Investigating the capability of two hybrid intelligence methods to predict bedform dimensions of alluvial channels

An alluvial channel is a water channel made up of loose sediments called alluvium. The sediments move as bedload and suspended load, and produce bedforms. The principles of creating different bedforms in different alluvial channels (sand-bed, gravel-bed, etc.) have been introduced in different references, e.g. van Rijn (1984). So, alluvial channels have mobile beds which is the most important parameter in creating bedforms. In fixed bed rivers like concrete channels, no sediment moves in the bed and no bedform is created. River bedforms play a significant role in friction and sediment transport. The accurate prediction of the geometric characteristics of bedforms is an essential component for estimating the flow resistance and the consequent flow conditions. In the past half-century, a number of empirical methods have been presented to relate bedform dimensions to flow characteristics. However, the presence of different kinds of variables has made the study of this problem very complicated. The method of van Rijn (1984) is one of the most widely used and applicable methods for prediction of bedform dimensions. Van Rijn's method is based on a regression analysis of 84 flume and 22 field data points. This model reveals a large variability in the prediction of bedform dimensions in the different conditions of natural channels (Julien & Klaassen 1995), showing underprediction of bedform dimensions under different morphodynamic conditions (Julien 1992). For this reason, other researchers have tried to achieve better methods to predict bedform dimensions (Yalin 1992; Julien & Klaassen 1995; Karim 1995; Talebbeydokhti et al. 2006). These methods are derived based on different river or flume data which are not appropriate for all conditions such as van Rijn's method. This shows that attempts to achieve a new relationship using a different data series will have similar results and the new predictor would not have a global use because of the complex behavior of bedform formation during different flow conditions.

Due to the uncertainty and complexity of the variables in prediction of bedform dimensions, it is difficult to present a unique empirical method for prediction of bedform dimensions. During recent years, many system identification techniques have been developed in various fields of engineering to predict the unknown behavior of complex systems based on a given input-output data set. One of these approaches is based on artificial intelligence (AI) methods. AI methods can be categorized into two major groups. The first includes data driven methods (DDM) such as artificial neural network (ANN), support vector machine (SVM), group method of data handling (GMDH), etc. The second group contains knowledge-based methods such as genetic algorithm (GA), shuffled complex evolution (SCE), harmony search (HS), etc. In recent decades, AI methods have become increasingly popular in hydrology and water resources research and engineering.

GMDH is an AI technique which belongs to the self-organizing modeling approach. In this method, the number of neurons, the number of layers and the behavior of each neuron is adjusted during the process of training, therefore, the prediction of system modeling in GMDH is more complex than in other artificial methods. GMDH has been reported to have good results in predicting debris flow (Zhang et al. 2013), significant wave height (Shahabi et al. 2016) and the discharge coefficient of rectangular sharp-crested side weirs (Ebtehaj et al. 2015).

Gaining better predictions by using AI methods rather than empirical models, has not been sufficient in recent decades, so scientists have tried to optimize the predictions. For this purpose, they have combined different DDM techniques with different optimization methods called hybrid intelligence systems. Many successful applications of these hybrid intelligence systems have been reported. For example, GA-ANN for flood forecasting (Wu & Chau 2006), GMDH-LSSVM, which is the combination of GMDH with least squares support vector machines (LSSVM) for river flow forecasting (Samsudin et al. 2011), GA-SVM for prediction of pollution in reservoirs (Su et al. 2015) and GMDH-HS for uplift capacity prediction of suction caissons (Masoumi Shahr-Babak et al. 2016).

According to the above-mentioned applications of AI and hybrid methods, it seems that using these methods can provide an accurate prediction of bedform dimensions which is an important factor in most river engineering problems. According to the literature, only Javadi et al. (2015) have used AI methods including ANN and SVM to predict bedform dimensions using 257 datapoints from the Rhine and Meuse rivers. Using data from just two rivers and no flume data plus a small range of variables increases doubts about using these methods with other natural or flume data sets. To solve this problem, a wider range of data sets and novel hybrid models are required.

The main objective of this research is to develop and apply two new hybrid intelligence methods called GMDH-HS and GMDH-SCE, based on a combination of GMDH with HS and SCE algorithms, to predict bedform dimensions from 447 river and flume data points with a wide range of variables. HS and SCE were used as subroutines for calibrating weights. Also, codes have been written in MATLAB to use the models not the tools in MATLAB. Finally, the results are compared to conventional methods of GMDH with two kinds of transfer functions called GMDH1 and GMDH2. The advantage of the GMDH model is that the number of neurons and layers is determined while running the model, so GMDH is faster than ANN, GP, etc. for predicting different parameters. Parameters will be defined as inputs which have the following characteristics: (1) easy to measure, (2) easy to calculate, (3) similar to the parameters extended from dimensional analysis, and (4) similar to the parameters that other researchers in the literature have used.

The computational procedure of HS is inspired by a group of musicians searching for a musically pleasing harmony (Ayvaz 2009). This process was first adapted into engineering optimization problems by Geem et al. (2001). In the HS algorithm, each musician is a decision variable and the collection of notes in the musicians’ memory is the value of the decision variables. Adaptation of the musical rules to the optimization problems is (1) generating a new solution vector from harmony memory (HM), (2) replacing a decision variable with a new one which is close to the current one (pitch adjusting) and (3) generating a solution vector from the possible random range (random selection). Combined utilization of these rules allows identification of the optimal or near optimal solutions. Although HS searches for the optimal solution by considering multiple solution vectors as in GA, its reproduction process is different from GA. While GA generates a new offspring from two parents in the population, HS generates it from all the existing vectors stored in HM (Ayvaz 2009).

The SCE method was developed at the University of Arizona to deal with the peculiarities encountered in environmental model calibration. It combines the best features of multiple complex shuffling and competitive evolution based on the simplex search method. The use of multiple complexes and their periodic shuffling provides an effective exploration of different promising regions of attraction within the search space (Muttil & Jayawardena 2008). The method is based on a synthesis of four concepts: (1) combination of deterministic and probabilistic approaches, (2) systematic evolution of a ‘complex’ of points spanning the parameter space, in the direction of global improvement, (3) competitive evolution and (4) complex shuffling. The synthesis of these elements makes the SCE method effective and robust, and also flexible and efficient (Kan et al. 2016). A detailed presentation of the theory underlying the SCE algorithm can be found in Duan et al. (1993).

Several hybrid intelligent systems have been developed based on a hybrid of DDM with optimization algorithms in various ways. Optimization algorithms are generally used to determine the structure of DDM, calibrate unknown weights or determine both of them. In this study, the HS and SCE are used to calibrate the weights of each neuron in GMDH rather than using the LSE method. Since GMDH is a self-organizing method with an unknown structure, its structure can be determined by HS and SCE algorithms. So, a hybrid integration of GMDH and HS or SCE algorithms may have a better performance by taking advantages of the characteristics of both methods together. In these algorithms, HS and SCE are employed to train and optimize the initial parameters or weights of transfer function in each neuron of the GMDH structure. The objective of the HS and SCE submodels is to determine optimal weights in order to attain the optimum structure of the GMDH model and minimum cumulative errors between the measured and predicted data sets.

Bedforms such as river dunes are rhythmic bed features which are developed by the interaction between water flow and sediment transport (van der Mark et al. 2008). River dunes are often schematized as a train of regular triangular features. Figure 2 illustrates characteristics of bedforms. The purpose of this study is to predict the length and height of dune bedforms in rivers and flumes.

K_s is the reference bed level or grain roughness where water velocity at that depth is equal to zero. Scientists have presented different values for K_s in the form of aD_b in which a and b take different values. By comparing Equations (7) and (8), it is found that if K_s= 0.25D₉₀. can also be simply calculated by use of the logarithmic distribution of velocity profile or boundary-layer characteristics method presented by Afzalimehr & Anctil (2000).

After calculating input variables, the ratio of bedform height to bedform length and flow depth are used as output variables. The 446 data points were taken from the Klaassen study of different flumes and rivers around the world (Klaassen 1990). In this study, data sets were divided into training and testing data. The data set of 312 data points (70%) was used randomly for training and the rest (30%) was used for testing, according to specific indices including: mean, skewness, minimum and maximum of training and testing data. In addition, to select data randomly, introducing some noisy data into the training set has reduced the chances of overfitting. Also, when the number of training examples is small, their discrepancies are big, causing a serious overtraining problem. To avoid this, a large number of training data with a wide range of statistical variables is chosen. The range of statistical parameters for the training data set is shown in Table 1. SD and CV are standard deviation and coefficient of variation, respectively.

It is important that both training and testing data sets have the same statistical indices. It is observed from Table 1 that the statistical indices are approximately the same for all parameters.

In order to assess the accuracy of each model, various statistics have been used. The best known and most widely used ones are presented below. These statistics were appropriately used in the calibration phase to determine the parameters and structures.

• 1.

  Nash-Sutcliffe Efficiency Coefficient (E)

• 2.

  Root Mean Square Errors (RMSE)

• 3.

  Mean Square Relative Error (MSRE)

• 4.

  Mean Absolute Percentage Error (MAPE)

• 5.

  Relative Bias (RB)

E is used to assess the predictive power of models (Masoumi Shahr-Babak et al. 2016). The range of this criterion is in which E = 1 corresponds to a ‘perfect’ fit of predicted data to the observed data. The value of is a ‘very satisfactory’ prediction, whereas is ‘fairly good’ and is ‘unsatisfactory’ (Masoumi Shahr-Babak et al. 2016). RMSE reflects the performance of the prediction model. Generally, the smaller the RMSE, the better the performance. MSRE and MAPE indicate the relative absolute accuracy of the models, while RB indicates whether a model is overpredicting or underpredicting. The values of MSRE for a perfect and acceptable model is and ⁠, respectively. Ranges of MAPE for perfect and acceptable models are similar to MSRE.

The range of RB is (negative values indicate a general overestimation, while positive values indicate a general underestimation of the model).

The combination of five dimensionless parameters which are used as inputs, is shown in Table 2.

The outputs are also and ⁠. It means that there are 10 runs. The first five runs are the different combinations presented in Table 2 as inputs and one output named and the other five runs are with the output named ⁠. Figure 3 shows a comparison of measured bedform dimensions and the predicted values from applied methods in the training and testing period for the first combination of inputs (C1). The lines +25% and −25% present the ratio of calculated parameters to measured ones; for example, when a dot is placed between these two lines, it means that the ratio of calculated parameters to measured ones lies in the range between +0.25 and −0.25. It can be observed from Figure 3 that GMDH-HS and GMDH-SCE have better performances during the training and testing period. Regarding the integration of GMDH and HS and SCE, it is reasonable to attain a better performance by taking advantages of the self-organization of GMDH and the global optimization of the HS and SCE methods.

Although GMDH-HS and GMDH-SCE have acceptable performances in predicting bedform dimensions, in order to find the best combination and method for predicting bedform dimensions, the ranking system is applied (Das & Suman 2015). The ranking system for all combinations and applied methods in and prediction are presented in Tables 3 and 4, respectively. In these tables, CE is the ‘coefficient of efficiency’ and can be calculated as in which n is the number of data series, x and y are the observed and predicted outputs respectively and x is the mean of observation outputs, R² is the determination coefficient, μ and σ are the mean and standard deviation of the ratio of predicted bedform dimensions to measured ones. P50 and P90 are numbers that 50 and 90 percent of relative outputs are larger than, respectively. Relative outputs is the ratio of predicted to observed outputs. For example, for training GMDH1 with C1, P50 is 1.563, which means that 50 percent of relative outputs are larger than 1.563. Lognormal and histogram values (Tables 3 and 4) are also related to the relative data with accuracy in lognormal distribution and histogram values, respectively, which are referred to the relative data, i.e. the ratio of predicted to observed outputs. Finally the indices R1, R2, R3 and R4 are ranking numbers that indicate the ranking of methods based on the best fit calculation, arithmetic calculation, cumulative probability of the ratio of predicted bedform dimensions to measured ones and prediction of bedform dimensions with accuracy, respectively. For example, in Table 3 in C1, R1 of GMDH1 was calculated to be 20. This R1 indicates that GMDH1 in C1 is 20th based on the fit calculation. Also R1, R2, R3 and R4 are defined to compare several methods. In total the best method has the minimum final rank. More details about this ranking can be found in Abu-Farsakh & Titi (2004) and Das & Suman (2015).

It can be observed from Table 3 that GMDH-HS has the best performance of all applied methods and is more capable of predicting ⁠. Also, based on the value of final rank, the fifth combination (C5) yields better answers for predicting ⁠.

In order to assess the applied methods in this research, the performance and outcome of these methods are compared with the empirical method of van Rijn (1984). Table 5 presents the results of bedform dimension prediction during training and testing periods in terms of various statistical indices. C5 and C3 are used in Table 5.

It can be observed from Table 5 that various AI methods have good performance during both training and testing periods. For prediction of ⁠, in the training period, GMDH-SCE obtained the best E, RMSE and MAPE statistics of 0.773, 0.011 and 31.303, respectively, while GMDH-HS and van Rijn (1984) obtained the best MSRE and RB statistics of 0.289 and −0.025, respectively. In the testing period, GMDH-HS obtained the best E, RMSE, MSRE and MAPE with the values of 0.819, 0.01, 0.542 and 34.232, respectively, while van Rijn (1984) obtained the best RB with the value of −0.027.

On the other hand, for predicting ⁠, in the training period, GMDH-SCE obtained the best E, RMSE, MSRE and MAPE statistics of 0.88, 0.039, 0.104 and 17.279, respectively, while GMDH2 obtained the best RB statistic of −0.06. The results in the testing period are also similar to the training period and the statistics are 0.839, 0.045, 0.172 and 23.427, respectively, while van Rijn (1984) obtained the best value of RB with −0.115. It can be seen from Table 5 that all AI methods outperform the empirical method of van Rijn (1984) and also the performances of GMDH-SCE and GMDH-HS are better than other AI methods in both training and testing periods. In addition, the performance of GMDH-HS during training and testing periods is shown in Figures 4 and 5.

Although empirical formulae often provide useful predictions of bedform dimensions in alluvial channels, the complexity of the interaction between flow characteristics and development of bedforms is such that these formulae cannot provide the accuracy required. In this study, two hybrid intelligence methods were developed using GMDH, HS and SCE. In the prediction of bedform dimensions, unlike empirical methods, there are no limitations in the ranges of inputs using AI techniques. For this reason, different combinations of the most frequently used dimensionless parameters in the literature were examined. Results reveal the following:

• (1) The combination of T, ⁠, and z is more accurate for predicting ⁠, while the combination of T, ⁠, and z has a better performance in predicting ⁠. Although these combinations have the best performances in predicting and ⁠, other combinations also have acceptable performances. So, in situations where researchers lack data, using other combinations can also yield appropriate answers.

• (2) For calculation of T, the logarithmic distribution of velocity profile or the boundary-layer characteristics method can be easily used in order to calculate with regard to the effects of height of roughness caused by bedforms.

• (3) The accuracy of all four AI methods in predicting the parameters and in flumes and rivers with each combination of dimensionless parameters is extremely high and acceptable.

• (4) The performance of GMDH-SCE for predicting bedform dimensions is better than other methods and GMDH-HS is in second place.

• 5) All AI methods have much better performances than the empirical method of van Rijn (1984). However, GMDH-HS and GMDH-SCE outperform all other methods for predicting bedform dimensions.