# High-Precision Combined Tidal Forecasting Model
## 3 Combined Tidal Forecasting Model Based on Harmonic Analysis and ARIMA-SVR  
### Prediction Steps  
As mentioned above, tides can be considered as the superposition of astronomical tidal levels and non-astronomical tidal levels. The astronomical tidal level is strongly periodic, being governed by the tide-generating force, whereas the non-astronomical variations in water level largely depend on climate, hydrology, wind, and other environmental factors, and exhibit a strong randomness [21]. Based on the above analysis, this paper establishes a combined tidal prediction model based on harmonic analysis and ARIMA-SVR. The prediction steps are described below. The first step is to preprocesses the sample data. While acquiring tidal data, many factors may cause the data to become inaccurate and incomplete, resulting in noise interference. The preprocessing step removes the noise from the tidal sequence, thus restoring the tidal motions and improving the prediction accuracy. The preprocessing is performed by wavelet analysis theory. Second, the height of the astronomical tide level is calculated via harmonic analysis. The harmonic constants are calculated by analyzing the historical observations of tidal heights, and the tidal heights are subsequently calculated. Meanwhile, the residual series generated by this method are collected as the sample data of the non-astronomical tidal level component. Third, the nonlinear variations in water level are predicted by the ARIMA-SVR model. As the astronomical characteristics of the data have been processed in the astronomical tidal level component, the non-astronomical tidal levels change is reflected in the prediction residue. This prediction step is divided into several sub-steps: Exploiting the strong processing ability of ARIMA for time series, a single-step ARIMA model of non-astronomical tidal level sequences is established, which determines the input in the SVR forecasting model according to the lagged rank of the time series \(p\) in ARIMA. The analysis of the time sequence in the ARIMA model confirms that the residual values from \(t-1\) to \(t-p\) moment have a noticeable relevance with the values of moment \(t\), which can be chosen as the input of the SVR forecasting model. That is, the residual values from \(t-1\) to \(t-p\) moment are used to predict the value at time \(t\). The second sub-step establishes a non-astronomical tidal level prediction model based on SVR, which has a strong nonlinear processing ability. As mentioned above, the input variable of SVR mode is  
\[\text{Input}=\left\{Y_{t-1},Y_{t-2},Y_{t-3},\dots,Y_{t-p^{\prime}}\right\} \tag{10}\]  
where \(Y_{t-p^{\prime}}\) represents the value at time \(t-p^{\prime}\). Additionally, the value of \(p^{\prime}\) is determined by the value of the lagged rank of the time series \(p\) in the ARIMA model. Without considering other factors (\(p^{\prime}\) value is different only, other factors are the same), the prediction model error is the smallest when \(p^{\prime}=p\), which is verified in Section 3.2.4. Furthermore, the output variable is  
\[\text{Output}= \left\{Y_{t}\right\} \tag{11}\]  
The residual sequence generated by the harmonic analysis is normalized to avoid computational saturation. The data are normalized to the interval (0, 1) as follows:  
\[Y_{t}=\frac{x_{t}-x_{\text{min}}}{x_{\text{max}}-x_{\text{min}}}\ (x\in R^{n},\ y\in(0,1)) \tag{12}\]  
where \(x_{\text{min}}=\text{min}(x)\) and \(x_{\text{max}}=\text{max}(x)\). The optimal kernel function type, penalty factor \(C\), and kernel function parameter \(\delta\) are foundThe error in the prediction model was measured by the mean absolute error (\(E_{\text{MA}}\)), the mean squared error (\(E_{\text{MS}}\)), the root mean squared error (\(E_{\text{RMS}}\)), and the correlation coefficient \(r\).