# Strength Estimation and Feature Interaction of Carbon Nanotubes-Modified Concrete Using Artificial Intelligence-Based Boosting Ensembles

^{1}

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## Abstract

**:**

^{2}, gradient boosting, light gradient boosting machine, and extreme gradient boosting models proved their accuracy. Extreme gradient boosting had the highest R

^{2}of 0.97, followed by light gradient boosting machine and gradient boosting with scores of 0.94 and 0.93, respectively. This type of research may help both academics and industry forecast material properties and influential elements, thereby reducing lab test requirements.

## 1. Introduction

_{2}is released into the environment by the cement industry [11]. Thus, the latter is responsible for around 7% of global CO

_{2}emissions [12]. Modern construction trends, such as the building of modern bridges, high-rise buildings, and huge water accumulation systems, are also driving the rising demand for concrete [13,14]. Nonetheless, the appearance of nanoscale voids and cracks is a major disadvantage that lowers concrete’s performance and reduces its lifespan [15]. Consequently, incorporating nanoparticles into a cementitious matrix increases mechanical strength and makes the material very resistant to cracking [16,17]. Researchers all around the globe are interested in nanotechnology because of its promising applications in a wide range of industries [18]. Particles of diameters between 1 and 100 nm are considered to be in this category [19]. Nanoparticles are employed in cementitious concrete to improve the material’s conventional strengths.

^{2}) and errors. Experimental-based research for the CS evaluation and mix design optimization of building materials requires significant effort, expenses, and time. This proposed research may aid both academics and industry in predicting the material attributes and influential factors, thus eliminating repeated test trials in the laboratory.

## 2. Research Methods

#### 2.1. Data Description

^{3}), curing (days), fine aggregate (FA) (kg/m

^{3}), coarse aggregate (CA) (kg/m

^{3}), w/b, and nanomaterial CNTs (%). The parameters of aggregates, such as their fineness modulus, maximum diameters, and saturated surface dry (SSD), were either unchanging or varied only very slightly. As a result, these parameters were not included in this particular research. Since all of the data points acquired for this study adhered to the American Society for Testing and Materials (ASTM) criteria, it was presumed that the preparation of concrete was the same in each and every instance. In addition to this, the output of the model is substantially impacted by the quantity of data points as well as the numerous input parameters. This study made use of 282 different data points in order to make a prediction regarding the CNTs-based compressive strength of concrete. The dataset was split into training (70% of the data) and testing (30% of the data) for the modelling phase. This allocation was taken considering pertinent literature in the field [30,31,32]. All of the algorithms needed to run the models were crafted in the programming language Python, and the software that was employed was the Spyder (version 5.4.3) accessed through the Anaconda Navigator. Figure 2 gives a visual depiction of the relative frequency distribution of all parameters used in creating the models. Furthermore, for the database, Pearson’s correlation matrix was produced, as illustrated in Figure 3. A correlation matrix presents the correlation coefficients between pairs of variables in the form of a symmetrical matrix. The intensity and direction of the linear relationship between two variables are quantified by correlation coefficients. The correlation matrix is a frequently employed instrument in the fields of statistics and multivariate analysis for examining the connections between numerous variables [33]. Potential results comprise a perfect negative correlation (denoted by −1), a perfect positive correlation (represented by +1), and the absence of any correlation (represented by 0). A positive correlation denotes that when one variable increases, the other variable also increases proportionally; conversely, a negative correlation suggests that when one variable increases, it generally results in a decrease in the other [34]. The most significant input was the cement quantity, which exhibited a strong positive correlation (as indicated by a correlation coefficient of 0.59) with the output (CS). The correlation between the output and CT was also positive, while the impact of w/b, CA, FA, and CNTs was shown to be negative. Statistics used for describing the data are listed in Table 1.

#### 2.2. Machine Learning Algorithms

#### 2.2.1. Gradient Boosting

#### 2.2.2. Light Gradient Boosting Machine

#### 2.2.3. Extreme Gradient Boosting

#### 2.3. Model Assessment and Validation Methods

^{2}, were utilized when assessing a model’s performance on a training or testing set. R

^{2}, also known as the determination coefficient, is a statistic that may be used to measure how well a model can predict future outcomes [43,44]. The advancements that have been made in AI modelling techniques have made it possible to provide more accurate forecasts of the mechanical characteristics of concrete. The GB, LGBM, and XGB models are statistically evaluated and contrasted with one another in this research by means of the determination of error criteria. There are a large number of data points that might perhaps shed light on the inaccuracy of the model. In order to determine whether or not the model is reliable and valid, the coefficient of determination can be utilized. The findings that are produced by models with R

^{2}values that are above 0.50 are disappointing, whereas the results that are produced by models with R

^{2}values that lie within the range of 0.65 and 0.75 are encouraging. The value of R

^{2}may be found by utilizing Equation (1). The MAE uses the same unit system for both its input and its output. A model with an MAE that falls within a certain range has the potential to produce mistakes of a significant nature on occasion. Equation (2) is what we use to compute MAE. The RMSE is the measure of how accurate estimates and measurements are. Error squared is determined by adding up all of the individual error squares. The new method gives more consideration to extreme circumstances than the older computations did, which results in squared differences that are larger in some cases but lower in others. The RMSE may be estimated using Equation (3). The model’s ability to reliably anticipate incoming data increases in proportion to the reduction in the value that represents the RMSE. The root-mean-square error, or RMSE, is a useful metric for contrasting models of differing degrees of complexity. Incorporating a larger logarithmic error into the RMSLE enables it to compute the proportional difference between the outcome that was predicted and the one that was actually seen. Because the log conversion presents the intended distribution in a very straightforward manner, it is helpful for analysing results that are right-skewed. The RMSLE can be computed with Equation (4).

## 3. Results and Discussions

#### 3.1. Gradient Boosting Model

^{2}value of 0.93, which indicates its capacity to explain a significant percentage of the variation included within the dataset. This demonstrates the model’s ability to forecast the future accurately. The linear regression fit of R

^{2}is also shown in Figure 7. As can be seen in Figure 8, a comprehensive examination of prediction errors has been carried out in order to evaluate the accuracy of the model in terms of providing estimates of actual values. The findings indicate that the bulk of the predictions are astonishingly near to the actual values, with 47.1% of them having an error of less than 1.5 MPa. This is demonstrated by the fact that there is a remarkable correlation between the two. In addition, 35.2% of the model’s predictions fall within the range of 1.5 to 4 MPa, which demonstrates the model’s reliability in providing correct forecasts throughout a wider spectrum. The fact that just 17.7% of the forecasts are off by more than 4 MPa from the values that really occurred demonstrates the robustness of the model in its ability to deal with difficult circumstances. When looking at the error distribution, we can see that the biggest forecast error is 8.2 MPa, while the smallest error is 0.1 MPa. This shows that the model is able to handle a diverse set of data points. The fact that the model has an error rate of 2.19 MPa on average underlines the robust prediction skills it possesses over a wide range of data sets.

#### 3.2. Light Gradient Boosting Machine Model

^{2}value of 0.94, which suggests that it has an exceptional level of predictive accuracy. This score is a reflection of how well the model is able to explain a sizeable percentage of the variability present in the data. In addition, a comprehensive error analysis has been carried out in order to assess how well the model performs in terms of its ability to forecast real values, as can be seen in Figure 10. According to the findings, the bulk of the forecasts are quite close to the actual values, with 52.9% of them having an error that is lower than 1.5 MPa. In addition, 29.4% of forecasts fall within the range of 1.5 to 4 MPa, which demonstrates the model’s constant accuracy over a larger range of values. The fact that just 17.7% of forecasts are off by more than 4 MPa from the actual values demonstrates how resilient the model is when it comes to dealing with severe circumstances. The dependability of the model is shown by the error distribution, which reveals that the model’s biggest forecast error measures 7.33 MPa, while the model’s lowest mistake measures just 0.05 MPa. The model has high prediction skills for a variety of data points, as seen by its moderate error of 1.99 MPa, which is calculated as an average over all of the data points.

#### 3.3. Extreme Gradient Boosting Model

^{2}value of 0.97. This indicates that the model is very successful in explaining approximately 97% of the variation in the data, showing its resilience in capturing underlying patterns, as illustrated in Figure 11. In addition, as can be shown in Figure 12, an in-depth investigation of the error distribution demonstrates that the model has an excellent level of accuracy. Its precision in forecasting values within a tight margin is demonstrated by the fact that 61.1% of the forecasts have errors that are less than 1.5 MPa. This is a major fraction of the predictions. In addition, 36.5% of forecasts come within the range of 1.5 MPa to 4 MPa, which demonstrates its adaptability in managing a wider range of value ranges. Even when errors are present that are more than 4 MPa, the model still maintains an acceptable degree of performance, with just 2.4% of predictions falling into this category. The reliability of the model’s forecasts is shown by the fact that its biggest mistake was calculated to be 5.5 MPa, while its lowest error was calculated to be a meagre 0.005 MPa. The model has been shown to have an inaccuracy of 1.44 MPa on average, which further demonstrates the outstanding precision and dependability with which it can estimate target values.

#### 3.4. K-Fold Cross-Validation Outcomes

^{2}value of 0.981 on a constant basis, which indicates that it has a remarkable capacity to explain data variation. In addition, it had the lowest MAE of 3.499 MPa, which indicates that it is extremely accurate, and the lowest RMSE of 2.605 MPa, which demonstrates that it can make accurate predictions. In addition, XGB showed the lowest RMSLE values, with 0.071 MPa being the lowest possible value. This demonstrates how well it can handle data on a variety of scales. Despite the fact that both GB and LGBM performed wonderfully, with impressive R

^{2}values of 0.935 and 0.940, respectively, as well as competitive MAE, RMSE, and RMSLE scores, XGB continually beat both, making it the best option for this specific CNTs dataset.

#### 3.5. Statistical Performance Indicators

#### 3.6. Taylor Diagrams

#### 3.7. Interaction Analysis Outcomes

## 4. Discussions

## 5. Conclusions

^{2}, MAE, RMSLE, MAPE, and RMSE). This study led to the following conclusions:

- In terms of the coefficient of determination (R
^{2}), each of the three ML models, GB, LGBM, and XGB, presented convincing evidence of their accuracy. XGB surpassed the other models with the greatest R^{2}of 0.97, while GB and LGBM also attained R^{2}of 0.93 and 0.94, respectively. - When compared to LGBM and GB, XGB had a low error distribution, and its average error was just 1.44 MPa. This indicated that XGB’s forecasts were more accurate and had lower variability than other models’ predictions.
- During KFCV, the performance of XGB was consistently superior to that of GB and LGBM since it produced lower values for MAE, RMSE, and RMSLE. The excellence of XGB’s predictions is further shown through measurements. Similar results are also shown using statistical checks.
- The findings were verified using the Taylor diagram, which demonstrated that the values predicted by XGB were closer to the actual values than those predicted by GB and LGBM. This also demonstrated that XGB was accurate in predicting the CS of CNTs-based concrete.
- The interaction pattern indicated that at higher values of CT and cement, strength improves. To attain maximum strength, w/b, CA, and FA need to be kept low, while CNTs’ percentage needs to be kept around 1%.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**An illustration of the LGBM method [38].

**Figure 13.**Statistics pertaining to KFCV for each model: (

**a**) R

^{2}, (

**b**) MAE, (

**c**) RMSE, and (

**d**) RMSLE.

Name | Abbreviation | Total Data | Mean | Standard Deviation | Sum | Minimum | Kurtosis | Skewness | Range | Median | Maximum |
---|---|---|---|---|---|---|---|---|---|---|---|

Curing time (d) | CT | 282 | 45.30 | 32.69 | 12,776 | 1 | 3.2 | 1.5 | 179.0 | 28 | 180 |

Cement (kg/m^{3}) | C | 282 | 397.82 | 45.40 | 112,186.6 | 250 | −0.3 | −0.1 | 225.0 | 400 | 475 |

Water to binder ratio | w/b | 282 | 0.50 | 0.08 | 140.62 | 0.4 | 5.1 | 1.9 | 0.5 | 0.49 | 0.87 |

Coarse aggregate (kg/m^{3}) | CA | 282 | 1031.25 | 163.87 | 290,813.5 | 498 | 1.7 | −0.9 | 968.8 | 1068.75 | 1466.8 |

Fine aggregate (kg/m^{3}) | FA | 282 | 638.64 | 163.44 | 180,096.4 | 175.5 | 3.2 | 1.2 | 1109.5 | 608.375 | 1285 |

Carbon nanotubes (%) | CNT | 282 | 0.51 | 1.89 | 145.21 | 0 | 18.4 | 4.4 | 10.0 | 0 | 10 |

Compressive strength (MPa) | CS | 282 | 45.14 | 10.57 | 12,730.26 | 14.7 | 0.6 | −0.9 | 52.0 | 46.5 | 66.7 |

K-Fold | GB Model | LGBM Model | XGB Model | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

R^{2} | MAE (MPa) | RMSE (MPa) | RMSLE (MPa) | R^{2} | MAE (MPa) | RMSE (MPa) | RMSLE (MPa) | R^{2} | MAE (MPa) | RMSE (MPa) | RMSLE (MPa) | |

1 | 0.82 | 3.65 | 2.08 | 0.036 | 0.71 | 2.47 | 2.57 | 0.068 | 0.88 | 3.33 | 2.52 | 0.033 |

2 | 0.92 | 4.02 | 4.60 | 0.072 | 0.91 | 3.95 | 3.89 | 0.017 | 0.94 | 3.25 | 3.78 | 0.012 |

3 | 0.83 | 2.43 | 2.82 | 0.042 | 0.89 | 3.28 | 3.21 | 0.014 | 0.87 | 3.01 | 1.04 | 0.035 |

4 | 0.89 | 3.29 | 3.80 | 0.044 | 0.87 | 3.19 | 3.02 | 0.022 | 0.96 | 3.50 | 3.71 | 0.029 |

5 | 0.91 | 2.27 | 2.96 | 0.078 | 0.94 | 2.73 | 1.16 | 0.071 | 0.95 | 2.49 | 2.52 | 0.071 |

6 | 0.83 | 3.09 | 3.23 | 0.017 | 0.93 | 2.14 | 3.96 | 0.027 | 0.90 | 3.36 | 2.13 | 0.015 |

7 | 0.68 | 3.86 | 3.95 | 0.036 | 0.83 | 3.02 | 3.68 | 0.069 | 0.98 | 1.17 | 3.75 | 0.034 |

8 | 0.85 | 1.53 | 2.46 | 0.047 | 0.86 | 2.31 | 3.11 | 0.078 | 0.88 | 1.31 | 1.65 | 0.012 |

9 | 0.88 | 3.17 | 3.53 | 0.092 | 0.86 | 3.03 | 1.74 | 0.036 | 0.94 | 2.03 | 2.38 | 0.060 |

10 | 0.94 | 1.36 | 3.84 | 0.057 | 0.85 | 1.74 | 4.42 | 0.090 | 0.93 | 1.41 | 2.57 | 0.009 |

Max | 0.94 | 4.02 | 4.60 | 0.092 | 0.94 | 3.95 | 4.42 | 0.090 | 0.98 | 3.50 | 3.78 | 0.071 |

Min | 0.68 | 1.36 | 2.08 | 0.017 | 0.71 | 1.74 | 1.16 | 0.014 | 0.87 | 1.17 | 1.04 | 0.009 |

Mean | 0.85 | 2.87 | 3.33 | 0.052 | 0.87 | 2.79 | 3.08 | 0.049 | 0.92 | 2.48 | 2.60 | 0.031 |

Technique | GB | LGBM | XGB |
---|---|---|---|

MAE (MPa) | 2.195 | 2.0 | 1.445 |

MAPE | 5.70% | 5.70% | 3.80% |

RMSE (MPa) | 2.863 | 2.844 | 1.798 |

RMSLE (MPa) | 0.065 | 0.064 | 0.041 |

Ref. | Material Studied | Properties Predicted | ML Techniques Employed | Optimum ML Model Noted | R^{2} of the Best ML Model |
---|---|---|---|---|---|

Current study | CNTs-based concrete | CS | GB, LGBM, and XGB | XGB | 0.97 |

[32] | Eggshell-based cement mortar | Reduction in CS after acid attack | GB, AdaBoost, and XGB | XGB | 0.94 |

[46] | Steel fibre reinforced concrete | Flexural strength | GB, random forest, and XGB | XGB | 0.94 |

[47] | Geopolymer concrete | CS | Support vector machine, GB, and XGB | XGB | 0.98 |

[48] | Nano-silica modified concrete | Ultrasonic pulse velocity | GB, AdaBoost, and XGB | XGB | 0.90 |

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© 2024 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 (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Zhu, F.; Wu, X.; Lu, Y.; Huang, J.
Strength Estimation and Feature Interaction of Carbon Nanotubes-Modified Concrete Using Artificial Intelligence-Based Boosting Ensembles. *Buildings* **2024**, *14*, 134.
https://doi.org/10.3390/buildings14010134

**AMA Style**

Zhu F, Wu X, Lu Y, Huang J.
Strength Estimation and Feature Interaction of Carbon Nanotubes-Modified Concrete Using Artificial Intelligence-Based Boosting Ensembles. *Buildings*. 2024; 14(1):134.
https://doi.org/10.3390/buildings14010134

**Chicago/Turabian Style**

Zhu, Fei, Xiangping Wu, Yijun Lu, and Jiandong Huang.
2024. "Strength Estimation and Feature Interaction of Carbon Nanotubes-Modified Concrete Using Artificial Intelligence-Based Boosting Ensembles" *Buildings* 14, no. 1: 134.
https://doi.org/10.3390/buildings14010134