更改

=====Method Based on Mutual Information Approximation=====

=====Method Based on Mutual Information Approximation=====

Although [[Rosas's causal emergence theory]] has given a strict definition of causal emergence. However it involves the combinatorial explosion problem of many variables in the calculation, makingo it difficult to apply this method to actual systems. To solve this problem, Rosas et al. bypassed the exact calculation of unique information and synergistic information <ref name=":5" /> and proposed an approximate formula that only needs to calculate [[mutual information]]. They also derived a sufficient condition for determining the occurrence of causal emergence.

Although [[Rosas's causal emergence theory]] has given a strict definition of causal emergence. However it involves the combinatorial explosion problem of many variables in the calculation, making it difficult to apply this method to actual systems. To solve this problem, Rosas et al. bypassed the exact calculation of unique information and synergistic information <ref name=":5" /> and proposed an approximate formula that only needs to calculate [[mutual information]]. They also derived a sufficient condition for determining the occurrence of causal emergence.

Here <math>X_t^j</math> represents the microscopic variable at time t in the j-th dimension, and <math>V_t ; V_{t + 1}</math> respectively represent macroscopic state variables at two consecutive times. Rosas et al. defined that when <math>\mathrm{\Psi}>0</math>, emergence occurs in the system; but when <math>\mathrm{\Psi}<0</math>, we cannot determine whether <math>V</math> has emergence, because this condition is only a sufficient condition for the occurrence of causal emergence.

Here <math>X_t^j</math> represents the microscopic variable at time t in the j-th dimension, and <math>V_t ; V_{t + 1}</math> respectively represent macroscopic state variables at two consecutive times. Rosas et al. defined that when <math>\mathrm{\Psi}>0</math>, emergence occurs in the system; but when <math>\mathrm{\Psi}<0</math>, we cannot determine whether the system has emergence, because this condition is only a sufficient condition for the occurrence of causal emergence.

To verify that the information related to macaque movement is an emergent feature of its cortical activity, Rosas et al. conducted an experiment using the electrocorticogram (ECoG) of macaques as the observational data of microscopic dynamics. To obtain the macroscopic state variable <math>V</math>, the authors chose the time series data of the limb movement trajectory of macaques obtained by motion capture (MoCap). The ECoG consisted of 64 channels, while the MoCap data consisted of 3 channels. Since the original MoCap data does not satisfy the conditional independence assumption of the supervenience feature, they used [[partial least squares]] and [[support vector machine]] algorithms to infer the part of neural activity encoded in the ECoG signal related to predicting macaque behavior. They then speculated that this information is an emergent feature of potential neural activity. Finally, based on the microscopic state and the calculated macroscopic features, the authors verified the existence of causal emergence.

To verify that the information related to macaque movement is an emergent feature of its cortical activity, Rosas et al. conducted an experiment using the electrocorticogram (ECoG) of macaques as the observational data of microscopic dynamics. To obtain the macroscopic state variable <math>V</math>, the authors chose the time series data of the limb movement trajectory of macaques obtained by motion capture (MoCap). The ECoG consisted of 64 channels, while the MoCap data consisted of 3 channels. Since the original MoCap data does not satisfy the conditional independence assumption of the supervenience feature, they used [[partial least squares]] and [[support vector machine]] algorithms to infer the part of neural activity encoded in the ECoG signal related to predicting macaque behavior. They then speculated that this information is an emergent feature of potential neural activity. Finally, based on the microscopic state and the calculated macroscopic features, the authors verified the existence of causal emergence.

=====Machine Learning-based Method=====

=====Machine Learning-based Method=====