Predictive Modelling of Pavement Quality Fibre-Reinforced Alkali-Activated Nano-Concrete Mixes through Artificial Intelligence

GGBS was supplied by JSW, Thoranagallu, India, and was acquired from a local supplier. The properties of GGBS, including its chemical and physical properties, are obtained from the research work of Sheshadri et al. [29] as this research work is a continuation of the previous research of Sheshadri et al. and are in accordance with the necessary BIS and BS standards (IS 12089-1987; BS 15167-1:2006).

We procured NCA from local vendors. Crushed, well-graded, angular, clean granite aggregates up to the highest aggregate size of 20 mm in accordance with IRC:44-2017 [32] were employed. RSFA belonging to zone II as per IS:383, 2016 [33] constituted the fine aggregates used in the research work. The properties of the aggregate are obtained from research work of Sheshadri et al., [29] as it is an extension of the previous research performed.

The alkaline activator solution is employed to activate the cementing property of pozzolanic binders in PQAC mixes. In the current study, LSS, that is, Na₂SiO₃, along with NaOH, that is, NaOH chips (98% purity), served as the alkaline activator solution. These chemicals were procured from a local chemical supplier. The composition of LSS can be described as follows: It consists of 14.70% sodium oxide (Na₂O), 32.80% silicon dioxide (SiO₂) and 52.50% water (H₂O) as per IS:14212, 1995 [34]. The ‘activator modulus’ (Ms) defined as the ‘ratio of SiO₂ to Na₂O’ in LSS was 2.23. At the same time, LSS and NaOH had specific gravities of 2.10 and 1.57, respectively. The dosage of NaOH in the LSS was adjusted to generate a Ms value of 1.25 when LSS + NaOH were combined with water; this ratio was maintained throughout the PQAC investigations. To obtain the desired Ms value of 1.25, the alkaline activator solution was made at least 24 hours prior to the concrete production by combining the NaOH chips with LSS and water. At this point, the water-to-binder (w/b) ratio was 0.20; later, while preparing the PQAC concrete, more water was added to raise the w/b ratio to 0.4.

NS particles are utilised, which are spherical and nano-sized. They are in the form of a white, odourless powder with an average particle size of 17 nm and a specific gravity of 2.2. The NS particles, owing to their larger specific surface area and superior fineness, enhance the pore structure of the concrete matrix. NS is composed of SiO₂ content of 99.88%, carbon content of 0.06%, chloride of 0.009%, Al₂O₃ of 0.005%, TiO₂ of 0.004%, and Fe₂O₃ of 0.001%.

NA is characterised by a spherical shape and is found in the form of a white, odourless and crystalline powder. Their mean particle size is 40 nm with a specific gravity of 3.4. The nanoparticles, because of their superior fineness and specific surface area, enhance the pore structure of the concrete matrix. Nano-alumina comprises pure Al₂O₃, which is in a crystalline form. The NA utilised in the experiments is made up of spherical particles with an Al₂O₃ concentration of 99.9%, Fe₂O₃ of 0.0012%, SiO₂ of 0.015% and NaO₂ of 0.45%.

PVAF was procured by Fibre Region supplier, Chennai, India, with 12 mm length, 40 μm diameter, 1600 MPa tensile strength, a specific gravity of 1.1, an elastic modulus of 40 GPa and a density of 1.29 g/cm³.

PPF was procured by Fibre Region supplier, Chennai, India, with a 6 mm length, 20 μm diameter, 500 MPa tensile strength, a specific gravity of 0.9, an elastic modulus of 4 GPa and a density of 0.91 g/cm³.

A total of 19 PQAC mixes were developed. The initial reference mix was developed based on the available literature studies [35], [36] to meet desired standards for high quality concrete pavements (28 days) and a slump value of 25–75 mm. For each mix ID, a total of 30 concrete cylinders were cast for testing. Therefore, the total number of samples that were cast was 540. Two nano-additives, that is, NS and NA, were adapted in the mix design to improve the strength of the PQAC. NS was used as a nano-additive at intervals of 0.5 starting from 0% to 2.0% by weight of the binder. NA was added at regular intervals of 0.25% starting from 0% to 1.25% by weight of the binder.

Furthermore, in order to overcome the brittleness of PQAC, fibres were incorporated into the concrete mix. The two fibres which were individually incorporated in the PQAC mixes are PVAF and PPF. Both the fibres were added at regular intervals of 0.4% starting from 0% to 2.0% by volume of the binder. The details of the mix design and mix IDs are presented in Tables 1 and 2. The two letters of the mix ID represent the nano-additive adapted in the mix design (Table 1). The three letters of the mix ID represent the fibre adapted in the mix design (Table 2).

The quantities of GGBS, RSFA, NCA, NS, NA, PVAF and PPF were measured by weight. The base reference mix was designed to meet desired standards for high quality concrete pavements with a water-to-binder (w/b) ratio of 0.4. NS and NA were added in increments of 0.5% and 0.25%, respectively, by weight of the binder. PVAF and PPF were added in increments of 0.4% by volume of the binder. GGBS, RSFA, NCA and the respective percentages of NS/NA were thoroughly mixed in a mechanical mixer for 2 minutes to ensure even distribution of dry components. The alkaline activator solution was slowly introduced to the dry mix while the mixer operated at low speed followed by additional water content. Mixing continued for 3 minutes to achieve homogeneity. PVAF or PPF fibres were gradually added over 1 minute to avoid clumping. The mixing was continued for additional 2 minutes to ensure uniform fibre dispersion. Cylindrical moulds (100 mm diameter × 200 mm height) were filled in three layers. Each layer was compacted using a vibrating table for 30 seconds to eliminate entrapped air. Specimens were demoulded after 24 hours and immediately transferred to curing conditions. The samples were water cured for 7 days and then air cured for the remaining period up to 28 days before the testing.

A flow chart representing the experimental framework is shown in Figure 2. Various tests were performed for the PQAC mixes developed as per the prescribed standards, the details of which are provided below.

Figure 2:

The detailed flowchart for the experimental programme and analysis.

The compaction factor test was performed as per IS 1199 (Part II) [37] in order to assess the workability of each PQAC mix, the results of which are presented in Figure 3

Figure 3:

Compaction factor values of PQAC mixes: (a) PQAC+NA and PQAC+NS and (b) PQAC+PVA and PQAC+PPF.

The STS test was performed on 30 cylindrical samples for each mix having a dimension of ‘200 mm height and 100 mm dia’ as per the Indian Standard Codes IS 5816-1999 [38] for the PQAC mixes. The obtained results are presented in Figure 4

Figure 4:

Split tensile strength and percentage variation in split tensile strength: (a) PQAC+NS, (b) PQAC+NA, (c) PQAC+PVAF and (d) PQAC+PPF.

The area of ML is widely applied across several domains and is now undergoing extensive study to enhance its practicality. The present study use 399 samples, with 70% randomly designated as the training set and the remaining 171 samples, constituting 30%, utilised as the test set for the trained model. The STS has been forecasted utilising ML techniques, such as MLR, DT, RF, AdaBoost, SVR and GB. This study employs many tools, including DT, MLR, RF, ADA, SVR and GB, within the Sci-kit-Learn software package using Python.

A representative dataset is essential and crucial for ML models. Hence, this study gathered a grand total of 570 samples through its experimental works. Table 3 shows the statistical features of the variables found in these samples. The input variables include GGBS, NaOH, LSS, RSFA, NCA, Water, NS, NA, PVAF and PPF. It can be observed that the content of NS, NA, PVAF and PPF varies in the entire data set. The aim of this computational modelling work is to utilise ML methods to create metamodels that represent the STS as a mathematical function of 10 variables.

The compiled database is examined to reveal the statistical attributes, as shown in Table 3, of all input and output parameters. The STS of the evaluated concrete mixes ranges from 3.69 to 8.2 MPa, reflecting the variability across the test samples in the dataset. Figure 5 presents pair plots illustrating the distribution of data points across different levels. The pair plot of the dataset provides a comprehensive visualisation of the relationships between the various input variables and the output variable, STS. The diagonal elements of the pair plot illustrate the distribution of individual variables, revealing that certain variables, such as GGBS, water content (W), RSFA, LSS, NaOH and NCA, exhibit low variability because of their categorical nature or restricted range in the experimental design. The off-diagonal scatter plots showcase the relationships between variable pairs, emphasising the sparse distribution of many variables, which suggests limited unique values or discrete levels within the dataset. This sparsity is especially evident in the integer-based variables. While some scatter plots display distinct patterns suggesting potential correlation such as clustering many variable pairs do not show strong correlations, implying that certain factors may have a more pronounced influence on STS than others. Overall, the pair plot serves as a preliminary exploratory tool, providing insights into the underlying data structure and guiding subsequent detailed analyses. Research has demonstrated that PVAF and PPF contribute positively to the STS of PQAC. The findings suggest that optimising PVAF and PPF content could lead to enhanced STS, as they not only serve as a reinforcement but also interact synergistically with other components, improving the overall mechanical properties of the concrete. In contrast, parameters such as GGBS, NaOH, LSS, Water, RSFA and NCA remain constant across the dataset, indicating limited variability and thus a negligible impact on STS. The lack of variation in these parameters suggests that they are unlikely to exhibit significant relationships with STS, as their fixed values do not allow for the exploration of potential correlations. The scatter plots involving NA/NS and STS suggest a non-linear relationship, where higher dosages of NA/NS may lead to specific changes in STS. However, the overall weak correlation indicates that while NA/NS can influence STS, its effect may not be as pronounced as that of PVAF/PPF. This aligns with literature that discusses the complex interactions between various nano-additives and fibres and their contributions to the mechanical properties of concrete. Furthermore, the variability of nano-additives and fibres in the dataset may indicate their potential significance as input parameters. These parameters show distinct clustering or trends in relation to STS, and they are also considered as important factors influencing STS. In conclusion, the analysis underscores PVAF as the most significant parameter affecting STS in PQAC, with PPF, NA and NS also showing potential influence depending on their variability and interactions with other materials.

Figure 5:

Pair plots for the input parameters and observed responses.

The co-relation matrix is a square matrix that displays each dataset’s pairwise co-relation coefficients between variables [73], [74], [75]. Every value in the matrix corresponds to the co-relation between two variables, whereas the diagonal components indicate the co-relation between each variable and itself, which is 1 [76]. Data co-relation analysis is a valuable technique for examining the connections between variables in a dataset and detecting patterns or trends. It can also be employed to detect multicollinearity, a problem that arises in statistical models when two or more predictor variables exhibit a high co-relation with each other.

Figure 6 displays the graphical representation of the correlation matrix between the input and output parameters, known as a Heatmap. Evidently, out of the 11 input characteristics mentioned, the correlation coefficients of 0.54 and 0.38 for ‘PVAF’ and ‘PPF’, respectively, constitute the most robust correlation. The parameter PVAF shows a strong positive correlation (0.54) with STS. This suggests that increases in PVAF are associated with increases in STS, highlighting a potentially significant relationship worth further exploration. Several parameters exhibit moderate negative correlations, such as STS and NS (−0.26) or NA and STS (−0.22). These relationships indicate that as one parameter increases, the other tends to decrease, which could be relevant for understanding input–output dynamics in the system being studied. Nevertheless, the association remains feeble, suggesting that making predictions using MLR for STS is not dependable because of the presence of intricate non-linear connections between these factors and the result.

Figure 6:

Correlation heatmap of input parameters and observed responses.

Figure 7 illustrates the relative frequency distribution of the predictions-to-test ratio across six predictive modelling techniques; each panel (a–f) includes histograms that depict the frequency of observed prediction ratios, accompanied by vertical lines indicating mean values (in orange), median values (in green) and standard deviation (SD) (in blue) for each model. Desirable circumstances are characterised by a lower standard deviation and a mean value approaching 1. A mean number above 1.0 defines over-prediction, whereas a mean value below 1.0 shows under-prediction. The AdaBoost model consistently delivers the most precise predictions, as seen by its mean value of 0.99 and matching standard deviation of 0.084. The DT model follows closely with a mean value of 1.0026 and an SD of 0.0855, followed by RF with a mean value of 1.0027 and an SD of 0.0858. The MLR model has a mean value of 1.003 and a standard deviation of 0.108. The SVR and GBR models exhibit comparable performance, with mean values of 1.0048 and 1.0045, respectively, and SD values of 0.109 and 0.087, respectively. The greatest mean values and SD values in the SVR model indicate the weakest performance. The histograms show that the predictions-to-test ratios vary significantly across different models. Each model’s predictions cluster around different ranges, reflecting their distinct predictive capabilities. The vertical lines representing the mean highlight the central tendency of each model. Notably, RF and GB demonstrate a higher mean prediction ratio, indicating better performance in comparison to other models. The width of the distributions suggests variability in the models’ predictions. For instance, SVR exhibits a narrower distribution range, implying more consistent predictions, while models like DT and AdaBoost show broader distributions, indicating variability in their performance. Certain models, such as AdaBoost, reveal potential outliers in the predictions, suggesting instances where the model performs significantly better or worse than the average. The mean prediction ratio varies across models, with GB exhibiting the highest mean (1.004576), followed closely by AdaBoost (0.999269). This suggests that GB consistently produces higher predictions relative to test scores. The median values also reflect this trend, aligning closely with the mean, indicating a symmetrical distribution for most models. The standard deviation values indicate the variability of the predictions. MLR (0.108407) and DT (0.085500) show lower variability, while AdaBoost (0.0845896) and GB (0.08731) have higher dispersion, suggesting greater variability in prediction accuracy. The skewness values highlight the asymmetry of the distributions. SVR shows a moderate positive skew (0.076785), while GB has a more moderate negative skew (−0.416545), indicating that a greater number of predictions are clustered around lower values with a tail extending towards higher values. Based on the relative frequency distribution observed in Figure 7, GB emerges as the best method among the compared models. Its higher mean prediction ratio, coupled with a reasonably spread distribution, reflects a strong balance between accuracy and consistency in predictions. This finding underscores the effectiveness of GB in capturing complex patterns in the data while minimising prediction error, making it a robust choice for achieving high predictive performance in this context.

Figure 7:

Relative frequency distribution of the prediction-to-test STS ratio.

Figure 8 displays a violin graph of the different models’ relative error percentages for various predictive models, including MLR, DT, RF, SVR, AdaBoost, and GBR. It is evident that in the test dataset, the GBR model shows a nearly zero relative prediction error and a higher concentration than other models. The mistakes’ statistical analysis emphasises the positive prediction performance even more. The plot reveals that the MLR model exhibits the widest distribution of errors, with a large spread from approximately −35% to 30%, indicating significant variability in its predictions. Simpler models, such as MLR, struggle with the complex, non-linear relationships inherent in the dataset. This limitation is evident from the wider distribution of errors, indicating that MLR fails to capture significant variances in the data effectively. Such deficiencies make MLR unsuitable for accurately predicting mixes with non-linear behaviours. The SVR presents a slightly wider distribution than the ensemble models but remains more stable than MLR. In contrast, models like RF and DT show more compact distributions, suggesting that they produce more consistent and accurate predictions with errors generally confined between −25% and 25%. These are better equipped to handle non-linearity. However, they can still face challenges. For instance, RF, with its narrower error distribution and a median close to zero, demonstrates consistent performance across various mixes, effectively managing both bias and variance. However, it may exhibit larger errors for mixes with unique characteristics not adequately represented in the training data. This issue is attributed to overfitting to the training set. Certain mixes, acting as outliers in the dataset, contribute significantly to larger errors in models like DT. Mixes with unusual or extreme properties skew the model’s ability to generalise effectively. This sensitivity to outliers is reflected in the variability of error distributions across mixes. The GBR and AdaBoost models also demonstrate a relatively narrow error distribution, though slightly broader than that of DT. These exhibits increased errors for underrepresented or imbalanced mixes within the dataset. These models, while robust in general, perform less accurately for mixes lying outside common data patterns. Overall, the ensemble methods, particularly DT, appear to offer more reliable performance, as evidenced by their tighter error distributions, while the MLR shows greater susceptibility to prediction errors.

Figure 8:

The violin plot illustrating the relative error percentages of different models.

Mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), R squared (R²) and cross-validation (CV) mean were calculated for all the models and are presented in Table 5. A greater R² value denotes a better match between the projected values of the analytical and gradient-boosting regressions, while lower values of MAE, MSE, RMSE and CV mean suggest a more accurate prediction result [28].

The variations between the projected outcomes and the observed outcomes of each sample for various models are depicted in Figure 9. Based on the coefficient of determination (R²), all six models have strong predictive accuracy. Within the models, the DT model attained the greatest correlation value of 0.7979, with RF following at 0.7967, AdaBoost at 0.7956, GBR at 0.7914, MLR at 0.608 and SVR at 0.599. Generally, depending on a solitary measure for assessment may result in unreliability. Hence, five error metrics were computed for the predictions of each model, as presented in Table 4. By comparison to the other five models, it is evident that the DT model attains lower error metrics. The DT model has an MSE of 0.204, RMSE of 0.452, MAE of 0.382, and CV mean of 0.713. In this study, the best-performing models (DT, RF, AdaBoost) achieved R² values around 0.79, indicating that these models explain approximately 79% of the variability in STS. A high R² ensures that predictions of STS are closely aligned with actual experimental results. For pavement applications, this accuracy means engineers can confidently optimise mix designs without extensive physical testing and saving time and resources while ensuring material performance meets design requirements. The MSE values for the best-performing models ranged from 0.202 to 0.211. This low value reflects the squared difference between predicted and observed STS values. For a typical range of STS values (3.69–8.2 MPa), an MSE of 0.202 implies that the average squared prediction error is minor compared to the overall STS range. In practical terms, such precision ensures that pavement mixes designed using these models will perform as expected, avoiding under- or over-designed materials that could lead to failures or inefficiencies. MAE values for the top models (e.g., DT, RF, AdaBoost) were in the range of 0.382–0.384, indicating the average absolute deviation between predicted and actual STS. In real-world scenarios, an MAE of 0.38 MPa means that on average, the predicted STS differs from the true STS by less than 0.4 MPa. For pavement engineers, this level of accuracy is sufficient to ensure that materials meet safety standards without excessive conservatism, which could inflate costs. RMSE values for these models were 0.45–0.46, slightly higher than MAE but still low relative to the STS range. RMSE penalises larger errors more than smaller ones. A low RMSE confirms that significant deviations between predicted and observed values are minimal, which is crucial in pavement design to prevent localised weaknesses or overestimated capacities in the final structure. Models showed CV mean values around 0.71, indicating consistency across different data subsets. High CV consistency suggests the model’s generalisability to new mix designs beyond those tested in the study. This robustness ensures that engineers can use these models for diverse pavement projects without retraining or overfitting concerns. The high predictive accuracy and low error rates of these models translate to reduced experimental testing and improved confidence in mix designs, especially in large-scale pavement projects. Additionally, the adoption of these predictive tools aligns with sustainable engineering practices by reducing material wastage and the carbon footprint associated with extensive experimental trials.

Figure 9:

Comparative analysis between actual and predicted values: a) MLR, b)DT, c) RF, d) SVR, e) AdaBoost and f) GBR.

Figure 10 displays the values derived from the six models together with the experimental data. Using six ML models, it displays scatter graphs showing the expected STS value against the experimental value. In general, the plots display a positive correlation, indicating that the models are capable of capturing the underlying trends in the data. MLR and DT tend to exhibit a closer clustering of points around the line of equivalence, suggesting a more accurate fit to the data. Conversely, the RF and GBR, while exhibiting some non-linear patterns, show greater scatter, indicating that they may struggle with certain instances, potentially leading to higher prediction errors in those cases. The SVR and AdaBoost models also demonstrate moderate scatter but showcase the ability to capture complex relationships in subsets of the data. All of the established ML models show a strong connection between the observed and anticipated STS with R² >79%. It may be deduced from this figure that employing the DT model may be enough and provide satisfactory accuracy with 11 input variables for estimating the STS. Ultimately, it is evident that the DT model offers superior performance accuracy compared to other models. This work may help engineers choose suitable supervised learning models and parameters for the manufacturing of PQAC. DT excels in capturing non-linear relationships and interactions between input variables. It segments the data into smaller, more homogeneous groups through a hierarchical structure, which is particularly effective for datasets with complex patterns like those in PQAC mixes. The simplicity of DT allows it to make precise splits in regions of the dataset where relationships between variables are most apparent. The dataset in this study had discrete and continuous variables, with some non-linear dependencies between inputs (e.g., fibre proportions, nano-material dosages) and the output (STS). DT effectively identified these dependencies. RF reduces overfitting, a common limitation of standalone DTs, by averaging the outputs of multiple trees trained on random subsets of the data. This process increases robustness and generalisation. The ensemble nature of RF enables it to smooth out inconsistencies caused by noisy or sparse data points, yielding better predictions. The dataset featured variability in STS based on diverse input combinations. RF leveraged bootstrapping to create diverse training subsets, improving its ability to generalise and handle potential outliers or noise. It consistently predicted STS due to its capability to aggregate diverse ‘opinions’ from individual trees. AdaBoost combines multiple weak learners (often simple decision stumps) to create a strong learner, adaptively focusing on errors in previous iterations. It is particularly effective for imbalanced data or data with some challenging-to-fit regions. In the PQAC dataset, certain input combinations (e.g., specific fibre proportions and nano-materials) have been harder to predict accurately. AdaBoost adapted by assigning higher weights to these harder-to-predict instances in subsequent iterations improves overall accuracy. Its iterative nature allowed it to fine-tune predictions, particularly for the tail-end cases where errors were higher for other models. While GB also showed competitive performance, it is more prone to overfitting than RF due to its sequential approach to reducing residuals. SVR, with its reliance on kernel functions to model non-linear relationships, struggled due to the high-dimensional feature space and sparse relationships in some input combinations. MLR assumes linear relationships, which limited its effectiveness given the complex and non-linear nature of the input–output relationships in the PQAC dataset. The superior performance of DT, RF and AdaBoost can be attributed to their ability to capture non-linear relationships and interactions effectively and handle variability and noise in the data through ensemble averaging (RF, AdaBoost). Adapt to challenging regions of the dataset where traditional models like MLR and SVR failed to generalise. These characteristics made ensemble methods (especially RF and AdaBoost) well-suited to the complexity and variability of predicting STS in PQAC mixes.

Figure 10:

Correlation between expected and experimental values of STS for PQAC.

The ML metamodels are initially calibrated to optimise the primary hyperparameters. This is accomplished by adjusting the quantity of estimators and training the ML metamodels.

Figure 11 presents that the MAE, MSE, R², CV and RMSE decrease as the number of estimators increases, indicating that the model’s accuracy improves with a higher number of estimators in RF. As the number of estimators increases, a notable decrease in MAE, MSE and RMSE is observed, indicating improved predictive accuracy and reduced error in the model’s predictions. The MAE reaches a plateau of around 300–400 estimators, suggesting that further increasing the number of estimators beyond this point may not significantly improve the model’s performance. Specifically, the MAE and MSE metrics show a marked decline up to a certain threshold, after which the improvements plateau, suggesting diminishing returns with additional estimators. Conversely, R² values exhibit an initial increase, reflecting enhanced model fit, before stabilising, which further supports the notion of optimal estimator count. The R² value, which represents the proportion of the variance in the target variable that is explained by the model, increases as the number of estimators increases. This indicates that the model’s ability to capture the underlying patterns in the data improves with a higher number of estimators. The cross-validation mean, which represents the average performance of the model across multiple validation folds, also shows a similar trend, decreasing as the number of estimators increases and reaching a plateau of around 300–400 estimators. These findings underscore the importance of selecting an appropriate number of estimators to balance model complexity and performance, ultimately leading to more robust and generalisable predictions in RF.

Figure 11:

Effect of the number of estimators on RF’s performance in terms of (a) MAE, (b) MSE, (c) RMSE, (d) R² and (e) cross-validation mean.

Figure 12 presents the performance of the AdaBoost algorithm using various metrics, including MAE, Mean MSE, RMSE, R² and CV, as the number of estimators varied. The results, as depicted in the plots, demonstrate that all the error metrics significantly decrease as the number of estimators increases from 0 to approximately 50, after which they stabilise and show little to no further improvement. This indicates that the model’s accuracy increases with more estimators up to a certain point, beyond which adding more estimators does not significantly enhance the model’s performance. The R² metric shows a corresponding increase as the number of estimators rises, reaching a plateau around the same point, suggesting a limit to the explanatory power of additional estimators. Similarly, the CV initially shows an improvement with an increase in the number of estimators but stabilises after approximately 50 estimators, indicating consistent model performance beyond this point. These findings suggest that while adding estimators improves model performance initially, the benefit diminishes, highlighting the importance of selecting an optimal number of estimators to balance performance and computational efficiency.

Figure 12:

Effect of the number of estimators on AdaBoost’s performance in terms of (a) MAE, (b) MSE, (c) RMSE, (d) R² and (e) cross-validation mean.

Figure 13 presents the performance of the GBR algorithm by varying the number of estimators, with the results captured across multiple performance metrics: MAE, MSE, RMSE, R² and CV mean. As shown in the plots, the MAE, MSE and RMSE metrics all demonstrate a sharp decline as the number of estimators increases from 0 to approximately 100, indicating significant improvements in model accuracy. Beyond this point, the error metrics stabilise, suggesting diminishing returns from adding more estimators. The R² metric, which measures the proportion of variance explained by the model, shows a marked increase as the number of estimators increases, plateauing around 100 estimators, indicating that most of the model’s predictive power is captured early on. Similarly, the cross-validation mean improves with the number of estimators, reaching its peak around 100 estimators and stabilising afterwards, reflecting consistent model performance across different folds of the data. These results suggest that while increasing the number of estimators initially leads to better model performance, there is a threshold beyond which additional estimators provide negligible benefits, emphasising the importance of choosing an optimal number of estimators for Gradient Boosting to balance accuracy and computational cost.

Figure 13:

Effect of the number of estimators on GBR’s performance in terms of (a) MAE, (b) MSE, (c) RMSE, (d) R² and (e) cross-validation mean.