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=====Quantification of causal emergence without coarse-graining=====

=====Quantification of causal emergence without coarse-graining=====

However, the greatest value of this theory lies in the fact that emergence can be directly quantified without a coarse-graining strategy. If the rank of <math>P</math> is <math>r</math>, that is, starting from the <math>r + 1</math>th singular value, all singular values are 0, then we say that the dynamics <math>P</math> has '''clear causal emergence''', and the numerical value of causal emergence is:

However, the greatest value of this theory lies in the fact that emergence can be directly quantified without a coarse-graining strategy. If the rank of <math>P</math> is <math>r</math>, that is, starting from the <math>(r + 1)</math>th singular value, all singular values are 0, then we say that the dynamics <math>P</math> has '''clear causal emergence''', and the numerical value of causal emergence is:

<math>

<math>

</math>

</math>

If the matrix <math>P</math> is full rank, but for any given small number <math>\epsilon</math>, there exists <math>r_{\epsilon}</math> such that starting from <math>r_{\epsilon}+1</math>, all singular values are less than <math>\epsilon</math>, then it is said that the system has a degree of '''vague causal emergence''', and the numerical value of causal emergence is:

If the matrix <math>P</math> is full rank, but for any given small number <math>\epsilon</math>, there exists <math>r_{\epsilon}</math> such that starting from <math>(r_{\epsilon}+1)</math>, all singular values are less than <math>\epsilon</math>, then it is said that the system has a degree of '''vague causal emergence''', and the numerical value of causal emergence is:

<math>\Delta \Gamma_{\alpha}(\epsilon) =  \frac{\sum_{i = 1}^{r} \sigma_{i}^{\alpha}}{r} -  \frac{\sum_{i = 1}^{N} \sigma_{i}^{\alpha}}{N} </math>

<math>\Delta \Gamma_{\alpha}(\epsilon) =  \frac{\sum_{i = 1}^{r} \sigma_{i}^{\alpha}}{r} -  \frac{\sum_{i = 1}^{N} \sigma_{i}^{\alpha}}{N} </math>