# Modeling the Nonlinearity of Sea Level Oscillations in the Malaysian Coastal Areas Using Machine Learning Algorithms

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Dataset

#### 2.2. Support Vector Machine

_{i}, x

_{j}) is the kernel function. The kernel value is equal to the internal value of two vectors x

_{i}and x

_{j}in the characteristic spaces ϕ(x

_{i}) and ϕ(x

_{j}), i.e., K(x

_{i}, x

_{j}) = ϕ(x

_{i}) × ϕ(x

_{j}).

#### 2.3. Genetic Programming

#### 2.4. Data Normalization and Model Performance

_{predicted}= (SST + MSL)

_{obs}

_{predicted}= (MSL + SST + Rainfall + MCC)

_{obs}

_{predicted}= (MSL + SST + MCC)

_{obs}

- (i)
- The root mean square errors (RMSEs) of the observed and predicted values were compared. The mean absolute error (MAE) is always small or equal to the RMSE. The variance of the individual errors in a sample will increase as long as the difference between the two values increases. Furthermore, all the errors in the sample have the same magnitude if the RMSE is equal to the MAE.$$\mathrm{RMSE}=\sqrt{\frac{{{\displaystyle \sum}}_{i=1}^{n}{(MS{L}_{p}-MS{L}_{o})}^{2}}{N}}$$
- (ii)
- The correlation coefficient (R) was applied to evaluate the relation between variables.$$\mathrm{R}=\frac{{{\displaystyle \sum}}_{i=1}^{n}(MS{L}_{o}-{\overline{MSL}}_{o})(MS{L}_{p}-{\overline{MSL}}_{p})}{\sqrt{{{\displaystyle \sum}}_{i=1}^{n}{(MS{L}_{o}-{\overline{MSL}}_{o})}^{2}}{{\displaystyle \sum}}_{i=1}^{n}{(MS{L}_{p}-{\overline{MSL}}_{p})}^{2}}$$
- (iii)
- The scatter index (SI) was calculated by dividing the RMSE with the mean of the observations.$$\mathrm{SI}=\frac{RMSE}{\overline{x}}$$
- (iv)
- The MAE measures the accuracy of continuous variables.$$\mathrm{MAE}=\frac{1}{n}{{\displaystyle \sum}}_{i=1}^{N}|MS{L}_{p}-MS{L}_{o}|$$
- (v)
- The mean absolute percentage error (MAPE) is the mean or average of the absolute percentage errors associated with forecasts.$$\mathrm{MAPE}=100\times \left[\frac{\mathrm{MAE}}{{\overline{\mathrm{MSL}}}_{\mathrm{p}}}\right]$$
- (vi)
- AI will be used to measure the significance of the proposed SVM2 over GP2 and can be expressed as follows$$\mathrm{AI}=\frac{SVM2-GP2}{GP2}\times 100$$
- (vii)
- The error percentage is used to determine the prediction precision and can be expressed as follows$$\mathrm{EP}=\frac{MS{L}_{p}-MS{L}_{o}}{MS{L}_{o}}\times 100\%$$

## 3. Results

#### 3.1. Model Performances of the SVM Model

#### 3.2. Optimal Kernel Functions with the Input Design of SVM2 for the Cross-Validation Process

#### 3.3. Model Performances of GP

#### 3.4. Optimal Selection Function with the Input Design of GP2 in the Crossover Process

#### 3.5. Comparison of the Average Error Percentages in SVM2 and GP2 at the Study Locations

#### 3.6. Comparison of the Accuracy Improvement (AI) in SVM2 and GP2 at the Study Locations

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 4.**Flow chart of the prediction methodology of historical monthly mean sea level (MMSL) using the support vector machine (SVM) and genetic programming (GP) algorithms.

**Figure 5.**Scatter plot of the actual and predicted monthly mean sea levels obtained from the PUK of SVM modeling during the training and testing periods at (

**a**) Kerteh, (

**b**) the Tioman Island, and (

**c**) Tanjung Sedili.

**Figure 6.**Model performances of the PUK in the SVM model at (

**a**) Kerteh, (

**b**) the Tioman Island, and (

**c**) Tanjung Sedili.

**Figure 7.**Scatter plots of the actual and predicted monthly mean sea levels obtained from ramped half–half (RHH) and the fitness-proportional selection of the GP model during the training and testing periods at (

**a**) Kerteh, (

**b**) the Tioman Island, and (

**c**) Tanjung Sedili.

**Figure 8.**Model performances of GP2 with RHH and the fitness-proportional selection model at (

**a**) Kerteh, (

**b**) the Tioman Island, and (

**c**) Tanjung Sedili.

**Figure 9.**Comparison of the average error percentages in case of SVM2 and GP2 at the study locations.

**Figure 11.**MSL prediction at different prediction horizons with the PUK in SVM2: (

**a**) Kerteh, (

**b**) Tioman Island, and (

**c**) Tanjung Sedili.

**Figure 12.**MSL prediction at different prediction horizons with the RHH fitness-proportional selection in GP2: (

**a**) Kerteh, (

**b**) the Tioman Island, and (

**c**) Tanjung Sedili.

**Table 1.**Arrangement of the statistical data obtained from the study locations between January 1, 2007 and December 31, 2017.

Statistics/Study Location | Kerteh | Tioman Island | Tanjung Sedili | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Rainfall Amount (mm) | Mean Cloud Cover (Okta) | Mean Sea Level (mm) | SST (°C) | Rainfall Amount (mm) | Mean Cloud Cover (Okta) | Mean Sea Level (mm) | SST (°C) | Rainfall Amount (mm) | Mean Cloud Cover (Okta) | Mean Sea Level (mm) | SST (°C) | |

Maximum | 1645.20 | 7.40 | 7411.00 | 31.00 | 880.40 | 7.40 | 7398 | 31.0 | 574.20 | 7.40 | 7415 | 31.0 |

Minimum | 2.00 | 6.38 | 6836.00 | 26.40 | 2.00 | 6.60 | 6839 | 26.9 | 10.00 | 6.60 | 6872 | 27.0 |

Sum | 73,322.47 | 2770.40 | 2,806,123.00 | 11,494.27 | 27,816.70 | 923.49 | 935,843 | 3,835.20 | 12,798.98 | 923.55 | 934,635 | 3844.93 |

Average | 185.16 | 7.00 | 7086.17 | 29.03 | 210.73 | 6.99 | 7,089.7 | 29.05 | 96.96 | 7.00 | 7080.57 | 29.13 |

Mean Standard deviation | 192.23 | 0.109 | 132.80 | 0.89 | 162.80 | 0.08 | 129.16 | 0.85 | 119.80 | 0.09 | 133.55 | 0.79 |

Type of Kernel Functions | Tuning or Affecting Parameters |
---|---|

Normalized polynomial kernel (NP) | d(exponent), C, and ϵ |

Radial basis kernel (RBF) | ɣ, C, and ϵ |

Pearson universal kernel (PUK) | ω, σ, C, and ϵ |

**Table 3.**Summary of the SVM model performance with different kernel types and input designs at Kerteh.

Input Design | SVM1 | SVM2 | SVM3 | |||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Kernel Type/ Model Performance | NP | RBF | PUK | NP | RBF | PUK | NP | RBF | PUK | |||||||||

Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | |

R | 0.341 | 0 | 0.510 | 0.443 | 0.523 | 0.228 | 0.751 | 0.724 | 0.635 | 0.647 | 0.863 | 0.861 | 0.778 | 0.697 | 0.512 | 0.462 | 0.766 | 0.708 |

RMSE (mm) | 144.00 | 140.40 | 124.79 | 140.16 | 117.63 | 157.12 | 86.89 | 74.81 | 118.24 | 134.40 | 69.17 | 83.06 | 110.6 | 103.14 | 124.63 | 139.41 | 88.65 | 106.15 |

SI | 1.20 | 1.17 | 1.03 | 1.16 | 0.98 | 1.3 | 0.72 | 0.62 | 0.98 | 1.12 | 0.57 | 0.69 | 0.92 | 0.85 | 1.03 | 1.16 | 0.73 | 0.88 |

MAE (mm) | 112.72 | 133.14 | 124.79 | 140.16 | 117.63 | 157.12 | 95.1 | 98.7 | 95.3 | 99.9 | 46.5 | 68.1 | 84.9 | 88.0 | 99.8 | 102.4 | 62.3 | 83.9 |

MAPE (%) | 68.2 | 85.1 | 48.9 | 52.3 | 46.7 | 76.1 | 32.2 | 35.3 | 45.7 | 46.9 | 25.6 | 25.2 | 36.8 | 43.7 | 49.9 | 51.8 | 37.5 | 41.7 |

Study Locations | Kerteh | Tioman Island | Tanjung Sedili | Cross-Validation | No. of Support Vector | Capacity | |||
---|---|---|---|---|---|---|---|---|---|

Model Performances | Train | Test | Train | Test | Train | Test | |||

R | 0.771 | 0.757 | 0.772 | 0.796 | 0.699 | 0.786 | 10 | 118 | 1.0 |

R | 0.777 | 0.766 | 0.779 | 0.805 | 0.706 | 0.795 | 9 | 118 | 1.0 |

R | 0.771 | 0.757 | 0.772 | 0.796 | 0.699 | 0.786 | 8 | 118 | 1.0 |

R | 0.764 | 0.749 | 0.765 | 0.787 | 0.692 | 0.777 | 7 | 118 | 1.0 |

R | 0.757 | 0.740 | 0.758 | 0.778 | 0.685 | 0.768 | 6 | 118 | 1.0 |

R | 0.750 | 0.731 | 0.751 | 0.769 | 0.678 | 0.759 | 5 | 118 | 1.0 |

R | 0.743 | 0.722 | 0.745 | 0.760500 | 0.672 | 0.750 | 4 | 118 | 1.0 |

R | 0.737 | 0.713 | 0.738 | 0.751 | 0.665 | 0.741 | 3 | 118 | 1.0 |

R | 0.730 | 0.704 | 0.731 | 0.742 | 0.658 | 0.732 | 2 | 118 | 1.0 |

**Table 5.**Summary of the GP model performances with different kernel types and input designs at Kerteh.

Input Design | GP1 | GP2 | GP3 | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Selections/Model Performance | RHH and Fitness Proportionate Selection | RHH and Rank Selection | RHH and Fitness Proportionate Selection | RHH and Rank Selection | RHH and Fitness Proportionate Selection | RHH and Rank Selection | ||||||

Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | Train | Test | |

R | 0.769 | 0.571 | 0.682 | 0.215 | 0.78 | 0.748 | 0.78 | 0.45 | 0.756 | 0.57 | 0.73 | 0.40 |

RMSE (mm) | 88.62 | 117.16 | 87.39 | 185.68 | 86.69 | 89.2 | 88.63 | 120.32 | 90.94 | 124.41 | 86.58 | 120.27 |

SI | 1.2 | 0.67 | 0.75 | 0.325 | 1.3 | 1.02 | 1.3 | 0.52 | 1.08 | 0.67 | 0.93 | 0.49 |

MAE (mm) | 125.6 | 135.9 | 120.3 | 159.7 | 103.2 | 106.5 | 121.3 | 144.7 | 115.5 | 138.2 | 128.9 | 140.2 |

MAPE (%) | 35.9 | 43.5 | 38.2 | 85.2 | 22.9 | 25.0 | 23.0 | 59.7 | 29.2 | 49.6 | 33.6 | 53.7 |

**Table 6.**Crossover processing for the RHH and fitness-proportional selection at the study locations.

Study Locations | Kerteh | Tioman Island | Tanjung Sedili | Crossover | Generation | |||
---|---|---|---|---|---|---|---|---|

Model Performances | Train | Test | Train | Test | Train | Test | ||

R; Last Change | 0.758;375 | 0.452;375 | 0.689;192 | 0.578;192 | 0.735;371 | 0.73;371 | 0.2 | 300 |

R; Last Change | 0.708;377 | 0.55;377 | 0.697;394 | 0.524;394 | 0.719;270 | 0.487;270 | 0.4 | 300 |

R; Last Change | 0.762;369 | 0.748;369 | 0.702;265 | 0.591;265 | 0.722;87 | 0.776;87 | 0.6 | 300 |

R; Last Change | 0.682;341 | 0.498;341 | 0.722;345 | 0.718;345 | 0.71;251 | 0.703;251 | 0.8 | 300 |

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## Share and Cite

**MDPI and ACS Style**

Lai, V.; Ahmed, A.N.; Malek, M.A.; Abdulmohsin Afan, H.; Ibrahim, R.K.; El-Shafie, A.; El-Shafie, A.
Modeling the Nonlinearity of Sea Level Oscillations in the Malaysian Coastal Areas Using Machine Learning Algorithms. *Sustainability* **2019**, *11*, 4643.
https://doi.org/10.3390/su11174643

**AMA Style**

Lai V, Ahmed AN, Malek MA, Abdulmohsin Afan H, Ibrahim RK, El-Shafie A, El-Shafie A.
Modeling the Nonlinearity of Sea Level Oscillations in the Malaysian Coastal Areas Using Machine Learning Algorithms. *Sustainability*. 2019; 11(17):4643.
https://doi.org/10.3390/su11174643

**Chicago/Turabian Style**

Lai, Vivien, Ali Najah Ahmed, M.A. Malek, Haitham Abdulmohsin Afan, Rusul Khaleel Ibrahim, Ahmed El-Shafie, and Amr El-Shafie.
2019. "Modeling the Nonlinearity of Sea Level Oscillations in the Malaysian Coastal Areas Using Machine Learning Algorithms" *Sustainability* 11, no. 17: 4643.
https://doi.org/10.3390/su11174643