Prediction of time-dependent bearing capacity of concrete pile in cohesive soil using optimized relevance vector machine and long short-term memory models

Abstract

The present investigation employs relevance vector machine (RVM) and long short-term memory (LSTM) models to predict the time-dependent bearing capacity of concrete piles. Each RVM model (SRVM) is configured by each linear, polynomial, gaussian, sigmoid, laplacian, and exponential kernel function. Each SRVM model has been optimized by each genetic (GA_SRVM) and particle swarm optimization (PSO_RVM) algorithm. Moreover, the double kernel-based RVM models (DRVM) have been employed and optimized by each GA (GA_DRVM) and PSO (PSO_DRVM) algorithm. Thus, an extensive comparison among 33 RVM (6SRVM + 6GA_RVM + 6PSO_RVM + 5DRVM + 5GA_DRVM + 5PSO_DRVM) has been carried out. Conversely, the Adam, root mean squared propagation and stochastic gradient descent with momentum algorithms have optimized the LSTM model. Each optimized RVM and LSTM model has been trained and tested by 100 and 26 datasets. In addition, the effect of structural and database multicollinearities has been analyzed on models’ prediction capabilities. The performance index (PI), the variance accounted for (VAF), performance (R), mean absolute error (MAE), normalized mean bias error (NMBE), and root mean square error (RMSE) matrices have analyzed the prediction capabilities of each model. The comparison of 33 RVM and 3 LSTM models reveals that the genetic algorithm-optimized Gaussian kernel function-based SRVM model, i.e., UBC7, has been recognized as the optimal performance model with the RMSE = 146.3962 kPa, PI = 1.85, VAF = 94.60, NMBE = 30.1379 kPa, MAE = 105.7009 kPa, and R = 0.9727, close to the ideal values. Furthermore, the score (= 56), Wilcoxon (= 94.95% confidence), uncertainty (= 1^st rank), generalizability (= close to ideal values), and Anderson Darling (= 9.435 ≈ 9.336) tests confirm the superiority of model UBC7. Still, structural and database multicollinearity has drastically impacted dual kernel-based RVM and stochastic gradient descent optimized LSTM models.

Introduction

The pile foundation is a deep foundation driven into the ground when the ground has insufficient load-bearing capacity. The pile foundation transfers the superstructure loads to the deeper level within the ground. The pile foundation’s load-bearing capacity in civil engineering projects represents the superstructures’ stability. Therefore, it becomes necessary to understand the pile-bearing capacity during a project’s design and construction phases. It helps designers and engineers select the pile type & parameters, installation technique, pile spacing, strata condition, embedment depth, and environmental conditions and ensure the overall stability and performance of the structures. A precise evaluation of bearing capacity helps minimize the risk of excessive settlement or structural failure while facilitating an economical and effective foundation design. The standard penetration test (SPT), cone penetration test (CPT), pressuremeter test (PT), and plate load test (PLT) are the most popular tests performed for determining the bearing capacity of the pile. However, performing these tests is time-consuming, costly, and cumbersome. Therefore, the geotechnical engineers used different empirical and advanced computer techniques to determine the pile-bearing capacity.

Artificial intelligence is the most popular advanced computer technique used to estimate pile-bearing capacity (Pu). The artificial intelligence approaches have been categorized under the umbrella of conventional, machine, advanced machine, deep, and hybrid learning methods. Fattahi et al.¹ performed 50 dynamic tests on precast concrete piles to implement the invasive weed optimization (IWO) and fruit fly optimization (FFO) algorithms and assess the Pu. The researchers found that the IWO algorithm assesses Pu better than FFO using pile length (L), cross-sectional area (A), hammer weight (W), drop height (H), and pile set (S). Gu et al.² configured the adaptive neuro-fuzzy inference system (ANFIS) and estimated Pu with a test performance of 0.9935. The investigators used pile diameter (D), thickness of the first, section & third soil layers with pile embedded (T₁, T₂, T₃), pile top elevation (P_T), natural ground elevation (NGE), elevation of extra segment pile top (P_TE), pile depth (d), average SPT blow count along the pile shaft (SPT_S) and pile tip (SPT_T) as independent variables. Khatti et al.³ employed hybrid LSTM and RVM models to estimate the ultimate bearing capacity of shallow foundations. Shen⁴ optimized multilayer perceptron (MLP) using fox optimization (FO) and crystal structure (CS) algorithms for assessing Pu. In addition, the authors found that the k-fold value plays a significant role in achieving a better performance. Finally, the authors concluded that the combined MLP_CS_FO model is better than the MLP_FO and MLP_CS models. Sun et al.⁵ implemented the hybrid support vector regressor (SVR) models using particle swarm optimization (PSO), sparrow search algorithm (SSA), Cuckoo search (CS), grey wolf optimization (GWO), and artificial bee colony (ABC) algorithms to assess the Puof rock-socketed piles. The performance analysis showed that the optimization algorithm enhances the base model’s performance, i.e., SVR. Hence, model GWO_SVR outperformed the ABC_SVR, CS_SVR, SSA_SVR, PSO_SVR, and SVR with a performance of 0.9978. Duan and Xiao⁶ evolved two hybrid decision tree (DT) models using gold rush optimizer (GRO) and Tasmanian devil optimizer (TDO) for assessing the Pu. Comparing the performance of models, the DTTD model performed better than DTGR, with a performance of 0.9979. Karakaş et al.⁷ reconfigured the machine learning (ML) models using Shapley and joint Shapley methods. Thottoth et al.⁸ estimated the Pu of single and double under-reamed piles in clay and sand strata. For that purpose, the researchers used artificial neural network (ANN) and random forest (RF) models and estimated Puof under-reamed piles with a performance of more than 0.8. Shoaib and Abu-Farsakh⁹ used a gradient boosting tree to compute the Pu based on a cone-penetration test. Nguyen et al.¹⁰ concluded that the genetic algorithm optimized ANN (GA_ANN) model estimates the Pu with high performance, i.e., 0.99. The probability analysis was performed by Silveira et al.¹¹ to analyze the failure of a small diameter bored pile in the unsaturated sandy soil. The researchers observed a significant impact of season on the bearing capacity estimation. Yang¹² developed Aquila-optimized (AO) and prairie dog-optimized (PDO) least square support vector machine (LSSVM) models and found an improvement in the prediction performance of LSSVM. The comparison showed that model PDO_LSSVM outperformed the AO_LSSVM model. Ozturk et al.¹³ analyzed the performance of the ML model using 481 and concluded that selecting suitable input variables plays an important role in predicting Pu. The investigators also concluded that the support vector regression model is the most potent tool for predicting the Pu. Yousheng et al.¹⁴ analyzed the prediction capabilities of the ML model, i.e., multilayer perceptron (MLP) and kernel ridge regression (KRR), in predicting the Pu of saline soils in cold regions.

Jie et al.¹⁵ analyzed the capabilities of slap swarm algorithm (SSA), biogeography-based (BBO), and equilibrium (EO) optimized ANN models in estimating the friction capacity of driven piles. The SSA_ANN, BBO_ANN, and EO_ANN attained performance of 0.9395, 0.9940, and 0.9385, respectively. Jin and Ji¹⁶ evolved an improved radial movement-optimized backpropagation neural network (IRMO_BPNN) model to estimate the axial bearing capacity of a pile. In addition, the researchers compared the IRMO_BPNN model to GA_ANN, PSO-optimized group method data handling integrated with ANFIS (PSO_GMDH_ANFIS), and radial basis function neural network (RBFANN). In the published work, the researchers used three databases (DB) and it was found that each IRMO_BPNN (DB1 = 0.9948, DB2 = 0.9969, DB3 = 0.9900), GA_ANN (DB1 = 0.9949), PSO_GMDH_ANFIS (DB3 = 0.9600), and RBFANN (DB2 = 0.9992) model achieved testing performance more than 0.95 in the testing phase. The researchers used L, D, cohesion (c), and internal friction angle (ϕ), ultimate tip resistance (Q_t) as input variables in the reported study. Mihálik et al.¹⁷ assessed the bearing capacity of continuous flight auger (CFA) piles using cone penetration test results. The authors used Unicode, laboratoire central des ponts et chaussees (LCPC), and eurocode7-2 methods to assess the Pu. Moghadama and Khanmohammadi¹⁸ estimated the time-dependent pile-bearing capacity using gene expression programming (GEP), GMDH, and multiple linear regression (MLR). In the published work, the researchers established the GEP, GMDH, and MLR models using L, D, plasticity index (PI), undrained shear strength (Su), effective stress (qv), initial bearing capacity (Qi), and time interval after driving (T) of 169 piles. The researchers concluded that the GEP model gives the most promising results of time-dependent Pu. Nguyen et al.¹⁹ implemented the genetic algorithm-optimized ANFIS (GA_ANFIS) model to estimate the Pu. The authors used D, L, settlement of piles (S), modulus of dynamic elasticity (Ed), cross-sectional area at the pile top (At) & tip (Ab), average SPT number along the pile shaft (SPT_NS) & the pile tip (SPT_NB), cohesion angle along the pile shaft (Cs) & tip (Cb), and friction angle along the pile shaft (ϕ_S) and tip (ϕ_B). The authors reported a performance of 0.9620 in the published work. Ren and Sun²⁰ optimized the BPNN model with GA and PSO algorithms to predict the Pu. The authors obtained 0.9532, 0.9860, and 0.9927 performances for BPNN, GA_BPNN, and PSO_BPNN models. The authors established models using L, pile area (A), pile type (PT), C, and ultimate end resistance of pile end soil (qp) of 72 piles. Vural et al.²¹ examined the test results of 8 piles and obtained a strong correlation of 0.984 using the Chin-Kondner method. Still, the authors achieved excellent Pu using the Ozkan-Alku method. Xiao et al.²² analyzed the Pu in soft soil under gravitational action. The authors concluded that Randolp’s and Janbu’s methods are the most reliable for assessing the Pu in soft soil. Moreover, Tra et al.²³ analyzed the load-settlement (Q-S) curve from a static load test for long-bored piles. The researchers concluded that the developed extrapolated curves using the Vander Veen, Chin-Kondner, Decourt, Hansen 80% criterion, 3D finite element analysis, and four newly proposed functions agree well with the Q-S curve. Conversely, Kumar et al.²⁴, Nguyen et al.²⁵, and Ibrahim et al.²⁶ found that hybrid machine learning approaches are reliable in estimating the pile-bearing capacity. Amâncio et al.²⁷ concluded that the most significant pile parameters, i.e., loading type (LT), soil type, pile type (PT), water level (WL) soil resistance index (NF), play a vital role in predicting the shaft and tip bearing capacity of piles. Also, the ANN approach can achieve an excellent estimation if it is well configured. Amjad et al.²⁸ reported that the extreme gradient boosting approach (XGBoost) is superior to adaptive boosting (ADBoost), RF, DT, and SVM approaches in predicting the Pu of concrete piles. In addition, the authors found that the SPT number is the most significant variable in assessing the Pu. Hoang et al.²⁹ concluded that the opposition-based differential flower pollination optimized LSSVR (ODFP-LSSVR) model is better than the relevance vector machine (RVM), regression tree (RT), and BPNN models in predicting Pu. Karkush et al.³⁰ derived a polynomial equation, i.e., Eq. 1, using the borehole coordinate and SPT values to assess the pile’s allowable bearing capacity (Q). Where ${K}_{0}, {K}_{1}, and {K}_{2}$ are the factors, E is Eastern, and N is Northen coordinate.

$$Q\left( {kN} \right) = K_{0} + K_{1} \left( {E - 47.5} \right) + K_{2} \left( {N - 30.2} \right)$$

(1)

Khanmohammadi et al.³¹ implemented a genetic programming (GP) model using the L, D, initial bearing capacity (Qi) time after the end of driving (T), and undrained shear strength (Su) of 256 pile results and estimated the Pu with a performance of 0.9187. Also, the authors concluded that the GWO is better than the ABC algorithm in maximizing pile capacity. Nguyen et al.³² assessed the Puof pre-stressed precast high-strength concrete (PHC) nodular pile using a feedforward neural network (FFNN) with a root mean square error (RMSE) of 0.04915. Pham and Tran³³ constructed GA and PSO-optimized RF models using 472 pile results to compute the Pu. The authors concluded that the GA_RF model (R = 0.9646) outperformed the PSO_RF (R = 0.9613) and conventional RF (R = 0.9613) models with an RMSE of 0.08847 in the testing phase. Pu et al.³⁴ computed the Pu of uplift piles in mixed strata using the empirical method. Thai et al.³⁵ used L, D, SPT_S, SPT_T, compressive strength of concrete (fc), and tensile strength of longitudinal reinforcement (fy) as input variables to assess the Pu by implementing ANFIS, SVM, and ANN models. The authors found that the SVM model is a better approach for the 75 pile test results to achieve an excellent prediction of Pu. In addition, Armaghani et al.³⁶, Heidarie et al.³⁷, Cao et al.³⁸, Arjomand et al.³⁹, Zhang & Xue⁴⁰, Moayedi et al.⁴¹, and Chen & Zhang⁴² reported that the hybrid approaches, such as imperialism competitive optimized GMDH_ANFIS (ICA_GMDH_ANFIS), multi-criteria decision-making (MCDM), multivariate neural network inference model (IMNNIM), PSO_ANN, multi expression programming (MEP), ICA_ANN, PSO_ANN, Invasive weed optimized ANN (IWO_ANN), and League championship optimized ANN (LCA_ANN), gives reliable and most promising results in predicting the pile-bearing capacity.

Ahmad et al.⁴³ estimated the ultimate bearing capacity of shallow foundations using GPR with a testing performance of 0.9697. Azimi et al.⁴⁴ mapped a relationship between cone penetration (CPT) and standard penetration (SPT) test parameters using the conventional method. The authors converted all CPT data into SPT N-values in the reported work. Benali et al.⁴⁵ employed teaching–learning-based-optimized neural network models (TLBONN) using pile head diameter (DH), internal diameters (DI), pile base diameter (DB), external diameter (DE), pile material, shaft SPT number (SPTS), pile SPT number (SPTP), load (P), and embedment length (EL) of 479 pile tests. The investigators reported that the TLBONN model predicted the ultimate bearing capacity with a performance of 0.9701. Benbouras et al.⁴⁶ compared deep neural network (DNN), ELM, Lasso, PLS, RF, Kridge, Ridge, LS, Step, SVM, and GP models in predicting pile bearing capacity. The published work was carried out for the concrete and steel piles using the L, pile area (A), flap number (F_N), average pile-soil friction angle (PS⁰), average soil unit weight (UW), average friction angle (ϕ), c, and pile material (P_M) as input variables for computational models. Dehghanbanadaki et al.⁴⁷ estimated the ultimate bearing capacity of the pile using c and ϕ parameters for hybrid soft computing models, i.e., GWO_MLP and GWO_ANFIS. Gomes et al.⁴⁸ concluded that the RF model estimates the Pu with a performance of 0.8746 (for the Décourt database) and 0.8706 (for the Meyerhof database). Still, the kNN model predicted Pu with a relative error of 24.12 better than RF. On the other hand, RF outperformed the ANN with a relative error of 31.72 for the Meyerhof database. Harandizadeh et al.⁴⁹ compared the conventional ANN, FPNN_GMDH, and PSO_ANFIS_GMDH models and found that PSO_ANFIS_GMDH model predicted the Pu with a testing performance of 0.9600. In the reported work, the authors used two new parameters, i.e., cone sleeve friction resistance (F_S) and cone tip resistance (Qc). Heidari & Ghazavi⁵⁰, Huat et al.⁵¹, Luo et al.⁵², Moayedi et al.⁵³, and Yong et al.⁵⁴ found that the hybrid computational models estimate the bearing capacity of pile accurately. Furthermore, Chen et al.⁵⁵reported that the neuro-genetic model outperformed the neuro imperialism, ANN, and GP approaches with a performance of 0.9939. Harandizadeh & Toufigh⁵⁶and Harandizadeh⁵⁷ successfully employed the hybrid computational models using the CPT database to estimate the Pu. Heidarie et al.⁵⁸ obtained a better prediction of Pu using the first-order second-moment method (FOSM) than the first-order reliability method (FORM), Monte-Carlo simulation (MCS), and modified FOSM. Kardani et al.⁵⁹ implemented PSO-optimized DT, kNN, MLP, RF, SVR, and XGB models to assess the Pu using 59 pile cases. The performance parameters, root mean square error (RMSE), the variance accounted for (VAF), and performance (R) showed that model PSO_XGB obtained a higher rank in the reported work. Also, Moayedi et al.⁶⁰ employed hybrid models, i.e., GA_ANN, ANFIS, DEA, FFNN, GRNN, and PSO_ANN, and found that PSO_ANN predicted Pu with a RMSE of 0.01 and VAF of 99.90. Momeni et al.⁶¹ employed the GPR model using squared exponential, exponential, Matern 5/2, and rational quadratic and compared it with the GA_ANN model in estimating the Pu. The statistical comparison revealed that the GPR model outperformed the GA_ANN model with a VAF of 86.41 and a performance of 0.9165. Murlidhar et al.⁶² stated that the PSO_ANN model outperformed the GA_ANN model. Wang et al.⁶³ estimated the pullout bearing capacity of helical piles using optimized neural network models. Pham et al.^64,65 concluded that the soft computing approaches are a reliable tool for estimating the pile-bearing capacity. Table 1 summarizes the published computational model used in estimating the pile-bearing capacity.

Table 1 Summary of the published computational models.

Gap Identification A thorough literature study reveals that many investigators employed several soft computing approaches to estimate pile-bearing capacity. In the published work, the ANFIS, MLP_CS_FO, IRMO_BPNN, PSO_ANFIS, ODFP-LSSVR, XGBoost, PSO_ANN, PSO_XGB, GWO_MLP, GA_SVR, PSO_ANFIS_GMDH, and GA_RF models were estimated the pile-bearing capacity. In addition, Moghadama and Khanmohammadi (2023) implemented the GEP, GMDH, and MLR models for computing the time-dependent bearing capacity of the pile and compared them. The researchers utilized the concrete square and steel piles database. Still, the models based on the relevance vector machine (single kernel-based, dual kernel-based, genetic algorithm optimized single kernel-based, particle swarm optimized single kernel-based, genetic algorithm optimized dual kernel-based and particle swarm optimized single dual-based) and long short-term memory (optimized by adaptive moment estimation, root mean squared propagation and stochastic gradient descent with momentum optimizer) have not been employed to estimate the time-dependent bearing capacity of the concrete pile. Also, it was noted that the impact of the database multicollinearity on the performance of hybrid models was not determined and analyzed.

Novelty of the Present Work Considering the gap identified in the literature study, the present research has the following novelty:

• This research compares Gaussian, Exponential, Linear, Laplacian, Sigmoid, and Polynomial kernel-based RVM (mentioned by SRVM) to find the best kernel function for the RVM model for the first time. Moreover, the five combinations of kernel functions have been made with the best kernel function to develop dual kernel-based (called parallel computing) RVM models (mentioned by DRVM) for the first time.

• This investigation optimizes each SRVM and DRVM model using each genetic and particle swarm optimization algorithm. Thus, this research compares thirty-three RVM models to find the optimal performance model for predicting the time-dependent pile-bearing capacity for the first time.

• This work also constructs and analyzes the adaptive moment estimation (Adam), root means squared propagation (RMSProp), and stochastic gradient descent with momentum (SGDM) optimized recurrent neural network model, i.e., long short-term memory.

• This investigation also analyzes the impact of database and structural multicollinearity on the hybrid RVM and LSTM models for the first time in predicting the time-dependent pile-bearing capacity.

• This research compares the capabilities of hybrid RVM and RNN_LSTM by performing score analysis, regression error characteristics curve, Wilcoxon test, uncertainty analysis, generalizability analysis, cost computing, and Anderson–darling test to identify the optimal performance model for estimating the time-dependent bearing capacity of concrete piles.

Research Significance Determining the time-dependent pile-bearing capacity by plate load test, cone penetration test, standard penetration test, pressuremeter tests, and vane shear test is time-consuming and cumbersome for geotechnical engineers and designers. Therefore, it is essential to introduce a robust and reliable method to estimate the time-dependent pile-bearing capacity. The present research will help geotechnical engineers and designers estimate the time-dependent pile-bearing capacity using a hybrid computational model.

Methodology

The present study introduces an optimal performance model by comparing the hybrid RVM and RNN_LSTM models. A database has been compiled from the published article by Moghadama and Khanmohammadi¹⁸ to learn and analyze each model. The database consists of pile length (L in m), pile diameter (D in m), pile initial capacity (Qi), time after the end of the drive (T in days), undrained shear strength (Su in kPa), plasticity index (PI in %), effective vertical stress (qv in kPa), and ultimate bearing capacity (Qu in kPa) of 126 concrete piles. For determining the multicollinearity of L, D, Qi, T, Su, PI, and qv, the variance inflation factor (VIF) method has been used. The database is also analyzed using distance correlation, ANOVA, and Z tests. In addition, a nonlinear sensitivity analysis has been performed to find the most influential input variables in predicting the time-dependent bearing capacity of concrete piles. The 80% and 20% data points have been randomly selected from the 126 data points to create training (= 100) and testing (= 26) databases. The min–max function normalizes the complete database. The correlation coefficient (R), root mean square error (RMSE), mean absolute error (MAE), performance index (PI), ratio of RMSE to the standard deviation of the observations (RSR), variance accounted for (VAF), normalized mean bias error (NMBE), and mean absolute percentage error (MAPE) metrics have measured the performance of each model.

The Gaussian, Exponential, Linear, Laplacian, Sigmoid, and Polynomial kernel functions have developed the single kernel function-based RVM models, called SRVM models. In addition, every SRVM model has been optimized by each GA (mentioned by GA_SRVM) and PSO (mentioned by PSO_SRVM) algorithm. Thus, eighteen single kernel function-based SRVM models (6SRVM + 6GA_SRVM + 6PSO_SRVM) have been developed, trained, tested, and analyzed. The performance of SRVM models has been compared to find the higher-performance SRVM model to select kernel 1 for developing the dual kernel function-based RVM models (mentioned by DRVM). Thus, five DRVM models have been developed. Furthermore, these five DRVM models have been optimized by each GA (mentioned by GA_DRVM) and PSO (PSO_DRVM) algorithm. Thus, fifteen DRVM models (5 DRVM + 5GA_DRVM + 5PSO_DRVM) have been developed, trained, tested, and analyzed. In addition, three RNN_LSTM models have been optimized and analyzed by the Adam, RMSProp, and SGDM algorithms. Thus, this research has employed, trained, tested, and analyzed thirty-six models. Seven best-performing models have been identified from the comparison of the performance parameters.

For identifying the optimal performance model from the seven best-performing models, the regression error characteristics (REC) curve, score analysis, Wilcoxon test, uncertainty analysis, Taylor plot, accuracy metrics, generalizability, cost computation, objective function criterion, and Anderson–darling test have been performed. In addition, the scatter, agreement, and a20 index metrics have been implemented to check the optimum performance model’s reliability in estimating the concrete pile’s time-dependent bearing capacity. Furthermore, the prediction capability of the optimal performance model has been analyzed using Pearson’s product-moment correlation coefficient for each input variable. Figure 1 depicts the methodology of the present research.

Fig. 1

Flow chart of the present research.

Data analysis and modelling

Data analysis

This research has been carried out using the pile length (L in m), pile diameter (D in m), pile initial capacity (Qi), time after the end of the drive (T in days), undrained shear strength (Su in kPa), plasticity index (PI in %), effective vertical stress (qv in kPa), and ultimate bearing capacity (Qu in kPa) of 126 concrete piles. The descriptive statistics of the database have been drawn to analyze the L, D, Qi, T, Su, PI, qv, and Qu, as summarized in Table 2. It is noted that the concrete pile database consists of L, D, Qi, T, Su, PI, qv, and Qu in the range of 12–64 m, 0.24–0.91 m, 45–1771 kPa, 0.03–216 days, 22.65–282.49 kPa, 30–45%, 82.30–284.05 kPa, and 152.4–3960.0 kPa, respectively. Based on the database’s statistical output, each variable’s frequency distribution has been plotted, as shown in Fig. 2.

Table 2 Statistics of the concrete pile database.

Fig. 2

Illustration of the frequency distribution of concrete pile parameters (a) L, (b) D, (c) Qi, (d) T, (e) Su, (f) PI, (g) qv, and (h) Qu Fig. 2 depicts that each variable has an excellent bell-shaped distribution curve, presenting a good range of concrete pile parameters. Hence, the distance correlation coefficient method has been used to map the relationship among the L, D, Qi, T, Su, Pi, qv, and Qu variables, as shown in Fig. 3.

The correlation coefficient presents the different levels of the relationship between two variables. The correlation ± 0.81 to ± 1.00, ± 0.61 to ± 0.80, ± 0.41 to ± 0.60, ± 0.21 to ± 0.40, and ± 0.00 to ± 0.20 presents the very strong, strong, moderate, weak, and no relationship between two variables^66,67. Here, the positive correlation reveals two variables increase continuously. Conversely, the negative correlation presents one variable increasing and another variable decreasing. Figure 3 illustrates that (a) L very strongly correlates with qv (= 0.9853), (b) L strongly correlates with D (= 0.8059), Qi (= 0.7279), Su (= 0.6597), PI (= 0.6870), (c) D strongly correlates with Qi (= 0.7440), T (= 0.6802), Su (= 0.7008), PI (= 0.8003), qv (= 0.8030), (d) L moderately correlates with T (= 0.5517), (e) Qi very strongly correlates with PI, (f) Qi strongly correlates with Su (= 0.7464), qv (= 0.7370), (g) Qi moderately correlates with T (= 0.5374), (h) T moderately correlates with Su (= 0.5287), qv (= 0.5509), (i) T strongly correlates with PI (= 0.6390), (j) T moderately correlates with Su (= 0.5287), qv (= 0.5509), (k) Su very strongly correlates with PI (= 0.8133), (l) Su strongly correlates with qv (= 0.6625), (m) PI strongly correlates with qv (= 0.6820), (n) Qu very strongly correlates with Qi (= 0.9577), PI (= 0.8723), (o) Qu strongly correlates with L (= 0.6939), D (= 0.7212), Su (= 0.7930), qv (= 0.7006), and (p) Qu moderately correlates with T = (0.5202). The correlation coefficient is one of the multicollinearity methods. The correlation results reveal that the following variables have multicollinearity: (i) L with PI & Qu, (ii) D with PI & qv, (iii) Qi with Su & qv, (iv) T with Su & Qu, (v) Qu with L and qv. Still, the correlation coefficient does not discuss the different multicollinearity levels. Therefore, Khatti and Grover⁶⁸ introduced five multicollinearity levels using the variance inflation factor ($VIF=1/\left(1-{R}^{2}\right)$), where R² is the coefficient of determination. The investigators introduced no (VIF = 0), weak (0 < VIF ≤ 2.5), considerable (2.5 < VIF ≤ 5.0), moderate (5.0 < VIF ≤ 10.0), and problematic (10 < VIF) multicollinearity levels. This study determines the multicollinearity levels for L, D, Qi, T, Su, PI, and qv variables. Figure 4 illustrates the computation of R² to find the multicollinearity levels for each variable, along with the results in Table 3.

Fig. 3

Depiction of relationship by distance correlation coefficient method.

Fig. 4

Illustration of computation of multicollinearity levels for (a) L, (b) D, (c) Qi, (d) T, (e) Su, (f) PI, (g) qv.

Table 3 Multicollinearity levels of input variables.

Table 3 shows that the pile length (VIF = 24.80) and effective vertical stress (VIF = 23.56) have problematic multicollinearity. Still, the pile diameter (VIF = 1.81), the initial capacity of the pile (VIF = 1.67), time after the end of the drive (VIF = 1.22), undrained shear strength (VIF = 2.04), and plasticity index (VIF = 2.24) have weak multicollinearity in predicting the time-dependent bearing capacity of concrete piles.

Based on the multicollinearity analysis, the research hypothesis (R_H) statements have been drawn as follows:

• The problematic multicollinearity may affect the performance of the hybrid models, i.e., GA_DRVM, PSO_DRVM, DRVM, and Adam_LSTM.

• The time-dependent bearing capacity of concrete piles is affected by the initial capacity (Qi) and undrained shear strength (Su) variables.

The analysis of variance (ANOVA) and Z tests have been performed in this research to analyze the research hypothesis. ANOVA is a group of statistical models used to examine mean differences and the corresponding estimation processes⁶⁹. The results of the ANOVA test are summarized in Table 4. The statistical clause for the research hypothesis is (i) F is greater than F crit and (ii) the p-value is less than the significance level, i.e., 0.05. Table 4 presents that L (156.7955 > 3.878724, 3E-28 < 0.05), D (169.8884 > 3.878924, 5.59E-30 < 0.05), Qi (24.85387 > 3.878924, 1.16E-06 < 0.05), T (165.0666 > 3.878924, 2.39E-29 < 0.05), Su (134.1665 > 3.878924, 4.03E-25 < 0.05), PI (154.9642 > 3.878924, 5.29E-28 < 0.05), and qv (92.77432 > 3.878924, 7.05E-19 < 0.05) follows the statistical clause. Hence, the ANOVA analysis accepts the R_Hin estimating the time-dependent bearing capcity of the concrete piles. Furthermore, the Z test, "Any statistical test for which a normal distribution may be used to approximate the test statistic’s distribution under the null hypothesis," is performed for the research hypothesis⁷⁰, and the results are summarized in Table 5.

Table 4 Results of ANOVA test for concrete pile bearing capacity database.

Table 5 Results of Z test for concrete pile bearing capacity database.

The statistical clauses for the z test are (i) z critical two tail is higher than z critical one tail and (ii) p one and two tails are less than 0.05. Table 5 presents that each input variable accepts the research hypothesis in estimating the time-dependent bearing capacity of the concrete pile. The negative z value demonstrates that the data point is below the average. In addition, sensitivity analysis is performed to find the most significant input variable in estimating the time-dependent bearing capacity of the concrete pile. The cosine amplitude method (CAM) has computed the sensitivity of L, D, Qi, T, Su, PI, and qv input variables. Equation 2is the mathematical expression of the cosine amplitude method^71,72,73,74. Here, ${x}_{ik}$ and ${x}_{jk}$ is the input and output variables.

$$CAM=\frac{\sum_{k=1}^{m}{x}_{ik}{x}_{jk}}{\sqrt{\sum_{k=1}^{m}{x}_{ik}^{2}\sum_{k=1}^{m}{x}_{jk}^{2}}}$$

(2)

The values of CAM are close to 1 and 0, which presents the high and low sensitivity of variables. According to sensitivity analysis, the sequence of the input variables is Qi (= 0.9636) > Su (= 0.8404) > PI (= 0.7967) > D (= 0.7196) > qv (= 0.6907) > L (= 0.6639) > T (= 0.2798), as presented in Fig. 5. Hence, the Qi and Su variables are the most significant variables in estimating the time-dependent bearing capacity of the concrete piles.

Fig. 5

Illustration of sensitivity of input variables.

Soft computing models

The present research develops, trains, tests, and analyzes the models based on the relevance vector machine and recurrent neural network-long short-term memory. The relevance vector machine is advanced to support vector machines based on the kernel function⁷⁵. The most popular kernel functions for developing the relevance vector machine are the Gaussian, Linear, Laplacian, Polynomial, Exponential, and Sigmoid. A Relevance Vector Machine (RVM) is a machine learning method that derives parsimonious regression and probabilistic classification solutions by applying Bayesian inference. This study employs six single kernel function-based RVM models (SRVM).

Moreover, these SRVM models have been optimized by each genetic (GA) and particle swarm optimization (PSO) algorithm. Comparing the SRVM models UBC1 to UBC6, the higher performance model has been recognized, and the kernel has been selected to develop the dual kernel function-based RVM models (DRVM). Therefore, five DRVM models have been developed by five combinations of kernel functions. For example, the exponential kernel-based SRVM model outperformed Gaussian, linear, laplacian, sigmoid, and polynomial kernel function-based SRVM models. Therefore, the exponential kernel function will be used as kernel 1 (K1), and the following kernel combinations will be made: exponential + Gaussian, exponential + linear, exponential + laplacian, exponential + sigmoid, and exponential + polynomial. Again, each GA and PSO algorithm has optimized the developed DRVM models. Table 6 presents the model ID of 33 RVM models.

Table 6 Designation ID of RVM models.

On the other hand, recurrent neural networks, mainly long short-term memory (LSTM), have been employed in this research. The LSTM model is frequently utilized for sequential data tasks like speech recognition, natural language processing, and time series prediction^76,77,78. Figure 6 illustrates the computational diagram of the LSTM network.

Fig. 6

Computational diagram of the LSTM network.

where β is a nonlinear function, Q is a function at the input/update gate, P is a function at the forget gate, R is a function at the output gate, $\overline{S }$ is a cell vector, U is a hidden vector, S is a cell state vector, and y is output. The LSTM model has been optimized by Adam, RMSProp, and SGDM optimizers and is designated UBC34, UBC35, and UBC36, respectively. Table 7 presents the hyperparameter configuration of RVM and LSTM models.

Table 7 Hyperparameter configuration of RVM and LSTM models.

Results and discussion

This investigation employs, trains, tests, and analyzes thirty-six soft computing models (33 RVM + 3 LSTM) for estimating the time-dependent bearing capacity of the concrete piles. Hundred and twenty-six data points have trained (TRN) and tested (TST) the soft computing models, respectively. The performance of all models has been measured by implementing the R, RMSE, MAE, PI, RSR, VAF, NMBE, and MAPE metrics. Table 8shows the mathematical formulation with ideal values of each metric^{79,80,81,82,83,84,85,86,95,96}.

Table 8 Mathematical formulation and ideal values of performance metrics.

Simulation of Results

Each model’s training and testing performance is summarized in Appendix, Tables A and B. The comparison of single kernel function-based SRVM models, i.e., UBC1 (gaussian kernel-based), UBC2 (exponential kernel-based), UBC3 (linear kernel-based), UBC4 (laplacian kernel-based), UBC5 (sigmoid kernel-based), and UBC6 (polynomial kernel-based), showed that model UBC2 achieved R of 0.9874, PI of 1.92, NMBE of 14.5440, RSR of 0.1582, RMSE of 108.1510 kPa, MAE of 82.7299 kPa, MAPE of 0.1249%, and VAF of 97.50 in the TRN phase. Furthermore, model UBC2 outperformed the UBC1, UBC3, UBC4, UBC5, and UBC6 with an RSR of 0.2425, NMBE of 32.2802, PI of 1.85, VAF of 94.28, MAPE of 0.1750%, R of 0.9710, MAE of 106.2227 kPa, and RMSE of 151.5102 kPa, close to the ideal values. Hence, model UBC2, an exponential kernel-based SRVM model, is recognized as the best-performing model. Figure 7a illustrates the QQ’ plot between the actual and predicted time-dependent bearing capacity of the concrete pile using the UBC2 model. Furthermore, each SRVM model was optimized by a genetic algorithm, and the bearing capacity of the concrete pile was estimated. The comparison of GA_SRVM models, i.e., UBC7, UBC8, UBC9, UBC10, UBC11, and UBC12, revealed that the UBC7 model estimated the bearing capacity with higher performance in TRN (RSR = 0.0376, NMBE = 0.8199, PI = 1.99, VAF = 99.86, MAPE = 0.0352%, R = 0.9993, MAE = 17.5099 kPa, RMSE = 25.6781 kPa) and TST (RSR = 0.2344, NMBE = 30.1379, PI = 1.85, VAF = 94.60, MAPE = 0..1203%, R = 0.9727, MAE = 105.7009 kPa, RMSE = 146.3963 kPa) phase. Hence, model UBC7, a genetic algorithm optimized Gaussian kernel-based SRVM, is the best-performing model. Figure 7b depicts the regression plot between the actual and predicted time-dependent pile-bearing capacity using model UBC7. It has also been observed that the TST performance of UBC2 (R = 0.9710), UBC5 (R = 0.9645), and UBC6 (R = 0.9667) was decreased after optimizing, i.e., models UBC8 (R = 0.8907), UBC11 (R = 0.7366), and UBC12 (R = 0.7024), Therefore, it can be stated that the genetic algorithm enhances the performance of gaussian kernel-based SRVM model. Interestingly, no effect of the genetic algorithm was observed for the linear kernel-based SRVM model. The linear SRVM (R = 0.9645, RMSE = 167.5416 kPa, MAE = 122.4959 kPa) and GA-optimized linear SRVM (R = 0.9645, RMSE = 167.5416 kPa, MAE = 122.4959 kPa) models achieved equal performance in estimating the time-dependent bearing capacity of the concrete pile. Moreover, models UBC1 to UBC6 were optimized by PSO algorithm and developed UBC13, UBC14, UBC15, UBC16, UBC17, and UBC18 models. The performance comparison of UBC13, UBC14, UBC15, UBC16, UBC17, and UBC18 models demonstrates that model UBC13 gained RMSE of 25.6781 kPa, MAE of 17.5099 kPa, R of 0.9993, MAPE of 0.0352%, VAF of 99.86, PI of 1.99, NMBE of 0.8199, and RSR of 0.0376 in the TRN phase. Also, model UBC13 achieved higher TST performance (RSR = 0.2682, NMBE = 39.4620, PI = 1.82, VAF = 93.02, MAPE = 0.2074%, R = 0.9645, MAE = 122.4944 kPa, RMSE = 167.5186 kPa) in predicting the time-dependent bearing capacity of the concrete pile. Hence, model UBC13 was recognized as the best-performing model. Figure 7c demonstrates the 1:1 plot between the actual and predicted time-dependent bearing capacity of the concrete piles using model UBC13. Furthermore, comparing UBC7 (R = 0.9993) and UBC13 (R = 0.9993) reveals that both models achieved the same performance in the TRN phase. Still, the comparison of UBC7 and UBC13 presents that GA_SRVM model UBC7 estimated time-depended bearing capacity better than PSO_SRVM model UBC13. In addition, the comparison of GA and PSO-optimized exponential kernel-based SRVM models showed that the PSO-optimized SRVM model UBC14 (R = 0.9570) gained higher performance than GA-optimized SRVM model UBC8 (R = 0.8907). On the other hand, the performance of GA-optimized laplacian kernel-based SRVM model UBC10 (R = 0.9607) decreased for PSO-optimized laplacian kernel-based SRVM model UBC16 (R = 0.9487). The comparison of SRVM, GA_SRVM, and PSO_SRVM models demonstrates that the UBC2, UBC7, and UBC13 models have been recognized as the best-performing models.

Fig. 7

Illustration of QQ’ plot between actual and predicted time-dependent bearing capacity of the concrete pile using models (a) UBC2, (b) UBC7, (c) UBC13, (d) UBC21, (e) UBC24, (f) UBC31, (g) UBC35.

The performance comparison revealed that the exponential kernel function-based SRVM model UBC2 is recognized as the best architectural model. Therefore, the exponential kernel was selected as kernel 1 (K1) to develop the dual kernel function-based DRVM models. Thus, dual kernel function-based models, i.e., UBC19, UBC20, UBC21, UBC22, and UBC23, were developed, trained, tested, and analyzed. Models UBC19, UBC20, UBC21, UBC22, and UBC23 were configured with exponential + Gaussian, exponential + linear, exponential + laplacian, exponential + sigmoid, and exponential + polynomial combined kernel functions. The performance comparison reveals that the UBC21 outperformed the UBC19, UBC20, UBC22, and UBC23 models with an excellent TRN (RSR = 0.1640, NMBE = 15.6354, PI = 1.92, VAF = 97.31, MAPE = 0.1314%, R = 0.9865, MAE = 84.9091 kPa, RMSE = 112.1352 kPa) and TST (RSR = 0.2425, NMBE = 32.2724, PI = 1.85, VAF = 94.28, MAPE = 0.1765%, R = 0.9710, MAE = 106.5387 kPa, RMSE = 151.4918 kPa) performance. Figure 7d depicts the relationship between the actual and predicted time-dependent bearing capacity of the concrete pile using model UBC21. The comparison of non-optimized SRVM and DRVM models presents that the performance of model UBC1 (R-value from 0.9496 to 0.9657), UBC3 (R-value from 0.9645 to 0.9705), and UBC4 (R-value from 0.9602 to 0.9710) was increased by implementing secondary kernel function, called DRVM model UBC19, in the testing phase. Conversely, the performance of UBC5 (R-value from 0.9645 to 0.9480) and UBC6 (R-value from 0.9667 to 0.9586) was decreased by implementing a secondary kernel function for the DRVM model UBC23 and UBC23. Based on the significant improvement in the performance of the SRVM model after implementing the secondary kernel function, each DRVM model was optimized by GA (models UBC24, UBC25, UBC26, UBC27, UBC28) and PSO (models UBC29, UBC30, UBC31, UBC32, UBC33) optimization algorithm. The comparison of the UBC24, UBC25, UBC26, UBC27, and UBC28 presents that model UBC24 estimated time-dependent bearing capacity of the concrete pile with an RSR of 0.2449, NMBE of 32.9088, PI of 1.85, VAF of 94.34, MAPE of 0.1751%, R of 0.9703, MAE of 108.4635 kPa, and RMSE of 152.9782 kPa in the TST phase. Similarly, model UBC24 outperformed the UBC25, UBC26, UBC27, and UBC28 models with a PI of 1.93, NMBE of 11.9803, RSR of 0.1436, VAF of 97.94, MAPE of 0.1227%, R of 0.9896, MAE of 75.8815 kPa, and RMSE of 98.1573 kPa. Hence, the UBC24 model is the best-performing model. Figure 7e illustrates the QQ’ plot between the actual and predicted time-dependent bearing capacity of the concrete pile using model UBC24.

The comparison of SRVM models (UBC1, UBC3, UBC4, UBC5, UBC6) and GA_DRVM models (UBC24, UBC25, UBC26, UBC27, UBC28) showed that the performance of UBC1 (R-value from 0.9496 to 0.9703) and UBC3 (R-value from 0.9645 to 0.9702) was increased after implementing exponential kernel and optimized by genetic algorithm. In the case of models UBC4 (R-value from 0.9602 to 0.9546), UBC5 (R-value from 0.9645 to 0.9566), and UBC6 (R-value from 0.9667 to 0.9662), the performance was decreased by implementing the exponential kernel and optimized by GA algorithm. In addition, the comparison of GA_SRVM (UBC7, UBC9, UBC10, UBC11, UBC12) And GA_DRVM (UBC24, UBC25, UBC26, UBC27, UBC28) models revealed that the exponential kernel function enhanced the performance of UBC9 (R-value from 0.9645 to 0.9702), UBC11 (R-value from 0.7366 to 0.9566), and UBC12 (R-value from 0.7024 to 0.9662). Furthermore, the comparison of PSO-optimized DRVM models (UBC29, UBC30, UBC31, UBC32, UBC33) demonstrates that model UBC31 achieved better RSR (TRN = 0.1327, TST = 0.2423), NMBE (TRN = 10.2277, TST = 32.2032), PI (TRN = 1.94, TST = 1.85), VAF (TRN = 98.24, TST = 94.35), MAPE (TRN = 0.1122%, TST = 0.1915%), R (TRN = 0.9912, TST = 0.9713), MAE (TRN = 67.5939 kPa, TST = 109.6489 kPa), and RMSE (TRN = 90.6936 kPa, TST = 151.3293 kPa) in predicting the time-dependent bearing capacity of the concrete pile, close to the ideal values. Hence, model UBC31 is recognized as the best-performing model. Figure 7f shows a relationship between the actual and predicted time-dependent bearing capacity of the concrete pile using model UBC31. The performance of SRVM (UBC1, UBC3, UBC4, UBC5, UBC6), GA_SRVM (UBC7, UBC9, UBC10, UBC11, UBC12), PSO_SRVM (UBC13, UBC15, UBC16, UBC17, UBC18), GA_DRVM (UBC24, UBC25, UBC26, UBC27, UBC28), and PSO_DRVM (UBC29, UBC30, UBC31, UBC32, UBC33) was compared to analyze the effect of optimization technique. From the comparison, the following observations were made: (i) the performance of SRVM model UBC1 (R value from 0.9496 to 0.9612) was increased by implementing secondary kernel and PSO optimization algorithm, called model UBC29, (ii) the PSO_DRVM model UBC29 (R = 0.9612) achieved less performance than GA_SRVM model UBC7 (R = 0.9727) and PSO_SRVM model UBC13 (R = 0.9645), and GA_DRVM model UBC24 (R = 0.9703), (iii) PSO_DRVM model UBC31 (R = 0.9713) achieved higher performance than SRVM model UBC4 (R = 0.9602), GA_SRVM model UBC10 (R = 0.9607), PSO_SRVM model UBC16 (R = 0.9487), and GA_DRVM model UBC26 (R = 0.9546), (iv) again, PSO_DRVM model UBC32 (R = 0.9612) gained higher performance than GA_SRVM model UBC11 (R = 0.7366), PSO_SRVM model UBC17 (R = 0.7366), and GA_DRVM model UBC27 (R = 0.9566).

The performance comparison of RNN_LSTM models UBC34 (Adam-optimized), UBC35 (RMSProp-optimized), and UBC36 (SGDM-optimized) showed that model UBC35 predicted the time-dependent bearing capacity of the concrete piles with a TRN (RSR = 0.1187, NMBE = 8.1819, PI = 1.95, VAF = 98.59, MAPE = 0.1080%, R = 0.9929, MAE = 63.0781 kPa, RMSE = 81.1175 kPa) and TST (RSR = 0.3724, NMBE = 76.0986, PI = 1.69, VAF = 87.42, MAPE = 0.3840%, R = 0.9367, MAE = 170.8586 kPa, RMSE = 232.6279 kPa) performance, better than models UBC34 and UBC36. Model UBC36, optimized by the SGDM algorithm, was performed very poorly. Figure 7g illustrates the regression plot between the actual and predicted time-dependent bearing capacity of the concrete pile using the UBC35 model.

Thus, seven best-performing models, UBC2, UBC7, UBC13, UBC21, UBC24, UBC31, and UBC35, have been obtained in estimating the time-dependent bearing capacity of the concrete pile. The summary of training and testing performance of the seven best-performing models has been mapped in Table 9, along with error histograms, as shown in Fig. 8.

Table 9 Performance summary of the best-performing models.

Fig. 8

Illustration of error histogram of the best-performing models in (a) TRN and (b) TST phase.

Regression error characteristics curve

Receiver operating characteristic (ROC) curves help compare and visualize classification outcomes. Regression Error Characteristic (REC) curves generalize ROC curves to regression. REC curves display the percentage of predicted points falling inside the tolerance on the y-axis and the error tolerance on the x-axis^87,88. The regression curve estimates the cumulative distribution function of the error. An estimate of the skewed predicted error is called the area-over-the-curve (AOC). The minimum AOC value presents the most accurate prediction. Figure 9a, b demonstrates the REC curve of the seven best-performing models in the TRN and TST phases. Also, the AOC of these best-performing models has been calculated and summarized in Table 10. Table 10 shows that the UBC7 outperformed the UBC2, UBC13, UBC21, UBC24, UBC31, and UBC35 and was identified as the optimal performance model for estimating the time-dependent bearing capacity of the concrete piles.

Fig. 9

Illustration of comparison of REC curve for seven best-performing models in (a) training and (b) testing phase.

Table 10 AOC results.

Score analysis

Score analysis is a conventional statistical approach for finding the optimum performance model. This analysis ranks the models’ performance in the TRN and TST phases. The maximum score for each performance metric is defined by the number of models used, i.e., N (present research has seven best-performing models). Each model’s score for each performance metric is added to obtain the total score in each TRN and TST phase. The grand score is determined by adding the total score of the TRN and TST phase for each model. Table 11 presents the total score of each model in each phase, along with the grand score, shown in Fig. 10.

Table 11 Score of RVM and LSTM models in TRN and TST phase.

Fig. 10

Illustration of the grand score of the best-performing models.

Table 11 and Fig. 10 present that the UBC7 model scored 48 and 56 in the TRN and TST phases, respectively. It was observed that model UBC13 obtained a total score of 48, equal to the UBC7 model, in the TRN phase. Still, model UBC13 did not score better than UBC7 in the TST phase. Analyzing the grand score of the best-performing models, it was observed that the model UBC7 obtained a grand score of 104, followed by UBC31, UBC13, and UBC24 models. Model UBC21 performed poorly in estimating the time-dependent bearing capacity of the concrete pile. Hence, the UBC7 model is the optimal performance model.

Wilcoxon test

A non-parametric statistical hypothesis test called the Wilcoxon signed-rank test is used to assess if there are any significant differences between two related groups. It is applied when the homogeneity of variances assumption fails, or the data are not normally distributed. Table 12 presents the Wilcoxon test results for the best-performing models in estimating the time-dependent bearing capacity of the concrete pile. It can be noted that the UBC7 model estimated the time-dependent bearing capacity with (i) a lower confidence level (LCL) of 0.1088, (ii) an upper confidence level (UCL) of 0.1646, and (iii) a confidence level (CL) difference of 0.0558 in the TRN phase, close to the CL of the actual model. Similarly, model UBC7 obtained an LCL of 0.0689, UCL of 0.1661, and CL of 0.0972 in the TST phase, close to the CL of the actual model. Therefore, model UBC7 has been recognized as the optimal performance model in estimating the time-dependent bearing capacity of the concrete pile.

Table 12 Wilcoxon test results for the best-performing models.

Uncertainty analysis (UA)

Any predictive model’s trustworthiness must be evaluated to accurately and reliably estimate predictive output^89,90. This study uses uncertainty analysis to assess the quantitative prediction of the best-performing models’ error in calculating the concrete piles’ time-dependent bearing capacity. The UA was performed for the training and testing phase, which consists of 100 and 26 data points of the time-dependent bearing capacity of the concrete piles. For the UA, the absolute error, mean of error, standard deviation (SD), the margin of error (MOE) at 95% confidence level (ME), the width of confidence bound (WCB), standard error, upper bound (UB), and lower bound (LB) parameters was computed for each best-performing model. The width of the confidence bound (WCB) is the difference between the upper and lower bound. The results of UA obtained in the TRN and TST phases for the best-performing models are summarized in Table 13 and the graphical presentation, as shown in Fig. 11.

Table 13 Uncertainty analysis results for the best-performing models.

Fig. 11

Illustration of uncertainty bandwidth of the best-performing models in (a) TRN and (b) TST phase.

Figure 11 demonstrates that model UBC7 estimated the time-dependent bearing capacity of the concrete pile with the lowest uncertainty bandwidth. Still, model UBC35 has the highest bandwidth. Therefore, the UBC7 model has been recognized as the optimal performance model.

Taylor plot

The Taylor plot or diagram is a graphical method for visualizing multiple variables or databases^91,92,93. In machine learning, the Taylor plot compares the actual and predicted databases. This study compares the seven best-performing models in the TRN and TST phases of estimating the time-dependent bearing capacity of the concrete pile, as shown in Fig. 12. From Fig. 12, it is noted that the UBC2, UBC7, UBC13, UBC21, UBC24, UBC31, and UBC35 models estimated the time-dependent bearing capacity of concrete pile with a standard deviation of 678.8375 kPa, 687.8094 kPa, 687.8094 kPa, 678.117 kPa, 679.9016 kPa, 681.0256 kPa, and 684.9422 kPa, respectively, in the TRN phase. It can be seen that models UBC7 and UBC13 estimated the time-dependent bearing capacity of concrete piles with the same standard deviation, close to the actual, i.e., 687.0062. In the testing phase, the UBC2, UBC7, UBC13, UBC21, UBC24, UBC31, and UBC35 models computed the time-dependent bearing capacity of the concrete pile with a standard deviation of 621.2001 kPa, 623.4622 kPa, 618.9903 kPa, 621.2775 kPa, 622.9781 kPa, 616.8252 kPa, 560.8191 kPa, respectively. The comparison of the standard deviation reveals that the UBC7 model estimated the time-dependent bearing capacity of the concrete pile with a standard deviation of 623.4622 kPa, close to the actual, i.e., 637.0381 kPa. Hence, model UBC7 is the optimal performance model in estimating the time-dependent bearing capacity of concrete piles.

Fig. 12

Illustration of Taylor plot for the best-performing model in (a) TRN and (b) TST phase.

Accuracy metrics

The accuracy (AC) metric is another statistical analysis performed to compare the soft computing models and identify the optimal performance model. In this study, RMSE_AC ($=1-(RMSE*{10}^{-3})$), MAE_AC ($=1-(MAE*{10}^{-3})$), R_AC ($=R*100$), MAPE_AC ($=1-(MAPE*{10}^{-3})$), VAF_AC (= VAF), PI_AC ($=\left(PI/2\right)*100$), NMBE_AC ($=1-(NMBE*{10}^{-3}$), and RSR_AC ($=\left(1-RSR\right)*100$) metrics computed the accuracy of the best-performing model for training and testing phase, as shown in Fig. 13. The accuracy comparison presented in Fig. 13 reveals that the UBC7 model achieved higher TRN (RMSE_AC = 97.43%, MAE_AC = 98.25%, R_AC = 99.93%, MAPE_AC = 100.00%, PI_AC = 99.52%, VAF_AC = 99.86%, NMBE_AC = 99.92%, RSR_AC = 96.24%) and TST (RMSE_AC = 85.36%, MAE_AC = 89.43%, R_AC = 97.27%, MAPE_AC = 99.99%, PI_AC = 92.68%, VAF_AC = 94.60%, NMBE_AC = 96.99%, RSR_AC = 76.56%) accuracies in estimating the time-dependent bearing capacity of the concrete pile. It was also noted that model UBC35 achieved poor accuracy in testing (RMSE_AC = 76.74%, MAE_AC = 82.91%, R_AC = 93.67%, MAPE_AC = 99.96%, PI_AC = 84.53%, VAF_AC = 87.42%, NMBE_AC = 92.39%, RSR_AC = 62.76%) phase. Hence, model UBC7 has been recognized as the optimal performance model for predicting the time-dependent bearing capacity of the concrete piles.

Fig. 13

Illustration of comparison of accuracy metrics of the best-performing models in (a) TRN and (b) TST phase.

External validation

An external validation has been conducted to check the generalizability of the best-performing models. A model’s generalizability is evaluated, and external validation is carried out to make sure the model isn’t just overfitting the training set. Finding the most accurate model for estimating the time-dependent bearing capacity of the concrete piles is made easier by the findings of external validation. Accuracy is the capacity of the model to correctly identify patients as having or not having the desired outcome. External validation checks for overfitting and guarantees that models are reliable. When a model is too tightly suited to the training data and does not generalize effectively to new data, it is said to be overfit. By contrasting the model’s performance on the training data with the test data, external validation can help to spot overfitting. The Golbraikh and Tropsha⁹⁴ theory, which was proposed, is an accurate model in this investigation. Table 14 provides a summary of the theory’s various mathematical expression-related aspects.

Table 14 Mathematical expression and condition of validation factors.

Where ${d}_{i}$ denotes the experimental time-dependent bearing capacity and ${y}_{i}$ denotes the time-dependent bearing capacity, $k$ and $k{\prime}$ represent the slopes of the predicted versus actual time-dependent bearing capacity and actual versus predicted time-dependent bearing capacity concerning the origin. ${R}_{o}^{2}$ and ${R{\prime}}_{o}^{2}$ denotes the coefficients of determination of the predicted versus actual time-dependent bearing capacity and actual versus predicted time-dependent bearing capacity. $m$ and $n$ represent the factors for estimating the predictive power of the proposed models. The external validation results are presented in Table 15 for the best-performing models in the TRN and TST phases. The comparison presented in Table 15 demonstrates that model UBC7 showed superiority in both the training and testing phases of estimating the time-dependent bearing capacity of the concrete piles.

Table 15 Results of generalizability for the best-performing models.

Internal validation

The present investigation was carried out using an HP pavilion G6 machine, configured with Intel Core i3-2350 M (2^nd gen), 2.3Ghz processor, 4 GB RAM, 240SSD, Intel HD 3000GPU, and AMD Radeon 7450 M. The internal validation was carried out by comparing the computational cost of the best-performing models, i.e., UBC2, UBC7, UBC13, UBC21, UBC24, UBC31, and UBC35. The best-performing models were configured with a k-fold of 5. Again, the best-performance models were configured with a k-fold of 10, and computational cost was measured. Figure 14 shows the computational cost comparisons. The comparison showed that model UBC2 (5 k = 0.3734 s, 10 k = 0.5624 s), UBC7 (5 k = 0.8460, 10 k = 0.9714), UBC13 (5 k = 0.4914, 10 k = 0.6092), UBC21 (5 k = 0.4173, 10 k = 0.4954), UBC24 (5 k = 0.2324, 10 k = 0.2389), UBC31 (5 k = 0.3065, 10 k = 0.3468), and UBC35 (5 k = 0.1745, 10 k = 0.3051) achieved highest computational cost in case of 15 k-fold. It can be noted that model UBC7 showed superiority with the computational cost of 0.8460 (for 5 k-fold) and 0.9714 (for 10 k-fold). Similarly, model UBC7 estimated the time-dependent bearing capacity of the concrete pile with the computational cost of 0.0134 (for 5 k-fold) and 0.0230 (for 10 k-fold) in the testing phase, showing superiority. Hence, the UBC7 model is the optimal performance model.

Fig. 14

Illustration of comparison of the computational cost of the best-performing models in (a) TRN and (b) TST phase.

Objective function (OBJF) criterion

The objective function (OBJF) criterion is another statistical method for finding the optimal performance model in computational mechanics. The OBJF criterion is calculated using (i) MAE_TR in the TRN phase, (ii) MAE_TS in the TST phase, (iii) ${R}_{TR}^{2}$ in the TRN phase, (iv) ${R}_{TS}^{2}$ in the TST phase, (v) total data points (TP = 126, in this work), (vi) training data points (TP_TR = 100, in this work), and (vii) testing data points (TP_TS = 26, in this work). The mathematical formula for OBJF is:

$$OBJF = \left( {\frac{{TP_{TR} - TP_{TS} }}{TP}} \right)*\left( {\frac{{MAE_{TR} }}{{R_{TR}^{2} }}} \right) + \left( {\frac{{2TP_{TS} }}{TP}} \right)*\left( {\frac{{MAE_{TS} }}{{R_{TS}^{2} }}} \right)$$

(3)

The lowest value of the OBJF represents the accurate estimation. Figure 15 illustrates the results of the OBJF criterion for the seven best-performing models in estimating the time-dependent bearing capacity of the concrete pile. The comparison of OBJF reveals that model UBC7 obtained the lowest OBJF, i.e., 0.0160, followed by model UBC13. It is noted that model UBC35 has an OBJF of 0.0322, higher among the UBC2, UBC13, UBC21, UBC24, and UBC31, presenting poor prediction. Hence, the UBC7 model is the optimal performance model.

Fig. 15

Illustration of comparison of OBJF values for the best-performing models.

Anderson darling test

A statistical test called the Anderson–Darling test determines whether a given sample of data is representative of a particular distribution, most commonly the normal distribution. The present work uses the Minitab statistical tool to perform the Anderson–Darling (AD) test for seven best-performing models. Figure 16a–h illustrates the AD test results for models UBC2, UBC7, UBC13, UBC21, UBC24, UBC31, and UBC35, along with the results summary in Table 16.

Fig. 16

Illustration of AD test results for models (a) actual (b) UBC2, (c) UBC7, (d) UBC13, (e) UBC21, (f) UBC24, (g) UBC31, and (h) UBC35.

Table 16 AD test results for the best-performing models.

Table 16 presents that model UBC7 estimated the time-dependent bearing capacity of the concrete pile with an AD value of 9.435, close to 9.336, showing superiority over other best-performing models. Also, the standard deviation of the estimated time-dependent bearing capacity using model UBC7 is 0.9978, close to the standard deviation of actual data, i.e., 1. Hence, the UBC7 model is the optimal performance model.

Discussion on results

This investigation introduces an optimal performance model for estimating the time-dependent bearing capacity of the concrete pile. To achieve this, an extensive comparison was drawn between the 36 models (33 RVM + 3 LSTM) by analyzing the RMSE, MAE, R, PI, RSR, VAF, NMBE, and MAPE performance metrics. The performance comparison revealed that the genetic algorithm-optimized Gaussian kernel-based RVM model UBC7 achieved higher testing performance, i.e., MAPE = 0.1203%, NMBE = 30.1379, VAF = 94.60, RSR = 0.2344, PI = 1.85, MAE = 105.7009 kPa, RMSE = 146.3963 kPa, and R = 0.9727. Moreover, the UBC7 model achieved an overall score of 104 (TRN = 48, TST = 56) in estimating the time-dependent bearing capacity of the concrete piles. Wilcoxon test also demonstrated the superiority of the UBC7 model with a confidence level of 94.95% in the testing phase. Also, the uncertainty analysis revealed that model UBC7 achieved first rank in both phases. The comparison of accuracy metrics presented that model UBC7 predicted time-dependent bearing capacity of the concrete piles with the RMSE_AC = 76.74%, MAE_AC = 82.91%, R_AC = 93.67%, MAPE_AC = 99.96%, PI_AC = 84.53%, VAF_AC = 87.42%, NMBE_AC = 92.39%, and RSR_AC = 62.76% in the testing phase. The analysis of the Taylor plot demonstrated that the UBC7 model predicted bearing capacity with a standard deviation of 623.4622 kPa, close to the actual, i.e., 637.0387 kPa. Hence, the UBC7 model is the optimal performance model. Still, the performance indexes, i.e., a20, agreement, and scatter, were implemented to analyze the reliability of the UBC7 model. The mathematical formulation of indexes is as follows:

$$a20 index = \frac{m20}{H}$$

(4)

$$IOA = 1 - \frac{{\mathop \sum \nolimits_{i = 1}^{n} \left( {\omega - \alpha } \right)}}{{2\mathop \sum \nolimits_{i = 1}^{n} \left( {\alpha - \beta } \right)}}$$

(5)

$$IOS = \frac{RMSE}{{Avg. of Actual Values}}$$

(6)

where m20 is the ratio of the experimental to the predicted value (0.8 to 1.2). The ideal values of the a20 index, index of agreement (IOA), and index of scatter (IOS) are 100, 1, and 0, respectively. Figure 17a–c illustrates the comparison of a20, IOA, and IOS results for the best-performing models, i.e., UBC2, UBC7, UBC13, UBC21, UBC24, UBC31, and UBC35. Figure 17a presents that model UBC7 obtained the a20 index of 99.00 (in the TRN phase) and 73.85 (in the TST phase), comparatively higher than other best-performing models. On the other side, model UBC35 achieved a20 index of 94.00 in the training phase but poorly performed in the testing phase with a20 index of 62.31 (poorer than models UBC2, UBC13, UBC21, and UBC24). Figure 17b illustrates that model UBC7 estimated the time-dependent bearing capacity of the concrete piles with IOA of 0.9811 (in the TRN phase) and 0.8867 (in the TST phase), close to the ideal values. Still, model UBC35 obtained the lowest IOA in the testing phase, i.e., 0.7962. The comparison of IOS presented in Fig. 17c demonstrates that model UBC7 gained an IOS of 0.0319 (in the TRN phase) and 0.2059 (in the TST phase), which is close to the ideal value. Hence, the UBC7 model is the most reliable in estimating the time-dependent bearing capacity of the concrete piles.

Fig. 17

Illustration of comparison of the best-performing models in terms of (a) a20 index, (b) agreement index, and (c) scatter index.

Furthermore, the overfitting of the best-performing models has been calculated and compared to find the effect of the multicollinearity on the model’s performance. The overfitting is a ratio of test RMSE to train RMSE. Figure 18 shows the comparison of overfitting of the best-performing models. Figure 18 presents that model UBC13 predicted the time-dependent bearing capacity of the concrete piles with the overfitting of 6.52, followed by model UBC7, i.e., 5.70. It is also observed that model UBC21 achieved an overfitting of 1.35, close to the best fit. Still, model UBC21 did not perform well because of the database multicollinearity. In addition, it can be stated that the model UBC21 was configured with dual kernels, i.e., exponential kernel (K1) and laplacian kernel (K4). The implementation of two kernel functions generated the structural multicollinearity. Therefore, model UBC21 did not estimate the time-dependent bearing capacity of the concrete piles. In addition, the combined effect of structural and database multicollinearity was observed for the LSTM model UBC36, optimized by the SGDM optimizer. Still, model UBC35 predicted the time-dependent bearing capacity of the concrete piles with an overfitting of 2.87.

Fig. 18

Illustration of comparison of overfitting of the best-performing models.

Finally, this research introduces an optimal performance model UBC7 (configured by Gaussian kernel and optimized by genetic algorithm) to estimate the time-dependent bearing capacity of the concrete piles. This research uses L, D, Qi, T, Su, PI, and qv parameters for the first time to develop the models. Therefore, the performance of the UBC7 model has been compared to the models available in the literature study, as given in Table 17.

Table 17 Comparison of model UBC7 and available models in the literature.

Analysis of results

The investigation introduces the genetic algorithm-optimized Gaussian kernel-based RVM model UBC7 as an optimal performance model in estimating the time-dependent bearing capacity of the concrete piles. This investigation was carried out using a bearing capacity of 126 concrete piles. The database consists of L, D, Qi, T, Su, PI, and qv parameters of concrete pile. To validate and check the estimation capability of the UBC7 model, the regression analysis has been performed for 126 data points using the following parameters, given in Table 18.

Table 18 Details of parameters for results analysis.

Pearson’s product-moment correlation coefficient method has calculated the linear relationship between the L, D, Qi, T, Su, PI, qv, and Qu to analyze the results obtained by the UBC7 model, as shown in Fig. 19.

Fig. 19

Illustration of linear relationship using correlation coefficient method.

Figure 19 illustrates that (i) L (= −0.0489), D (= −0.0619), T (= −0.0143), and qv (= 0.0226) have no relationship with Qu, (ii) Su (= 0.5609) has a moderate relationship with Qu, and (iii) Qi (= 0.9129) has a very strong relationship with Qu. Here, a negative correlation represents that one variable increases and another decreases. It can be seen that a few pairs of variables have positive and negative gradients, i.e., T with Qu, Su with Qu, and qv to Qu. The bifurcation of the T, Su, and qv revealed that the gradient of the Qu changes with the range of T, Su, and qv. The relationships between L, D, Qi T, Su, PI, qv, and Qu have been drawn using the variables given in Table 18 and presented in Fig. 20. From Fig. 20, the following points have been observed and mapped with Fig. 19.

• The time-dependent bearing capacity of the concrete pile (Qu) increases with pile length (L); refer to Fig. 20a–c.

• The time-dependent bearing capacity of the concrete pile (Qu) is directly proportional to the initial capacity of the pile (Qi); refer to Fig. 20d–f.

• The Qu increases with driven time up to 14 days; refer to Fig. 20g, h. Further, the Qu starts decreasing.; refer to Fig. 20i.

• The undrained shear strength (Su) significantly affects the time-dependent bearing capacity of the concrete pile (Qu); refer to Fig. 20j–l.

• The effective vertical stress also significantly affects the Qu; refer to Fig. 20m–o.

Fig. 20

Illustration of analyzed results for (a–c) pile length, (d–f) initial capacity of the pile, (g–i) time after the end of the drive, (j–l) undrained shear strength, (m–o) effective vertical stress.

The regression method demonstrates that model UBC7 has predicted the time-dependent bearing capacity of the concrete pile with higher accuracy, and the predicted values follow the pattern of the actual values. Hence, the UBC7 model is the optimal performance model for estimating the time-dependent bearing capacity of the concrete pile (Qu).

Summary and conclusions

The estimation of the time-dependent bearing capacity of the concrete pile using in situ and laboratory procedures is time-consuming and cumbersome. Therefore, the current investigation introduces an optimal performance soft computing model to estimate the time-dependent bearing capacity of the concrete piles. To achieve this, thirty-six soft computing models (33 RVM and 3 LSTM) were developed, trained, tested, and analyzed using 126 concrete pile results. The database consists of L, D, Qi, T, Su, Pi, qv, and Qu of the concrete piles. The R, RMSE, MAE, PI, RSR, VAF, NMBE, and MAPE metrics computed the performance of each model. Based on the overall analysis, the following conclusions have been mapped:

• Comparison of SRVM and DRVM Model: Implementing a secondary kernel, i.e., exponential, enhanced the performance of Gaussian, Linear, and Laplacian kernel-based SRVM models. Still, the performance of the sigmoid and polynomial kernel-based SRVM models was decreased because (i) the sigmoid kernel function is less suitable for regression problems, (ii) the polynomial kernel function is susceptible to the input variables and the input variables used in this work has problematic multicollinearity.

• Effect of GA and PSO Algorithms: The results of optimized models reveal that the optimization algorithm enhances the performance of models in estimating the time-dependent bearing capacity of the concrete pile. The Gaussian kernel-based SRVM model performs better if a genetic algorithm optimizes it. On the other hand, the PSO-optimized exponential kernel-based SRVM model gains higher performance than the GA-optimized exponential kernel-based SRVM model. In the case of DRVM, (i) the GA-optimized exponential + Gaussian DRVM and GA-optimized exponential + linear DRVM models achieved higher performance. The complete analysis shows that the genetic algorithm-based model, i.e., UBC7, outperformed the other models.

• Effect of Adam, RMSProp, & SGDM Optimizer: The comparison of Adam, RMSProp, and SGDM optimized LSTM models showed that the RMSProp outperformed the Adam and SGDM optimized LSTM models with a R of 0.9367, RMSE of 232.6279 kPa, and PI of 1.69. The RMSProp is highly stable in the training phase and can be verified by R obtained in training, i.e., 0.9929. Also, it is very effective for non-stationary objectives. Hence, model UBC35 was identified as the best-performing model in predicting the time-dependent bearing capacity of the concrete piles.

• Impact of Database Multicollinearity: The VIF analysis revealed that pile length (L) and effective vertical stress (qv) have problematic multicollinearity. Still, the variables D, Qi, T, Su, and PI have weak multicollinearity. A significant effect of database multicollinearity has been observed for models UBC11, UBC12, UBC17, UBC18, UBC22, UBC26, UBC27, UBC32, and UBC35. These models estimated the time-dependent bearing capacity of the concrete piles with a residual error of more than 200 kPa.

• Impact of Structural Multicollinearity: The present investigation implements dual kernels to develop DRVM models, creating the structural multicollinearity. Also, optimizing these DRVM models develops structural multicollinearity. In this study, the performance of models UBC11 (GA-optimized sigmoid kernel-based SRVM), UBC12 (GA-optimized polynomial kerne-based SRVM), UBC17 (PSO-optimized sigmoid kernel-based SRVM), UBC18 (PSO-optimized polynomial kerne-based SRVM), UBC26 (GA-optimized exponential + laplacian kernel-based DRVM), UBC27 (GA-optimized exponential + sigmoid kernel-based DRVM), UBC29 (PSO-optimized exponential + gaussian kernel-based DRVM), UBC32 (PSO-optimized exponential + sigmoid kernel-based DRVM), UBC36 (SGDM-optimized LSTM) were highly influenced by the structural multicollinearity.

• Optimal Performance Model: This work concluded that the UBC7 (GA-optimized Gaussian kernel function-based SRVM) model was recognized as an optimal performance model with RSR of 0.2344, NMBE of 30.1379, PI of 1.85, VAF of 94.60, MAPE of 0.1203%, R of 0.9727, RMSE of 146.3963 kPa, and MAE of 105.7009 kPa. In addition, the score analysis (TRN = 48, TST = 56), AOC (TRN = 4.21E-05, TST = 1.22E-03), Wilcoxon test (TRN = 95.02% CL, TST = 94.95% CL), uncertainty analysis (rank 1 in both phase), accuracy metrics (R_AC = 97.27%, RMSE_AC = 85.36%, MAE_AC = 89.43%, PI_AC = 92.68%), generalizability test (m = −0.06, n = −0.05), cost computation (TRN = 0.8460, TST = 0.0134), OBJF criterion (= 0.0160), and AD test (= 9.435) confirmed the superiority of the UBC7 over the rest of the RVM and LSTM models in estimating the time-dependent bearing capacity of the concrete piles.

To sum up, this investigation introduces an optimal performance model UBC7 to estimate the time-dependent bearing capacity of the concrete piles. The prediction capability of the GA-optimized Gaussian kernel-based model UBC7 demonstrates that model UBC7 may be used to solve other geotechnical problems. The present work uses only 126 data points, which is the limitation of this research. This study may be extended using a large database. In addition, more studies and analyses are required to determine the effect of database and structural multicollinearities on the performance of hybrid models. To achieve this, the RVM and LSTM models may be optimized by different metaheuristic algorithms, such as swarm-based, nature-inspired, biogeographic-stimulated, evolutionary, and physics-based algorithms.