Figure 6.

The dependence of the dimension averaged $\mathit{CausalEmergence}$ (dCE) on different scales (q) of the Markov dynamics (a), the learned mapping between micro states and macro states on the optimal scale (q) (b), and the learned macro-dynamics the mapping from ${\mathbf{y}}_t$ to $\mathbf{y}(t+1)$ (c). There are two clear separated clusters on the y-axis in (b) which means the macro states are discrete. We found that the two discrete macro states and the mapping between micro and discrete macro states are identical as the example in Ref. [6] which means the correct coarse-graining strategy can be discovered by our algorithm automatically under the condition without any prior information. In (d), $I(x_t,{\widehat{x}}_t)\le I(x_t,y_t^{q=3})\le I(x_t,y_t^{q=4})\le I(x_t,y_t^{q=5})\le I(x_t,y_t^{q=6})$ is reflected. In order to make the data clearer, we have taken a moving average for each group of data. This result can be regarded as the verification of Theorem 6.