Indirect Estimation of Swelling Pressure of Expansive Soil: GEP versus MEP Modelling

Jalal, Fazal E.; Iqbal, Mudassir; Ali Khan, Mohsin; Salami, Babatunde A.; Ullah, Shahid; Khan, Hayat; Nabil, Marwa

doi:https://doi.org/10.1155/2023/1827117

Advances in Materials Science and Engineering

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Abstract Introduction Results and Discussion Conclusions Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2023 | Article ID 1827117 | https://doi.org/10.1155/2023/1827117

Indirect Estimation of Swelling Pressure of Expansive Soil: GEP versus MEP Modelling

Fazal E. Jalal,¹Mudassir Iqbal,²Mohsin Ali Khan,^3,4Babatunde A. Salami,⁵Shahid Ullah,²Hayat Khan,⁶and Marwa Nabil⁷

Academic Editor: Sharanabasava Ganachari

Received24 Aug 2022

Revised26 Dec 2022

Accepted31 Dec 2022

Published23 Jan 2023

Abstract

In this article, detailed trials were undertaken to study the variation in genetic parameters in order to formulate more robust predictive models using gene expression programming (GEP) and multigene expression programming (MEP) for computing the swelling pressure of expansive soils (P_s-ES). A total of 200 datasets with ten input parameters (i.e., clay fraction C_F, liquid limit , plastic limit , plasticity index I_P, specific gravity G_s, swell percent S_p, sand content, silt content, maximum dry density ρ_dmax, and optimum water content ) and one output variable, i.e., P_s-ES are collected from the literature, which comprises 120 internationally publications. The effect of input parameters in contributing to P_s-ES has been validated using Pearson correlation (r), sensitivity analysis (SA), as well as a parametric study. The results reveal that the GP-based techniques correctly characterize the swelling characteristics of the ES, thus leading to reasonable prediction performance; however, the MEP model yielded relatively better performance. Also, the proposed predictive models were compared with widely used AI models (ANN, ANFIS, RF, GB-T, DT, and SVM). The ANN performed relatively better; however, it is recommended to use the GEP and MEP due to the blackbox nature of the ANN. Other models exhibited inferior performance. The SA revealed different importance by the GEP and MEP models, however, its confirmed that the maximum dry density and optimum moisture content significantly affect the P_s-ES. The variation in P_s-ES with changes in input attributes is further corroborated from literature. Hence, it is recommended that the proposed GEP and MEP models can be deployed for computing the P_s-ES which efficiently lessens the laborious and time-consuming testing.

1. Introduction

Expansive soils (ES) are troublesome because water causes volumetric and structural changes, putting overlying constructions at risk of upheaval, fracturing, and collapse. Building of engineering structures on these problematic soft soils is often unavoidable due to growing urbanization [1–3]. Strong hydrophilic clay minerals, significant plasticity, a higher proportion of clay particles, and a high void ratio characterize these problematic soils. They possess poor strength and compressibility, which have a detrimental impact on the surrounding environment [4–7]. As a result of the damages brought by ES, they are considered ecologically susceptible, as well as dangerous. Thus, the engineering community is driven to apply sustainability concepts in infrastructure building so as to achieve economical solutions [4, 8–10]. Annual losses in the USA due to distress and the shrink-swell propensity of the swelling soils are expected to approach approximately $13 billion [11], whereas it is more than €150 million in the UK [12]. Additionally, the damages have been reported in many regions across the world [6, 13]. These ES are extensively found in six of the seven continents across more than 35 countries, i.e., Africa (Algeria, Chad, Egypt, Ethiopia, Ghana, Kenya, Morocco, South Africa, Sudan [14], Zimbabwe); Asia (Burma, China, India, Iran, Iraq, Israel, Jordan [15], KSA, Oman, Pakistan, Turkey); Australia; Europe (Cyprus, Greece, Norway, Poland [16], Romania, Spain, UK); North America (Canada, Cuba, Mexico, Puerto Rico, USA); and South America (Argentina, Brazil [17], Venezuela) [3, 18–21]. The percentage land covered in different countries with abundant presence of ES can be seen in Figure 1.

One of the key concerns is dealing with the hydrophilic nature of the ES in the subgrade of road pavements [23]. A neutral moisture layer adsorbs on the clay particles due to the interaction between ES particles and water, thus causing the soil to swell. The total volume of the ES grows as the film thickness develops; that is regulated by the criteria of dry density alongside the water absorption level [24, 25]. The untreated, compacted ES efficiently inhibits moisture access due to smaller values of hydraulic conductivity [26]. Note that the ES are typical multicrack fine-grained soils in which a high amount of compaction results in less fracturing in case the swelling pressure (P_s) is regulated [27]. Furthermore, the ES are also frequently utilized as barrier materials in waste containment facilities. The performance of dense ES has received a lot of attention in recent decades as it relates to their utilization as built barriers in order to separate the nuclear waste burial over greater depths [28–32].

It is critical to examine the swell characteristics of the ES to avoid postconstruction disasters of infrastructures coupled with rising urbanization as well as industrialization [33–36]. Note that the swell behaviour is primarily categorized by free swell (FS) and the P_s of ES (from here on referred to as P_s-ES) which can be determined in the laboratory with the help of an oedometer. Because it is laborious and time-taking test (utilizes >01 week because of incremental loading condition) owing to the delayed saturation, a number of empirical formulae for the reliable forecasting of P_s-ES have been devised beforehand [37, 38]. The P_s-ES is generally characterized by amended soil bulk density and lesser macroporosity [39]. The swell-strength characteristics of ES have been broadly studied in the past years, with a focus on the effects of particle gradation, plasticity, and compaction, alongside other factors on the development of P_s-ES and unconfined compression strength (P_s-UCS) [2, 40–42]. The estimation of the P_s-ES is necessary to rapidly predict the swelling characteristics during construction pertaining to geotechnical engineering [43–45]. Numerous estimation techniques have been presented to compute the P_s-ES in a reliable and efficient manner. More realistic and explicit models must be developed to forecast the P_s-ES, particularly, that cover the entire range of the ES from lower to higher I_P values alongside pertinent governing factors.

Artificial intelligence (AI) approaches are gaining popularity as a result of their improved predictive capability compared to older methods, and they are being utilized to simulate the complicated behaviour of a range of geotechnical engineering systems [46–48]. To address the issue with the help of a relatively larger database having more robustness, numerous soft computing approaches were used for predicting the P_s-ES to enable better use of soil for geotechnical engineering purposes. In this regard, multiple regression (MLR), the artificial neural network (ANN) approach [2, 49] using Levenberg–Marquardt (LM) and scaled conjugate gradient (SCG) [50], the adaptive neural fuzzy inference system (ANFIS) [51–53], and gene expression programming (GEP) [2] have been previously used to predict the P_s-ES. Among these methods, the GEP technique is an efficient approach that exhibits the capability to estimate complex as well as highly nonlinear problems [2, 54, 55]. Moreover, there exist certain drawbacks to applying the GEP, including the fact that (a) it is computationally complex, (b) the generated solutions may be overfit, and (c) the GEP modelling requires additional complex heuristics to achieve optimal results [56]. Therefore, multiexpression programming (MEP) has been applied here alongside the GEP approach to compute the P_s-ES in the current study to compare the two genetic programming approaches.

It can be seen from Figure 2, that the year-wise research articles on expansive soils after 2000 are given for various keywords, such as, “Expansive Soils” (ES: 4256), and “Properties of Expansive Soils” (p-ES: 1603),“AI or ML in Expansive Soils” (AM-ES: 100), and “AI in UCS of Expansive Soils” (AI-UCS-ES: 98) in Figures 2(a)–2(d), respectively. Note that the number in parenthesis represents the cumulative number of research articles published from 2000 to date. It can be seen that only two articles are available which consider the AI application to evaluate the P_s-ES.

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Traditional statistical studies were used to obtain prior correlations for the P_s-ES, which had drawbacks, such as (a) fewer data points, (b) lower correlation among governing factors, and (c) the absence of integrated comparative evaluation, etc. [55]. Previously, myriad studies have employed simple geotechnical indices combined with AI approaches to evaluate the swelling pressure characteristics [57–59]. The goal of this study is to formulate models that accurately predict the swelling behaviour of virgin ES with the help of simple and economically computable basic geomechanical characteristics. Previously, [2] considered 168 datapoints with 9 input parameters to propose a GEP equation for determining P_s-ES. In the current research, two genetic-based approaches (i.e., GEP and MEP) were deployed to derive simple mathematical expressions for predicting the P_s-ES. To date, no such study exists to estimate the P_s-ES, despite their practical significance, using the above-stated two genetic-based computational methods. Nine soil parameters were utilized as input variables, i.e., clay fraction (CF), plasticity index (PI), specific gravity (G_s), maximum dry density (ρ_dmax), optimum moisture content (), swell percent (SP), natural water content (), percent sand and percent silt. All the input characteristics were attained from a database comprising 200 datapoints of P_s-ES from the available literature. The major objectives include: (a) formulation of GEP and MEP-based prediction formulae, (b) investigation of the feasibility of the two genetic programming approaches, and (c) comparison of the performance of the GEP and MEP models to forecast the P_s-ES. The statistical performance criteria, for instance, mean absolute error (MAE), root squared error (RSE), root mean square error (RMSE), Nash–Sutcliffe efficiency (NSE), coefficient of correlation (R), coefficient of determination (R²), as well as relative root mean square error (RRMSE) were used to evaluate appropriateness of the GEP and MEP models. In addition, parametric study was performed followed by sensitivity analysis in order to categorize the most positive as well as negative influencing input factors.

2. Genetic Programming Approaches

2.1. GEP

Genetic algorithms (GA) are evolutionary algorithms (EA) incorporating a stochastic approach to find and optimize a specific problem solution based on the genetic process [60]. Cramer introduced genetic programming (GP) in 1985, whereas Koza enhanced it by employing a range of sizes and shapes [61–63]. Later, Ferreira [64] suggested the GEP, which is a genotype-phenotype-based computer programming approach. It builds tree-like complex computer models that learn and evolve by changing their shapes, sizes, and structures, much like a natural biological creature. The GP approach uses the most attractive computational intelligence philosophy, which resembles the principles of Mendel’s genetic theory as well as Darwin’s evolutionary theory [65]. Furthermore, it contains basic linear chromosomes having fixed length to encode a program for addressing a complex problem. The GEP model has a key benefit in that it forecasts via a simple mathematical equation, which results in reliability and improved prediction efficiency [66]. This simple mathematical expression can be practically used and has evolved rapidly in geotechnical engineering, particularly after 2010. In recent years, this framework has been effectively applied to a wide range of particular geotechnical design challenges, including forecasting of soil–water characteristic curves [67], compression strength of highly plastic limestone [68, 69], permeability and compressibility of unsaturated soils [70], uniaxial compressive strength of rocks [71], peak ground acceleration [72], driven pipe piles set-up [73], and efficiency of tunnel boring machines (TBMs) [74], among others. Additionally, GEP simulation is frequently employed in modelling functions, solving regressions, detecting, predicting, as well as in the process of data mining [54, 75–78].

The functioning mechanism of the GEP model can be visualized in Figure 3(a). In reference to the flow diagram, the GEP prediction starts with the randomized creation of chromosomes for defined values following the Karva language (symbolic representation) to introduce the chromosomes (Figure 3(b)). For finding a solution to a problem, four distinct components must be defined, which include the function and terminal set (constants and input variables), GEP controlling parameters (crossover, population size, and mutation), fitness function, and the termination conditions [80, 81]. As stated by Garg et al. [82]; the GEP functioning process can be subdivided into several phases. In the first phase (initialization), the population is produced based on function and terminal setting parameters. A conventional chromosome is made up of a head (connecting the function or terminal symbols) and a tail (connecting only terminal symbols). Because of the fixed length of chromosomes, it can be easily translated into an algebraic equation (Figure 3(b)) [83]. Following that, the chromosomes are transformed into different shapes and sizes of expression trees (ETs). Knowing the pattern or syntax of a gene language is like knowing ET’s language [64]. The genes forming the chromosome can be integrated using four different linking functions, i.e., multiplication, addition, subtraction, and division [61].

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In the second phase (evolution), the productivity of the developed population is evaluated using two fundamental criteria, namely, the maximum number of iterations (or generations) and the fitness function. The criterion for increasing the number of ETs is dependent on the termination of the iterative process or the indication of overfitting. Following that, approaches, namely, the roulette wheel approach, ranking selection, greedy overselection, elite strategic approach, and tournament selection [46, 56], are deployed to choose feasible chromosomes of the current iteration and transfer them to the new generation. In the event that the performance of the population is compatible with the requirement, or the maximum number of iterations is reached, the program will be terminated. Conversely, a fresh population is produced continuously with three different genetic operations, i.e., (i) reproduction, (ii) crossover, and (iii) mutation (Figure 3(b)). The process continues untill the threshold condition is met and the best solution is generated ([84, 85]).

2.2. MEP

The MEP is a cutting-edge and linear-based advanced GP approach that uses linear chromosomes to encode solutions. The operational mechanism of MEP is analogous to that of the GEP method and is a comparatively recent extension of the GP technique. The key advantages include encoding numerous computer models (solutions to problems) with the help of a single chromosome. The best-suited chromosome is chosen via comparison of a variety of fitness values. The most-suited chromosome is then chosen by comparing the fitness scores, which is considered to be the optimal solution [60, 86]. In accordance with the findings of Oltean and Grosan [87], the mechanism of binary environment selection results in the identification of two parents, which are then integrated to produce two distinct offspring. The offspring are mutated, and the procedure is continued till the finest code before reaching the termination criterion (Figure 4). The tuning of different parameters can be adopted in the MEP modelling just like in the GEP modelling. The main attributes in the MEP approach are; (i) the size of the subpopulations, (ii) the number of subpopulations, (iii) the code length, (iv) the functions set, and (v) the crossover probability [88]. The size of the population relates to the number of programs and the subpopulation size. The greater number of subpopulations leads to more complexity and higher computational time. Furthermore, the length of the derived mathematical equation is strongly affected by code length. Table 1 shows the MEP setting parameters that were used to generate a reliable and robust solution of P_s-ES by deploying the MEP approach.

2.3. GEP vs MEP

Past data sets are often employed in the assessment and modelling phases of both GP techniques [55, 89]. For instance, the GEP and MEP approaches are recognized as the most prevalent linear-based GP approaches that accurately describe the soil deformation modulus [90, 91]. Grosan and Abraham [92] investigated the effectiveness of MEP and linear genetic programming (LGP) methodologies, finding that the integration of LGP and MEP outperforms conventional neural network strategies. In comparison to MEP, the GEP approach possesses a highly advanced operating process [88]. However, despite GEP, the MEP representation is acceptable for code reuse. The noncoding portions in the MEP can exist at any place on the chromosome, and the call argument references are recorded directly. Therefore, the MEP is less packed as compared to the GEP [93]. Additionally, it is associated with such a conventional GEP chromosome, which consists of a head and tail, and both include special symbols that give an efficient manner of recording syntactically valid computer programs [87], implying that the GEP executes significantly well. Thus, it is highly needed to further investigate and evaluate the applicability of these two GP-based approaches to particular engineering challenges.

3. Model Development

3.1. Geological Database

A detailed dataset of the myriad ES properties was compiled from an extensive literature review, as shown in Table 2. Statistical analysis was conducted to investigate the potential of P_s-ES and to select an optimized dataset. Due to the missing values of other geotechnical engineering properties they were omitted in the current research. The properties of ES, which include clay fraction C_F, liquid limit , plastic limit , plasticity index I_P, specific gravity G_s, swell percent S_p, sand content, silt content, maximum dry density ρ_dmax, and optimum water content , are shown in Table 2. They were accounted while establishing the GEP and MEP models for P_s-ES as expressed in in the following equation:

Among the most frequently utilized indices of correlation is Pearson’s correlation coefficient, which is denoted by r [94]. It is commonly understood that multicollinearity and interdependence must be examined since they pose impediments in the inference of a certain machine learning models [94, 95]. In this regard, Pearson’s correlation matrix consists of “r” entities was generated for the dataset investigated in this work, with ten different input variables for optimally evaluating the P_s-ES (Table 3). The correlation matrix is a squared-shape symmetric matrix of order M × M with the (ab)^th entity equaling the coefficient of correlation r_ab between the (a)^th and (b)^th variables. The diagonal entities showing the correlations of variables with each other must always be constant, equaling one [96].

So, the eleven columns on the left-hand side of the correlation matrix indicate qualitatively the r values among the soil explanatory variables (C_F, , I_P, G_s, S_p, , , ρ_dmax, sand, and silt) and the model output parameter i.e., P_s-ES (Table 3). The correlation coefficients range from −1 (indicating a strong negative correlation) to 1 (indicating strong positive correlation). Zero indicates that there is no correlation. As shown in Table 3, the and C_F had the most significant negative (r = −0.5008) and positive (r = +0.508) association with P_s-ES, respectively. The degree of correlation (irrespective of negative and positive) between inputs variables and output (P_s) follows the trend: > C_F > silt > I_P > ρ_dmax > > S_p > > G_s > sand (r = 0.508, −0.5008, −0.388, 0.356, 0.333, 0.266, 0.209, −0.188, 0.1707, and −0.080, respectively). A significant correlation of the output (P_s-ES) exists for most of the variables considered. The sand content is the only less governing input variable showing a lesser correlation with P_s-ES (i.e., −0.080). As stated in Table 3, all coefficient values depict that the P_s-ES is observed to have pronounced impact on all the considered input variables.

Furthermore, the descriptive statistical analysis of the input variables and output parameter are mentioned in Table 1. The statistical analysis demonstrates the minimum and maximum values representing the range of all the considered parameters. It should be noted that more than 84% of records had a plasticity index higher than 20%, confirming the existence of lower and very higher expansive soils in the current database. The standard deviation, skewness, and Kurtosis scores are also provided. A lower standard deviation indicates that the majority of the points are nearer to the mean (ρ_dmax, , S_p, , sand), while a higher standard deviation indicates that the distribution of readings exhibits more spread (C_F, , I_P, silt, P_s) [105]. Skewness measures the asymmetries of the probability distribution of the considered parameters in reference to the mean value. It can be negative, positive, zero, or undefined [106]. The negative skewness indicates that the one-tail is stretched to the left side with respect to the normal distributions curve (ρ_dmax, ), whereas the positive skewness indicates that the one-tail is stretched to the right side (S_p, , sand, C_F, , I_P, silt, P_s), which is mirrored and confirmed by the frequency histograms provided in Figures 5(a)–5(j), respectively. Consequently, Kurtosis reveals the pattern of a probability distribution, showing the peakedness or tailedness of the data [107]. It determines whether the data is light-tailed or heavy-tailed in comparison to a particular normal distribution. It can be categorized as platykurtic (Kurtosis < 3: ρ_dmax, , , C_F,, I_P, silt, P_s), mesokurtic (Kurtosis = 3: sand) and leptokurtic distribution (Kurtosis > 3: S_p), which is also displayed and verified from Figure 5 [108]; Brown and Greene [109, 110].

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3.2. Model Overview

All the 200 swelling pressure records (shown in Table 2) were randomly distributed by GeneX tool (GEP) and MEPX software (MEP) into three distinct sets, i.e., training set (70% records), and testing set (30% records) [110]. To evaluate the performance accuracy of developed GP models, both portioned sets were subjected to different statistical indicators, i.e., mean absolute error (MAE), Nash–Sutcliffe efficiency (NSE), root mean square error (RMSE), coefficient of regression (²), coefficient of correlation () and performance index () [55, 80, 112, 113]. The equations for the aforementioned metrics have been provided as equations (2)–(6).

In the above equation, and represent the actual observations and estimated outcomes, respectively; and denote the average of the actual observations and estimated outcomes, respectively; and is the total number of records. Lower the error metrics (MAE, RMSE, and ρ) and higher the value of NSE, R, and R² and the predefined significance (α = 0.05) from t-test and F-test and, presents effective performance and higher accuracy of developed models [114, 115]. It also signifies that the observed and forecasted P_s-ES outcomes are highly correlated. Moreover, the values of R above 0.8, R² values approaching 1.0, RMSE values close to or equal to 0, and ρ values nearer to 0 would result in superior calibrated model [48, 90, 116]. Despite RMSE, MAE is a useful evolving measure once the dataset is continuous and smooths [117]. While the NSE ranges from −∞ to 1.0, with 1.0 representing an optimal solution. However, NSE lying between 0 and 1 is normally recognized as an adequate degree of predictive performance [118].

4. Results and Discussion

4.1. Details of Trials Performed for GEP and MEP

A total of 27 GEP trials were undertaken to identify optimal combinations for models with higher performance as shown in Figures 6(a) and 6(b). The GEP algorithm is kept running indefinitely so that the correlation and fitness functions (RMSE in this case) do not change their values. The process is performed multiple times until the best model is found. Table 4 shows the best settings for the GEP model utilized in this study. It can be observed from Figure 6(a) that the GEP27 trial is the most optimum trial having best fitness (721.1 and 661.54), RMSE (91.37 and 95.25), MAE (70.85 and 75.77), and R (0.49 and 0.813) for the training and validation datasets, respectively. This model was kept running for 690 minutes. It is also pertinent to mention that the fitness function was “subset selection” instead of the default “optimal evolution” setting. Finally, the aforementioned optimal combination of the GEP parameters (Table 4) was employed in finalizing of the proposed model to attain simple mathematical formula to forecast the P_s-ES.

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The effect of increasing number of chromosomes, head size, and genes were also studied on the coefficient of determination (R²) values of the respective trials (Figure 6(b)). Only one particular GEP parameter was altered while keeping the remaining constant for evaluating the behaviour of the aforementioned GEP code parameters on the R² value. It was revealed that the R² decreased with increasing number of chromosomes (30 to 150), whereas, it increased at higher number of head size and number of genes. Similar observations were obtained while evaluating the compaction parameters of ES via GEP modelling [46]. The constants per genes were also increased from 10 to 15 alongside changing the linking functions (+, −, ÷) and the best results were obtained for the default value of 10 and for “plus” linking function, respectively.

Similarly, the details of all the 32 MEP trials undertaken to evaluate the P_s-ES are depicted in Figures 6(c) and 6(d). It can be seen from Figure 6(c) that the most optimal model is MEP 32 obtained in the last trial. This model possesses an average MSE (2432.17) and R² (0.754 and 0.728) for the training and validation datasets, respectively. This model was kept running for 142 minutes, after which it terminated. Table 4 shows the best settings for the MEP model utilized in this study (i.e., MEP 32 trial). Furthermore, the effect of increasing the number of generations, number of subpopulations, and code length was also studied on the coefficient of determination (R²) values of the respective trials (Figure 6(d)). On the basis of these MEP code parameters, the effect of a particular MEP parameter on the robustness of the estimation of the P_s-ES can be clearly recorded. It was found that the R² increased with increasing number of chromosomes (100 to 1400), number of subpopulations (1 to 100), as well as the code length (20 to 100). However, the length of the final mathematical expression in cases of higher code length (i.e., 100) is very complex; therefore, the code length of 50 was selected for the optimal trial. Similar results in case of MEP modelling were found when determining the compaction parameters of soft soils [46].

4.2. Prediction Models

In obtaining the best predictive model, the GEP and MEP models for predicting the P_s-ES were chosen bearing in mind the strategy of multiple objectives. These objectives are built on building simple model that would yield easily employed mathematical expression, providing the best fitted values for the learned data and also for the validated data [90]. To propose a GEP model with optimal parameters, the initial parameters were selected based on previously recorded parameters in published works that employed GEP modelling technique [55, 90, 119]. The original optimized combination of GEP parameters was established using the trial and error method and with this method, many parametric combinations were generated whose performances were evaluated using R² values [80]. The data training process by the two genetic programming approaches continued until the correlation and evaluation functions reach a point of approximately no change in their values. Using an arithmetic function to join them, the number of genes and their individual sizes are steadily increased. This process goes through several circles of iteration in a bid to reach the optimal models having robust performance. The parameter settings to reach the optimum models are shown in Table 4. Karva notations can be converted to ETs and decoded to provide an explicit mathematical formula (using the GEP approach) capable of estimating the P_s-ES as a function of specified input variables, in the form of (7), using the mentioned parameter settings:

In addition, with the help of the settings shown in Table 4, the simplified mathematical expression (using the MEP approach) was derived after extracting the C++ code. The expression given in (8) is capable of predicting the target values P_s-ES:where

C: clay fraction; P: plasticity index; G: specific gravity; M: maximum dry density; O: optimum moisture content; S_p: swell percent; W: natural water content; S_n: sand content; and S_l: silt content.

The results of MEP attained in (8) to determine the P_s-ES are more reliable as compared with those obtained from the GEP equation (7) It is to say that, the MEP approach exhibits good predictive abilities alongside greater generalization performance. Similar results were achieved by [60]. However, the results obtained from the GEP equation are also reliable, and the robust performance of the GEP has been documented in previous literature by various researchers as well [46, 95].

The graphical representation of the cross-plots for the experimental and predicted P_s-ES data by the proposed models developed using GEP and MEP techniques shown in Figure 7. For the sake of comparison of genetic programming approaches with the neural network-based approaches, i.e., artificial neural network (ANN) and adaptive neuro fuzzy inference system (ANFIS), the experimental versus predicted values alongside variety of performance indices are given in Figure 8. The plots revealed the accuracy of the proposed models and the influence of the input parameters in estimating the P_s-ES. The result of prediction provided by the aforementioned equations (7) and (8) found from the GEP and MEP models, respectively, are shown Figure 7. Note that Figure 7(a) to Figure 7(d) illustrate the comparison between the predicted values of the GEP and MEP models and the experimental values of the P_s-ES in addition to their respective R and RMSE values. As compared to the ideal fit (1 : 1), there exists a strong correlation between the training and the validation datasets as revealed in the regression line slope for the proposed models. The proposed GEP and MEP models' validity is confirmed by the closeness of the points to the ideal fit and the inclusion of the majority of the points within the acceptable confidence interval. As stated in previous works, R values more than 0.8 [80] indicate that the suggested models for the P_s-ES perform adequately well. Comparatively, using the R² values obtained for the training and validation sets, the prediction values using the MEP equation performed better than the GEP model using its equation. Figure 7(e) and Figure 7(f) revealed the scattered plot between the experimental and predicted data alongside the models’ prediction error. Visually, lower error values were recorded with the proposed MEP model in comparison with the GEP model, which implies that the MEP model can be satisfactorily deployed in testing new datasets. Unlike the genetic programming approaches, the R values in the training dataset of ANN and ANFIS modelling are 0.885 and 0.853 (as shown in Figure 8(a) and Figure 8(c), respectively), which also suggests their better performance; however, both of these methods do not yield simple mathematical expressions and generally possess the problem of overfitting. Moreover, upon comparing the results of the residual error plots for the P_s-ES using GEP (Figure 7(e)), MEP (Figure 7(f), ANN (Figure 8(e)), and ANFIS (Figure 8(f)), it can be observed that the increasing trend of accuracy followed the order: ANN > MEP > GEP > ANFIS.

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4.3. Models Validity

This section discusses the validation of the developed GEP and MEP models in terms of the geological database (Table 2) used for this study. It also illustrates the statistical evaluation of accuracy obtained for the validation dataset as the “first level of validation.” Furthermore, sensitivity and a parametric study are also conducted to reflect the variation of P_s-ES versus changing attributes and compare it with the experimental studies existing in the literature [120, 121].

4.3.1. 1^st Level Validation

The employed database comprising 200 data points for P_s-ES was divided into 70% training and 30% validation. This distribution makes 134 records in the training and 66 observations for the validation of the superior GEP and MEP models. Table 5 lists the summary of previous studies reported on the basis of the number of observations employed for developing AI models. Le et al. [122] investigated the nonlinear capabilities of the ANN model for predicting unconfined compression strength (UCS) of soil using 6 input variables and 118 observations. Pham et al. [123] developed the GEP model for UCS of soil employing 1138 observations. Similarly, Mozumder and Laskar [125] and Tinoco et al. [127] developed ANN and SVM models taking into account 283 and 444 observations, respectively. It can be observed from Table 5 that the ratio of observations to number of input variables ranges between 10 and 103.45. It infers that there is no strict rule for number of observations used for developing AI models; however, Gandomi and Roke [111] and Mousavi et al. [78] recommended the minimum ratio of experimental observations to the number of input variables as five. In this study, 10 inputs have been used to predict P_s-ES, making this ratio equal to 20. Therefore, the authors believe that number of observations used for the development of current models is sufficient to reflect the robustness of the developed models. Most of the previous studies reported in Table 5 used a 70/30 split between the training and validation data, which has been exercised in this study as well.

Based on the results achieved in Figure 7, concerning the comparison plots of experimental and predicted P_s-ES, it can be inferred that the performance indices (R, RMSE, MAE, and NSE) indicate high accuracy with R values of 0.813 and 0.853 for the validation data of both the GEP and MEP models, respectively. Similarly, the values of MAE for the validation data for the GEP and MEP models manifest 28.4% and 14.4% of average MAE, reflecting the robustness of the MEP model in comparison to the GEP model. Smith [129]; in his logical hypothesis, stated that the models yielding a high correlation coefficient (R > 0.8) and low error values (e.g., MAE and RMSE) represent the prediction relationship among input and output variables is accurate and reliable. The developed models having the value of R > 0.8 and average MAE values lesser than 20% show that the P_s-ES is strongly correlated with the input variables, and they are in good agreement [128–130]. The models presented in comparison to the GEP and MEP models revealed values of R and MAE were recorded as 0.874, 34.86 and 0.649, 54.30 for ANN and ANFIS, respectively, for the validation data (Figures 7 and 8, respectively). Furthermore, Figure 9 depicts the comparison of the R values using genetic programming approaches (GEP and MEP), neural network approaches (ANN and ANFIS), and other widely used AI techniques (random forest regression (RF), gradient boosted trees (GBT), decision trees (DT), and support vector machines (SVM)). It can be seen that the GEP, MEP and ANN interpreted reliable performance; however, the RF, GBT, DT, and SVM depict inferior performance compared to the models investigated in the current study. Note that the details of parameters settings along with various performance indices in case of the RF, DT, SVM and GB trees are tabulated in Table 6.

4.3.2. Second Level Validation

This study presented a sensitivity and parametric study on the basis of the developed GEP and MEP models as the “2^nd level validation”. It is important to know the effect of contributing parameters on the target variable in order to compare with the experimental studies. Sensitivity and parametric studies are intended to validate further the robustness of the proposed MEP and GEP models that whether or not they support the same physical process achieved from experiments [95]. The sensitivity analysis (SA) on the simulated database tells how sensitive a developed model is to a certain change in the specific parameters considered [80, 130, 131]. A simulated dataset was obtained such that one parameter was changed between its extreme values and the other parameters were maintained at their average values. The relative contribution of each parameter in yielding the P_s-ES was obtained by employing the SA. For a particular input variable X_i, the SA is determined using equations (10) and (11):where f_max (X_i) and f_min (X_i) denotes, respectively, the maximum and minimum of the estimated output on the basis of i^th input domain, while the remaining input parameters are unchanged at their average values.

The SA employing the MEP model revealed that is the most influencing parameter in determining the P_s-ES whereas for the GEP model, ρ_dmax interpreted the highest contribution (Figure 10(a)). The order of importance was observed as ρ_dmax > C_F > silt > > > S_P > sand > LL for the GEP model, whereas for the MEP model, it was recorded as > ρ_dmax > PI > C_F > > silt > S_P > sand. Though the order of importance was observed to be different for the GEP and MEP models; however, it is confirmed from both the models that ρ_dmax and significantly affect the P_s-ES. While investigating the parametric evaluation (Figures 10(b)–10(i)), an increase in P_s-ES was noticed by increasing the amount of C_F. The increasing swell potential with the rise in the C_F is due to silicates of clay minerals [135]. An increase in I_p also depicted an increase in the P_s-ES evident from previous literature [136, 137].

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

The MEP models captured no significant changes in the P_s-ES with the rise in ρ_dmax; however, the GEP models manifested a slight increase. The P_s-ES decreased initially with the change in up to 20%, beyond which no appreciable change was noticed in the P_s-ES. It is also pertinent to mention that S_p directly increased the P_s-ES as observed from both models. The developed models manifested a pattern of variation in the P_s-ES with change in input attribute as corroborated from the existing literature.

5. Conclusions and Recommendations

This research presents a novel application of genetic programming approaches (i.e., gene expression programming (GEP) and multi expression programming (MEP)) for the development of estimation models of swelling pressure of expansive soils (P_s-ES). The input parameters of the proposed model consist of clay fraction C_F, liquid limit , plastic limit , plasticity index I_P, specific gravity G_s, swell percent S_p, sand content, silt content, maximum dry density ρ_dmax, and optimum water content . A database comprising the results of 200 swelling pressure tests of the ES was utilized to formulate the GEP as well as the MEP models. 134 datapoints (70%) were used for training whereas the remaining 66 points (30%) were utilized as validation sets. The P_s-ES can easily be forecasted with the help of easily determinable soil physical characteristics using these GP models. The following conclusions can be drawn:(1)While investigating the nonlinear capability of the generated models, the entire data was initially subjected to linear analysis using the Pearson correlation matrix which interpreted a strong positive correlation between the C_F and P_s-ES (i.e., substantially higher than 0.5). On the other hand, yielded strong negative correlation with the P_s-ES. The trend achieved from the parametric analysis further validated the Pearson correlation values such that rise in C_F increased the P_s-ES, whereas, increasing the (greater than 20%) negatively influenced the P_s-ES. Moreover, the Pearson correlation values obtained in case of other input attributes were significantly smaller, which suggest that there exists a nonlinear relationship among these attributes and the swell characteristics of the ES.(2)A total of 27 trials were undertaken for GEP modelling of the P_s-ES which revealed that the performance of the developed models increases significantly with varying number of chromosomes (30–150). The head size equaling 12 yielded optimal performance of the GEP model (when it was changed from 8 to 12), whereas the number of genes had no significant impact on the performance of the GEP models. Similarly, MEP modelling for 32 trials was conducted and various genetic variations of the MEP parameters (number of subpopulations, number of generations, and code length) alongside the maximum time duration was studied for obtaining sufficient accuracy while training. It was found that, the R² increased with increasing number of chromosomes (100 to 1400), number of subpopulations (1 to 100), as well as the code length (20 to 100).(3)GEP27 trial was recorded as the most optimum trial having best fitness (721.1 and 661.54), RMSE (91.37 and 95.25), MAE (70.85 and 75.77), and R (0.49 and 0.813) for the training and validation datasets, respectively. This GEP model was kept running for 690 minutes. On the other hand, the most optimal model was MEP32 which possesses an averaged MSE (2432.17), and R² (0.754 and 0.728) for the training and validation datasets, respectively. This MEP model was kept running for 142 minutes after which it terminated.(4)Simple mathematical equations were formulated based on MATLAB models achieved in the case of both the GP-based approaches, which are used conveniently to predict the P_s-ES based on a wider range of input attributes. Note that the performance of MEP model (R = 0.868 and 0.853) exceled than that of GEP model (R = 0.849 and 0.813) for the training and validation datasets, respectively, in terms of the correlation coefficient and other statistical error indices, namely, MAE, RMSE, and NSE. However, the significance of the GEP model is also reasonable owing to the simplified short-length equation.(5)The performance of genetic models was compared with other AI models, namely, artificial neural network (ANN) and adaptive neuro fuzzy inference system (ANFIS). The increasing order of performance was: ANN > MEP > GEP. The performance of the ANN model interpreted robust perform compared to the GP-based methods, however, the associated blackbox nature inhibits its application. Other models (i.e., random forest regression (RF), gradient boosted trees (GBT), decision trees (DT), and support vector machines (SVM)) were also investigated for comparison purposes; however, their performance was below the acceptable limit for evaluating the P_s-ES.(6)The developed GP-based models were validated using 2 stage validation process. In the 1^st stage, the ratio of the number of observations to the number of input attributes (in this article) was compared with the experimental observations of other AI models in the existing literature which interpreted that the current ratio lies in acceptable range. Also, the 30% dataset used different validation of models yielded reliable performance suggests robustness and applicability of the developed models. In the 2^nd stage, the developed models were tested using simulated dataset for investigating the contribution of each input parameter in yielding the P_s-ES and its variation with the change in the attribute. The variation reflected by modelling was compared with the physical processes involved in experimental studies, which showed corroboration with the past literature.

It is suggested to validate the results of this study on P_s-ES on a wider and/or experimental data, by deploying other AI methods, such as, gradient boost trees, random forest regression, generalized regression neural networks, and convolutional neural network, among others. Also, the optimization algorithms (for example, particle swarm optimization, marine predators’ algorithm, grey wolf optimization, slime mould algorithms, etc.) can be incorporated to further enhance the accuracy of the models. Moreover, similar equations for the waste-treated expansive soils are suggested to be formulated by conducting extensive experimental tests.

Data Availability

The data used to support the findings of this study are available from the authors upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Key Project of the National Natural Science Foundation of China (Grant no. 41630633), National Key Research and Development Project (Grant no. 2019YFC1509800), and Key Special Project of the Ministry of Science and Technology of the People’s Republic of China for Monitoring, Warning, and Prevention of Major Natural Disasters are acknowledged for the financial support.

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Copyright © 2023 Fazal E. Jalal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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Advances in Materials Science and Engineering

Indirect Estimation of Swelling Pressure of Expansive Soil: GEP versus MEP Modelling

Abstract

1. Introduction

2. Genetic Programming Approaches

2.1. GEP

2.2. MEP

2.3. GEP vs MEP

3. Model Development

3.1. Geological Database

3.2. Model Overview

4. Results and Discussion

4.1. Details of Trials Performed for GEP and MEP

4.2. Prediction Models

4.3. Models Validity

4.3.1. 1st Level Validation

4.3.2. Second Level Validation

5. Conclusions and Recommendations

Data Availability

Conflicts of Interest

Acknowledgments

References

Copyright

4.3.1. 1^st Level Validation