluation via a Lanczos method[J]. Applied Mathematics Letters, 1994, 7(5): 75-80.</ref> and approximation methods based on Krylov <ref name=":15" /><ref name=":17">CHRISTIAN DE VILLEMAGNE & ROBERT E. SKELTON (1987) Model reductions using a projection formulation, International Journal of Control, 46:6, 2141-2169, DOI: 10.1080/00207178708934040 </ref><ref>Boley D L. Krylov space methods on state-space control models[J]. Circuits, Systems and Signal Processing, 1994, 13: 733-758.</ref>. The former is based on singular value decomposition, and the latter is based on moment matching. Although the former has many ideal properties, including error bounds, it cannot be applied to systems with high complexity. On the other hand, the advantage of the latter is that it can be implemented iteratively and is therefore suitable for high-dimensional complex systems. Combining the advantages of these two methods gives rise to a third type of approximation method, namely the SVD/Krylov method <ref>Gugercin S. An iterative SVD-Krylov based method for model reduction of large-scale dynamical systems[J]. Linear Algebra and its Applications, 2008, 428(8-9): 1964-1986.</ref><ref>Khatibi M, Zargarzadeh H, Barzegaran M. Power system dynamic model reduction by means of an iterative SVD-Krylov model reduction method[C]//2016 IEEE Power & Energy Society Innovative Smart Grid Technologies Conference (ISGT). IEEE, 2016: 1-6.</ref>. Both methods evaluate the model reduction effect based on the error loss function of the output function before and after coarse-graining. Therefore, the goal of model reduction is to find the reduced parameter matrix that minimizes the error.

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 defaults that the system reduction process will lose information. 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 is also the biggest difference between the coarse-graining strategy in causal emergence research and model reduction in control theory. When the dynamical system is a stochastic system <ref name=":17" />, directly calculating the loss function will lead to unstable due to the existence of randomness, so the effectiveness of reduction cannot be accurately measured. The effective information and causal emergence index based on stochastic dynamical systems can increase the effectiveness of evaluation indicators to a certain extent and 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===

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 and find the data mapping according to the flow field changes of different frequencies. This method is based on transforming nonlinear infinite-dimensional dynamics into finite-dimensional linear dynamics, and adopts the ideas of 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>. Dynamic mode decomposition sorts the system according to frequency and extracts the eigenfrequency of the system to observe the contribution of flow structures of different frequencies to the flow field. At the same time, the dynamic mode decomposition modal eigenvalue can predict the flow field. Because the dynamic mode decomposition algorithm has the advantages of theoretical rigor, stability, and simplicity. While being continuously applied, the dynamic mode decomposition algorithm is also continuously improved on the original basis. For example, it is combined with the SPA test to verify the strong effectiveness of the stock price prediction comparison benchmark point and by connecting the dynamic mode decomposition algorithm and spectral research. The way to simulate the vibration mode of the stock market in the circular economy. These applications can effectively collect and analyze data and finally obtain 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 by frequency, extracting eigenfrequencies to analyze how flow structures at different frequencies contribute to the flow field. At the same time, the dynamic mode decomposition modal eigenvalue can predict the flow field because the dynamic mode decomposition algorithm has the advantages of theoretical rigor, stability, and simplicity. While being continuously applied, the dynamic mode decomposition algorithm is also continuously improved on the original basis. For example, it is combined with the SPA test to verify the strong effectiveness of the stock price prediction comparison benchmark point. Additionally, the dynamic mode decomposition algorithm is connected with spectral research to explore further applications. The way to simulate the vibration mode of the stock market in the circular economy. These applications can effectively collect and analyze data and finally obtain 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 is another method similar to the coarse-graining strategy in causal emergence, which takes minimizing error as the main goal for optimization. Although both model reduction and dynamic mode decomposition are very close to model coarse-graining, they are not optimized based on effective information. In essence, they both default to a certain degree of information loss and will not enhance causal effects. In the literature <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>, the authors proved that in fact the error minimization solution set contains the optimal solution set of 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 model coarse-graining, they are not optimized based on effective information. In essence, they both default to a certain degree of information loss and will 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.

===Simplification of Markov chains===

===Simplification of Markov chains===

The [[simplification of Markov chains]] (or called [[coarse-graining of Markov chains]]) is also importantly related to causal emergence. The coarse-graining process in causal emergence is essentially the simplification of Markov chains. Model simplification of 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 in state transition system modeling. It reduces the complexity of Markov chains by merging multiple states into one state.

The [[simplification of Markov chains]], also referred to as [[coarse-graining of Markov chains]], is closely related to causal emergence. The coarse-graining process in causal emergence is essentially the simplification of Markov chains. Model simplification of 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 in state transition system modeling. It reduces the complexity of Markov chains by merging multiple states into one state.

There are mainly three meanings of simplification. First, when we study a very large-scale system, we will not pay attention to the changes of each microscopic state. Therefore, in coarse-graining, we hope to filter out some noise and heterogeneity that we are not interested in, and summarize some mesoscale or macroscopic laws from the microscopic scale. Second, some state transition probabilities are very similar, so they can be regarded as the same kind of state. Clustering this kind of state (also called partitioning the state) to obtain a new smaller Markov chain can reduce the redundancy of system representation. Third, in reinforcement learning using Markov decision processes, coarse-graining the Markov chain can reduce the size of the state space and improve training efficiency. In many literatures, coarse-graining and dimension reduction are equivalent <ref>Coarse graining. ''Encyclopedia of Mathematics.'' URL: <nowiki>http://encyclopediaofmath.org/index.php?title=Coarse_graining&oldid=16170</nowiki></ref>.

There are mainly three meanings of simplification. First, when we study a very large-scale system, it is unnecessary to focus on the changes of each microscopic state. Coarse-graining filters out noise and heterogeneity to derive mesoscale or macroscopic laws from microscopic data. Second, some state transition probabilities are very similar, so they can be regarded as the same kind of state. Clustering this kind of state (also called partitioning the state) to obtain a new smaller Markov chain can reduce the redundancy of system representation. Third, in reinforcement learning using Markov decision processes, coarse-graining the Markov chain can reduce the size of the state space and improve training efficiency. In much of the literature, coarse-graining and dimension reduction are equivalent <ref>Coarse graining. ''Encyclopedia of Mathematics.'' URL: <nowiki>http://encyclopediaofmath.org/index.php?title=Coarse_graining&oldid=16170</nowiki></ref>.

Among them, there are two types of coarse-graining of the state space: hard partitioning and soft partitioning. Soft partitioning can be regarded as a process of breaking up the microscopic state and reconstructing some macroscopic states, and allowing the superposition of microscopic states to obtain macroscopic states. Hard partitioning is a strict grouping of microscopic states, dividing several microscopic states into one group without allowing overlap and superposition (see [[coarse-graining of Markov chains]]).

Among them, there are two types of coarse-graining of the state space: hard partitioning and soft partitioning. Soft partitioning involves breaking up microscopic states, reconstructing macroscopic states, and allowing their superposition. Hard partitioning is a strict grouping of microscopic states, dividing several microscopic states into one group without allowing overlap and superposition (see [[coarse-graining of Markov chains]]).

The coarse-graining of Markov chains not only needs to be done on the state space, but also on the transition matrix, that is, to simplify the original transition matrix according to the state grouping to obtain a new smaller transition matrix. In addition, the state vector needs to be reduced. Therefore, a complete coarse-graining process needs to consider the coarse-graining of state, transition matrix, and state vector at the same time. Thus, this leads to a new problem, that is, how should the transition probability in the new Markov chain obtained by state grouping be calculated? At the same time, can the normalization condition be guaranteed?

The coarse-graining of Markov chains not only needs to be done on the state space, but also on the transition matrix. This involves simplifying the original transition matrix according to the state grouping to obtain a new, smaller transition matrix. In addition, the state vector needs to be reduced. Therefore, a complete coarse-graining process needs to consider the coarse-graining of state, transition matrix, and state vector at the same time. Thus, This raises the question of how to calculate transition probabilities in the new Markov chain after state grouping. Another question is whether the normalization condition can be guaranteed.

In addition to these basic guarantees, we usually also require that the coarse-graining operation of the transition matrix should be commutative with the transition matrix. This condition can ensure that the one-step evolution of the state vector after coarse-graining through the coarse-grained transition matrix (equivalent to macroscopic dynamics) is equivalent to first performing one-step transition matrix evolution on the state vector (equivalent to microscopic dynamics) and then performing coarse-graining. This puts forward requirements for both the state grouping (the coarse-graining process of the state) and the coarse-graining process of the transition matrix. This requirement of commutativity leads people to propose the requirement of [[clustering property of Markov chains]].

In addition to these guarantees, the coarse-graining operation of the transition matrix is typically required to be commutative with the matrix itself. This condition ensures that the one-step evolution of the state vector through the coarse-grained matrix (equivalent to macroscopic dynamics) is preserved. It is equivalent to first performing one-step transition matrix evolution on the state vector (equivalent to microscopic dynamics) and then applying coarse-graining. This puts forward requirements for both the state grouping (the coarse-graining process of the state) and the coarse-graining process of the transition matrix. This requirement of commutativity leads people to propose the requirement of [[clustering property of Markov chains]].

For any hard partition of states, we can define the so-called concept of lumpability. Lumpability is a measure of clustering. This concept first appeared in Kemeny, Snell's Finite Markov Chains in 1969 <ref name=":33">Kemeny, John G., and J. Laurie Snell. ''Finite markov chains''. Vol. 26. Princeton, NJ: van Nostrand, 1969. https://www.math.pku.edu.cn/teachers/yaoy/Fall2011/Kemeny-Snell_Chapter6.3-4.pdf</ref>. Lumpability is a mathematical condition used to judge "whether a certain hard-blocked microscopic state grouping scheme is reducible to the microscopic state transition matrix". No matter which hard-blocking scheme the state space is classified according to, it has a corresponding coarse-graining scheme for the transition matrix and probability space <ref>Buchholz, Peter. "Exact and ordinary lumpability in finite Markov chains." ''Journal of applied probability'' 31.1 (1994): 59-75.</ref>.

For any hard partition of states, we can define the concept of lumpability. Lumpability is a measure of clustering. This concept first appeared in Kemeny, Snell's Finite Markov Chains in 1969 <ref name=":33">Kemeny, John G., and J. Laurie Snell. ''Finite markov chains''. Vol. 26. Princeton, NJ: van Nostrand, 1969. https://www.math.pku.edu.cn/teachers/yaoy/Fall2011/Kemeny-Snell_Chapter6.3-4.pdf</ref>. Lumpability is a mathematical condition to determine whether a hard-blocked microscopic state grouping scheme is reducible to the microscopic state transition matrix. No matter which hard-blocking scheme the state space is classified according to, it has a corresponding coarse-graining scheme for the transition matrix and probability space <ref>Buchholz, Peter. "Exact and ordinary lumpability in finite Markov chains." ''Journal of applied probability'' 31.1 (1994): 59-75.</ref>.

For specific methods of coarse-graining Markov chains, please refer to [[coarse-graining of Markov chains]].

For specific methods of coarse-graining Markov chains, please refer to [[coarse-graining of Markov chains]].

==References==

==References==