Assessment of novel nature-inspired fuzzy models for predicting long contraction scouring and related uncertainties

Abstract

The scouring phenomenon is one of the major problems experienced in hydraulic engineering. In this study, an adaptive neuro-fuzzy inference system is hybridized with several evolutionary approaches, including the ant colony optimization, genetic algorithm, teaching-learning-based optimization, biogeographical-based optimization, and invasive weed optimization for estimating the long contraction scour depth. The proposed hybrid models are built using non-dimensional information collected from previous studies. The proposed hybrid intelligent models are evaluated using several statistical performance metrics and graphical presentations. Besides, the uncertainty of models, variables, and data are inspected. Based on the achieved modeling results, adaptive neuro-fuzzy inference system–biogeographic based optimization (ANFIS-BBO) provides superior prediction accuracy compared to others, with a maximum correlation coefficient (R_test = 0.923) and minimum root mean square error value (RMSE_test = 0.0193). Thus, the proposed ANFIS-BBO is a capable cost-effective method for predicting long contraction scouring, thus, contributing to the base knowledge of hydraulic structure sustainability.

Graphical abstract

Keywords

long contraction scour / prediction / uncertainty / ANFIS model / meta-heuristic algorithm

1 Introduction

The scouring phenomenon occurs due to the high flow intensity of water and the creation of topical bubbling flow downstream of the hydraulic structures. Contraction scouring is a crucial concern in hydraulic engineering, which occurs by increasing the flow velocity due to the reduction of the river width influenced by hydraulic structures, i.e., bridge abutments and piles. The accurate estimation of contraction scouring plays a vital role in the stability rate of hydraulic structures. Various physical models and field investigations have been conducted to approximate the long contraction scouring [1–11]. Empirical relationships were developed based on the linear regression associated with uncertainties. In the last two decades, computer models have increased rapidly, which can predict more accurately compared to empirical relations [12]. Different numerical models such as Hydraulic engineering circular No. 18 (HEC-18), Sediment erosion rate flume (SERF), and Fast 3D were explored in this subject [13–16]. However, soft computing (SC) models are alternative approaches for contraction scour depth estimation [12]. SC models are closed-form approaches, which map the relations between inputs and outputs to solve complex problems [17–19]. Since 2005, the rate of using these approaches for scour depth estimation has increased significantly [20–24]. Different variety of methods are used to assess the scour depth, such as Artificial neural networks(ANN), adaptive neural-fuzzy inference system (ANFIS), Support vector machine (SVM), Group method of data handling(GMDH), etc [20–29]. Among several existing SC models for scouring simulation, the ANFIS model, which was developed by [30], is one of the efficient models in the field of hydraulic engineering. This model uses the practical benefits of fuzzy logic and neural network (NN) to find the best solution in nonlinear problems. Besides, its capacity to solve the uncertainty associated with stochastic and nonlinear problems is remarkable. Several attempts have been made to apply the ANFIS model for assessing local scour depth [26,31–39]. The main limitation associated with the ANFIS model is the hyper parameter optimization problem [40]. Hence, several attempts have been made to tackle this limitation using different nature-inspired optimization algorithms. Sharafati et al. [41] tuned ANFIS parameters for increasing the accuracy of standalone ANFIS using particle swarm optimization, genetic algorithm (GA), differential evolution (DE), and ant colony optimization (ACO). They computed the local scouring in downstream of sluice gates and concluded that new tuned ANFIS could generate more precise results in the assessment of scour depth.

Few surveys have been conducted to determine the scour depth in the long contraction. Raikar et al. [42] provided NN and GA models using laboratory data to assess the depth of contraction scouring. The results revealed that the GA provides superior estimation performance than the NN. Najafzadeh et al. [43] estimated contraction scouring using laboratory data from previous studies [4,44–46]. Their findings show that the support vector machine technique and ANFIS have similar performance, although the support vector machine model performs slightly better in training and testing phases. Najafzadeh et al. [47] investigated the abilities of evolutionary polynomial regression, model tree (MT), and gene-expression programming in estimating contraction scouring. They concluded that MT was superior to others in determining scouring depth. It is worth mentioning that the hybridized ANFIS with several natural optimization approaches attained a variable capacity for solving the regression problems (prediction problems) [48]. Hence, the hybridization of the ANFIS model with different optimization algorithms can give insightful results irrespective of the simulated problem. The primary aim of the current investigation was to understand the capability of different types of hybrid ANFIS models in assessing the long contraction scour depth. Similarly, this study aims to improve the efficiency of the standalone ANFIS model using meta-heuristic algorithms in estimating a long contraction scour depth. Different meta-heuristic algorithms are hybridized with the ANFIS model to attain a reliable and robust predictive model. The main objective of the proposed models is to provide a generalized predictive model that can mimic the actual mechanism between the input parameters and the targeted scour depth.

2 Analysis of main effective parameters on the contraction scouring depth

The experimental studies indicated that the most effective parameters (Eq. (1)) in the contraction-scouring process are sedimentation specifications, flow hydraulic conditions, and geometrical characteristics of the flume or canal [4,44–46].

(1)d_{\mathrm{s}}=f(d_{\mathrm{50}}\cdot U_{\mathrm{1}}\cdot U_{\mathrm{C}}\cdot v\cdot g\cdot {\sigma }_{\mathrm{g}}\cdot h_{\mathrm{1}}\cdot b_{\mathrm{1}}\cdot b_{\mathrm{2}}\cdot {\rho }_{\mathrm{w}}\cdot {\rho }_{\mathrm{s}}),

where ρ_s and ρ_w denote the density of sediments and water density. Flow width at restricted section and flow width at the upstream section is indicated with b₂ and b₁ respectively. h₁ is the water depth at the upstream, σ_g is the geometric standard deviation of sediment particles, υ is flow viscosity, U_C is critical velocity and U₁ is the velocity at the upstream section. d₅₀ and d_s are the mean sediment size and contraction scouring depth, respectively. Figure 1 shows the schematic view of a long contracted channel at equilibrium scour condition.

Dimensional analysis was performed to investigate the dimensionless variables involved in the depth of contraction scouring. By applying the Buckingham π theory, the relationship between the depth of contraction scouring and the affecting parameters is expressed as follows: [43]

(2){\displaystyle \frac{d_{\mathrm{s}}}{b_{\mathrm{1}}}}=f\left({\displaystyle \frac{d_{\mathrm{50}}}{b_{\mathrm{1}}}}\cdot {\displaystyle \frac{h_{\mathrm{1}}}{b_{\mathrm{1}}}}\cdot {\displaystyle \frac{b_{\mathrm{2}}}{b_{\mathrm{1}}}}\cdot {\sigma }_{\mathrm{g}}\cdot {\displaystyle \frac{U_{\mathrm{1}}}{U_{\mathrm{C}}}}\cdot {\displaystyle \frac{{\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{w}}}}\cdot F{r}_{\mathrm{0}}\right).

In Eq. (2), Fr₀ is the densitometric Froude number which can be calculated using the following expression:

(3)F{r}_{\mathrm{0}}={\displaystyle \frac{U_{\mathrm{1}}}{\sqrt{g\left({\displaystyle \frac{{\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{w}}}}-\mathrm{1}\right)d_{\mathrm{50}}}}}.

3 Methodology

3.1 Adaptive neuro-fuzzy inference system

Mapping a nonlinear relationship between inputs and outputs is a major ANFIS capability, which is given by membership functions (MFs). Several forms of MF exist such as trapezoidal, triangular, and Gaussian membership (GMF) function. The GMF is determined as follows:

(4)f(x,\sigma ,c)={\mathrm{e}}^{\frac{-{(x-c)}^{\mathrm{2}}}{\mathrm{2}{\sigma }^{\mathrm{2}}}},

where σ and c are the GMF parameters.

ANFIS includes the premise and consequent sections. Besides, it comprises five layers: Fuzzification, rule, normalization, defuzzification, and output layers. The initial Fuzzy Inference system (FIS) is provided in the fuzzification layer, and the MF parameters (premise parameters) are determined. The firing degrees are computed through the rule layer. The outputs of the previous layer are normalized through the normalization layer. The output of each fuzzy rule (if-then rules) is computed in the defuzzification layer. Ultimately, the weighted average of overall outputs from the previous layer is determined in the output layer.

In this study, the proposed evolutionary algorithms (EAs) are tuned to the ANFIS parameters (e.g., premise and consequent parameters) to enhance the capability of standalone ANFIS for predicting the long contraction scour depth. The following sections introduce EA models for tuning the classical ANFIS used in this study.

3.2 Ant colony optimization

Dorigo [49] proposed the ACO model for solving complex problems, which was inspired by the method of food finding in the ant colony. The Ants pass different paths to find food. In iterative behaviors, the ants explore the shortest (optimized) path to their food source. The chance of choosing a specific path by ants is computed as follows: [49]

(5)p_{i,j}^k=\left\{{\displaystyle \frac{{({\tau }_{i,j})}^{\alpha }{({\eta }_{i,j})}^{\beta }}{\sum {({\tau }_{im})}^{\alpha }{({\eta }_{im})}^{\beta }}}\right\},

where p_{i,j}^k,{\tau }_{i,j}, α, β, {\eta }_{im}, and {\tau }_{im} present the probability of selecting the route from point i to j by an ant (k), amount of pheromone in the current iteration, weighting coefficients quantities of pheromone, and fitness function depending on preceding ant experience. The mathematical expression for computing the amount of pheromone in every passed route is described as follows

(6){\tau }_{i,j}^{\ast }+\sum \mathrm{\Delta }{\tau }_{i,j}^k\to {\tau }_{i,j},

where {\tau }_{i,j}^{\ast } is the amount of pheromone in the current iteration, \sum \mathrm{\Delta }{\tau }_{i,j}^k is a summation of all pheromones from previous passing routes.{\tau }_{i,j} stands for the total existing amount of pheromone in every passed route between point i and j.

After several iterations, the best path can be obtained. The governing process on finding the shortest (best) paths by ants described in Fig. 2(a).

3.3 Biogeographic based optimization (BBO)

Simon [50] mathematically described the process of species migration between different ecosystems. He implemented the habitat-suitability index (HSI) and the suitability index variable (SIV) indicators. The HSI identifies the quality of living in a specific habitat, while the SIV quantifies the habitat quality using multiple stated variables. The tendency for immigration (λ) and emigration (μ) is related to the HSI values in habitats. The mathematical expressions of immigration and emigration rates are determined as follows:

(7){\lambda }_i=I\left(\mathrm{1}-{\displaystyle \frac{K}{S_{\max }}}\right),

(8){\mu }_i=E\left({\displaystyle \frac{K}{S_{\max }}}\right),

where I, E, K, and S_max represent the maximum possible value of immigration rate, maximum possible value emigration rate, number of living individuals of ith habitat, and maximum number of living individuals, respectively. The BBO uses a roulette wheel selection operator. Improving the quality of low HSI habitat is one of the beneficiaries of this operator. Equation (9) describes the operator:

(9)u(z)={\displaystyle \frac{\mathrm{1}-a(z)}{A}}\times U,

where u(z), z, a(z), A, and U represent the mutation, the number of species, solution probability of species, maximum value of a(z), and maximum mutation rate, respectively. The BBO model is based on an iterative computational procedure to converge to a solution. The general optimization procedure of BBO is presented in Fig. 2(b).

3.4 Genetic algorithm

GA employs the Darwin theory to explain the gene distribution of the species [51]. GA uses five main functions to solve complex problems. The first function generates an initial population. In this paradigm, solutions that include the variable values are determined as chromosomes. The following functions are applied in the iterative procedure if the best solution is obtained. The second function selects the chromosomes or solutions having high fitness function values. Two types of selection functions exist, such as the selection of tournaments and the roulette wheel. In this study, the roulette wheel selection is used. The mathematical expression of the roulette wheel selection is shown as follows:

(10)P_i={\displaystyle \frac{f_i}{{\displaystyle \sum _{i=\mathrm{1}}^N}{f}_i}},

where P_i, f_i, and N represent the probability of selecting the ith chromosome, fitness function, and population, respectively. New offspring are produced in the third stage, which is called the crossover process, by swapping the parents' chromosomes. Suppose X and Y and K present gene chromosome of parents and random point for a crossover (K\in [\mathrm{1},n]), therefore their offspring (X_new, Y_new) are generated as follows:

X=[X_{\mathrm{1}},X_{\mathrm{2}},\dots ,X_n]

(11)Y=[Y_{\mathrm{1}},Y_{\mathrm{2}},\dots ,Y_n]],

(12)\left\{\begin{array}{l}{X}_{\mathrm{new}}=[X_{\mathrm{1}},X_{\mathrm{2}},\dots ,X_K,Y_{K+\mathrm{1}},Y_{K+\mathrm{2}},\dots ,Y_n]\hfill \\ Y_{\mathrm{new}}=[Y_{\mathrm{1}},Y_{\mathrm{2}},\dots ,Y_K,X_{K+\mathrm{1}},X_{K+\mathrm{2}},\dots ,X_n]\hfill \end{array}\right\},

The fourth function is the mutation function, which avoids getting stuck in a local optimum. Besides, this function helps increase the probability of selecting a weak gene in the next generation. The mutation function is mathematically expressed as

(13)X^{\prime }=[X_{\mathrm{1}},\dots ,X_K^{\prime },\dots ,X_n],

where X_K^{\prime } presents random gene features in the range [X_K^{\mathrm{L}}-X_K^{\mathrm{U}}]. The values of X_K^{\mathrm{L}} and X_K^{\mathrm{U}} identify the lower and upper values of the gene variable, respectively. Also, the X_K^{\prime } gene can be substituted by either X_K^{\mathrm{L}} or X_K^{\mathrm{U}}with the same probability. The final function is the evaluation function, which calculates the fitness function for each solution obtained during calculation. The general optimization procedure of GA is presented in Fig. 2(c).

3.5 Invasive weeds optimization (IWO)

Mehrabian and Lucas [52] provided mathematical patterns to explain the process of colonizing weeds and seeking suitable space for cultivation. The algorithm consists of four operators:

Initial population operator: The random dispersion of seeds is also called the initial solution for finding the best solution to solve problems.

Reproduction Operator: This operator is like GA in the crossover, although it operates differently. IWO uses colony fitness to the lowest and highest values of weeds. This operator is specified as follows:

(14)\mathrm{}weed=\mathrm{}floor\left({\displaystyle \frac{f-f_{\min }}{f_{\max }-f_{\min }}}\right)(s_{\max }-s_{\min })+s_{\min },

where the floor (f), f_min, f_max, s_min and s_max represent the round down operator, fitness, lowest and highest values of fitness, and minimum and maximum numbers of seeds, respectively.

Spatial dispersal operator: This operator randomizes IWO. This randomness propagates seeds over parental plants. This operator depends on the standard deviation (STD) for each iteration of newly created results and can be computed as follows:

{\sigma }_{\mathrm{iter}}={\displaystyle \frac{{(\mathrm{}iter{\mathrm{}}_{\max }\mathrm{}-iter)}^n}{{(\mathrm{}iter{\mathrm{}}_{\max })}^n}}({\sigma }_{\mathrm{inital}}-{\sigma }_{\mathrm{final}})+{\sigma }_{\mathrm{final}},

(15){\sigma }_{\mathrm{iter}}={\displaystyle \frac{{(\mathrm{}iter{\mathrm{}}_{\max }\mathrm{}-iter)}^n}{{(\mathrm{}iter{\mathrm{}}_{\max })}^n}}({\sigma }_{\mathrm{inital}}-{\sigma }_{\mathrm{final}})+{\sigma }_{\mathrm{final}},

where σ_iter, iter_max, σ_initial, and σ_final are STDs of the current iteration, maximum iteration times, and primary and last computed STDs.

Competitive exclusion operator: This operator increases the survival probability of weak plants with low fitness values. To this end, after attaining maximum population, the members are sorted by their fitness values, and colonies with a maximum population are chosen for the next round of computations in further iterations. The general optimization procedure of the IWO is presented in Fig. 2(d).

3.6 Teaching-learning-based optimization (TLBO)

Rao et al. [53] introduced a mathematical expression to identify the interaction between students and teachers in a classroom. This model includes the teacher and learner phases. In the first phase, the teacher assists students (learners) to improve their grades. In the next phase, the learners collaborate to achieve higher grades. Both phases are stated as follows:

1) Teacher phase

The main task here is selecting the person with the highest knowledge to solve the problem and then sharing the knowledge of the chosen person to enhance the learners’ grades.

In this regard, the average grade of the learners is calculated, and the difference between the teacher and the average grade can be obtained as Eq. (16):

(16)\mathrm{Difference\; mean}=r\times (\mathrm{Teacher}-TF\times \mathrm{Mean}),

where r and TF represent random vector and teaching coefficient, respectively, and r varies between zero and one.

The teaching coefficient is determined as follows:

(17)TF=round[\mathrm{1}+rand(\mathrm{0},\mathrm{1})],

In each iteration, the existing solution is recalculated repeatedly using the difference mean as follows:

(18)X_{i,\mathrm{new}}=X_{i,\mathrm{old}}+\mathrm{Difference\; mean,}

where X_i,new is the recalculated value of X_i,old using the mean difference.

2) Learner phase

Here, all learners try improving their grades by collaborating. A learner (X_i) chooses another learner (X_j) randomly for sharing their knowledge. The interaction process, which improves a learner’s knowledge, is determined by Eq. (19).

(19)\mathrm{new}{X}_i=X_i+r\times (X_i-X_j),

where r is a random value, ranging between zero and one. In the iterative procedure, a new fitness value is achieved for learner X_i as long as the stopping criterion is obtained. The general optimization procedure for TLBO is presented in Fig. 2(e).

3.7 ANFIS-heuristic algorithms

The ANFIS parameters in premise and consequent phases need to be tuned for achieving the best results. Several studies have reported the unsuitability of standalone ANFIS in generating relevant results due to the un-tuned parameters in the stated phases [54]. The hybridization process steps are as follows.

• The train and test data are loaded. The data are randomly broken down into two sets (train and test). 75% of the data are used as the training set, while the remainder is used as the test set. (25%)

• A basic ANFIS is created, as mentioned in previous sections.

• The ANFIS parameters are tuned using the mentioned algorithms and performance indices are computed.

• The best ANFIS model with the highest accuracy is selected as a superior model.

• The hybridization procedure used here is presented in Fig. 3.

The ANFIS parameters optimized by the BBO algorithm are illustrated in Tables 1 and 2.

4 Description of the proposed predictive models

To obtain the input combinations, the correlation values between the predictive variables (e.g., {\displaystyle \frac{d_{\mathrm{50}}}{b_{\mathrm{1}}}}\cdot {\displaystyle \frac{h_{\mathrm{1}}}{b_{\mathrm{1}}}}\cdot {\displaystyle \frac{b_{\mathrm{2}}}{b_{\mathrm{1}}}}\cdot {\sigma }_{\mathrm{g}}\cdot {\displaystyle \frac{U_{\mathrm{1}}}{U_{\mathrm{c}}}}\cdot {\displaystyle \frac{{\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{w}}}}\cdot F{r}_{\mathrm{0}}) and target variable ({\displaystyle \frac{d_{\mathrm{s}}}{b_{\mathrm{1}}}}) are presented in Table 3.

The highest and the lowest correlations with the target variable are observed in {\displaystyle \frac{b_{\mathrm{2}}}{b_{\mathrm{1}}}} (R = − 0.578) and {\displaystyle \frac{{\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{w}}}} (R≈0) (Table 3).

Table 4 represents the defined input combinations. The first input combination (Model-1) comprises the main predictive variables, while the insignificant predictive variables are eliminated through the other input combinations. For instance, the last input combination (Model-7) is characterized based on the best predictive variable ({\displaystyle \frac{b_{\mathrm{2}}}{b_{\mathrm{1}}}}).

5 Description of the employed data sets

To examine the proposed models, an experimental data set with 167 observations was obtained from several previous studies [4,44–46]. The data set was divided into 124-43 observations for the training-testing phases. Table 5 demonstrates the ranges of non-dimensional parameters.

6 Model performance indicators

To explore the accuracy of each model in predicting the contraction scouring and the capability of the improved ANFIS models for assessing the scour depth, the Willmott-Index (W-index), Legates and McCabe index (LMI), correlation coefficient (R), mean absolute error (MAE) and root mean square error (RMSE) are computed as follows [55,56]:

(20)RMSE=\sqrt{{\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^n}{(X_{\mathrm{Observed}}-X_{\mathrm{Predicted}})}^{\mathrm{2}}}{n}}},

(21)MAE={\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^n}\left|X_{\mathrm{Predicted}}-X_{\mathrm{Observed}}\right|}{n}},

(22)R={\left({\displaystyle \frac{n(\sum X_{\mathrm{Observed}}\cdot X_{\mathrm{Predicted}})-(\sum X_{\mathrm{observed}})(\sum X_{\mathrm{Predicted}})}{\sqrt{[n\sum (X_{\mathrm{Observed}}^{\mathrm{2}})-{(\sum X_{\mathrm{Observed}})}^{\mathrm{2}}]-[n\sum (X_{\mathrm{Predicted}}^{\mathrm{2}})-{(\sum X_{\mathrm{Predicted}})}^{\mathrm{2}}]}}}\right)}^{\mathrm{0.5}},

(23)LMI=\mathrm{1}-{\displaystyle \frac{\sum \left|X_{\mathrm{Predicted}}-X_{\mathrm{Obserbed}}\right|}{\sum \left|X_{\mathrm{Observed}}-\overline{X_{\mathrm{Observed}}}\right|}},

(24)W-\mathrm{index}=\mathrm{1}-\left[{\displaystyle \frac{{\displaystyle \sum _{i=\mathrm{1}}^N}{(X_{\mathrm{Observed}}-X_{\mathrm{Predicted}})}^{\mathrm{2}}}{{\displaystyle \sum _{i=\mathrm{1}}^N}{\left(\right|X_{\mathrm{Predicted}}-{\bar{X}}_{\mathrm{Observed}}|+|X_{\mathrm{Observed}}-{\bar{X}}_{\mathrm{Observed}}\left|\right)}^{\mathrm{2}}}}\right],

7 Uncertainty analysis

The uncertainty related to the input combination and model formulation in the simulation of long contraction scouring was determined. The following steps were performed to quantify the model formulation uncertainty.

• A set of scouring values simulated by the superior predictive models (e.g., ANFIS- ACO, ANFIS- GA, ANFIS- TLBO, ANFIS- BBO, and ANFIS- IWO) was prepared for each measured scouring depth.

• A normal function was fitted to each simulated set.

• Many scouring depths (1000 generations) were obtained using the Monte Carlo simulation (MCS) for each measured scouring depth.

• The 95% uncertainty band (distance between the 2.5% and the 97.5% percentiles), which is called the 95 PPU was obtained using the generated scouring depths for each measured scouring depth.

• The uncertainty was then quantified using the R-factor index as follows:

(25)R\mathrm{-factor}={\displaystyle \frac{Xm}{Xs}},

where Xs is the standard deviation of the observed scouring depths and the Xm value computed as follows:

(26)Xm={\displaystyle \sum _{i=\mathrm{1}}^n}(U{L}_L^i-L{L}_L^i)/n,

where n denotes the total number of observed scouring depths, and U{L}_{\mathrm{L}}^i and L{L}_{\mathrm{L}}^i describe the upper (97.5%) and lower (2.5%) limits of the ith value, respectively.

To assess the uncertainty related to the input variables, the scour depths obtained from various input combinations through the superior predictive model were used for each measured scouring depth. Then, steps ii to v (uncertainty of models) were performed to quantify the uncertainty associated with the input variables.

8 Results and discussion

The performance results of the standalone ANFIS model and its hybrid models (ANFIS-TLBO, ANFIS-GA, ANFIS-BBO, ANFIS-ACO, and ANFIS-IWO) are presented in this section. Several dimensionless variables were used to assess the depth of long contraction scouring. The variables can be expressed mathematically as follows:

(27){\displaystyle \frac{d_{\mathrm{S}}}{b_{\mathrm{1}}}}=f\left({\displaystyle \frac{d_{\mathrm{50}}}{b_{\mathrm{1}}}}\cdot {\displaystyle \frac{h_{\mathrm{1}}}{b_{\mathrm{1}}}}\cdot {\displaystyle \frac{b_{\mathrm{2}}}{b_{\mathrm{1}}}}\cdot {\sigma }_{\mathrm{g}}\cdot {\displaystyle \frac{U_{\mathrm{1}}}{U_{\mathrm{c}}}}\cdot {\displaystyle \frac{{\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{w}}}}\cdot F{r}_{\mathrm{0}}\right),

Multiple statistical performance metrics (R, RMSE, SRMSE, MAE, and LMI) were computed to evaluate the predictability performance. The prediction results of the developed hybrid predictive models and all constructed seven-input combinations are tabulated in Table 6. For all models, the training and testing modeling phases were consistent for the best prediction accuracy, which is evidenced by the correct learning process initiated during the training phase for all predictive models. Although seven-input combinations were constructed based on the assumption of the input parameters' influence, the best results were attained for the first three input combinations for all models. This trend proves that the simulated long contraction depth scouring is highly correlated with all physical parameters. It was also observed that in some cases, the best prediction accuracy for the ANFIS-GA model was attained for the third input combination where the Froude number and the ratio of density are excluded. Thus, this trend shows the credibility of the developed ANFIS-GA model in obtaining considerable predictions with less associated input parameters.

For better prediction visualization, the best performance results of each model used over the training and testing phases are tabulated in Table 7. The error measurement for the training phase shows that the ANFIS-BBO model predicts the depth of scouring prediction better than other models using the 2nd input combination. The values of the performance indices in the training phase for the ANFIS-BBO model indicate that this model has the lowest RMSE and highest R (RMSE of 0.0158 and R = 0.9619), respectively. Thus, the model is more powerful than the classic ANFIS model. Next to the ANFIS-BBO model in predicting the contraction scour depth is the ANFIS-IWO model. Generally, the best performance prediction is found in the ANFIS-BBO model, while the ANFIS-TLBO model provides the lowest accuracy in the training phase. A similar condition is observed in the testing phase, where the ANFIS-BBO indicated the highest accuracy (RMSE, MAE, and R of 0.0193, 0.128, and 0.9232, respectively) in predicting the long contraction scouring.

To better the predictive models, the heat map diagram was employed using the normalized performance indices. The best performance values tend to dark red cells, while the dark blue cells are assigned to the lowest performance prediction. Figure 4 shows the performance of each model in different phases. The superior model (ANFIS-BBO) tends to dark red color, meaning more accurate result generation compared to other models.

The scatter diagrams of the proposed models are shown in Fig. 5. The ANFIS-TLBO (R_train = 0.92, R_test = 0.88) model has the lowest linear correlation between the observed and predicted scouring depth, while the ANFIS-BBO (R_train = 0.96, R_test = 0.92) model has the highest.

The Taylor diagram was also generated for better graphical examination. The results of the Taylor diagram show good agreement with those obtained by the scatter charts. Figure 6 shows a slightly better consistency between the ANFIS-BBO predictive model (black circle) and the observed data in the training phase. Conversely, a significant consistency is observed between the ANFIS-BBO model and observed data compared to the other models. Furthermore, the ANFIS-TLBO predictive model (red triangle) provides the lowest consistency.

To compare the variability of the scouring depths predicted by the proposed models against the observations, a box plot diagram was used in both the training and testing phases (Fig. 7). All predictive models showed an acceptable accuracy to capture the variability of observed data in both the training and testing phases. However, better consistency is observed between the upper percentile of the observed values (0.23) and those predicted by the ANFIS-BBO model (0.203) in the testing phase. A similar consistency is also found in the lower percentile, where the closest value to the observed data (0.029) is provided by the ANFIS-BBO model (0.028) in the testing phase.

To evaluate the influence of the randomized data set used in prediction modeling on the obtained results, ten training and testing data sets were considered randomly. The mean values of the input variables of the different samples are presented in Table 8.

Using the samples defined in Table 8, the scouring depth was predicted by the ANFIS-BBO model (superior model). Table 9 represents the performance indices obtained using different samples.

The mean values of the relative difference between the metrics obtained using the samples and those obtained using the primary data set in the training and testing phases are 7.5% and 6.5%, respectively (Table 8). Thus, it can be concluded that the influence of data sampling on prediction performance is negligible.

The impact of the sample size and data set division on prediction performance is of crucial concern in prediction modeling [57,58]. Hence, different sample sizes (120, 130, 140, and 150), and different data set divisions (60%–40%, 65%–35%, 70%–30%, and 75%–25%) were examined by the ANFIS-BBO model to predict the long contraction scouring (Table 10).

R varied from 0.927 to 0965 (relative change of 3.9%) for different data set divisions and 0.9% for different sample sizes in the training phase (Table 10). Those values are 2.9% and 7.7%, respectively, in the testing phase. Generally, the different data set divisions and sample sizes slightly affect the prediction performance. Besides, the performance obtained in the testing phase is more sensitive to sample size compared to those obtained in the training phase.

Assessing the uncertainty associated with the obtained results is crucial in prediction modeling. Several approaches can be used to quantify the uncertainty sources, such as approximation methods, analytical methods, and sampling approaches [59–63]. Besides, several uncertainty sources are associated with prediction modeling. Thus, the uncertainty assessment provides a comprehensive perspective to determine the significant factors through prediction modeling. The uncertainty of model formulation was quantified using the employed predictive models, including ANFIS-ACO, ANFIS-GA, ANFIS-TLBO, ANFIS-BBO, and ANFIS-IWO models for their best input combination. To assess the uncertainty of input variables, the prediction results of the different input combinations (Model 1-7) through the best predictive model (ANFIS- BBO) were considered.

Figure 8 represents the 95 PPU band (prediction confidence interval) and different prediction percentiles for model formulation and input variables uncertainties in the testing phase. The R-factor (0.73) for the model formulation is greater than that for the input variable (1.5). Thus, the prediction results are less sensitive to the model formulation compared to the input variables.

9 Conclusions

Here, the ANFIS was tuned by five evolutionary models (GA, BBO, TLBO, ACO, and IWO) to predict the long contraction scouring depth. A data set obtained from previous studies [4,44–46] was used for the prediction modeling. Seven dimensionless input variables ({\displaystyle \frac{d_{\mathrm{50}}}{b_{\mathrm{1}}}},{\displaystyle \frac{h_{\mathrm{1}}}{b_{\mathrm{1}}}},{\displaystyle \frac{b_{\mathrm{2}}}{b_{\mathrm{1}}}},{\sigma }_{\mathrm{g}},{\displaystyle \frac{U_{\mathrm{1}}}{U_{\mathrm{c}}}},{\displaystyle \frac{{\rho }_{\mathrm{s}}}{{\rho }_{\mathrm{w}}}},\mathrm{and}F{r}_{\mathrm{0}}) were employed through several combinations to obtain the best predictive model using several metrics (WI, LMI, R, MAE, and RMSE). The results obtained show that the ANFIS-BBO model (RMSE = 0.0193 and R = 0.923) provides the highest prediction performance while the ANFIS-TLBO model (RMSE = 0.0243 and R = 0.879) offers the lowest compared with other models.

This study also assessed the uncertainty sources associated with prediction modeling’s such as model formulation and input variable uncertainties. Thus, many scour depths were predicted using the MCS technique, and the 95% prediction uncertainty of the generated data was evaluated through the R-factor index. From the results, the uncertainty due to the input variable combination is larger than that of the model formulation. In general, the ANFIS model integrated with the BBO algorithm provides a robust model for predicting the long contraction scour depth.

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