# Chaotic Features of Decomposed Time Series from Tidal River Water Level
## 4 Chaotic Features Analysis of Decomposed Time Series
### Estimation of Correlation Dimension  
Correlation dimensions were estimated to identify the hydrological factors that affect the water level data of the water level station. The correlation estimation requires calculation of the slope of the local area and the delay time. If the correlation dimension and embedding dimension diverges in the graph, the data can be considered as stochastic. On the other hand, when it converges to a value, the data can be considered as chaotic. Figure 8 shows the correlation integral described by the relation between log C(r) and log r for each data when the embedding dimension (\(m\)) changes from 2 to 25.  
[PERSON] et al. (1995) divided the graphs of logC(r) and log r into three regions (noise, scaling, and saturation regions), and suggested that the correlation dimension can be estimated using the slope of the scaling region that has the linearity [35]. In this study, we estimated the relationship between the correlation dimension and the embedding dimension values for each dimension in the scaling region of Figure 8, where the accurate estimation of the correlation dimension can be made, and the results are shown in Figure 9. Especially, the correlation integral of rainfall-runoff component shows different characteristics from others. As mentioned above, the rainfall-runoff component usually shows a value of 0 and has a certain value when the water level rises rapidly. Therefore, the number of time series existing in the circle is not large, even though the dimension and the radius of circle are increased (see Figure 8e).  
Figure 7: Attractors of each component: (**a**) water level, (**b**) tide component, (**c**) wave component, (**d**) noise component.  
The result shows that the tide component has chaotic characteristics, given the convergence to a certain value of 1.30. However, the water level data and wave, noise, and rainfall-runoff components have stochastic characteristics as it showed a diverging pattern. [PERSON] et al. (2018) confirmed that correlation dimension of rainfall data at a specific rainfall station is diverging even though the dimension increases [36]. Through the above study, we can estimate that the water level time series shows stochastic characteristics because the correlation dimension diverged without converging to a specific value. As for the tide component, the correlation dimension of 1.30 indicates the two factors having influence on the component. They are likely to have the periodic characteristic caused by the moon that orbits around the Earth in approximately 30 days and by the Earth's changing position as it orbits around the sun in approximately one year. Tide component shows fluctuations in the short-cycle (orbit of the earth), the mid-cycle (gravity of Earth and moon), and the long-cycle (gravity of earth and sun). However, since this study used daily data, only the  
Figure 8: Correlation function versus radius r for (**a**) water level data, (**b**) tide component, (**c**) wave component, (**d**) noise component, and (**e**) rainfall-runoff component.  
Figure 9: Correlation dimension of each component.  
effects on the mid-cycle and long-cycle components could be considered. In this case, the tide component has aperiodic and nonlinear characteristics as they are gradually deviating from the periodicity, rather than having an accurate period.