# High-Precision Combined Tidal Forecasting Model
## 3 Combined Tidal Forecasting Model Based on Harmonic Analysis and ARIMA-SVR
### Model Checking
#### 3.2.3 Analysis of Prediction Results of Non-Astronomical Tide Level  
The model parameters of the non-astronomical part were determined by the single-step ARIMA model. To determine the difference term \(d\), the stationarity of the sequence must be determined by visual processing. To this end, the 744 nonlinear data throughout October 2018 were processed by the augmented Dickey-Fuller (ADF) stationarity test. The parameters are shown in Table 2. Here, the p-value is the probability of significance test in statistics, which is different from the order of the autoregressive model \(p\) mentioned in Formula (8). The \(t\)-statistics were below the critical values at the 1%, 5%, and 10% significance levels, and the p-values were close to 0, confirming that the tidal level sequence was a stationary sequence with \(d=0\)[22]. To determine the factors \(p\) in the autoregressive term and \(q\) in the moving average term, the sequence was evaluated by an autocorrelation function and a partial autocorrelation function [23], respectively. The evaluation results are shown in Figures 9 and 10, respectively. In the partial autocorrelation function plot (Figure 9), the sequence is mainly located in the confidence interval after the third order; i.e., it begins to truncate after the third order, and the autoregressive term \(p\) is thus 3. After establishing \(p\), the independent variables of the sample set in the SVR model are set to the tide levels at times \(t-i\) (\(i=1,2,3\)), and the dependent variable is the tide level at time \(t\). Figure 8: Linear regression of the harmonic analysis predictions versus the observed results. The red line is the best-fit line of the data. Along the black line, the predicted and observed values are equal, and the error is zero. Figure 7: Error distribution map of the harmonic analysis prediction. As is evident in the autocorrelation plot (Figure 8), the tidal level time series is tailing, and the moving average term \(q\) is 0. In summary, the sequence establishes the ARIMA \((3,0,0)\) model. The prediction process for the non-astronomical tidal part is shown in Figure 11.  
To reduce the influence of the order of magnitude of the sample on the prediction accuracy, the data in the training and test sets were normalized to the interval (0,1), and the RBF was selected as the kernel function. The parameters (penalty factor \(C\) and kernel function parameter \(g\)) were optimized by the PSO algorithm. The group termination algebra was set to 200; the population number was set to 20; and the learning factors \(C_{1}\) and \(C_{2}\) were set to 1.5 and 17, respectively. The optimization results are shown in Table 3. The searched optimal parameters were input to the SVM model as the training data. After training, the test data were input to the model, and the predicted results were compared with the real results. The comparisons and their relative errors are displayed in Figures 12 and 13, respectively. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter Type** & **t-Statistic** & \(p\)**-Value** & **1\% Level** & **5\% Level** & **10\% Level** \\ \hline Parameter values & \(-3.679002\) & \(0.0002\) & \(-2.568196\) & \(-1.941266\) & \(-1.616402\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters determined in the ADF unit root test. Figure 10: Partial autocorrelation analysis of the residual tidal level data. The area between the two blue lines and the coordinate axis is the confidence interval. Figure 9: Autocorrelation analysis of the residual tidal level data. The area between the two blue lines is the confidence interval. Figure 11: Schematic of the non-astronomical tidal level prediction part. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Penalty Factor \(C\)** & **Kernel Function Parameter \(g\)** \\ \hline SVR model & 20.4906 & 798.0125 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Optimal parameters detected by PSO. Figure 12: Comparison of the observations and predicted non-astronomical tidal level computed by the SVR model.