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Here [math]\mathcal{U}([-\frac{L}{2},\frac{L}{2}]^n)[/math] represents the uniform distribution on the cube [math][-\frac{L}{2},\frac{L}{2}]^n[/math]. From this, it can be seen that although L is still implicitly included in the expectation, all terms in the equation that explicitly contain L are eliminated. In actual numerical calculations, the expected calculation can be manifested as taking the average of multiple samples on [math][-\frac{L}{2},\frac{L}{2}]^n[/math], and therefore is also independent of the size of L. This demonstrates the rationality of introducing dimension averaged EI.
Here [math]\mathcal{U}([-\frac{L}{2},\frac{L}{2}]^n)[/math] represents the uniform distribution on the cube [math][-\frac{L}{2},\frac{L}{2}]^n[/math]. From this, it can be seen that although L is still implicitly included in the expectation, all terms in the equation that explicitly contain L are eliminated. In actual numerical calculations, the expected calculation can be manifested as taking the average of multiple samples on [math][-\frac{L}{2},\frac{L}{2}]^n[/math], and therefore is also independent of the size of L. This demonstrates the rationality of introducing dimension averaged EI.
==随机迭代系统==
==Random Iterative Systems==
We can extend the above results to linear iterative dynamical systems. For an iterative system of the form:
<math>
<math>
</math>
</math>
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.
可以看出这一迭代系统可以看做是公式{{EquationNote|5}}的特例，其中[math]y[/math]对应这里的[math]x_{t+1}[/math]，[math]f(x_t)[/math]即是[math]A x_t[/math]。
可以看出这一迭代系统可以看做是公式{{EquationNote|5}}的特例，其中[math]y[/math]对应这里的[math]x_{t+1}[/math]，[math]f(x_t)[/math]即是[math]A x_t[/math]。
为定义EI，设干预空间大小为<math>L</math>，对于单步的映射我们可以得到维度平均有效信息
为定义EI，设干预空间大小为<math>L</math>，对于单步的映射我们可以得到维度平均有效信息
, the EI for a single step mapping is:
<math>
<math>
其中，[math]W[/math]为粗粒化矩阵，它的阶数为n*m，m为宏观状态空间的维度，它的作用是把任意的微观态[math]x_t[/math]映射为宏观态[math]y_t[/math]。[math]W^{\dagger}[/math]为W的伪逆运算。式中第一项是由确定性引发的涌现，简称'''确定性涌现'''(Determinism Emergence)，第二项为简并性引发的涌现，简称'''简并性涌现'''。更详细的内容参看[[随机迭代系统的因果涌现]]。
其中，[math]W[/math]为粗粒化矩阵，它的阶数为n*m，m为宏观状态空间的维度，它的作用是把任意的微观态[math]x_t[/math]映射为宏观态[math]y_t[/math]。[math]W^{\dagger}[/math]为W的伪逆运算。式中第一项是由确定性引发的涌现，简称'''确定性涌现'''(Determinism Emergence)，第二项为简并性引发的涌现，简称'''简并性涌现'''。更详细的内容参看[[随机迭代系统的因果涌现]]。
The effective information of a random iterative system can be decomposed into two terms: determinism and degeneracy.
* '''Determinism''' describes the predictability of the system's future state based on the current state.
* '''Degeneracy''' describes the ability to trace back the previous state from the current state.
The stronger the determinism and the weaker the degeneracy, the greater the effective information, leading to stronger causal effects.
==前馈神经网络==
==前馈神经网络==
针对复杂系统自动建模任务，我们往往使用神经网络来建模系统动力学。具体的，对于前馈神经网络来说，[[张江]]等人推导出了前馈神经网络有效信息的计算公式<ref name="zhang_nis">{{cite journal|title=Neural Information Squeezer for Causal Emergence|first1=Jiang|last1=Zhang|first2=Kaiwei|last2=Liu|journal=Entropy|year=2022|volume=25|issue=1|page=26|url=https://api.semanticscholar.org/CorpusID:246275672}}</ref>，其中神经网络的输入是<math>x(x_1,...,x_n)</math>，输出是<math>y(y_1,...,y_n)</math>，并且满足<math>y=f(x)</math>，<math>f</math>是由神经网络实现的确定性映射。然而，根据公式{{EquationNote|5}}，映射中必须包含噪声才能够体现不确定性。
针对复杂系统自动建模任务，我们往往使用神经网络来建模系统动力学。具体的，对于前馈神经网络来说，[[张江]]等人推导出了前馈神经网络有效信息的计算公式<ref name="zhang_nis">{{cite journal|title=Neural Information Squeezer for Causal Emergence|first1=Jiang|last1=Zhang|first2=Kaiwei|last2=Liu|journal=Entropy|year=2022|volume=25|issue=1|page=26|url=https://api.semanticscholar.org/CorpusID:246275672}}</ref>，其中神经网络的输入是<math>x(x_1,...,x_n)</math>，输出是<math>y(y_1,...,y_n)</math>，并且满足<math>y=f(x)</math>，<math>f</math>是由神经网络实现的确定性映射。然而，根据公式{{EquationNote|5}}，映射中必须包含噪声才能够体现不确定性。