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第77行： − + − + 第86行：
第86行： − + + − The G-emergence theory proposed by Seth is the first attempt to use causal measures to quantify emergence phenomena. However, the causal relationship used by the author is Granger causality, which is not a strict causal relationship. At the same time, the measurement results also depend on the regression method used. In addition, the measurement index of this method is defined according to variables rather than dynamics, which means that the results will depend on the choice of variables. These all constitute the drawbacks of the G-emergence theory. − − − The causal emergence framework also has similarities with the aforementioned G-emergence. The macroscopic states of both methods need to be manually selected. In addition, it should be noted that some of the above methods for quantitatively quantifying emergence often do not consider true interventionist causality. +
→G-emergence
==== G-emergence ====
==== G-emergence ====
The G-emergence theory was proposed by Seth in 2008 and is one of the earliest studies to quantitatively quantify [[emergence]] from a causal perspective <ref name=":4" />. The basic idea is to use nonlinear [[Granger causality|Granger causality]] to quantify [[weak emergence]] in complex systems.
The G-emergence theory was proposed by Seth in 2008 and is one of the earliest studies to quantify [[emergence]] from a causal perspective <ref name=":4" />. The basic idea is to use nonlinear [[Granger causality|Granger causality]] to quantify [[weak emergence]] in complex systems.
Specifically, if we use a binary autoregressive model for prediction, when there are only two variables A and B, [[autoregressive model]] has two equations, each equation corresponds to one of the variables, and the current value of each variable is composed of its own value and the value of the other variable within a certain time lag range. In addition, the model also calculates residuals. Here, residuals can be understood as prediction errors and can be used to measure the degree of Granger causal effect (called G-causality) of each equation. The degree to which B is a Granger cause (G-cause) of A is calculated by taking the logarithm of the ratio of the two residual variances, one being the residual of A's autoregressive model when B is omitted, and the other being the residual of the full prediction model (including A and B). In addition, the author also defines the concept of “G-autonomous”, which represents a measure of the extent to which the past values of a time series can predict its own future values. The strength of this autonomous predictive causal effect can be characterized in a similar way to G-causality.
Specifically, if we use a binary autoregressive model for prediction, it involves two variables, A and B. The [[autoregressive model]] comprises two equations, each corresponding to one variable. The current value of each variable is determined by its own past values and the past values of the other variable within a certain time lag range. In addition, the model also calculates residuals. Residuals, or prediction errors, measure the degree of Granger causal effect (G-causality) for each equation. The extent of the Granger causality of B on A is determined by the logarithm of the ratio of residual variances: one from A's model without B and the other from the full model including A and B. The concept of “G-autonomous” measures how well past values of a time series predict its future values. The strength of this autonomous predictive causal effect can be characterized in a similar way to G-causality.
As shown in the above figure, we can judge the occurrence of emergence based on the two basic concepts in the above G-causality (here is the measure of emergence based on [[Granger causality|Granger causality]], denoted as G-emergence). If A is understood as a macroscopic variable and B is understood as a microscopic variable. The conditions for emergence to occur include two: 1) A is G-autonomous with respect to B; 2) B is a G-cause of A. The degree of G-emergence is calculated by multiplying the degree of A's G-autonomous by the degree of B's average G-cause.
As shown in the figure, the occurrence of emergence can be judged based on two key concepts in the G-causality. This measure of emergence, derived from [[Granger causality|Granger causality]], is denoted as G-emergence. A can be understood as a macroscopic variable, while B represents a microscopic variable. The conditions for emergence to occur are as follows: 1) A is G-autonomous with respect to B; 2) B is a G-cause of A. The degree of G-emergence is calculated by multiplying the degree of A's G-autonomous by the degree of B's average G-cause.
gence theory proposed by Seth is the first attempt to use causal measures to quantify emergence phenomena. However, the causal relationship used by the author is Granger causality, which is not a strict causal relationship. At the same time, the measurement results also depend on the regression method used. In addition, the measurement index of this method is defined according to variables rather than dynamics, which means that the results will depend on the choice of variables. These limitations highlight the drawbacks of the G-emergence theory.
The causal emergence framework also has similarities with the aforementioned G-emergence. The macroscopic states of both methods need to be manually selected. In addition, it should be noted that some of the above methods for quantifying emergence often do not consider true interventionist causality.
==== Other theories for quantitatively characterizing emergence ====
==== Other theories for quantitatively characterizing emergence ====