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Where [math]A\in\mathcal{R}^{n\times n}[/math] is a full-rank n*n matrix representing the dynamics of the linear iterative system, and [math]\varepsilon_t\sim\mathcal{N}(0,\Sigma)[/math] is n-dimensional Gaussian noise with mean zero and covariance matrix [math]\Sigma[/math]. Among them, the covariance matrix [math]\Sigma[/math] is also full rank.

Where [math]A\in\mathcal{R}^{n\times n}[/math] is a full-rank n*n matrix representing the dynamics of the linear iterative system, and [math]\varepsilon_t\sim\mathcal{N}(0,\Sigma)[/math] is n-dimensional Gaussian noise with mean zero and covariance matrix [math]\Sigma[/math]. Among them, the covariance matrix [math]\Sigma[/math] is also full rank.

It can be seen that this iterative system can be regarded as a special case of formula 5, where [math]y[/math] corresponds to [math]x_{t+1}[/math] here, and [math]f(x_t)[/math] is [math]A x_t[/math].

It can be seen that this iterative system can be regarded as a special case of formula {{EquationNote|5}}, where [math]y[/math] corresponds to [math]x_{t+1}[/math] here, and [math]f(x_t)[/math] is [math]A x_t[/math].

To define EI, let the intervention space size be <math>L</math>. For a single step mapping, we can obtain the average effective information of the dimensions.

To define EI, let the intervention space size be <math>L</math>. For a single step mapping, we can obtain the average effective information of the dimensions.

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'''Determinism''' describes the predictability of the system's future state based on the current state. The stronger the certainty, the smaller the randomness, and the easier it is to predict the future trend of the system.

is the term of '''Determinism''', which describes the predictability of the system's future state based on the current state. The stronger the certainty, the smaller the randomness, and the easier it is to predict the future trend of the system.

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'''Degeneracy''' describes the ability to trace back the previous state from the current state. The weaker the degeneracy, the easier it is to infer the system's past evolutionary path. Among them, [math]\tilde{x}_{t+1}[/math] is a new [math]x_{t+1}[/math] variable obtained after intervention while keeping the causal mechanism unchanged.

is the term of '''Degeneracy''', which describes the ability to trace back the previous state from the current state. The weaker the degeneracy, the easier it is to infer the system's past evolutionary path. Among them, [math]\tilde{x}_{t+1}[/math] is a new [math]x_{t+1}[/math] variable obtained after intervention while keeping the causal mechanism unchanged.

The stronger the determinism and the weaker the degeneracy, the greater the effective information, leading to stronger causal effects.

The stronger the determinism and the weaker the degeneracy, the greater the effective information, leading to stronger causal effects.

After subtracting the macro effective information from the micro effective information, the causal emergence of the iterative system can be obtained as follows:

After subtracting the macro effective information from the micro effective information, the dimension averaged causal emergence of the iterative system can be obtained as follows:

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Among them, [math]W[/math] is a coarse-grained matrix with an order of n * m. m is the dimension of the macroscopic state space, and its function is to map any microscopic state [math]x_t[/math] to a macroscopic state [math]y_t[/math]. [math]W^{\dagger}[/math] is the pseudo inverse operation of W. The first term in the equation is the emergence caused by determinism, abbreviated as Deterministic Emergence, and the second term is the emergence caused by degeneracy, abbreviated as Degenerative Emergence. For more detailed information, please refer to the causal emergence of stochastic iterative systems.

Among them, [math]W[/math] is a coarse-grained matrix with an order of n * m (m is the dimension of the macroscopic state space), and its function is to map any microscopic state [math]x_t[/math] to a macroscopic state [math]y_t[/math]. [math]W^{\dagger}[/math] is the pseudo inverse operation of W. The first term in the equation is the emergence caused by determinism, abbreviated as Deterministic Emergence, and the second term is the emergence caused by degeneracy, abbreviated as Degenerative Emergence. For more detailed information, please refer to the causal emergence of stochastic iterative systems.

==Feedforward Neural Networks==

==Feedforward Neural Networks==