## 4 Analysis on the Features of 2D-DPRCML
### Correlation Analysis  
In traditional spatiotemporal chaotic systems, coupled processes cause high correlations among lattices. At the same time, the literature [44] points out that whether in temporal or spatial dimensions, the correlations of CMLs are sensitive to the coupling coefficient and control parameter. However, high correlations make it easy for an attacker to infer a particular lattice's output from some given lattices' outputs, causing the cryptosystem to be insecure when CML is applied to the cryptosystem. Therefore, analyzing the correlation of spatiotemporal chaotic systems is helpful in assessing their applicability to cryptosystems. In this paper, we let the system iterate 5000 times and obtain the output sequences of all lattices, after which we calculate the Pearson correlation coefficients between these sequences and derive the mean value. Similarly, the mean values of Pearson correlation coefficients are calculated for all pairs of parameters. We plotted these values to  
Figure 8: Distribution of sequences generated by the (a) 2D-CML, (b) 2D-NLCML, (c) 2D-MCPML and (d) 2D-DPRCML at \(\varepsilon=0.375\), \(\mu=4\).  
obtain the correlation analysis of the four systems under different parameter pairs, which are presented in Figure 9.  
In statistics, the correlation intensity between any two variables is judged by the Pearson correlation coefficient \(\rho\). The correlation intensities are weak and irrelevant when \(\rho\in\) [0.2, 0.4) and \(\rho\in\) [0, 0.2), respectively. In the two systems shown in Figure 9a,b, when \(\mu\in\) [3.5, 4] the correlation coefficients rapidly as \(\mu\) increases, indicating that the correlation coefficients of these two systems are sensitive to the coupling coefficients. The overall correlation coefficients of 2D-CML are even lower than those of 2D-NLCML when \(\mu\in\) [3.5, 4] and \(\varepsilon\in\) [0.5, 1]. In addition, the average of the correlation coefficients were below 0.4 for only 10.6% and 8.96% of the parameter pairs in the systems presented in Figure 9a,b, respectively. At the same time, 79.48% and 76.84% of the cases in the first two systems result in correlation coefficients higher than 0.8, respectively. In addition, almost all correlation coefficients in these systems are equal to 1 when \(\mu\in\) [3, 3.5], indicating that the sequences produced by different lattices in the first two systems are closely related to each other that make them insecure when applied to cryptosystems. By contrast, 2D-MCPML and 2D-DPRCML optimize the correlation between the lattices significantly, but the correlation coefficient of the 2D-MCPML system increases rapidly as \(\mu\) approaches four. Furthermore, 92.72% and 98.08% of the parameter pairs in Figure 9c,d, respectively, make the correlation coefficients less than 0.4, demonstrating that the correlation among any 2 sequences produced by various lattices is significantly decreased in the latter 2 systems. The low correlation coefficients of the 2D-DPCML make an attacker less likely to extrapolate a particular lattice's output from known lattices' outputs, demonstrating that it can enhance the security of cryptosystems when it is applied to them.  
Figure 9: The correlation analysis of (**a**) 2D-CML, (**b**) 2D-NLCML, (**c**) 2D-MCPML and (**d**) 2D-DPRCML.