# Improving Significant Wave Height Prediction Using a Neuro-Fuzzy Approach and Marine Predators Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Case Study

## 3. Methods

#### 3.1. Multivariate Adaptive Regression Splines (MARS)

_{k}is the explanatory variables or the equal elements of the Parajets coefficient determined by minimizing the sum of the squares of the residuals. Each basis function can be a form of a linear spline function or the product of two or more of them, indicating mutual effects. In the model MARS, the space of explanatory variables is divided into several separate regions by certain knots, which gives the greatest reduction of the sum of squared errors [24,25,26].

#### 3.2. Adaptive Neuro-Fuzzy Inference System (ANFIS)

_{t}and SWH

_{t−1}.

#### 3.3. Optimization Algorithms

#### 3.3.1. Genetic Algorithm (GA)

- 1.
- An initial set of composite functions indicating predictive models is considered randomly.
- 2.
- Each person in the above population is evaluated with appropriate functions.
- 3.
- For each production, the following steps are followed to select a new population:
- (a)
- One of the transitions, mutation, and copy operators is selected.
- (b)
- An appropriate number of individuals from the current community are selected.
- (c)
- The selection operator is used to generate the descendants.
- (d)
- The named descendant enters a new society.
- (e)
- The considered model is evaluated using various adjustments.

- 4.
- The third step is repeated until the maximum production is reached.

#### 3.3.2. Particle Swarm Optimization (PSO)

#### 3.3.3. Marine Predators Algorithm (MPA)

_{max}and y

_{min}are the upper and lower bounds of the design variables, respectively, and rand is also a random vector in the range [1 0]. In the MPA, there are two main matrices, the best-fit predator matrix (Best/Elite) and the prey matrix (Prey), given in Equations (5) and (6).

#### 3.4. Development of Hybrid ANFIS Methods

_{t}, SWH

_{t−1}, and SWH

_{t−2}, referring to the SWH at time t, t − 1, and t − 2, where t is hour and output is SWH at time t + 1 (one hour ahead) to t + 24 (one day ahead). Each input has Gaussian membership functions, and each membership function has two parameters: mean and standard deviation. Three metaheuristic algorithms, i.e., PSO, GA, and MPA, were used for optimizing linear (consequent) and nonlinear (premise) parameters of ANFIS. The procedure for developing hybrid ANFIS models, including ANFIS-PSO, ANFIS-GA, and ANFIS-MPA, is depicted in Figure 5.

#### 3.5. Accuracy Assessment

## 4. Development of Hybrid ANFIS-PSO, ANFIS-GA, and ANFIS-MPA Models

## 5. Results and Discussion

#### 5.1. Results

^{2}in both stations. The models with three inputs (best models) are compared using Taylor diagrams in Figure 8 and Figure 9. This type of graph is very useful for observing the accuracy of the models based on RMSE, standard deviation, and correlation. The plots show that the MPA-based ANFIS has the highest correlation and the lowest squared error in predicting the SWH of both stations. The violin charts in Figure 10 and Figure 11 compare the SWH predictions and observations distributions. The figures show that the mean, median, and distribution of the MPA-based ANFIS are more like the observed values. Figure 12 illustrates the average RMSE and MAE errors of all implemented models in predicting the SWH of both stations. It is clearly seen from the bar charts that the ANFIS-MPA has fewer RMSE and MAE errors in the short-term prediction of SWH in both sites.

#### 5.2. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Case study area for significant wave height modeling representing Station 1 (Cairns) and Station 2 (Palms Beach).

**Figure 2.**Description of the ANFIS method: (

**A**) fuzzy inference and (

**B**) corresponding ANFIS structure.

**Figure 6.**Scatterplots of the observed and predicted SWH by different models in the test period using the best input combination at Station 1.

**Figure 7.**Scatterplots of the observed and predicted SWH by different models in the test period using the best input combination at Station 2.

**Figure 8.**Taylor diagrams of the predicted SWH by different models using the best input combination at Station 1.

**Figure 9.**Taylor diagrams of the predicted SWH by different models using the best input combination at Station 2.

**Figure 10.**Violin charts of the predicted SWH by different models using the best input combination at Station 1.

**Figure 11.**Violin charts of the predicted SWH by different models using the best input combination at Station 2.

**Figure 12.**Average RMSE and MAE of the applied models in predicting SWH using all models for all input combinations during test period of both stations.

Mean | Min. | Max | Skewness | Std. Dev. | |
---|---|---|---|---|---|

Station 1 | |||||

Whole Dataset | 0.4483 | 0.0840 | 1.4230 | 0.5173 | 0.2066 |

Training Dataset | 0.4632 | 0.0840 | 1.4230 | 0.4570 | 0.2083 |

Testing Dataset | 0.4118 | 0.0890 | 1.1220 | 0.6722 | 0.1978 |

Station 2 | |||||

Whole Dataset | 1.2179 | 0.2600 | 4.0640 | 1.1955 | 0.5688 |

Training Dataset | 1.2467 | 0.2600 | 4.0640 | 1.1093 | 0.5725 |

Testing Dataset | 1.1441 | 0.2680 | 3.9960 | 1.4586 | 0.5524 |

**Table 2.**Training and test statistics of the models for multiple steps ahead SWH predictions—MARS for Station 1.

Time Horizon | Input Combination | Training Period | Test Period | ||||
---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | ||

t + 1 | SWH_{t} | 0.0317 | 0.0218 | 0.9728 | 0.0337 | 0.0240 | 0.9706 |

SWH_{t}, SWH_{t − 1} | 0.0312 | 0.0216 | 0.9752 | 0.0327 | 0.0235 | 0.9733 | |

SWH_{t}, SWH_{t − 1}, SWH_{t − 2} | 0.0310 | 0.0214 | 0.9756 | 0.0325 | 0.0232 | 0.9748 | |

t + 2 | SWH_{t}, SWH_{t − 1}, SWH_{t − 2} | 0.0510 | 0.0367 | 0.9358 | 0.0601 | 0.0453 | 0.9171 |

t + 4 | SWH_{t}, SWH_{t − 1}, SWH_{t − 2} | 0.0856 | 0.0637 | 0.8316 | 0.0750 | 0.0538 | 0.8569 |

t + 8 | SWH_{t}, SWH_{t − 1}, SWH_{t − 2} | 0.1190 | 0.0819 | 0.7086 | 0.1078 | 0.0909 | 0.6741 |

t + 12 | SWH_{t}, SWH_{t − 1}, SWH_{t − 2} | 0.1248 | 0.0971 | 0.6053 | 0.1256 | 0.0977 | 0.6084 |

t + 24 | SWH_{t}, SWH_{t − 1}, SWH_{t − 2} | 0.1341 | 0.1036 | 0.5855 | 0.1410 | 0.1076 | 0.5201 |

**Table 3.**Training and test statistics of the models for multiple steps ahead SWH predictions—ANFIS-PSO for Station 1.

Time Horizon | Input Combination | Training Period | Test Period | ||||
---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | ||

t + 1 | SWH_{t} | 0.0299 | 0.2110 | 0.9755 | 0.0332 | 0.0238 | 0.9739 |

SWH_{t}, SWH_{t−1} | 0.0295 | 0.0203 | 0.9768 | 0.0323 | 0.0230 | 0.9746 | |

SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0286 | 0.0198 | 0.9791 | 0.0312 | 0.0226 | 0.9753 | |

t + 2 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0482 | 0.0333 | 0.9421 | 0.0501 | 0.0353 | 0.9406 |

t + 4 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0788 | 0.0569 | 0.8560 | 0.0721 | 0.0518 | 0.8684 |

t + 8 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1130 | 0.0784 | 0.7427 | 0.1004 | 0.0858 | 0.7055 |

t + 12 | SWH_{t}, SWH_{t −1}, SWH_{t−2} | 0.1244 | 0.0950 | 0.6054 | 0.1148 | 0.0897 | 0.6464 |

t + 24 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1300 | 0.1005 | 0.5961 | 0.1391 | 0.1071 | 0.5422 |

**Table 4.**Training and test statistics of the models for multiple steps ahead SWH predictions—ANFIS-GA for Station 1.

Time Horizon | Input Combination | Training Period | Test Period | ||||
---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | ||

t + 1 | SWH_{t} | 0.0287 | 0.0209 | 0.9778 | 0.0326 | 0.0235 | 0.9755 |

SWH_{t}, SWH_{t−1} | 0.0284 | 0.0200 | 0.9789 | 0.0306 | 0.0219 | 0.9780 | |

SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0281 | 0.0196 | 0.9819 | 0.0302 | 0.0216 | 0.9787 | |

t + 2 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0462 | 0.0312 | 0.9493 | 0.0466 | 0.0318 | 0.9438 |

t + 4 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0777 | 0.0558 | 0.8607 | 0.0698 | 0.0496 | 0.8783 |

t + 8 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1104 | 0.0751 | 0.7549 | 0.0986 | 0.0834 | 0.7188 |

t + 12 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1132 | 0.0856 | 0.6675 | 0.1143 | 0.0895 | 0.6612 |

t + 24 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1267 | 0.0978 | 0.6297 | 0.1352 | 0.1022 | 0.5785 |

**Table 5.**Training and test statistics of the models for multiple steps ahead SWH predictions—ANFIS-MPA for Station 1.

Time Horizon | Input Combination | Training Period | Test Period | ||||
---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | ||

t + 1 | SWH_{t} | 0.0276 | 0.0202 | 0.9820 | 0.0312 | 0.0224 | 0.9782 |

SWH_{t}, SWH_{t−1} | 0.0262 | 0.0199 | 0.9834 | 0.0279 | 0.0198 | 0.9818 | |

SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0256 | 0.0188 | 0.9848 | 0.0277 | 0.0192 | 0.9831 | |

t + 2 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0415 | 0.0290 | 0.9603 | 0.0445 | 0.0300 | 0.9495 |

t + 4 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0722 | 0.0541 | 0.8911 | 0.0678 | 0.0481 | 0.8831 |

t + 8 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1051 | 0.0728 | 0.8035 | 0.0983 | 0.0736 | 0.7544 |

t + 12 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1092 | 0.0829 | 0.6766 | 0.1137 | 0.0879 | 0.6718 |

t + 24 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1147 | 0.0904 | 0.6480 | 0.1344 | 0.1019 | 0.5833 |

**Table 6.**Training and test statistics of the models for multiple steps ahead SWH predictions—MARS for Station 2.

Time Horizon | Input Combination | Training Period | Test Period | ||||
---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | ||

t + 1 | SWH_{t} | 0.1136 | 0.0844 | 0.9608 | 0.1180 | 0.0870 | 0.9570 |

SWH_{t}, SWH_{t−1} | 0.1070 | 0.0824 | 0.9633 | 0.1138 | 0.0842 | 0.9600 | |

SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1004 | 0.0808 | 0.9653 | 0.1067 | 0.0824 | 0.9635 | |

t + 2 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1135 | 0.0893 | 0.9606 | 0.1163 | 0.0846 | 0.9587 |

t + 4 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1309 | 0.0995 | 0.9448 | 0.1469 | 0.1038 | 0.9331 |

t + 8 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1658 | 0.1277 | 0.9120 | 0.1853 | 0.1238 | 0.8932 |

t + 12 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1991 | 0.1502 | 0.8506 | 0.2171 | 0.1465 | 0.8647 |

t + 24 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.2642 | 0.1935 | 0.7782 | 0.2928 | 0.2029 | 0.7303 |

**Table 7.**Training and test statistics of the models for multiple steps ahead SWH predictions—ANFIS-PSO for Station 2.

Time Horizon | Input Combination | Training Period | Test Period | ||||
---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | ||

t + 1 | SWH_{t} | 0.0821 | 0.0564 | 0.9769 | 0.0860 | 0.0584 | 0.9717 |

SWH_{t}, SWH_{t−1} | 0.0802 | 0.0542 | 0.9799 | 0.0809 | 0.0545 | 0.9786 | |

SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0717 | 0.0498 | 0.9815 | 0.0741 | 0.0514 | 0.9823 | |

t + 2 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0822 | 0.0580 | 0.9782 | 0.0960 | 0.0642 | 0.9713 |

t + 4 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1175 | 0.0807 | 0.9554 | 0.1252 | 0.0908 | 0.9515 |

t + 8 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1631 | 0.1154 | 0.9152 | 0.1688 | 0.1159 | 0.9048 |

t + 12 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1934 | 0.1435 | 0.8782 | 0.2061 | 0.1443 | 0.8724 |

t + 24 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.2637 | 0.1918 | 0.7816 | 0.2708 | 0.2005 | 0.7461 |

**Table 8.**Training and test statistics of the models for multiple steps ahead SWH predictions—ANFIS-GA for Station 2.

Time Horizon | Input Combination | Training Period | Test Period | ||||
---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | ||

t + 1 | SWH_{t} | 0.0794 | 0.0540 | 0.9804 | 0.0827 | 0.0568 | 0.9771 |

SWH_{t}, SWH_{t−1} | 0.0742 | 0.0509 | 0.9822 | 0.0804 | 0.0541 | 0.9798 | |

SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0692 | 0.0480 | 0.9845 | 0.0693 | 0.0479 | 0.9835 | |

t + 2 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0818 | 0.0574 | 0.9784 | 0.0954 | 0.0639 | 0.9716 |

t + 4 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1148 | 0.0880 | 0.9577 | 0.1209 | 0.0828 | 0.9524 |

t + 8 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1542 | 0.1040 | 0.9255 | 0.1619 | 0.1176 | 0.9178 |

t + 12 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1868 | 0.1359 | 0.8835 | 0.1999 | 0.1376 | 0.8792 |

t + 24 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.2568 | 0.1890 | 0.7828 | 0.2695 | 0.1987 | 0.7681 |

**Table 9.**Training and test statistics of the models for multiple steps ahead SWH predictions—ANFIS-MPA for Station 2.

Time Horizon | Input Combination | Training Period | Test Period | ||||
---|---|---|---|---|---|---|---|

RMSE | MAE | R^{2} | RMSE | MAE | R^{2} | ||

t + 1 | SWH_{t} | 0.0746 | 0.0510 | 0.9820 | 0.0785 | 0.0537 | 0.9808 |

SWH_{t}, SWH_{t−1} | 0.0670 | 0.0438 | 0.9860 | 0.0750 | 0.0523 | 0.9818 | |

SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0643 | 0.0428 | 0.9871 | 0.0689 | 0.0475 | 0.9847 | |

t + 2 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.0771 | 0.0503 | 0.9815 | 0.0913 | 0.0628 | 0.9731 |

t + 4 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1148 | 0.0880 | 0.9577 | 0.1209 | 0.0828 | 0.9524 |

t + 8 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1520 | 0.1105 | 0.9374 | 0.1599 | 0.1137 | 0.9218 |

t + 12 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.1859 | 0.1342 | 0.8890 | 0.1954 | 0.1363 | 0.8833 |

t + 24 | SWH_{t}, SWH_{t−1}, SWH_{t−2} | 0.2515 | 0.1842 | 0.7925 | 0.2640 | 0.1962 | 0.7735 |

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**MDPI and ACS Style**

Ikram, R.M.A.; Cao, X.; Sadeghifar, T.; Kuriqi, A.; Kisi, O.; Shahid, S.
Improving Significant Wave Height Prediction Using a Neuro-Fuzzy Approach and Marine Predators Algorithm. *J. Mar. Sci. Eng.* **2023**, *11*, 1163.
https://doi.org/10.3390/jmse11061163

**AMA Style**

Ikram RMA, Cao X, Sadeghifar T, Kuriqi A, Kisi O, Shahid S.
Improving Significant Wave Height Prediction Using a Neuro-Fuzzy Approach and Marine Predators Algorithm. *Journal of Marine Science and Engineering*. 2023; 11(6):1163.
https://doi.org/10.3390/jmse11061163

**Chicago/Turabian Style**

Ikram, Rana Muhammad Adnan, Xinyi Cao, Tayeb Sadeghifar, Alban Kuriqi, Ozgur Kisi, and Shamsuddin Shahid.
2023. "Improving Significant Wave Height Prediction Using a Neuro-Fuzzy Approach and Marine Predators Algorithm" *Journal of Marine Science and Engineering* 11, no. 6: 1163.
https://doi.org/10.3390/jmse11061163