# Chaotic Features of Decomposed Time Series from Tidal River Water Level
## 5 Summary and Discussions  
This study reconstructed the phase space, analyzed the correlation dimension, and a CRP to identify the characteristics of the water level time series of the tidal river. Because the water level time series of a tidal river shows complex hydrological characteristics affected by tides, waves, and other factors, this study decomposed the water level time series into four components to examine each of their hydrological characteristics. In quantifying the effects of the decomposed components on the water level time series, entropy information theory was used. As a result, the tide component was identified to have the most influence on the original Ulsan water level time series by 48%, followed by the wave component at 24%, the rainfall-runoff component at 17%, and the noise component at 11%. This indicates the possibility of the tide component with a periodic fluctuation, but not exact periodic, as the major factor of water level change that leads to chaotic characteristics. Furthermore, the delay time was calculated using the ACF and average mutual information to examine characteristics of each component. For data such as the tide component and the original time series of the water level, the same delay time was calculated. However, the data with random characteristics, such as the rainfall-runoff component, had different delay times due to the difference in the linearity and nonlinearity. The calculated delay times were used to draw the attractors on two-dimensional phase space. As a result, the water level time series, tide component, and wave component had the trajectory within a certain boundary. However, the noise component presented a random trajectory. In determining whether each component has stochastic or chaotic characteristics, the correlation dimension was estimated. With the correlation dimension of 1.30, the tide component was influenced by two variables (mid-period and long period). As shown in Figure 9, only the tide component had a chaotic characteristic as it converged into a certain value, and the other components showed stochastic characteristics as they diverged. These results are significant in determining whether a linear model or a nonlinear model should be used to predict tidal river water levels with complex hydrological characteristics. Last, this study analyzed the chaotic characteristics of each component using CRP to understand the recurring trajectory in a graph. When regular horizontal line segments repeatedly appear in the CRP, it can be assumed as the periodic data that the corresponding trajectory is continuously visited. On the other hand, the irregular horizontal line segments indicate chaotic characteristics, while random distribution without lines implies stochastic characteristics. As a result, the original water level time series showed aperiodic characteristics caused by tidal effects during no rainfall (Figure 11a). However, when rainfall occurs, it had oblique line segments indicating stochastic characteristics. Therefore, this time series might be mixed with chaotic and stochastic characteristics, which can be called stochastic-chaos. For the tide component, irregular horizontal line segments appear similar to those of the aperiodic function and so it indicated chaotic characteristics by the uncertainty of natural phenomena. For the other components, the segments are oblique lines, indicating their stochastic characteristics. Table 4 shows the characteristics of each of the components. The water level time series were influenced by various hydrological factors and showed stochastic characteristics in the correlation dimension, but the characteristics of aperiodic-stochastic chaos in CRP. [PERSON] et al. (2009) have suggested that the behaviors related to stochastic chaos characteristics can be identified using the white noise in the deterministic chaos system [37; 38; 39].