Analyzing the compressive performance of lightweight foamcrete and parameter interdependencies using machine intelligence strategies

2.1 Dataset assortment and scrutiny

The creation of effective and widely applicable ML models depends on the availability of accurate and dependable datasets. This study utilizes a dataset of 300 detailed records on compressive strength (C-S), sourced from a previously published article, to investigate the prediction of lightweight foamed concrete strength using GEP and MEP techniques [56]. The experimental data was used to create the model, and it captured the nonlinear correlations between the input variables and concrete strength exceedingly well. There was a deliberate strategy to the data collection process, with an emphasis on including pertinent attributes and using trustworthy sources. The databases utilized nine input factors to build foamcrete: density (Dn), cement (CM), sand (Sa), sand-to-cement ratio (SCR), water-to-cement ratio (WCR), sand size (SS), foaming agent (Ag), foam content (Fm), and age (A). The output variable was C-S. According to previous research, the ideal number of records per input variable for making accurate predictions is at least 5 [57]. The utilization of a dataset of 300 points for C-S, with nine distinct input variables (resulting in 300/9 = 33.33), markedly enhances the observed ratio in this study. The reliability of this database stems from two key factors: (i) the data were generated through experiments carried out in the same laboratory by the same personnel, following uniform international standards and environmental conditions; and (ii) the dataset is sufficiently large to encompass the full range of variables affecting concrete compressive strength. Furthermore, it is a globally recognized resource utilized by other researchers for the advancement of soft computing models, facilitating direct comparisons [58,59].

In building and refining M–L models, the preprocessing of data is an important and vital step. Typical data preprocessing operations encompass handling missing data, encoding, identifying and addressing outliers, and partitioning data [60]. A thorough data pretreatment procedure was implemented to ensure that no outliers were present, even though a multivariate outlier identification technique was not expressly used in the study. Every feature was subjected to univariate outlier identification before the M–L models were trained. Finding and removing data points that did not fall inside the specified acceptable ranges was part of this process. To further ensure that no outliers were missed, the dataset was subjected to extensive statistical and visual analysis. A brief synopsis of the input and output statistical data is provided in Table 1. The statistical measurements provide the ranges of values for the variables, including maximum and minimum. Included as well are the standard deviation, median, kurtosis, skewness, mode, and standard error. The cement content varies from 439.2 to 992.8 kg·m^‒3, while the density ranges from 1406.91 to 2009.48 kg·m^‒3. In all C-S situations, the maximum SCR and WCR were maintained at 2.0 and 0.45, respectively. The maximum foam content utilized was 357 kg·m^‒3, similarly. Additional statistical metrics include data on the mean, variability, kurtosis, and skewness of each input and output. The degree to which the probability distribution of a real-valued variable is asymmetric with regard to its mean is called its skewness. In most cases, an elongated left-hand side of the dispersion curve is indicated when negative numbers are present [61]. The likelihood distribution along the vertical axis can be better understood with the help of kurtosis, a statistical metric that shows if a dataset is appropriate for a specific normal distribution and has light or heavy tails [62].

Additionally, the complete compressive strength (C-S) dataset was randomly divided into two subsets: 70% (210 samples) was used for model training, while the remaining 30% (90 samples) was allocated for validation and testing. For ML model development and evaluation, data must be split into training and test sets. The model parameters are fitted using the training data, while the model’s performance on new data is evaluated objectively using the test set [62]. Avoiding overfitting and getting a good idea of the model’s generalizability are both made possible using a separate test set.

The possible impact of input variables on output was also assessed using the Pearson correlation (R) matrix. For C-S, this is graphically shown in Figure 1. R values between −1 and +1 show strong negative or positive connections, while R values close to 0 indicate weak relationships. The inputs exhibit a robust correlation with C-S, evidenced by the maximum positive R-value of 1.0, which clearly substantiates this connection. The proximity of R to zero, indicating a weak correlation, does not always imply that the two variables are entirely independent. This is a significant aspect to consider. Consequently, it is advisable to examine models derived from other studies, such as Shapley Additive exPlanations (SHAP) and sensitivity analysis, to gain a thorough comprehension of the relationship between inputs and outcomes.

Figure 1

C-S database correlation matrices.

The efficiency of the model is strongly correlated with the distribution of the input variables. 3D histogram prism charts, as shown in Figure 2(a)–(i), help to clarify the inputs’ proportionate distribution in extensive datasets for C-S. A really random distribution among the polyhedral forms is clearly indicated by the data points. Additionally, noteworthy data clusters, patterns, or outliers can be highlighted by 3D prism charts, allowing for the discovery of trends or possible problems. It is worth noting that the dataset does not contain any clusters or outliers, which suggests that the data points follow a normal distribution. When it comes to ML models, this aspect is paramount.

Figure 2

3D variable frequency histograms for the C-S database: (a) density; (b) cement; (c) sand; (d) S/C; (e) W/C; (f) sand size; (g) foaming agent; (h) foam content; (i) age.

2.2 ML modeling

To measure foamcrete’s compressive strength, a controlled setting was utilized. In order to get the output (C-S), nine inputs were needed. The C-S forecasts for foamcrete were created using cutting-edge ML algorithms like GEP and MEP. In the evaluation of ML algorithms, it is customary to juxtapose the outputs with the input data. Thirty percent of the data was allocated for testing, whereas seventy percent was used to train the ML models. The R ² value of the predicted outcome indicates the model’s effectiveness. R ² is small for a substantial disparity, indicating that the expected and actual values diverge only marginally [64]. The model’s accuracy is corroborated using many approaches, including statistical testing and error assessments. Figure 3 presents a scenario model, while Tables 2 and 3 detail the hyperparameters used for the GEP and MEP models, respectively.

Figure 3

Flow diagram of the approach used from data generation and model validation.

2.2.1 GEP ML technique

The genetic algorithm (GA), inspired by Darwin’s theory of evolution, was developed by J. H. Holland. In order to address optimization problems, this algorithm mimics the process of natural selection and the principle of survival of the fittest by gradually improving solutions over time [65]. A series of GAs denotes genomic progression, culminating in uniformly sized chromosomes. A novel GA termed “gene programming” was developed by Koza [66]. Genetic programming (GP) employs GAs to create an evolutionary model, serving as a universal approach for problem-solving [67]. In genetic programming, flexibility arises from the ability to utilize nonlinear structures, such as parsing trees, in lieu of fixed-length binary strings. The present artificial neural system aligns with Darwin’s theory [68] and utilizes naturally occurring genetic elements (such as procreation, crossovers, and modification) to address reproductive issues. Similar to the last case, the unsuitable trees were removed, and the remaining ones were utilized to replant the area according to our chosen method. However, the evolutionary process helps prevent premature convergence [68,69]. It is essential to ascertain the following five elements prior to using the GP: crucial domain tasks, fitness evaluation, fundamentally useful operators (such population size and crossover), and results from terminals that are specific to the methods used [68]. A crossover genomic processor manages the predominant growth of parse trees, even when the model creation of the GP is reiterated. In nonlinear GP, representations must function as both genotype and phenotype, leading to more complex expressions of desired traits [69].

The original proponent of GP was Candida Ferreira, who was also responsible for inventing GEP. An improvement on classical GP, this method allows for more efficient and capable evolution by capturing programs as linear chromosomes and then expressing them as tree topologies [69]. GEP, grounded in the concept of population-based modeling, utilizes linear chromosomes of fixed length and their corresponding parse trees. Often considered an extension of GP, GEP encodes intermediate-sized programs through simple, fixed-length chromosomes. This approach enables the formulation of predictive equations capable of addressing complex and nonlinear problems [70,71]. The termination criteria, final set, and fitness function are all supplied, similar to GP. The “Karva” dialect is used to designate the chromosomes before manufacture, even though the GEP process uses random numbers to generate them. The foundation of GEP is a line with a constant length. In contrast, the data processing of the GP produces parse trees of differing lengths. Individual cords with pronged morphologies of varying diameters depict chromosomes through nonlinear manifestation/parse trees after being defined as genomes of static length [68]. These genotypes and phenotypes can be differentiated by their distinct genetic representations [34]. GEP prevents costly structural modifications and replications by preserving the genome across generations. Because of their unusual “head” and “tail” arrangement, GEP chromosomes are able to generate intricate expressions of several genes from a single copy of DNA. This well-designed structure improves the technique’s ability to built complex solutions [68]. These genes encode instructions rooted in mathematics, statistics, logic, and Boolean algebra. Activator elements connect these genetic instructions to the specific computational processes they govern, much like how biological DNA regulates cellular functions. Due to the emergence of a novel language named Karva, which is capable of interpreting these chromosomes, equations derived from empirical data are now feasible. An illustrious revolutionary begins their voyage at Karva after the ET. The underneath layer is allocated to nodes by ET utilizing Eq. (1) [70]. It is possible that the total number of ETs is a good predictor of the degree of GEP gene K-expression as well as the duration of that expression

(1) ET GEP = log i − 3 j .

Unlike less sophisticated ML methods, GEP can learn from data even in the absence of labels. In Figure 4, we can see the numerous steps that go into developing GEP equations. At birth, every cell has the same number of chromosomes. To evaluate everyone’s health, these chromosomes must be certified as ETs. Only the fittest and healthiest individuals are able to breed. When the greatest people are involved in an iterative process, the outcome is optimal. Following three generations of breeding, mutation, and crossover, the final product is the result of all these operations

Figure 4

Gene-expression programming workflow diagram [72].

2.2.2 MEP ML technique

MEP is considered by some to be a cutting-edge linear variant of GP, distinguished by its use of linear chromosomes. What sets MEP apart from other modern GP approaches is its ability to encode multiple candidate solutions or expressions within a single chromosome. Fitness analysis is employed to choose the most optimum chromosome to accomplish the desired result [73,74]. This occurs when a bipolar system couples twice, leading to the formation of two new generations, as elucidated by Oltean and Grosan. Every generation secures a progenitor for itself [75]. Before the termination condition occurs, the procedure will continue until the ideal software is determined (as indicated in Figure 5). Fitness analysis plays a vital role in MEP for determining the dataset compatibility of emerging mathematical expressions. In order to find the optimal set of chromosomes to reproduce, the fitness function compares the program’s actual and anticipated results. Selection, crossover, and mutation are the tools used by MEP to promote fit programs. When the system hits a certain fitness level, generational threshold, or limit of improvement, the algorithm can cease its repetitive procedures and remain in charge. Mutations in MEP are a mechanism by which evolution modifies linear chromosomal components. Minor alterations to the genetic code increase genetic diversity in populations. Mutations, initiated early in the MEP optimization process, alter the genetic material of successive generations, enabling the exploration of diverse solutions. The algorithm’s functionality in searching solution spaces and adapting to fitness landscapes is enhanced with the introduction of mutations. The MEP model permits component merger, similar to other ML paradigms. Alphabet or code size, function number crossover frequency, and subpopulation count are crucial factors to think about when doing MEP [76]. Assessing the populace gets increasingly tedious and time-consuming as the number of individuals equals the number of packages. Code length plays a critical role in determining the computational output generated during the process. Keeping in mind the MEP properties mentioned in Table 3 is essential for building a dependable mechanical property model.

Figure 5

MEP process flow diagram [72].

When evaluating and modeling using the MEP technique, it is common practice to use datasets that comprise published literature [77,78]. Some researchers suggest that popular linear GP methods, such as MEP, may offer superior performance in predicting the real-world properties of concrete. For instance, Grosan and Abraham identified an optimal neural network approach by combining linear genomic programming with maximum likelihood estimation [79]. One notable distinction between the two is the operational approach of the GEP, which is undeniably more intricate than that of the MEP [76]. Notwithstanding the reduced density of MEP relative to GEP, significant differences are present between the two: (i) MEP facilitates the reprocessing of code; (ii) encased by chromosomes, non-coding constituents are not obligatory to be exhibited at a precise position; and (iii) it clearly encodes references to function arguments [80]. Due to the structured design of standard GEP genes, with defined “head” and “tail” regions that facilitate the generation of syntactically correct programs, many consider GEP to be a more powerful and capable modeling technique [75]. This finding calls for a more in-depth examination of the limitations and challenges associated with each GP technique.

2.3 Substantiation of models

The models built using GEP and MEP were subjected to statistical analysis using a test set. All of the models’ computed metrics consist of root mean square error (RMSE), Pearson’s correlation coefficient (r), mean bias error (MBE), Nash–Sutcliffe efficiency (NSE), mean absolute error (MAE), mean absolute percentage error (MAPE), and normalized root mean square error (NRMSE) [57,78,81,82,83]. The formulas for several statistical indicators are given as

In the context of the above equations, O i represents the observed value, P i denotes the predicted value, O ̅ signifies the mean of the observed values, P ̅ indicates the mean of the predicted values, and n represents the total number of data points.

With M being the dataset size and m20 the entry count, this accounts for an expected or experimental value ranging from 0.80 to 1.20, as stated in Eq. (8) [84]. The prediction model indicates that an a20-index of 1% would be optimal. This index reflects the performance of the physical engineering approach by measuring the proportion of samples that fall within a 20% uncertainty range of the experimental values. Another key metric for assessing a model’s predictive accuracy is the correlation coefficient (r), where higher values of r signify a stronger correlation between predicted and actual outcomes [85]. The value of component R remains unchanged regardless of whether it is divisible or multiplied. Since it takes into account both actual and predicted results, R ² provide a closer estimate of the true value. R ² values that are higher and approach 1 indicate a more accurate and robust model construction [86,87]. Like MAE and RMSE, the proposed model shows significant improvements as the number of errors increases, leading to even greater performance with fewer mistakes. The amount of errors grows, but both approaches eventually approach zero [88,89]. Closer inspection, however, showed that MAE truly excels in continuous and smooth databases [90]. In most cases, the model performs better when the previously computed error values are smaller.

Two effective approaches for assessing a model’s predictive performance are statistical validation and the use of a Taylor diagram. The Taylor diagram provides a visual means of evaluating the accuracy and reliability of models by comparing their deviations from a reference point or observed data [91,92]. At the actual value point, circular lines represent RSMEs, radial lines show correlation coefficients, and the x- and y-axes display standard deviations. With these three measures, you might be able to locate your model’s optimal point. The top model is the most reliable one if we look at its prediction accuracy [91].

3.1 C-S GEP model

Figure 6(a)–(d) present the expression trees (ETs) generated using the GEP technique for the C-S dataset of foamcrete. These ETs represent the mathematical relationships (Eqs. (10)–(14)) derived from the input parameters to predict the C-S of foamcrete. The ETs, built using a range of mathematical operations like addition, square roots, subtraction, division, multiplication, and exponentiation, play a key role in modeling compressive strength. By encoding these operations, the GEP method generates arithmetic formulas capable of estimating the future C-S based on the input data. These models, when provided with sufficient data, have the potential to outperform idealized models under optimal conditions. Figure 7(a) displays the scatter plot comparing the test and estimated C-S values generated by the GEP model, with the black line representing the perfect fit line. The GEP model exhibited significant accuracy, evidenced by a high R ² value of 0.94. The coefficient of determination (R ²) reflects how well the model’s predicted values correspond to the actual observed values. In ML models, a greater R ² value (approaching 1) signifies a superior fit, indicating that the model can account for a substantial percentage of the variance in the output variable, hence validating the model’s correctness and reliability.

Figure 6

C-S GEP expression tree schematic: (a) Sub-ET 1; (b) Sub-ET 2; (c) Sub-ET 3; (d) Sub-ET 4.

Figure 7

(a) Predicted vs test C-S relationships using the GEP model; and (b) error distribution analysis.

Figure 7(b) illustrates the error distribution between the test and estimated C-S values across all data points. The plot shows that the estimated C-S values align well with the test results, as depicted by the closeness of the data to the reference line. In terms of the error values, the maximum error recorded is 7.04 MPa, while the minimum is 0.004 MPa, with an average error of 1.99 MPa. The distribution of errors indicates as shown in Figure 8 that 32 data points have errors below 1.0 MPa, 42 points have errors between 1.0 and 3.0 MPa, and 26 points have errors greater than 3.0 MPa. This suggests that the majority of predictions fall within a reasonable acceptable error range for lightweight structural and nonstructural applications, where variations of ±2 MPa are generally considered tolerable [93]. Both the high R ² value (0.94) and the error distribution further validate the strong prediction capability of the GEP method for the C-S dataset of foamcrete, confirming the model’s reliability and accuracy in estimating the compressive strength of foamcrete. This confirms the reliability of the developed models for real-world foamcrete strength prediction

where D _n is the density, CM is the cement, S _a is the sand, SCR is the sand-to-cement ratio, WCR is the water-to-cement ratio, SS is the sand size, A _g is the foaming agent, F _m is the foam content, A is the age, and C-S is the compressive strength.

Figure 8

GEP error distribution frequency distribution plot.

3.2 CS-MEP model

A practical formula was established to ascertain the C-S of foamcrete, based on the results of the MEP technique and considering the influence of nine independent variables. The concluding set of mathematical equations produced during the modeling procedure is delineated as follows:

where D _n is the density, CM is the cement, S _a is the sand, SCR is the sand-to-cement ratio, WCR is the water-to-cement ratio, SS is the sand size, A _g is the foaming agent, F _m is the foam content, A is the age, and C-S is the compressive strength.

Figure 9(a) displays a scatter plot comparing the predicted and actual C-S values based on the MEP model, with the black line representing the line of perfect agreement. The close clustering of data points around this line highlights the excellent predictive accuracy of the MEP model. An R ² value of 0.97 further confirms its strong performance, indicating that the model explains nearly all the variability in the C-S data. In ML, such a high R ² and close alignment with the ideal fit line are key indicators of a model’s reliability and effectiveness in capturing real-world trends.

Figure 9

(a) Predicted vs test C-S relationships using the MEP model and (b) error distribution analysis.

Figure 9(b) illustrates the error distribution between the predicted and actual C-S values across all test data points. The plot demonstrates a strong alignment between the estimated and observed values, with most errors clustering near the reference line. The error analysis indicates a maximum deviation of 6.93 MPa, a minimum of 0.01 MPa, and an average error of 1.13 MPa, reflecting the model’s high predictive accuracy. Notably, 62 data points exhibit errors below 1.0 MPa, 32 points fall between 1.0 and 3.0 MPa, and only 6 points show errors above 3.0 MPa, as shown in Figure 10. This distribution indicates that the MEP model consistently delivers accurate predictions with minimal deviation falling within the acceptable range of ±2 MPa for lightweight structural and nonstructural applications [93]. Both the high R ² value of 0.97 and the error distribution validate the MEP method’s superior prediction capability for the foamcrete C-S dataset, exceeding the performance of the GEP method. MEP outperforms GEP due to its efficient chromosome structure, where multiple expressions are encoded within a single chromosome, allowing for the selection of the best-performing solution. This approach enhances convergence speed, as multiple candidate solutions are evaluated simultaneously [94]. MEP also shows greater robustness to overfitting by improving generalization across datasets. Its linear representation makes the model more interpretable and easier to analyze compared to the tree-based format of GEP. Additionally, MEP offers better computational efficiency due to its simpler decoding process [95]. These combined advantages explain MEP’s superior predictive performance observed in the study.

Figure 10

MEP error distribution frequency distribution plot.

3.3 Model’s validation

R, RMSE, MAE, RRMSE, RSE, NSE, and the results from Eqs. (2)–(9) are summarized in Figure 4, which also includes the outcomes of the efficacy and error evaluations. Reducing error levels often improve the accuracy of model predictions. This is especially true when comparing the C-S MEP model to the C-S GEP model: the former achieves a much lower MAE (from 1.993 to 1.121 MPa) and a far lower MAPE (from 10.10% to 5.00%). Following suit are other error-based measures, such as NRMSE, MBE, and RMSE. With a slightly higher Nash–Sutcliffe efficiency (0.969) than the latter’s 0.944, the C-S MEP model achieves better efficiency than the C-S GEP model. Both models produce findings that are similar when we use Pearson’s coefficient (r). Figure 11 shows the Taylor diagram, which compares all forecasting models. It shows that MEP models are close to one another when predicting the C-S of foamcrete, but GEP models are farther away. The MEP model is the best ML-based method for predicting the C-S of foamcrete because it has the lowest error, highest R ² value, and the fewest standard deviations, which is consistent with previous research (Table 4).

Figure 11

Validating C-S model precision with Taylor’s diagram.

3.4 Sensitivity analysis

The study examines how input parameters affect foamcrete’s C-S prediction. Strong correlation between input factors and anticipated output [96]. Figure 12, which displays how each input impacts the C-S of foamcrete, provides a glimpse into the future of building materials and concrete. The foamcrete density (Dn) is the most important factor, explaining 48% of the variation in C-S predictions. Other factors include cement content (11.0%), age (8.5%), WCR (8.0%), foaming agent (Ag) (7.5%), SS (7.0%), foam content (Fm) (5.0%), sand (Sa) (4.5%), and SCR (0.5%). The outcomes of the sensitivity analysis were significantly influenced by both the number of data points and the number of model parameters. Notably, certain input variables, such as the proportions of concrete mix components, had varying impacts on the results when analyzed using the ML approach. To determine the relative importance of each input parameter within the model, Eqs. (18) and (19) were employed

where f min ( x i ) is the lowest projected value through the ith outputs and f max ( x i ) is the maximum.

Figure 12

Sensitivity analysis pie chart.