*3.1. Analysis of Coupled Dynamic Model*

The validity of the proposed method was first tested on signals generated using two Henon map unidirectional coupled subsystems with one as the driver subsystem 1 and the other as the responder subsystem 2. The equations of the system are expressed as follows:

$$\begin{cases} \mathbf{x1}\_{t+1} = \mathbf{1.4} - \mathbf{x1}\_{t}^{2} + 0.3 \times \mathbf{y1}\_{t} \\ \mathbf{y1}\_{t+1} = \mathbf{x1}\_{t} \\ \mathbf{x2}\_{t+1} = \mathbf{1.4} - (\mathbf{C} \times \mathbf{x1}\_{t} + (1 - \mathbf{C}) \times \mathbf{x2}\_{t}) \times \mathbf{x2}\_{t} + 0.3 \times \mathbf{y2}\_{t} \\ \mathbf{y2}\_{t+1} = \mathbf{x2}\_{t} \end{cases} \tag{8}$$

The parameter *C* is a coupling parameter varying from 0 to 1. When *C* is 0, the two subsystems are entirely independent and there is no definite dynamical behavior between them. When *C* is 1, the two subsystems are completely synchronized and there is a definite dynamical relationship between them. The values of *x*11, *y*11, *x*21 and *y*21 are initialized randomly in the range from 0 to 1. Then, 50,000 points are calculated according to (8) and the first 20,000 are discarded as the transients.

From the definition of CBTN, we can see that the unique parameter relevant to the CBTN is the embedding dimension *d*. The parameter *d* defines the embedding spatial dimension of a given time series. Another noteworthy issue is the appropriate signal length in order to obtain reliable results. One fact is that the signal length is limited, and the other is that the calculation process can only be performed in one window. The values of these two parameters determine whether the results of the analysis can be described or not and whether it is possible to extract the deep relationships hidden between the two time series. Here, the determination process of these two parameters is explained by analyzing the unidirectional coupled Honen model with the deterministic coupling relationship. It was

selected because the theoretical value of each expected return can be calculated. Based on this theoretical value, it can be evaluated whether the result obtained with a specific parameter setting can converge reliably and stably to the expected value or not. In addition, in order to highlight the impact of the CBTN on the analysis result, the CPE results for the same objects were used as a comparison. When the CPE was used, the embedded dimension was 5 and the delay time was 2.

The appropriate signal length was determined by investigating the effect of the width of the analysis window on the results. For the time series *x*1 and *x*2 obtained using the unidirectional coupled Henon model under a certain coupling strength, the surrogate data *x*2*Surrogate* were first calculated by surrogating *x*2 using iAAFT (iterative amplitudeadjusted Fourier transform with five iterations) to mimic the random coupling state. Next, the sliding time window with a fixed moving step of 500 samples was chosen to segment the paired time series *x*1 and *x*2 and *x*1 and *x*2*Surrogate*. The purpose of using the sliding time window is to enhance the effect of data analysis. The width of the sliding window width was increased from 200 to 5000 samples with a step of 200 samples. With each window width, the coupling strength between *x*1 and *x*2*Surrogate* and between *x*1 and *x*2 was calculated using the CBTN for individual sliding windows, obtaining the *Hx*1−*x*2*Surrogate NOTE* and *Hx*1−*x*<sup>2</sup> *NOTE*, respectively. The differences *<sup>H</sup>x*1−*x*2*Surrogate NOTE* <sup>−</sup> *<sup>H</sup>x*1−*x*<sup>2</sup> *NOTE* were first calculated for individual windows and then averaged across windows as the measured difference. The same procedure was repeated for 30 times with each window width and then averaged across repetitions to obtain the average and standard deviation of the measured difference. The coupling strength between *x*1 and *x*2 was set to 0.1, 0.3, and 0.5, respectively, and the results are shown in Figure 2. Figure 2a shows the results of the CBTN method and Figure 2b shows the results of the CPE method. It can be found that the two methods can make a good distinction between different coupling strengths. The CPE method can reach a stable state when the window width is less than 1000 samples, and the CBTN method can reach a stable state when the window width is more than 2000 samples. It was speculated that the CPE was appropriate for the analysis of short time series when it was proposed. Therefore, the appropriate window width for the CBTN is 2000 samples. It should be noted that the differences *Hx*1−*x*2*Surrogate CPE* <sup>−</sup> *<sup>H</sup>x*1−*x*<sup>2</sup> *CPE* had a negative value with the CPE method when the coupling strength was 0.1. This is inconsistent with the theory and indicates that the CPE was not capable of differentiating the weak coupling state.

With the determined window width of 2000 samples, the impact of the embedding dimension *d* on the estimation of the coupling strength was further investigated using the same method as above. The dimension *d* varied from 3 to 15 with a step of 1. Figure 3 compares the results between the CBTN method and the CPE method. It can be seen that the results of CBTN method tended to be stable with the increase in the embedding dimension *d*. When the embedding dimension was greater than 10, the measured difference under three coupling strength levels basically reached a stable state. In contrast, the measured difference using the CPE method was greatly affected by the embedded dimension. Within the varying range of the embedded dimension, the measured difference under three coupling strength levels could reach a stable state. When the embedding dimension was greater than 11, the measured difference under the stronger coupling strength (C = 0.3) was even smaller than that under the weaker coupling strength (C = 0.1 and C = 0.2). These results demonstrated that the CBTN method can be less affected by the embedding dimension *d* compared with the CPE method. More specifically, the embedding dimension would have little influence when it is greater than a certain value. Accordingly, for the unidirectional coupled Henon model, the recommended embedding dimension *d* for the CBTN method was set to 10.

**Figure 2.** The results of the unidirectional coupled Henon model using CBTN and CPE for coupling analysis, respectively, at different coupling strengths *C* = 0.1, 0.3, 0.5 and when the sliding time window width is varied in steps of 200 samples within [200, 5000]. The values of the ordinate are *Hx*1−*<sup>x</sup> surrogate* <sup>2</sup> <sup>−</sup> *<sup>H</sup>x*1−*x*<sup>2</sup> . *<sup>x</sup> surrogate* <sup>2</sup> can be obtained by surrogating *x*<sup>2</sup> using the iAAFT method. (**a**) The results of coupling analysis using CBTN (30 repeated calculations); (**b**) The results of coupling analysis using CPE, *d* = 5, *τ* = 2 (30 repeated calculations).

**Figure 3.** At different coupling strengths *C* = 0.1, 0.3, 0.5, the sliding time window width is fixed at 2000 samples, and the embedding dimension *d* is taken from 3 to 15; the results of unidirectional coupled Henon model using CBTN and CPE for coupling analysis, respectively. The values of ordinate are *Hx*1−*<sup>x</sup> surrogate* <sup>2</sup> <sup>−</sup> *<sup>H</sup>x*1−*x*<sup>2</sup> . *<sup>x</sup> surrogate* <sup>2</sup> can be obtained by surrogating *x*<sup>2</sup> using the iAAFT method. (**a**) The results of coupling analysis using CBTN (30 repeated calculations); (**b**) The results of coupling analysis using CPE, *τ* = 2 (30 repeated calculations).

After the embedding dimension and the sliding window width were determined, the CBTN was used to analyze the unidirectional coupled Honen model under different coupling strengths *C*. Specifically, the coupling strength *C* increased from 0 to 0.9 with a step of 0.05. For a given coupling strength, the data of *x*1, *x*2 and *x*2*Surrogate* within individual sliding windows were analyzed using the CBTN and CPE methods, respectively, to obtain the *Hx*1−*x*<sup>2</sup> *NOTE*, *<sup>H</sup>x*1−*x*<sup>2</sup> *CPE* , *<sup>H</sup>x*1−*x*2*Surrogate NOTE* and *<sup>H</sup>x*1−*x*2*Surrogate CPE* . The procedure was also repeated 30 times under each coupling strength, and the mean value and the standard deviation across 30 repetitions were calculated. Figure 4 illustrates the average value of *Hx*1−*x*<sup>2</sup> *NOTE*, *<sup>H</sup>x*1−*x*<sup>2</sup> *CPE* , *<sup>H</sup>x*1−*x*2*Surrogate NOTE* and *<sup>H</sup>x*1−*x*2*Surrogate CPE* across all repetitions under different

coupling strength levels. In the figure, the dashed boxes indicated by the arrows are partial zooms of the analysis results. As can be seen from Figure 4, the greatest difference between the analytical results of CBTN and CPE was mainly in the part where the coupling strength was less than 0.2. In this part, the CBTN method gives correct analysis results, while the CPE calculation results are greater than the values under the random coupling state, which is inconsistent with the theory. The possible reason is the CPE method that is based on the probability distribution statistics of symbols cannot distinguish the interactions between time series under weak coupling conditions. In contrast, the proposed CBTN method has good detection capability of interactions between time series with a weak coupling strength.

**Figure 4.** Coupling analysis results for the CBTN-based unidirectional coupled Henon model when the sliding time window is fixed at 2000 samples and the coupling strength *C* is varied in steps of 0.05 in the range [0, 0.9]. The values of the ordinate are *HNOTE*. The blue curve is the NOTE between *x*<sup>1</sup> and *x*<sup>2</sup> and the red curve is the NOTE between *x*<sup>1</sup> and *x surrogate* <sup>2</sup> . *x surrogate* <sup>2</sup> can be obtained by surrogating *x*<sup>2</sup> using the iAAFT method (all values in the graph are the result of 30 repeated calculations). (**a**) The results of coupling analysis using CBTN. (**b**) The results of coupling analysis using CPE, *d* = 5, *τ* = 2.
