# High-Precision Combined Tidal Forecasting Model

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Related Concepts

#### 2.1. SVM and SVR

#### 2.2. PSO Algorithm

#### 2.3. The Harmonic Analysis Method

#### 2.4. ARIMA

## 3. Combined Tidal Forecasting Model Based on Harmonic Analysis and ARIMA-SVR

#### 3.1. Prediction Steps

#### 3.2. Model Checking

#### 3.2.1. Sample Data Preprocessing

#### 3.2.2. Analysis of Prediction Results of Astronomical Tide Level

#### 3.2.3. Analysis of Prediction Results of Non-Astronomical Tide Level

#### 3.2.4. Analysis of Prediction Results of the Combined Model

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 6.**Comparison of tidal levels predicted by the harmonic analysis method alone and the observed data.

**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 9.**Autocorrelation analysis of the residual tidal level data. The area between the two blue lines is the confidence interval.

**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 12.**Comparison of the observations and predicted non-astronomical tidal level computed by the SVR model.

**Figure 14.**Comparison of the (not de-noised) observations and the water levels predicted by the combined model.

**Figure 16.**Linear regression plot of the predicted versus observed tidal heights. The predicted results were calculated by the combined model.

**Table 1.**The harmonic constants of four tidal constituents of the Bay Waveland Yacht Club tidal station.

Constituent | Amplitude/m | Phase/° |
---|---|---|

M_{2} | 0.03 | 31.7 |

S_{2} | 0.026 | 36 |

K_{1} | 0.169 | 328.2 |

O_{1} | 0.154 | 325.1 |

Parameter Type | t-Statistic | p-Value | 1% Level | 5% Level | 10% Level |
---|---|---|---|---|---|

Parameter values | −3.679002 | 0.0002 | −2.568196 | −1.941266 | −1.616402 |

Parameter | $\mathbf{Penalty}\text{}\mathbf{Factor}\text{}\mathit{C}$ | $\mathbf{Kernel}\text{}\mathbf{Function}\text{}\mathbf{Parameter}\text{}\mathit{g}$ |
---|---|---|

SVR model | 20.4906 | 798.0125 |

Data | ${\mathit{E}}_{\mathbf{M}\mathbf{A}}/{\mathbf{m}}^{2}$ | ${\mathit{E}}_{\mathbf{M}\mathbf{S}}/\mathbf{m}$ | ${\mathit{E}}_{\mathbf{R}\mathbf{M}\mathbf{S}}/\mathbf{m}$ |
---|---|---|---|

With de-noising | 0.0152928 | 0.0005121 | 0.0226293 |

Without de-noising | 0.0245718 | 0.0015876 | 0.0398451 |

**Table 5.**Performance comparison of different models simulating the tidal behavior at Bay Waveland Yacht Club. The second column lists “${p}^{\prime}$”, which refers to the fact that when the prediction time step is 1, the residual values from $t-1$ to $t-{p}^{\prime}$ moment are used to predict the value at time t. The last column lists the processing time required for the various methods.

Model | ${\mathit{p}}^{\prime}$ | ${\mathit{E}}_{\mathbf{M}\mathbf{A}}/{\mathbf{m}}^{2}$ | ${\mathit{E}}_{\mathbf{M}\mathbf{S}}/\mathbf{m}$ | ${\mathit{E}}_{\mathbf{R}\mathbf{M}\mathbf{S}}/\mathbf{m}$ | $\mathit{r}$ | Time/s |
---|---|---|---|---|---|---|

Harmonic Analysis | - | 0.1439167 | 0.0326057 | 0.1805706 | 0.52892 | 32.547131 |

BP | 3 | 0.1381779 | 0.0194776 | 0.1395623 | 0.61248 | 7.404504 |

PSO-SVR | 3 | 0.0415324 | 0.0023897 | 0.0498613 | 0.97381 | 56.04157 |

Combined model | 3 | 0.0152928 | 0.0005121 | 0.0226293 | 0.99066 | 148.28581 |

**Table 6.**Sample-size comparison tidal prediction errors in the SVR model. The second column lists “${p}^{\prime}$”, which refers to the fact that when the prediction time step is 1, the total tidal level from $t-1$ to $t-{p}^{\prime}$ moment is used to predict the total tidal level at time t.

The Size of Sample Set | ${\mathit{p}}^{\prime}$ | ${\mathit{E}}_{\mathbf{M}\mathbf{A}}/{\mathbf{m}}^{2}$ | ${\mathit{E}}_{\mathbf{M}\mathbf{S}}/\mathbf{m}$ | ${\mathit{E}}_{\mathbf{R}\mathbf{M}\mathbf{S}}/\mathbf{m}$ | $\mathit{r}$ | Time /s |
---|---|---|---|---|---|---|

One month | 3 | 0.0415324 | 0.0023897 | 0.0498613 | 0.9738146 | 56.042 |

Three months | 2 | 0.0413146 | 0.0022134 | 0.0495362 | 0.9738124 | 37.312 |

Six months | 4 | 0.0412874 | 0.0021045 | 0.0494135 | 0.9738113 | 64.617 |

Twelve months | 3 | 0.0411763 | 0.0020983 | 0.0493245 | 0.9738137 | 143.982 |

**Table 7.**Sample-size comparison tidal prediction errors in the combined model. The second column lists “${p}^{\prime}$”, which refers to the fact that when the prediction time step is 1, the residual water level from $t-1$ to $t-{p}^{\prime}$ moment is used to predict the non-astronomical tidal level at time t.

The size of Sample Set | ${\mathit{p}}^{\prime}$ | ${\mathit{E}}_{\mathbf{M}\mathbf{A}}/{\mathbf{m}}^{2}$ | ${\mathit{E}}_{\mathbf{M}\mathbf{S}}/\mathbf{m}$ | ${\mathit{E}}_{\mathbf{R}\mathbf{M}\mathbf{S}}/\mathbf{m}$ | $\mathit{r}$ | Time /s |
---|---|---|---|---|---|---|

One month | 3 | 0.0152928 | 0.0005121 | 0.0226293 | 0.9906614 | 148.286 |

Three months | 2 | 0.0152994 | 0.0005184 | 0.0227172 | 0.9906592 | 174.217 |

Six months | 4 | 0.0152613 | 0.0005127 | 0.0226981 | 0.9906452 | 209.576 |

Twelve months | 2 | 0.0151472 | 0.0005094 | 0.0226242 | 0.9906601 | 251.880 |

**Table 8.**Tidal-type comparisons of prediction errors in the harmonic analysis, SVR, and combined models. The third column lists “p′”, which refers to the fact that when the prediction time step is 1, the value from $t-1$ to $t-{p}^{\prime}$ moment is used to predict the value at time t.

Tidal Stations | Tidal Type | ${\mathit{p}}^{\prime}$ | Combined Model | SVR Model | Harmonic Analysis | |||
---|---|---|---|---|---|---|---|---|

${\mathit{E}}_{\mathbf{R}\mathbf{M}\mathbf{S}}/\mathbf{m}$ | $\mathit{r}$ | ${\mathit{E}}_{\mathbf{R}\mathbf{M}\mathbf{S}}/\mathbf{m}$ | $\mathit{r}$ | ${\mathit{E}}_{\mathbf{R}\mathbf{M}\mathbf{S}}/\mathbf{m}$ | $\mathit{r}$ | |||

Nawiliwili | Semidiurnal mixed tide | 3 | 0.0157421 | 0.9967487 | 0.0637458 | 0.9267340 | 0.0517735 | 0.9760012 |

The Battery | Semidiurnal tide | 2 | 0.0345787 | 0.9945872 | 0.2547214 | 0.8998741 | 0.2661074 | 0.9014375 |

Texas Point | Diurnal mixed tide | 2 | 0.0424531 | 0.9857817 | 0.0845512 | 0.9459475 | 0.1908219 | 0.7382920 |

Bay Waveland Yacht Club | Diurnal tide | 3 | 0.0152928 | 0.9906615 | 0.0498613 | 0.9738146 | 0.1805706 | 0.7188010 |

**Table 9.**The relative magnitude of the astronomical tide and the un-astronomical tide parts of four tidal stations.

Tidal Stations | The Astronomical Tide | The Non-Astronomical Tide |
---|---|---|

Nawiliwili | 88.1535% | 11.8465% |

The Battery | 85.9369% | 14.0631% |

Texas Point | 74.2308% | 25.7692% |

Bay Waveland Yacht Club | 60.406% | 39.594% |

**Table 10.**The input of SVR model comparisons of prediction accuracy of the combined method at Bay Waveland Yacht Club Tidal Station. The first column lists “${p}^{\prime}$”, which refers to the fact that when the prediction time step is 1, the residual values from $t-1$ to $t-{p}^{\prime}$ moment is used to predict the value at time t.

${\mathit{p}}^{\prime}$ | ${\mathit{E}}_{\mathbf{M}\mathbf{A}}/{\mathbf{m}}^{2}$ | ${\mathit{E}}_{\mathbf{M}\mathbf{S}}/\mathbf{m}$ | ${\mathit{E}}_{\mathbf{R}\mathbf{M}\mathbf{S}}/\mathbf{m}$ | $\mathit{r}$ | Time /s |
---|---|---|---|---|---|

1 | 0.0218577 | 0.0009071 | 0.0301173 | 0.9916444 | 29.233 |

2 | 0.0163588 | 0.0005606 | 0.0236780 | 0.9948650 | 40.472 |

3 | 0.0152412 | 0.0005076 | 0.0225293 | 0.9953641 | 130.600 |

4 | 0.0153038 | 0.0005076 | 0.0225293 | 0.9953641 | 169.639 |

6 | 0.0154409 | 0.0005186 | 0.0227718 | 0.9952334 | 281.034 |

12 | 0.0793346 | 0.0073769 | 0.0858892 | 0.9904756 | 556.413 |

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**MDPI and ACS Style**

Liu, J.; Shi, G.; Zhu, K. High-Precision Combined Tidal Forecasting Model. *Algorithms* **2019**, *12*, 65.
https://doi.org/10.3390/a12030065

**AMA Style**

Liu J, Shi G, Zhu K. High-Precision Combined Tidal Forecasting Model. *Algorithms*. 2019; 12(3):65.
https://doi.org/10.3390/a12030065

**Chicago/Turabian Style**

Liu, Jiao, Guoyou Shi, and Kaige Zhu. 2019. "High-Precision Combined Tidal Forecasting Model" *Algorithms* 12, no. 3: 65.
https://doi.org/10.3390/a12030065