# Application of the Improved POA-RF Model in Predicting the Strength and Energy Absorption Property of a Novel Aseismic Rubber-Concrete Material

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## Abstract

**:**

^{2}: 0.9800 and 0.9108, VAF: 98.0005% and 91.0880%, RMSE: 0.7057 and 1.9128, MAE: 0.4461 and 0.7364; R

^{2}: 0.9857 and 0.9065, VAF: 98.5909% and 91.3652%, RMSE: 0.5781 and 1.8814, MAE: 0.4233 and 0.9913). In addition, the sensitive analysis results indicated that the rubber and cement are the most important parameters for predicting the strength and energy absorption properties, respectively. Accordingly, the improved POA-RF model not only is proven as an effective method to predict the strength and energy absorption properties of aseismic materials, but also this hybrid model provides a new idea for assessing other aseismic performances in the field of tunnel engineering.

## 1. Introduction

## 2. Research Significance

## 3. Experiment of a Novel Aseismic Concrete Material

## 4. Methodologies

#### 4.1. Random Forest

_{t}) and the random features (Maxdepth) are the main hyperparameters that affect its prediction performance. Although the increase of the N

_{t}will not cause the overfitting of the model, it is difficult to obtain satisfactory predictive performance through the time-consuming manual debugging of hyperparameters’ combinations [38,39].

#### 4.2. Improved Pelican Optimization Algorithm

#### 4.2.1. Pelican Optimization Algorithm

#### 4.2.2. Optimization Methods

#### Chaotic Mapping Method (CM)

#### Latin Hypercube Sampling Method (LHS)

#### 4.3. A Novel Combination of the IPOA and RF Model

_{t}and Maxdepth are (1, 100) and (1, 10), respectively; (c) Performance evaluation—four statistical indices were used to evaluate the predictive performance of the proposed hybrid RF models.

## 5. Performance Evaluation

^{2}) is also known as goodness of fit, which reflects the interpretation of the predicted value to the measured value from the fitting perspective. If R

^{2}is equal to 1, that means the prediction model is perfect. The variance accounted for (VAF) is often used to evaluate the degree to which the prediction model can explain the variance of the considered data. The function of the root mean square error (RMSE) is to evaluate the model performance by measuring the error between the predicted value and the measured value. In addition, the mean absolute error (MAE) can further reflect the real situation of error. These statistical indices have been considered to verify the performance of different prediction models for solving the regression problem [57,58,59,60,61,62].

_{t}and p

_{t}represent the t-th measured and predicted values, respectively. $\overline{m}$ is the average of the measured values.

## 6. Results and Discussion

#### 6.1. The Development Results of the Proposed Models

#### 6.2. Performance Evaluation for the UCS and the ETR Prediction

^{2}: 0.9800, RMSE: 0.7057, MAE: 0.4461 and VAF: 98.0005%) for predicting the UCS, but also obtained better performance indices (R

^{2}: 0.9108, RMSE: 1.9128, MAE: 0.7364 and VAF: 91.0880%) than other models in the ETR prediction. After this model, the performance indices of the CMPOA-RF model are also better than the POA-RF for predicting the UCS and ETR.

^{2}and RMSE in the same set are 4 and 1, respectively. As demonstrated in Table 6, the LHSPOA-RF model has achieved the highest values of ranking scores (12 and 12) for predicting the UCS and the ETR in the training phase. Figure 6 illustrates the regression diagrams of the proposed hybrid RF models using the train set. In each diagram, the blue line represents the perfect prediction function y = x, i.e., the predicted value is equal to the measured value. Therefore, the more points that locate on the blue line or close to it, the more superior the model’s performance compared with others. Of course, the regression lines with 10% have similar functions for evaluating the models’ performance. As can be seen in Figure 6b,e, the LHSPOA-RF model not only obtained the most data points close to or located at the blue line in the UCS prediction, but also showed the same excellent prediction performance in the ETR prediction resulting in the best performance indices. After the LHSPOA-RF, the CMPOA-RF model has achieved a more satisfying predictive performance than the POA-RF model resulting in more data points within the 10 % lines for the UCS prediction as shown in Figure 6a,c, and obtained the same performance for predicting the ETR (see Figure 6d,f).

^{2}(0.9857 and 0.9065) and VAF (98.5909% and 91.3652%), and the lowest values of RMSE (0.5781 and 1.8814) and MAE (0.4233 and 0.9913). After the LHSPOA-RF model, the CMPOA-RF model has better performance indices than the POA-RF model for predicting the UCS and the ETR.

_{h}) and the number of neurons in each hidden layer (N

_{e}). The penalty parameter (P

_{c}) and the RBF kernel parameter (k

_{1}) are the main parameters affecting the SVR model performance. The ELM model is a special ANN model with a single hidden layer, thus the number of neurons in this layer (N

_{n}) can control the performance of the ELM model. The KELM model has similar hyperparameters with the SVR model, i.e., regularization coefficient (R

_{c}) and the RBF kernel parameter (k

_{2}). After developing these four ML models, the performance indices of each model with the best hyperparameter combination are shown in Table 8. The performance comparison results of the proposed models and four common ML models in the UCS and the ETR prediction are shown in Figure 8. As can be seen in these pictures, the performance indices of the LHSPOA-RF model and the CMPOA-RF model are obviously better than that of the four ML models, i.e., higher values of R

^{2}and VAF, and lower values of RMSE and MAE. Meanwhile, it is obvious that the LHSPOA-RF model is the best model to predict the UCS and the ETR of the novel aseismic rubber-concrete material in this study.

#### 6.3. Sensitively Analysis

## 7. Conclusions

^{2}, RMSE, MAE and VAF (UCS: 0.9857, 0.5781, 0.4233 and 98.5909%; ETR: 0.9065, 1.8814, 0.9913 and 91.3652%). Furthermore, the comparison results of four common ML models and the proposed hybrid RF models showed that the LHSPOA-RF model is the best prediction performance among all models.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**The demonstration of parameter correlation analysis: (

**a**) UCS prediction and (

**b**) ETR prediction.

**Figure 5.**The development results of the proposed models for predicting the UCS and ETR of the novel aseismic concrete material.

**Figure 6.**The regression diagrams of the proposed models for predicting the UCS and ETR in the training phase.

**Figure 7.**The regression diagrams of the proposed models for predicting the UCS and ETR in the testing phase.

**Figure 8.**The performance comparison of the proposed models and four common ML models in the UCS and the ETR prediction.

Procedures | Description |
---|---|

Step 1—Dosing | Mixing the rubber and river sand with the cement in pre-designed proportions and thoroughly stirring. |

Step 2—Concreting | Stirring the mixture for five minutes and quickly pouring it into the mold. |

Step 3—Demolding | After 24 h, separating the specimen and polishing it to the specified specification. |

Step 4—Maintaining | Maintaining all specimens at required temperature (20 degrees) and humidity (95 %). |

Step 5—Testing | After 28 days, testing 140 specimens in the laboratory. |

RM (%) | SM (%) | CM (%) | RPS (mm) |
---|---|---|---|

0 | 100 | 40 | / |

10 | 90 | 30 | 1~2 |

10 | 90 | 40 | 0.5~1 |

10 | 90 | 40 | 0.075~0.25 |

10 | 90 | 50 | 0.25~0.5 |

10 | 90 | 60 | 0.075~0.25 |

30 | 70 | 30 | 0.5~1 |

30 | 70 | 40 | 1~2 |

30 | 70 | 40 | 0.075~0.25 |

30 | 70 | 50 | 0.075~0.25 |

30 | 70 | 60 | 0.25~0.5 |

50 | 50 | 30 | 0.25~0.5 |

50 | 50 | 30 | 0.075~0.25 |

50 | 50 | 40 | 1~2 |

50 | 50 | 40 | 0.5~1 |

50 | 50 | 40 | 0.25~0.5 |

50 | 50 | 40 | 0.075~0.25 |

50 | 50 | 50 | 1~2 |

50 | 50 | 50 | 0.075~0.25 |

50 | 50 | 60 | 0.5~1 |

50 | 50 | 60 | 0.075~0.25 |

70 | 30 | 30 | 0.075~0.25 |

70 | 30 | 40 | 0.25~0.5 |

70 | 30 | 40 | 0.075~0.25 |

70 | 30 | 50 | 0.5~1 |

70 | 30 | 60 | 1~2 |

100 | 0 | 40 | 0.075~0.25 |

Variables | Sign | Unit | Min | Max | Mean | Median | St. D |
---|---|---|---|---|---|---|---|

Rubber | R | g | 15.66 | 90.83 | 61.21 | 65.85 | 22.76 |

River sand | S | g | 33.26 | 207.63 | 92.82 | 74.99 | 50.52 |

Cement | C | g | 50.52 | 236.10 | 129.47 | 129.38 | 51.26 |

Rubber particle size | RPS | mm | 0.16 | 1.50 | 0.56 | 0.38 | 0.48 |

Specimen mass | M | g | 168.40 | 393.50 | 283.49 | 279.25 | 65.71 |

Specimen density | r | g/cm^{3} | 0.98 | 50.59 | 16.09 | 1.72 | 22.42 |

Specimen diameter | D | mm | 48.99 | 50.59 | 50.11 | 50.15 | 0.29 |

Specimen length | L | mm | 95.52 | 102.67 | 99.05 | 99.27 | 1.20 |

Uniaxial compressive strength | UCS | MPa | 0.47 | 18.52 | 5.82 | 4.11 | 5.02 |

Variables | Sign | Unit | Min | Max | Mean | Median | St. D |
---|---|---|---|---|---|---|---|

Rubber | R | g | 7.28 | 42.53 | 29.11 | 30.38 | 10.91 |

River sand | S | g | 16.60 | 97.96 | 44.35 | 36.90 | 23.64 |

Cement | C | g | 24.36 | 111.48 | 62.84 | 60.59 | 23.65 |

Rubber particle size | RPS | mm | 0.16 | 1.50 | 0.56 | 0.38 | 0.48 |

Specimen mass | M | g | 81.20 | 186.70 | 136.29 | 138.00 | 29.59 |

Specimen density | r | g/cm^{3} | 0.91 | 1.95 | 1.45 | 1.46 | 0.30 |

Specimen diameter | D | mm | 48.47 | 49.98 | 49.48 | 49.48 | 0.30 |

Specimen length | L | mm | 46.46 | 50.15 | 48.74 | 48.86 | 0.67 |

Energy transmission rate | ETR | % | 0.00 | 36.43 | 2.32 | 0.13 | 6.37 |

Population | Fitness (RMSE) | |||||
---|---|---|---|---|---|---|

UCS | ETR | |||||

POA-RF | LHSPOA-RF | CMPOA-RF | POA-RF | LHSPOA-RF | CMPOA-RF | |

20 | 0.07503 | 0.06054 | 0.06338 | 0.16103 | 0.13669 | 0.14078 |

40 | 0.08197 | 0.06505 | 0.06265 | 0.16081 | 0.13484 | 0.14024 |

60 | 0.07419 | 0.06329 | 0.06277 | 0.16199 | 0.13340 | 0.14158 |

80 | 0.08340 | 0.05929 | 0.06154 | 0.16085 | 0.13294 | 0.14239 |

100 | 0.08368 | 0.06324 | 0.06159 | 0.16083 | 0.13671 | 0.14238 |

Best hyperparameters combination | ||||||

N_{t} | 15 | 22 | 17 | 18 | 13 | 14 |

Maxdepth | 2 | 2 | 2 | 1 | 1 | 1 |

Models | UCS Prediction | |||||||

Performance Indices | ||||||||

R^{2} | Score | RMSE | Score | MAE | Score | VAF (%) | Score | |

POA-RF | 0.9654 | 1 | 0.9285 | 1 | 0.5913 | 1 | 96.5393 | 1 |

LHSPOA-RF | 0.9800 | 3 | 0.7057 | 3 | 0.4461 | 3 | 98.0005 | 3 |

CMPOA-RF | 0.9761 | 2 | 0.7710 | 2 | 0.4652 | 2 | 97.6378 | 2 |

Models | ETR Prediction | |||||||

Performance Indices | ||||||||

R^{2} | Score | RMSE | Score | MAE | Score | VAF (%) | Score | |

POA-RF | 0.8814 | 1 | 2.2052 | 1 | 0.8571 | 1 | 88.1907 | 1 |

LHSPOA-RF | 0.9108 | 3 | 1.9128 | 3 | 0.7364 | 3 | 91.0880 | 3 |

CMPOA-RF | 0.9062 | 2 | 1.9608 | 2 | 0.7732 | 2 | 90.6635 | 2 |

Models | UCS Prediction | |||||||

Performance Indices | ||||||||

R^{2} | Score | RMSE | Score | MAE | Score | VAF (%) | Score | |

POA-RF | 0.9663 | 1 | 0.8865 | 1 | 0.6060 | 1 | 96.6339 | 1 |

LHSPOA-RF | 0.9857 | 3 | 0.5781 | 3 | 0.4233 | 3 | 98.5909 | 3 |

CMPOA-RF | 0.9726 | 2 | 0.7995 | 2 | 0.5595 | 2 | 97.2639 | 2 |

Models | ETR Prediction | |||||||

Performance Indices | ||||||||

R^{2} | Score | RMSE | Score | MAE | Score | VAF (%) | Score | |

POA-RF | 0.8790 | 1 | 2.1400 | 1 | 1.2675 | 1 | 87.9153 | 1 |

LHSPOA-RF | 0.9065 | 3 | 1.8814 | 3 | 0.9913 | 3 | 91.3652 | 3 |

CMPOA-RF | 0.9047 | 2 | 1.8993 | 2 | 1.0537 | 2 | 90.5980 | 2 |

Models | UCS Prediction | Hyperparameter | |||

Performance Indices | |||||

R^{2} | RMSE | MAE | VAF (%) | ||

BPNN | 0.8782 | 1.6859 | 1.1049 | 88.1409 | N_{h} = 1; N_{e} = 8 |

SVR | 0.9406 | 1.1779 | 0.8091 | 94.0956 | P_{c} = 64; k_{1} = 0.5 |

ELM | 0.9334 | 1.2464 | 1.0643 | 93.6570 | N_{n} = 40 |

KELM | 0.9356 | 1.2257 | 0.8654 | 93.5694 | R_{c} = 32; k_{2} = 0.5 |

Models | ETR Prediction | Hyperparameter | |||

Performance Indices | |||||

R^{2} | RMSE | MAE | VAF (%) | ||

BPNN | 0.7926 | 2.8016 | 1.5106 | 79.2861 | N_{h} = 1; N_{e} = 6 |

SVR | 0.7838 | 2.8604 | 1.5438 | 78.4293 | P_{c} = 35; k_{1} = 0.25 |

ELM | 0.6641 | 3.5650 | 2.7334 | 69.6411 | N_{n} = 60 |

KELM | 0.8388 | 2.4700 | 1.3255 | 83.8904 | R_{c} = 55; k_{2} = 0.15 |

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## Share and Cite

**MDPI and ACS Style**

Mei, X.; Cui, Z.; Sheng, Q.; Zhou, J.; Li, C.
Application of the Improved POA-RF Model in Predicting the Strength and Energy Absorption Property of a Novel Aseismic Rubber-Concrete Material. *Materials* **2023**, *16*, 1286.
https://doi.org/10.3390/ma16031286

**AMA Style**

Mei X, Cui Z, Sheng Q, Zhou J, Li C.
Application of the Improved POA-RF Model in Predicting the Strength and Energy Absorption Property of a Novel Aseismic Rubber-Concrete Material. *Materials*. 2023; 16(3):1286.
https://doi.org/10.3390/ma16031286

**Chicago/Turabian Style**

Mei, Xiancheng, Zhen Cui, Qian Sheng, Jian Zhou, and Chuanqi Li.
2023. "Application of the Improved POA-RF Model in Predicting the Strength and Energy Absorption Property of a Novel Aseismic Rubber-Concrete Material" *Materials* 16, no. 3: 1286.
https://doi.org/10.3390/ma16031286