## 2 Methods
### Data Analysis
#### 2.3.2 Correlation Sum  
As an estimator of the correlation integral, the correlation sum indicates the mean probability that two points in the embedding space at two different times are close [17; 33]. It reflects self-similarity, and can be calculated as follows:  
\[C(m,\epsilon)=\frac{2}{N(N-1)}\sum_{i=1}^{N}\sum_{j=i+1}^{N}H(\epsilon-\| \mathbf{X}(i)-\mathbf{X}(j)\|)\, \tag{3}\]where \(m\) also expresses the embedding dimension, and the function is highly related to the recurrence plot.  
The local slope of the correlation sum, which is also called local scaling exponent (LSE) [12; 34], is calculated as follows:  
\[D(m,\epsilon)=\frac{\partial\log C(\epsilon)}{\partial\log\epsilon}\approx \frac{\log C(\epsilon_{2})-\log C(\epsilon_{1})}{\log\epsilon_{2}-\log \epsilon_{1}}. \tag{4}\]  
The LSEs on different scales \(\epsilon\) reveal the local scaling structure of the time series [35].  
The correlation dimension is a dynamic invariant measure that characterizes the dynamics of the time series. As the scale decreases, the correlation dimension can be estimated when the LSE reaches a plateau, and the value of the LSE for the plateau remains approximately constant in various embeddings. The existence of such plateaus indicates determinism. Regions where the LSE is constant and the same for all embedding dimensions are called scaling ranges. As suggested in [34], the LSEs of the gaussian white noise with different embedding dimensions in Figure 3a indicate that they are separated from each other and increase monotonically to the embedding dimension as the scale \(\epsilon\) decreases; such region is called the noise regime. For such data, the scaling regions are hardly discerned, and the corresponding correlation dimension could not be obtained. Figure 3b shows the LSEs for an Henon attractor (\(x_{i+1}=1-1.4x_{i}^{2}+0.3x_{i-1}\)) with different embedding dimensions and, with and without additive gaussian white noise. Such attractor has a correlation dimension of 1.26, which can be approximated according to the plateaus for noise free data (solid lines) when \(\log\epsilon<-4\). For large \(\epsilon\) (such as \(\log\epsilon>-1\) in the figure), the LSE does not exhibit scaling in this region (macroscopic regime), which only corresponds to the macroscopic structures. When the noise exists (dotted lines) and \(\log\epsilon<-1.5\), the LSE reaches a plateau (scaling range) in each embedding. However, the values of the LSE related to the plateau in these embeddings (\(m=2,3,4\)) are different from each other and are larger than 1.26. This implies that the true dimension of such data cannot be extracted when there is noise in the data, such as physiological time series, as also mentioned in [34]. The noise regime appears when \(\log\epsilon<-3\), where the LSE increases monotonically to the embedding dimension.  
Figure 3: (**a**) Local scaling exponents (LSEs) for gaussian white noise with different embedding dimensions, \(m=1,\dots,10\). (**b**) LSEs for Henon attractor of length 10,000 with and without 1.5% additive gaussian white noise; the embedding dimensions are set as \(m=2,\dots,4\); the solid and dotted lines correspond to the noise free data and data with additive noise, respectively.