更改

=====NIS+=====

=====NIS+=====

Although NIS took the lead in proposing a scheme to optimize EI to identify causal emergence in data, this method has some shortcomings: the author divides the optimization process into two stages, but does not truly maximize the EI, that is, Equation {{EquationNote|1}}. Therefore, Yang et al. <ref name=":6" /> further improved this method and proposed the [[NIS+]] scheme. By introducing reverse dynamics and [[reweighting technique]], the original maximization of EI is transformed into maximizing its variational lower bound. This is achieved using [[variational inequality]], enabling the direct optimization of the objective function.

NIS, although it was the first to propose a framework to optimize EI to identify causal emergence in data, has some shortcomings: the authors divide the optimization process into two stages, but does not truly maximize the EI directly, that is, Equation {{EquationNote|1}}. Therefore, Yang et al. <ref name=":6" /> further improved this method and proposed the [[NIS+]] scheme. By introducing reverse dynamics and [[reweighting technique]], the original maximization of EI is transformed into maximizing its variational lower bound. This is achieved using [[variational inequality]], enabling the direct optimization of the objective function.

======Mathematical principles======

======Mathematical principles======

</math>

</math>

Here <math>\tilde{p}(\boldsymbol{y}_{t})</math> is the target distribution and <math>p(\boldsymbol{y}_{t})</math> is the original distribution of the data.

Here <math>\tilde{p}(\boldsymbol{y}_{t})</math> is the target distribution and <math>p(\boldsymbol{y}_{t})</math> is the original distribution of the data, [math]\phi[/math] is the coarse-graining function.

======Workflow and model architecture======

======Workflow and model architecture======

The following figure shows the entire model framework of NIS+. Figure a is the input of the model: time series data, such as trajectory sequence, continuous image sequence, and EEG time series data; Figure c is the output of the model, which includes the degree of causal emergence, macroscopic dynamics, emergent patterns, and coarse-graining strategies; Figure b is the specific model architecture. Different from the NIS method, two parts of reverse dynamics and reweighting technology are added.

The following figure shows the entire model framework of NIS+. Figure a is the input of the model: time series data, such as trajectory sequence, continuous image sequence, and EEG time series data; Figure c is the output of the model, which includes the degree of causal emergence, macroscopic dynamics, emergent patterns, and coarse-graining strategies; Figure b is the specific model architecture. Different from the NIS method, the reverse dynamics training and the reweighting technology are added.

[[文件:NIS+.png|居左|600x600像素|替代=NIS模型框架图|NIS+模型框架图]]

[[文件:NIS+.png|居左|600x600像素|替代=NIS模型框架图|NIS+模型框架图]]

The following figure shows the experimental results of NIS+ learning the flocking behavior of the Boids model. (a) and (e) give the actual and predicted trajectories of bird flocks under different conditions. Specifically, the author divides the bird flock into two groups and compares the multi-step prediction results under different noise levels (<math>\alpha</math> is 0.001 and 0.4 respectively). The prediction is good when the noise is relatively small, and the prediction curve will diverge when the noise is relatively large. (b) shows that the mean absolute error (MAE) of multi-step prediction gradually increases as the radius r increases. (c) shows the change of the degree of [[causal emergence]] <math>\Delta J</math> and prediction error (MAE) under different dimensions (q) with the change of training epoch. The author finds that causal emergence is most significant when the macroscopic state dimension q = 8. (d) is the attribution analysis of macroscopic variables to microscopic variables, and the obtained significance map intuitively describes the learned coarse-graining function. Here, each macroscopic dimension can correspond to the spatial coordinate (microscopic dimension) of each bird. The darker the color, the higher the correlation. Here, the microscopic coordinates corresponding to the maximum correlation of each macroscopic state dimension are highlighted with orange dots. These attribution significance values are obtained by using the [[Integrated Gradient]] (referred to as IG) method. The horizontal axis represents the x and y coordinates of 16 birds in the microscopic state, and the vertical axis represents 8 macroscopic dimensions. The light blue dotted line distinguishes the coordinates of different individual Boids, and the blue solid line separates the two bird flocks. (f) and (g) represent the changing trends of the degree of causal emergence ΔJ and normalized error MAE under different noise levels. (f) represents the influence of changes in external noise (that is, adding observation noise to microscopic data) on causal emergence. (g) represents the influence of internal noise (represented by α, added by modifying the dynamics of the Boids model) on causal emergence. In (f) and (g), the horizontal line represents the threshold that violates the error constraint in Equation {{EquationNote|1}}. When the normalized MAE is greater than the threshold of 0.3, the constraint is violated and the result is unreliable.

The following figure shows the experimental results of NIS+ learning the flocking behavior of the Boids model. (a) and (e) give the actual and predicted trajectories of bird flocks under different conditions. Specifically, the author divides the bird flock into two groups and compares the multi-step prediction results under different noise levels (<math>\alpha</math> is 0.001 and 0.4 respectively). The prediction is good when the noise is relatively small, and the prediction curve will diverge when the noise is relatively large. (b) shows that the mean absolute error (MAE) of multi-step prediction gradually increases as the radius r increases. (c) shows the change of the degree of [[causal emergence]] <math>\Delta J</math> and prediction error (MAE) under different dimensions (q) with the change of training epoch. The author finds that causal emergence is most significant when the macroscopic state dimension q = 8. (d) is the attribution analysis of macroscopic variables to microscopic variables, and the obtained significance map intuitively describes the learned coarse-graining function. Here, each macroscopic dimension can correspond to the spatial coordinate (microscopic dimension) of each bird. The darker the color, the higher the correlation. Here, the microscopic coordinates corresponding to the maximum correlation of each macroscopic state dimension are highlighted with orange dots. These attribution significance values are obtained by using the [[Integrated Gradient]] (referred to as IG) method. The horizontal axis represents the x and y coordinates of 16 birds in the microscopic state, and the vertical axis represents 8 macroscopic dimensions. The light blue dotted line distinguishes the coordinates of different individual Boids, and the blue solid line separates the two bird flocks. (f) and (g) represent the changing trends of the degree of causal emergence ΔJ and normalized error MAE under different noise levels. (f) represents the influence of changes in external noise (that is, adding observation noise to microscopic data) on causal emergence. (g) represents the influence of internal noise (represented by α, added by modifying the dynamics of the Boids model) on causal emergence. In (f) and (g), the horizontal line represents the threshold that violates the error constraint in Equation {{EquationNote|1}}. When the normalized MAE is greater than the threshold of 0.3, the constraint is violated and the result is unreliable.

This set of experiments shows that [[NIS+]] can learn macroscopic states and coarse-graining strategies by maximizing EI. This maximization improves the model's generalization ability to situations outside the training data range. The learned macroscopic state effectively identifies the average [[group behavior]] and can be attributed to individual positions using the gradient integration method. The degree of causal emergence increases with external noise but decreases with increase of internal noise. This result indicates that the model reduces external noise through coarse-graining but cannot mitigate internal noise.

This set of experiments shows that [[NIS+]] can learn macroscopic states and coarse-graining strategies by maximizing EI. This maximization improves the model's generalization ability to situations outside the training data range. The learned macroscopic state effectively identifies the average [[group behavior]] and can be attributed to individual positions using the gradient integration method. The degree of causal emergence increases with external noise but decreases with increase of internal noise. This result indicates that the model reduces external noise through coarse-graining but cannot mitigate internal noise.

The brain experiment uses fMRI data collected from two experiments involving 830 human subjects (the results are shown in the following picture). In the first group, subjects performed a visual task by watching a short movie clip, during which recordings were made. In the second group of experiments, they were asked to be in a resting state and the recording was completed. Since the original dimension is relatively high, the authors reduced the original 14000-dimensional data to 100 dimensions using the [[Schaefer atlas]] method, with each dimension corresponding to a brain region. After that, the authors learned these data through NIS+ and extracted the dynamics at six different macroscopic scales. Figure a shows the multi-step prediction error results at different scales. Figure b shows the comparison of EI of NIS and NIS+ methods on different macroscopic dimensions in the resting state and the visual task of watching movies. The authors found that in the visual task, causal emergence is most significant when the macroscopic state dimension is q = 1. Through attribution analysis, it is found that the visual area plays the largest role (Figure c), which is consistent with the real scene. Figure d shows different perspective views of brain region attribution. In the resting state, one macroscopic dimension is not enough to predict the microscopic time series data. The dimension with the largest causal emergence is between 3 and 7 dimensions.

 

The brain experiment uses fMRI data collected from two experiments involving 830 human subjects. In the first group, subjects performed a visual task by watching a short movie clip, during which recordings were made. In the second group of experiments, they were asked to be in a resting state and the recording was completed. Since the original dimension is relatively high, the authors reduced the original 14000-dimensional data to 100 dimensions using the [[Schaefer atlas]] method, with each dimension corresponding to a brain region. After that, the authors learned these data through NIS+ and extracted the dynamics at six different macroscopic scales. Figure a shows the multi-step prediction error results at different scales. Figure b shows the comparison of EI of NIS and NIS+ methods on different macroscopic dimensions in the resting state and the visual task of watching movies. The authors found that in the visual task, causal emergence is most significant when the macroscopic state dimension is q = 1. Through attribution analysis, it is found that the visual area plays the largest role (Figure c), which is consistent with the real scene. Figure d shows different perspective views of brain region attribution. In the resting state, one macroscopic dimension is not enough to predict the microscopic time series data. The dimension with the largest causal emergence is between 3 and 7 dimensions.

[[文件:NIS+ 脑数据.png|居左|700x700像素|脑神经系统中的因果涌现]]

[[文件:NIS+ 脑数据.png|居左|700x700像素|脑神经系统中的因果涌现]]