# Stability Assessment of Rubble Mound Breakwaters Using Extreme Learning Machine Models

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

_{s}, which is commonly obtained from the Hudson formula [1] or the van der Meer formula [2]. Hudson’s formula is widely used in breakwater design because of its convenient calculation of the unit mass and stability number. Some important physical factors are not involved in the formula, such as the wave period, wave length and the water depth in front of the breakwater, so other researchers did further research. van der Meer [2] proposed the following formula based on more than 300 runs of breakwater experiments under irregular wave attack. Compared to the Hudson formula, more parameters are included in the van der Meer formula (VM formula), such as the number of wave attacks and the breakwater permeability.

_{s}can be expressed as follows in the study of van der Meer [2]:

_{s}is stable, and that it is a function of the wave height and the relative density and nominal diameter of the stone. Meanwhile, in different studies, the damage condition of the breakwater section used to compute the stability number has various definitions. In the studies of Thompson [10] and Hanzawa [6], a damage parameter was proposed to describe the damage condition of the breakwater section, which was a function of the stone density, stone size, wave height, wave number and erosion area in a cross section, while in the studies of van der Meer [2,3] and Kajima [5], a simple damage level was proposed. A summary of the definitions is listed in Table 1 [11].

_{s}can also be predicted by using the machine learning approaches. In the past two decades, a large and growing body of literature has investigated the machine learning approaches to assess the stability of rubble mound breakwaters, such as Artificial Neural Networks (ANN) [12,13,14,15], Fuzzy Neural Networks (FNN) [16,17], Model Tress (MT) [8,18], Support Vector Machine (SVM) [19,20,21,22], and Genetic Programing (GP) [23]. These studies have shown that the performance of machine learning approaches is better than that of the traditional formulas [23,24]. The study of Balas, Koç and Tür [13] provides new insights into improving the prediction accuracy of ANN models via the principal component analysis, which could reduce the needed amount of training data and transform the original data set into a set of uncorrelated variables that capture all of the variance of the original data set [25], but many methods still suffered from complex establishment procedures and large demands for the training sets’ size. Thus, reducing the parameter counts and the training data size, and simplifying the training process, should be a concern for further research.

_{s}can also be predicted by using the machine learning approaches. In the past two decades, a large and growing body of literature has investigated the machine learning approaches to assess the stability of rubble mound breakwaters, such as Artificial Neural Networks (ANN) [12,13,14,15], Fuzzy Neural Networks (FNN) [16,17], Model Tress (MT) [8,18], Support Vector Machine (SVM) [19,20,21,22], and Genetic Programing (GP) [23]. These studies have shown that the performance of machine learning approaches is better than that of the traditional formulas [23,24]. The study of Balas, Koç and Tür [13] provides new insights into improving the prediction accuracy of ANN models via the principal component analysis, which could reduce the needed amount of training data and transform the original data set into a set of uncorrelated variables that capture all of the variance of the original data set [25], but many methods still suffered from complex establishment procedures and large demands for the training sets’ size. Thus, reducing the parameter counts and the training data size, and simplifying the training process, should be a concern for further research.

## 2. Extreme Learning Machine Models

#### 2.1. Fundamental of Extreme Learning Machine Model

#### 2.2. Model Establishment

_{a}) are introduced as follows:

_{i}are the measured values, and their average is $\overline{X}$; Y

_{i}are the predicted values, and their average is $\overline{Y}$; and N is the number of observations.

## 3. Results and Discussion

#### 3.1. The Influence of Hidden Neurons on the Assessment Performance of ELM Models with Different Activation Functions

_{a}are increasing following the addition of hidden neurons when the number of the hidden neurons is within the range of 5 to 20; then, the values of CC and I

_{a}stay relatively constant when the number of the hidden neurons is above 20. The assessment performance criteria have a rapidly decreasing trend as the hidden neurons number increases in the range of 50 to 90, which indicates that too many hidden neurons lead to over-fitting. The models with the best assessment performance were the models built with 40–50 hidden neurons in the hidden layer, no matter what activation function was used in the models. What stands out in Figure 4 is that, for an ELM model with a randomly selected number of hidden neurons from 20 to 50, I

_{a}is no less than 0.9 and CC is no less than 0.85. The selection of the activation function has little influence on the best performance of each model, which is mainly because the training data has few noisy data sets and is pure enough. The simulation results show that the ELM algorithm has a good generalization performance for the stability assessment.

_{a}was the model built with 45 hidden neurons and the Sin function. The parameters for the model training and application were determined for these conditions.

#### 3.2. Predicted Performance Comparison of Different Methods

_{a}values among those machine learning approaches, which indicates the best prediction performance. The GPM3 formula was not obtained in [23], and the GPM1 model has a similar performance to the GPM3 model presented in [23], so the GPM1 model was selected. These assessment results, predicted by different methods, were prepared and are shown in Figure 5 and Figure 6, respectively.

_{a}. Lower values of BIAS and SI represent a better assessment performance, and higher values of CC and I

_{a}indicate a better prediction agreement. When the values of CC and I

_{a}are close to 1, this indicates a perfect agreement between the predicted and measured stability numbers. Table 3 lists the statistical index values of the three approaches. As shown in the table, the CC and I

_{a}values of the VM formula are the smallest among these three assessment approaches, while the BIAS and the SI values are the largest, which indicates that the performance of the VM formula has the lowest quality agreement. The CC and I

_{a}values of the EB formula, the GPM1 formula and the M1 model are nearly the same, while on the other hand the SI values of the two methods are also nearly the same. The evaluation indices show that the EB formula, the GPM1 formula and the M1 model have similar abilities for predicting the stability number of breakwaters with a low damage level, but that the M1 model was built based on a smaller size of training data, which indicates that the M1 model has a good generalization ability.

_{a}of the M2 model are the highest, while the BIAS and SI of the M2 model are the lowest, which indicates that the M2 model has the best performance for the stability number prediction of the breakwater sections with a wider damage level range.

_{a}values of the M1 and M2 models are higher than those in many of the previous studies, whose models were built using a larger size of training data and more input parameters. It should also be noted that the I

_{a}or CC values should not be the only evaluation criteria in comparing different methods, since the testing data for each model was not the same. A comparison between the MT2 model (EB formula), GPM1 formula and the ELM method was made in the previous section by using the same testing data, which presented the advantages of the ELM method. The testing data for the MT2 model and the GPM1 formula in [8,23] is not the same testing data used in this paper, which leads to the different I

_{a}and CC values presented in Table 3, Table 4 and Table 5. The CC values of HNN models in the study of Balas, Koç and Tür [13] are slightly higher than the CC values of the M1 and M2 models in the current paper, while the training data is pre-processed by using the principal component analysis (PCA), and the original data sets are 554 sets of experimental data. The PCA could remove the noisy data from the training data and extract the required information [13], so the use of PCA enhances the prediction ability of the machine learning models. It could also be expected that a PCA-ELM model will get a better prediction performance.

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

0.5241 | 26.4975 | 0.8109 | −0.4255 | 0.6935 | −0.2008 | |||

0.5979 | −3.6891 | 0.8853 | 0.6774 | 0.6278 | −0.9448 | |||

0.6609 | 4.1169 | 0.1518 | 0.2682 | 0.2991 | 0.0216 | |||

0.9402 | 1.1036 | −0.5524 | 0.7574 | 0.7449 | −0.6862 | |||

0.1974 | −32.1789 | −0.3415 | 0.2084 | −0.7634 | −0.1137 | |||

0.8710 | −3.2400 | −0.6313 | 0.8539 | −0.3247 | 0.9143 | |||

0.7430 | −29.8057 | 0.3799 | −0.2154 | −0.7326 | −0.9678 | |||

0.2418 | 5.5068 | −0.8844 | 0.7903 | 0.7482 | −0.7737 | |||

0.5977 | −22.8297 | 0.1486 | −0.9762 | 0.4362 | 0.0493 | |||

0.7125 | 1.7637 | 0.1595 | 0.5090 | −0.9902 | 0.1901 | |||

0.1448 | 21.5258 | −0.5829 | 0.0194 | −0.7497 | −0.6622 | |||

0.4441 | −4.4733 | −0.8343 | 0.7352 | 0.8387 | −0.2093 | |||

0.1918 | 6.0145 | −0.7073 | 0.0374 | 0.8523 | 0.7221 | |||

0.7374 | 52.4679 | −0.4033 | −0.0100 | 0.8501 | −0.7359 | |||

0.1496 | −9.1100 | 0.6958 | −0.2846 | 0.9258 | −0.7403 | |||

0.1726 | −3.6265 | 0.4105 | −0.9711 | 0.5676 | −0.2996 | |||

0.8718 | −8.6176 | −0.9463 | −0.0608 | −0.5387 | −0.6198 | |||

0.8638 | 32.5670 | 0.7170 | 0.0887 | −0.4715 | −0.5685 | |||

0.2632 | 13.7757 | 0.1976 | −0.0730 | −0.7203 | −0.0630 | |||

0.1091 | −9.6165 | 0.0072 | −0.6522 | 0.2843 | 0.5996 | |||

0.3324 | 3.5370 | 0.3308 | 0.2424 | −0.6404 | 0.6969 | |||

BHN_{1}= | 0.1969 | InW_{1}= | −30.1128 | InW_{2}= | −0.7382 | −0.5162 | −0.7029 | 0.4078 |

0.5033 | 40.9492 | −0.1724 | −0.1571 | 0.2812 | −0.6081 | |||

0.7217 | −2.1417 | 0.0027 | −0.1367 | 0.9792 | −0.1936 | |||

0.0935 | −4.5602 | 0.7380 | −0.4168 | 0.7734 | −0.7967 | |||

0.8949 | −7.4840 | −0.8876 | −0.7521 | 0.7573 | 0.1826 | |||

0.9296 | −51.8195 | −0.3970 | 0.0788 | 0.6631 | −0.9412 | |||

0.3114 | −32.1941 | 0.5991 | 0.3968 | −0.0596 | 0.2747 | |||

0.8365 | 0.7267 | 0.9239 | 0.6791 | 0.7207 | −0.1689 | |||

0.6055 | 35.3792 | −0.4155 | −0.4794 | −0.8263 | −0.0045 | |||

0.1465 | 5.8143 | −0.9828 | −0.4143 | 0.2699 | 0.9241 | |||

0.9326 | −14.1016 | 0.5911 | 0.8271 | 0.5772 | 0.0635 | |||

0.1928 | 9.7569 | −0.4223 | −0.3700 | 0.2338 | 0.7443 | |||

0.4138 | 2.4507 | −0.1683 | −0.2665 | −0.5608 | 0.6952 | |||

0.0855 | −7.5543 | −0.9139 | −0.9217 | −0.7361 | −0.3699 | |||

0.7125 | 8.2359 | −0.7147 | 0.3655 | −0.7379 | −0.7774 | |||

0.5891 | 3.9906 | 0.4442 | 0.7030 | 0.2163 | 0.0113 | |||

0.8273 | 1.0311 | 0.9852 | 0.9763 | −0.4108 | −0.4178 | |||

0.4677 | 11.6137 | −0.2928 | −0.8980 | −0.1545 | 0.3437 | |||

0.6765 | 6.5585 | 0.2751 | 0.9346 | 0.7867 | 0.6949 | |||

0.3229 | −7.2543 | −0.1302 | 0.1766 | 0.9851 | −0.9479 | |||

0.7244 | −8.8476 | −0.4926 | 0.8206 | 0.0350 | −0.9965 | |||

0.1206 | −10.8684 | 0.0382 | −0.5207 | 0.0727 | 0.9225 | |||

0.5268 | −5.1499 | −0.1425 | −0.2191 | 0.4494 | −0.9388 | |||

0.2891 | 8.4672 | 0.6724 | 0.1706 | −0.4620 | 0.9983 |

0.4319 | −42.0761 | −0.9071 | −0.2011 | 0.4272 | 0.8116 | |||

0.0320 | −20.9949 | −0.3291 | −0.0591 | −0.4020 | 0.8705 | |||

0.5944 | −32.7021 | −0.8404 | −0.6064 | 0.8841 | 0.6630 | |||

0.6627 | 43.8901 | −0.7591 | −0.2472 | 0.8186 | 0.9823 | |||

0.9264 | 13.2387 | 0.8394 | −0.8762 | −0.1618 | −0.3568 | |||

0.5949 | −20.2892 | 0.5871 | 0.8688 | −0.0913 | 0.7016 | |||

0.8525 | −63.1985 | 0.3422 | −0.7897 | −0.4640 | −0.2132 | |||

0.8806 | −116.6938 | 0.2035 | −0.5851 | −0.2849 | −0.8588 | |||

0.6270 | −13.5202 | 0.7838 | 0.9148 | −0.8121 | 0.6147 | |||

0.2328 | 35.2013 | −0.1258 | −0.3481 | −0.7869 | −0.1297 | |||

0.2941 | 3.1475 | −0.8012 | 0.0277 | −0.4674 | −0.6218 | |||

0.2577 | 12.4026 | −0.8559 | −0.6591 | −0.9608 | 0.2650 | |||

0.6162 | 0.2601 | −0.4507 | −0.2077 | −0.4970 | 0.7523 | |||

0.1584 | −3.0225 | 0.9716 | 0.8243 | −0.4446 | 0.6805 | |||

0.5654 | −7.7516 | −0.6291 | −0.5789 | −0.5272 | 0.3921 | |||

0.5730 | 5.6515 | −0.2855 | −0.5305 | 0.4384 | 0.8396 | |||

0.6728 | 27.5190 | 0.0217 | 0.4931 | −0.1090 | −0.5729 | |||

0.7424 | 0.7453 | −0.4198 | 0.1380 | 0.4327 | −0.9063 | |||

0.7593 | 36.1610 | 0.4848 | 0.2726 | 0.6648 | 0.6994 | |||

0.7122 | 44.1426 | −0.6639 | −0.4197 | 0.5753 | 0.2885 | |||

0.6100 | −16.2722 | −0.0565 | −0.0394 | 0.8366 | −0.8595 | |||

BHN_{2}= | 0.0537 | InW_{2}= | −1.1557 | InW_{2}= | −0.7270 | −0.1948 | −0.1188 | −0.1234 |

0.4458 | 22.3288 | 0.5387 | 0.8696 | −0.4597 | 0.9628 | |||

0.8475 | −1.6268 | 0.7513 | −0.9025 | −0.6607 | 0.6460 | |||

0.9733 | −83.7627 | 0.3622 | −0.6518 | −0.4596 | −0.5125 | |||

0.8544 | 22.4302 | 0.8799 | −0.2234 | −0.4083 | −0.7212 | |||

0.3858 | −21.6072 | −0.5399 | 0.1999 | 0.1068 | −0.4392 | |||

0.9096 | −2.8837 | −0.4029 | −0.6285 | 0.8447 | −0.2820 | |||

0.1069 | −28.1949 | −0.7637 | 0.7851 | −0.3326 | −0.1881 | |||

0.2582 | −18.8255 | −0.0014 | −0.1194 | 0.5881 | 0.2810 | |||

0.5765 | 47.8772 | 0.5480 | −0.4075 | −0.7139 | −0.6196 | |||

0.3990 | −5.0568 | 0.8476 | 0.1595 | −0.4691 | 0.1434 | |||

0.3779 | 9.6864 | 0.7929 | −0.4492 | −0.8683 | −0.4401 | |||

0.3411 | 10.6359 | −0.4233 | 0.5854 | −0.9226 | 0.3489 | |||

0.2897 | 3.4956 | 0.8980 | −0.7244 | 0.0454 | 0.2533 | |||

0.7287 | 45.6903 | −0.4528 | 0.5858 | 0.1254 | −0.0241 | |||

0.7738 | −45.6212 | 0.8116 | −0.2095 | −0.0985 | −0.6733 | |||

0.5252 | 25.9720 | 0.2493 | −0.7998 | −0.2112 | −0.3585 | |||

0.8545 | −50.5003 | −0.0441 | 0.5296 | −0.0523 | 0.2865 | |||

0.0416 | 3.4476 | −0.8948 | 0.9645 | −0.8378 | 0.9041 | |||

0.6695 | 8.8539 | −0.6859 | −0.9783 | −0.8757 | −0.9541 | |||

0.8819 | 20.7560 | 0.0010 | 0.8347 | −0.0483 | −0.2737 | |||

0.9352 | 133.8480 | 0.2844 | −0.8046 | −0.2266 | −0.8481 | |||

0.1300 | 5.0666 | 0.6626 | −0.6074 | 0.4537 | −0.5816 | |||

0.9134 | 14.3699 | −0.5942 | 0.5127 | −0.0012 | −0.3285 |

## Appendix B

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**Figure 2.**Parameters used in the M1 and M2 models: (

**a**) the permeability of M1; (

**b**) permeability of M2; (

**c**) damage level of M1; (

**d**) damage level of M2; (

**e**) slope angle of M1; and (

**f**) slope angle of M2.

**Figure 4.**Performances of the ELM models with different hidden nodes: (

**a**) Sigmoid function; (

**b**) Sin function; (

**c**) Hardlim function; (

**d**) Tribas function; and (

**e**) Radbas function.

**Figure 5.**A performance comparison of different methods: (

**a**) the van der Meer formula; (

**b**) Etemad-Bonakdar formula; (

**c**) GPM1 formula; and (

**d**) M1 model.

**Figure 6.**A performance comparison of different methods: (

**a**) the van der Meer formula; (

**b**) Etemad-Bonakdar formula; (

**c**) GPM1 formula; and (

**d**) M2 model.

Definition | Formula | Researcher |
---|---|---|

Damage parameter | ${N}_{\Delta}=\frac{A{\rho}_{r}9{D}_{50}}{{\rho}_{\alpha}{D}_{50}^{3}\frac{\pi}{6}}$ | Thompson and Shuttler (1975) [10] |

Damage parameter | ${N}_{0}={\left(\frac{{H}_{1/3}/\Delta Dn-1.33}{2.32}\right)}^{2}{N}_{w}^{0.5}$ | Hanzawa et al. (1996) [6] |

Damage level | $\mathrm{S}=\frac{A}{{D}_{50}^{3}}$ | van der Meer (1988) [2] |

Damage level | ${N}_{od}=\frac{M}{B/{D}_{n}}$ | van der Meer (1998) [3] |

Damage level | $\mathrm{S}=0.6{D}^{\prime}$ | Kajima (1994) [5] |

_{1/3}is the average wave height of the N

_{w}/3 highest waves reaching a rubble mound breakwater of a sea state composed of Nw waves, M is the number of stones removed from the structure in a strip, B is the length of the test section, and ${D}^{\prime}$ is the ratio of the number of displaced units to the total number of units.

Parameters | M1 Training Data | M1 Testing Data | M2 Training Data | M2 Testing Data |
---|---|---|---|---|

P | 0.1, 0.5, 0.6 | 0.1, 0.5, 0.6 | 0.1, 0.5, 0.6 | 0.1, 0.5, 0.6 |

S_{d} | 2–8 | 2–8 | 8–32 | 8–32 |

cot a | 1.5–6 | 1.5–6 | 1.5–6 | 1.5–6 |

N_{w} | 1000, 3000 | 1000, 3000 | 1000, 3000 | 1000, 3000 |

ξ_{m} | 0.67–6.83 | 0.67–6.83 | 0.7–5.8 | 0.7–6.4 |

Ns | 1.19–3.61 | 1.17–4.62 | 1.41–4.3 | 1.41–4.3 |

Methods | BIAS | SI | CC | I_{a} |
---|---|---|---|---|

VM | −0.0807 | 0.1400 | 0.8689 | 0.9293 |

EB | −0.0494 | 0.1032 | 0.9297 | 0.9582 |

GPM1 | −0.0378 | 0.1046 | 0.9272 | 0.9558 |

ELM(M1) | −0.0055 | 0.1066 | 0.9234 | 0.9604 |

Methods | BIAS | SI | CC | I_{a} |
---|---|---|---|---|

VM | −0.0394 | 0.1400 | 0.8462 | 0.8959 |

EB | −0.0676 | 0.1225 | 0.9057 | 0.9189 |

GPM1 | 0.0123 | 0.1102 | 0.9045 | 0.9434 |

ELM(M2) | 0.0030 | 0.1022 | 0.9186 | 0.9576 |

Researchers | CC | I_{a} | Training Data | Input Parameters | Testing Data | |
---|---|---|---|---|---|---|

Mase, Sakamoto and Sakai [24] | 0.91 | 100 | 6 | No | ||

Dong and Park [12] | I | 0.914 | 100 | 6 | 641 | |

II | 0.906 | 100 | 5 | 641 | ||

III | 0.902 | 100 | 6 | 641 | ||

IV | 0.915 | 100 | 7 | 641 | ||

V | 0.952 | 100 | 8 | 641 | ||

Kim, Dong and Chang [15] | I | 0.905 | 0.948 | 207 | 5 | 119 |

II | 0.913 | 0.954 | 201 | 5 | 114 | |

Erdik [16] | FL | 0.945 | 579 | 6 | 579 | |

Balas, Koç and Tür [13] | HNN-1 | 0.936 | 180 (PCA) | 5 | 76 | |

HNN-2 | 0.927 | 180 (PCA) | 4 | 76 | ||

Koç and Balas [17] | GA-FNN | 0.932 | 166 (PCA) | 5 | 42 | |

HGA-FNN | 0.947 | 166 (PCA) | 5 | 42 | ||

Etemad-Shahidi and Bonakdar [8] | MT1 | 0.931 | 0.97 | 386 | 5 | 193 |

MT2 | 0.968 | 0.976 | 386 | 6 | 193 | |

Koc, Balas and Koc [23] | GPM1 | 0.98 | 207 | 7 | 372 | |

GPM2 | 0.95 | 40 | 7 | 22 | ||

GPM3 | 0.989 | 207 | 7 | 372 | ||

GPM4 | 0.991 | 40 | 7 | 22 | ||

VM | 0.969 | 372 | ||||

VM | 0.65 | 22 | ||||

Current Study | ELM-M1 | 0.923 | 0.960 | 100 | 5 | 100 |

ELM-M2 | 0.919 | 0.958 | 100 | 5 | 100 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Wei, X.; Liu, H.; She, X.; Lu, Y.; Liu, X.; Mo, S.
Stability Assessment of Rubble Mound Breakwaters Using Extreme Learning Machine Models. *J. Mar. Sci. Eng.* **2019**, *7*, 312.
https://doi.org/10.3390/jmse7090312

**AMA Style**

Wei X, Liu H, She X, Lu Y, Liu X, Mo S.
Stability Assessment of Rubble Mound Breakwaters Using Extreme Learning Machine Models. *Journal of Marine Science and Engineering*. 2019; 7(9):312.
https://doi.org/10.3390/jmse7090312

**Chicago/Turabian Style**

Wei, Xianglong, Huaixiang Liu, Xiaojian She, Yongjun Lu, Xingnian Liu, and Siping Mo.
2019. "Stability Assessment of Rubble Mound Breakwaters Using Extreme Learning Machine Models" *Journal of Marine Science and Engineering* 7, no. 9: 312.
https://doi.org/10.3390/jmse7090312