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==Related research fields==

==Related research fields==

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]].

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]].

===Dynamical model reduction===

===Dynamical model reduction===

An important indicator of causal emergence is the selection of coarse-graining strategies. When the microscopic model is known, coarse-graining the microscopic state is equivalent to performing a '''model reduction''' on the microscopic model. Model reduction is an important subfield in control theory. Antoulas once wrote a related review article about model reduction for large-scale dynamical systems <ref name=":15">Antoulas A C. An overview of approximation methods for large-scale dynamical systems[J]. Annual reviews in Control, 2005, 29(2): 181-190.</ref>.

An important indicator of causal emergence is the selection of coarse-graining strategies. When the microscopic model is known, coarse-graining the microscopic state is equivalent to performing a '''model reduction''' on the microscopic model. Model reduction is an important subfield in control theory. Antoulas once wrote a related review article about model reduction for large-scale dynamical systems <ref name=":15">Antoulas A C. An overview of approximation methods for large-scale dynamical systems[J]. Annual reviews in Control, 2005, 29(2): 181-190.</ref>.

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.

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 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.

===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.

There are mainly three meanings of coarsening. 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(partitioning) this kind of states (also called lumping the states) 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>.

There are mainly three meanings of coarsening. 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(partitioning) this kind of states (also called lumping the states) 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 partitioning(clustering, grouping) 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]]).

Among them, there are two types of partitioning(clustering, grouping) 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. This involves simplifying the original transitional probability matrix (TPM) according to the state grouping to obtain a new, smaller TPM. In addition, the state vector needs to be reduced. Therefore, a complete coarse-graining process needs to consider the coarse-graining of states, TPM, and state vector at the same time. Thus, This raises the question of how to calculate TPM in the new Markov chain after states grouping. Another question is whether the normalization condition can 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 transitional probability matrix (TPM) according to the state grouping to obtain a new, smaller TPM. In addition, the state vector needs to be reduced. Therefore, a complete coarse-graining process needs to consider the coarse-graining of states, TPM, and state vector at the same time. Thus, This raises the question of how to calculate TPM in the new Markov chain after states grouping. Another question is whether the normalization condition can be guaranteed.

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

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

For any hard partition of states, we can define the concept of lumpability. Lumpability is a requirement 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 macroscopic TPM. No matter which hard-blocking scheme the state space is partitioned according to, it has a corresponding coarse-graining scheme for the TPM and distributions <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 requirement 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 macroscopic TPM. No matter which hard-blocking scheme the state space is partitioned according to, it has a corresponding coarse-graining scheme for the TPM and distributions <ref>Buchholz, Peter. "Exact and ordinary lumpability in finite Markov chains." ''Journal of applied probability'' 31.1 (1994): 59-75.</ref>.

Suppose a grouping method '''<math>S=A_1\cup A_2\cup ... ,A_r\}</math>''' is given for the state space '''<math>A</math>'''. Here [math]A_i[/math] is any subset of the state space '''<math>S</math>''' and satisfies [math]A_i\cap A_j= \Phi[/math], where [math]\Phi[/math] represents the empty set. [math]\displaystyle{ s^{(t)} }[/math] represents the microscopic state of the system at time [math]\displaystyle{ t }[/math]. The microscopic state space is [math]\displaystyle{ S=\{s_1, s_2,...,s_n\} }[/math] and the microscopic state '''<math>s_i\in S</math>''' are all independent elements in the Markov state space. Let the transition probability from microscopic state <math>s_k</math> to <math>s_m</math> be <math>p_{s_k \rightarrow s_m} = p(s^{(t)} = s_m | s^{(t-1)} = s_k)</math>, and the transition probability from microscopic state <math>s_k</math> to macroscopic state <math>A_i</math> be <math>p_{s_k \rightarrow A_i} = p(s^{(t)} \in A_i | s^{(t-1)} = s_k)</math>. Then the necessary and sufficient condition for lumpability is that for any pair <math>A_i, A_j</math>, <math>p_{s_k \rightarrow A_j}</math> of every state <math>s_k</math> belonging to <math>A_i</math> is equal, that is:

Suppose a grouping method '''<math>S=A_1\cup A_2\cup ... ,A_r\}</math>''' is given for the state space '''<math>A</math>'''. Here [math]A_i[/math] is any subset of the state space '''<math>S</math>''' and satisfies [math]A_i\cap A_j= \Phi[/math], where [math]\Phi[/math] represents the empty set. [math]\displaystyle{ s^{(t)} }[/math] represents the microscopic state of the system at time [math]\displaystyle{ t }[/math]. The microscopic state space is [math]\displaystyle{ S=\{s_1, s_2,...,s_n\} }[/math] and the microscopic state '''<math>s_i\in S</math>''' are all independent elements in the Markov state space. Let the transition probability from microscopic state <math>s_k</math> to <math>s_m</math> be <math>p_{s_k \rightarrow s_m} = p(s^{(t)} = s_m | s^{(t-1)} = s_k)</math>, and the transition probability from microscopic state <math>s_k</math> to macroscopic state <math>A_i</math> be <math>p_{s_k \rightarrow A_i} = p(s^{(t)} \in A_i | s^{(t-1)} = s_k)</math>. Then the necessary and sufficient condition for lumpability is that for any pair <math>A_i, A_j</math>, <math>p_{s_k \rightarrow A_j}</math> of every state <math>s_k</math> belonging to <math>A_i</math> is equal, that is:

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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]].