Development of expert systems for the prediction of

Applied Soft Computing 34 (2015) 51–59

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Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc

Development of expert systems for the prediction of scour depth under live-bed conditions at river confluences: Application of different types of ANNs and the M5P model tree Behnam Balouchi a,∗ , Mohammad Reza Nikoo a,1 , Jan Adamowski b a b

School of Engineering, Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran Department of Bioresource Engineering, Faculty of Agricultural and Environmental Sciences, McGill University, Canada

a r t i c l e

i n f o

Article history: Received 23 July 2014 Received in revised form 24 March 2015 Accepted 16 April 2015 Available online 28 April 2015 Keywords: River confluences Maximum scour depth Live-bed conditions Multi-layer perceptron (MLP) Radial basis function (RBF) M5P model tree

a b s t r a c t The three-dimensional structure of water flow at river confluences makes these zones of particular importance in the fields of river engineering, fluvial geomorphology, sedimentology and navigation. While previous research has concentrated on the effects of hydraulic and geometric parameters on the scour patterns at river confluences, there remains a lack of expert systems designed to predict the maximum scour depth (dsm ). In the present study, several soft computing models, namely multi-layer perceptron (MLP), radial basis function (RBF) and M5P model tree, were used to predict the dsm at river confluences under live-bed conditions. Model performance, assessed through a number of statistical indices (RMSE, MAE, MARE and R2 ), showed that while all three models could provide acceptable predictions of dsm under live-bed conditions, the MLP model was the most accurate. By testing the models at three different ranges of scour depths, we determined that while the MLP model was the most accurate model in the low scour depth range, the RBF model was more accurate in the higher range of scour depths. © 2015 Elsevier B.V. All rights reserved.

1. Introduction River confluences, the regions where two rivers merge, are important parts of river systems. The mixing of two water flows at such sites results in three-dimensional flow patterns and deep scour holes. These, in turn, can cause changes in river morphology and accelerate the rate of bank erosion. According to Best [1], six distinct zones exist at river confluences, namely stagnation, flow deflection, separation, maximum velocity, flow recovery and shear layer zones (Fig. 1). The flow separation zone, located near the left river bank just downstream of the river confluence is the main cause of a horizontal vortex in this region that leads to the deposition of sediment at the center of the zone and generates a point bar. The zone of maximum flow velocity can accelerate erosion and bank failure on the right river bank and lead to the development of a meander. The flow vortex and high flow velocity in this zone can also create problems for navigation. Because of these practical

∗ Corresponding author. Tel.: +98 71 36133497. E-mail addresses: behnamm [email protected] (B. Balouchi), [email protected] (M.R. Nikoo), [email protected] (J. Adamowski). 1 Tel.: +98 71 36133497; fax: +98 71 36133163. http://dx.doi.org/10.1016/j.asoc.2015.04.040 1568-4946/© 2015 Elsevier B.V. All rights reserved.

applications, the study of flow pattern, scour and sedimentation at river confluences has attracted the attention of many researchers. Despite extensive investigations of flow characteristics at river confluences at both the two-dimensional [2–8] and three dimensional level [9–14], few experimental studies have investigated sediment transport and scour at river confluences [15–23]. Mosley [15] conducted experimental tests for Y-shaped river confluences in a small flume, and showed that the dimensions of the scour hole increased rapidly as the river confluence angle () increased from 15◦ to 75◦ , but then declined at values of . Roy and Roy [16] conducted field measurements on 30 river confluences, and found that, at all locations, the flow area cross-section downstream of the river confluence generally decreased as flow velocity (vf ) increased. At the scour hole the vf was found to reach up to 1.6-fold that of the upstream flow. Biron et al. [17,18] studied bed morphology at the confluences of rivers of unequal depth and showed that, even if river bed morphology changed, scour hole dimensions remain constant. They also found that river morphology at confluences of unequal channel depth differed from that at confluences of equal channel depth. Similarly, Shafai Bejestan and Hemmati [19] showed reduced scour for a discordant bed confluence than a level bed confluence. Using a 90◦ junction pilot model, Borghei and Nazari [20] showed scour depth (ds ) increased with a rise in the

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Nomenclature Abbreviations/symbols ANFIS adaptive neuro-fuzzy inference system ANN artificial neural network depth of flow downstream from confluence (m) dd ds scour depth (m) dsm maximum scour depth (m) ∗ maximum scour depth ratio (=dsm /wd ) dsm d50 median size of river bed material (m) densimetric Froude number Fd FFBPN feed-forward back propagation network gravitational acceleration (m s−2 ) g Gs specific gravity MAE mean absolute error mean absolute relative error MARE MLP multi-layer perceptron, a type of ANN. M5P a tree type model is the number of observations in the measured or n predicted data set Oi the ith observed value Pi the ith model-predicted value p number of observation data inputs in RBF model Qb live-bed discharge (≡ sediment load) (m3 s−1 ) water discharge in channel downstream from conQd fluence (m3 s−1 ) Qm water discharge in main channel upstream of confluence (m3 s−1 ) Qt water discharge of the tributary upstream of confluence (m3 s−1 ) Qbd ratio of live bed sediment discharge to discharge in channel downstream from confluence ratio of tributary discharge to main branch disQtd charge linear correlation coefficient between measured R2 and estimated dsm Re Reynolds number RBF radial basis function, a type of ANN root mean square error RMSE S0 channel bed slope T set of examples in a model tree Uj center of the jth radial basis function f in a RBF function vf river flow velocity (m s−1 ) vfd river flow velocity downstream from confluence (m s−1 ) width of channel downstream from confluence (m) wd wij is the weight of the connection between the jth neuron in a layer with the ith neuron in the previous layer of an ANN width of main channel upstream of confluence (m) Wm wmj is the weight of the connection between the hidden and output nodes in a RBF network width of tributary channel upstream of confluence wt (m) wtd ratio of tributary channel width to channel width below the confluence We Weber number xi is the value of the ith neuron in the previous layer of an ANN is the output from the jth neuron in a given layer of yj an ANN Z bed elevation difference (m) Zm the network output in RBF model

 s   j  red

flow viscosity (kg m−1 s−1 ) density of sediment (kg m−3 ) tributary-main branch confluence angle (◦ ) force of surface tension (kg s−2 ) the size of the radius around the RBF center (Uj ) the standard deviation of the reduction in M5P model tree

ratio of tributary discharge to that in the channel downstream of the confluence (Qtd = Qt /Qd ), but decreased with increases in sediment particle size and relative tributary to width of the channel downstream from the confluence (wtd = wt /wd ). Ghobadian and Shafai Bejestan [21] showed that (Qtd ) was the most important parameter in river confluence studies. Based on their finding that as Qtd , , and the densimetric Froude number (Fd ) increased, so did ds , they developed a mathematical relation for the prediction of ds at river confluences. Borghei and Jabbari Sahebari [22] studied the scour patterns at the junction of two loose-bed channels under clearwater conditions and showed that the position of dsm temporally moved to the outer wall and upstream to the main channel, according to the values of , wtd , Qtd and the ratio of mean downstream velocity to threshold velocity. Despite the existing body of literature concerning flow patterns and river morphology at river confluences, knowledge regarding local scour under live-bed conditions is more limited. To address this knowledge gap, Balouchi and Shafai Bajestan [23] conducted experiments to determine the effects of the Qtd , Fd , and sediment load (live-bed discharge, Qb ) on the dsm at river confluences under live-bed conditions. They found that the quantity of bed load could reduce the dsm up to 35%. Moreover, they found that bed scour and sedimentation patterns under live-bed conditions differed from those which occurred with clear water, particularly with respect to the absence of a point bar in the region of river confluence in the latter case. The authors [23] also presented a regression equation for the prediction of the ds under live-bed conditions, and, through sensitivity analyses, showed that this equation was most sensitive to Fd . Having drawn considerable attention given their effectiveness at representing complex and nonlinear relationships, artificial intelligence methods such as artificial neural networks (ANN) and adaptive neuro-fuzzy inference system (ANFIS) have been extensively used in the prediction of ds and sedimentation (Table 1) [24–32]. In spite of the importance of predicting the dsm at river confluences, there has not yet been, to the best of the authors’ knowledge, any study evaluating the use of intelligent system methods for

Fig. 1. Flow characteristics at a river confluence. Adapted from Best [1]

B. Balouchi et al. / Applied Soft Computing 34 (2015) 51–59 Table 1 Use of artificial intelligence methods in the prediction of ds and sedimentation. Type of model

Model used to estimate

References

ANN ANN ANN, ANFIS radial basis function ANN ANN, ANFIS ANN, fuzzy logic ANN, ANFIS

Scour around ocean piles Scour pattern at bridge piers Scour depth around bridge piers Scour depth at bridge piers

24 27, 30 25 31

Scour depth below spillways Long-shore sediment transport rate Local scouring depth around concave and convex arch shaped circular bed sills, and ANN vs. ANFIS performance Volume of sediment deposition in a reservoir.

26 28 29

ANN, Model Tree, Genetic Programming

53

Start Experimental setup of river confluence under live-bed conditions

Dimensional analysis

Data set Training data

Validation data

Training the softcomputing models

32

RBF

MLP

estimating dsm at river confluences. To address this omission, data-driven simulation models based on data from experimental results were developed for the prediction of dsm at river confluences under live-bed conditions. The multi-layer perceptron (MLP) [24,25,27,33,34] and the radial basis function (RBF) [25,31], both ANN models, as well as the M5P tree model [35–38] have shown a strong performance in accurately predicting complex system responses in the engineering field. As a result, the present study used these three soft computing models to predict dsm at river confluences under live-bed conditions and evaluated and compared their performance on validation data sets of experimentally determined channel confluence scour depths. 2. Model framework Two real-life scenarios were investigated in the present study: (i) a river/canal confluence, where the hydraulic conditions of the confluence have changed, e.g., water levels, bed load, and/or discharges of either branch changed due to human and/or natural events, and (ii) a river/canal branch into which another river/canal is newly diverted, e.g. in constructing a dam, or adding a flood bypass canal to prevent flood damage to a residential or historic area. Since under both scenarios dsm at the river confluence will be altered, it is necessary and important to accurately predict dsm . This ability to predict dsm is also of great practical interest in the fields of river engineering, fluvial geomorphology, sedimentology, navigation and hydraulic structures (e.g., setup of a pump station, construction of a bridge pier at river confluences). Given the complexity of flow and scour patterns at river confluences, accurate prediction of ds for general scenarios can be facilitated by applying an expert system to the existing experimental or field data. However, expert systems for the estimation of dsm at river confluences are generally lacking. In this paper, several experimental data-driven models are proposed for the prediction of dsm at river confluences. The datasets used in the models are based on experimental input-output data of scour depth at channel confluences from Balouchi and Shafai Bajestan [23]. The Qtd , sediment (live-bed) load ratio (Qbd = Qb /Qd ) and densimetric Froude number (Fd ) were identified through dimensional analysis as appropriate inputs for the artificial intelligence models, while the maximum scour depth ∗ = d /w ) was used as the output variable. ratio (dsm sm d The experimental process began by gathering data from a physical model of river confluences under live-bed conditions and using dimensional analysis to identify the most important input variables (Fig. 2). Next, three data-driven simulation models (i.e., the RBF and MLP neural networks and the M5P model tree) were generated and trained based on experimental input-output data. Their relative performance was then evaluated through the use of a validation data set from the same experimental setup. These three main steps

M5P model tree

Validating the models using validation data set Several statistical indices used to compare the accuracy of the MLP, RBF and M5P treemodelsand select the superior model End Fig. 2. A flowchart of the proposed methodology for developing expert systems for the prediction of scour depth under live-bed conditions at river confluences.

of the methodology are explained in greater detail in the following sections. 2.1. Physical model setup Many parameters can affect the maximum ds at river confluences under live-bed conditions. In order to identify the most important non-dimensional parameters for the prediction of the dsm , the authors had previously carried out a dimensional analysis [23]. This analysis showed that the maximum scour depth (dsm ), total channel flow and width downstream of the confluence (Qd and wd , respectively), flow in the main and lateral or tributary channels upstream of the confluence (Qm and Qt , respectively), width of the main and lateral or tributary channels (wm and wt , respectively), angle of river confluence (), channel bed slope (S0 ), water flow velocity and depth downstream of the confluence (vfd and dd , respectively), median size of river bed material (d50 ), bed elevation difference (Z) and live-bed (sediment) load (Qb ) were the parameters of greatest impact on the sedimentation pattern at river confluences under live-bed conditions:



f dsm , d50 , s , Qb , Qt , Qm , Qd , dd , wt , wm , wd , S0 , , ␴, , g, Z) = 0

(1)

where,  and s are the flow viscosity and density of sediment particles, respectively; g and  denote the gravitational acceleration and the surface tension force, respectively. Based on the Buckingham theory [39], equation 1 can be rewritten in the following non-dimensional form: ds =f wd

Q

t

Qd

,

Qb wm wd dd s Z , , , , , S0 , , F , Re , We , Qd wt d50 d50  wt d



(2)

where, Re is Reynolds number, We is the Weber number, and Fd is the densimetric Froude number: Fd =



vfd g (Gs − 1) d50

where Gs denotes the specific gravity.

(3)

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Fig. 3. Schematic view of the experimental setup. Adapted from Balouchi and Shafai Bajestan [23]

As in the previous study by Balouchi and Shafai Bajestan [23], wm , wt and wd were kept constant, as were Z and  (zero and 60◦ , respectively) in this study. Given that Gurram et al. [3] showed that bed slope (S0 ) has no significant effect on flow pattern under subcritical conditions, that for high Re values and rough boundaries the Re had no effect on flow pattern, and that the dimensions of the flumes were large enough that the Weber number could be ignored, Eq. (2) could be reduced to: ds = f2 wd

Q

t

Qd

,

Qb ,F Qd d

 or

∗ dsm = f2 (Qtd , Qbd , Fd )

(4)

∗ , Q , and Q where dsm td bd are the maximum score depth ratio, the flow ratio and the sediment (live-bed) load ratio, respectively. The experimental setup used by Balouchi and Shafai Bajestan [23] consisted of a main (9 m length, 25 cm width, 60 cm depth) and lateral flume (3 m length, 25 cm width, 60 cm depth), both covered with a fine sand sediment (d50 = 0.6 mm). Water flow was measured with an electronic flow meter precise to the nearest 0.01 L s−1 . A general schematic of the experimental setup is shown in Fig. 3. To simulate live-bed conditions, specific ranges for the Qtd , Qbd and Fd were tested (Table 2). To investigate the dsm under various hydraulic conditions at river confluences, 38 experimental tests (with different flow ratios, live-bed load ratios and densimetric Froude numbers) were conducted. Fig. 4 shows an example of the sedimentation topography pattern in the main flume at Fd = 8.22, Qtd = 0.2 and Qbd = 0, plotted using Surfer® software.

Table 2 Ranges (lower and upper bounds) of variables for live-bed conditions (Adapted from Balouchi and Shafai Bajestan [23]). Interval bounds

Flow ratio (Qtd ) Live-bed load ratio (Qbd ) Densimetric Froude number (Fd )

2.2. Soft computing models The results of these experimental tests served to develop the three soft computing models (MLP, RBF and M5P) to determine the dsm at river confluences under live-bed conditions. These three soft computing models were developed and coded using MATLAB® software. The data was divided into training and validation data sets (70% and 30%, respectively), and the accuracy of the maximum scour depth predictions obtained using each of the models were compared. More details on applications of MLP, RBF and M5P models can be found elsewhere [34–38,40–53]. 2.2.1. Multi-layer perceptron network (MLP) Artificial neural networks (ANN) are mathematical models which were originally developed to mimic human brain neural systems and are generally used to predict the value of an output vector based on known values in an input vector, especially where the relationships between the two are complex or non-linear [34]. A typical configuration for a multi-layer perceptron (MLP) model, a special class of ANN, is shown in Fig. 5. In an MLP network, a set of data (x1 ; x2 ; . . .) is first fed into the network through the input layer, subsequently passes through one or more hidden layers, and is finally outputted as the predicted output y in the output layer [25]. The number of hidden layers determines the complexity of the network, as a greater number of hidden layers increases the number of connections in the ANN. In an MLP, neurons receive inputs from their upstream interconnections and generate outputs by the transformation of these inputs with an appropriate nonlinear transfer function [33]. In the case of the sigmoid transfer function, for example, the output yj from the jth neuron in a layer is determined by the function:

⎡

Variable Upper

Lower

0.1 0 5.04

0.3 6.5 × 10−4 8.22

yj = f

 i=n  i=1

 wij xi

⎢ ⎢

=⎢ ⎢1 − e

⎣

 ⎤−1

 i=n 

−

wij xi i=1

⎥ ⎥ ⎥ ⎥ ⎦

(5)

B. Balouchi et al. / Applied Soft Computing 34 (2015) 51–59

55

Fig. 4. An example of sediment patterns in the main flume under live-bed conditions.

where, wij is the weight of the connection joining the jth neuron in a layer with the ith neuron in the previous layer and xi is the value of the ith neuron in the previous layer. Training of neural networks includes a learning process during which input and known output data are provided to the model and the values of the many wij are adjusted to optimize the accuracy of the model output. The ANN model training process was continued until the error index (e.g., the sum of squared errors) dropped below a pre-determined value or until the number of training epochs exceeded a specific value. After successful training, the model was tested with a validation data set and the accuracy of the model’s prediction performance was evaluated by comparing the model output with observed values [27]. In this study, a three-layer feed-forward back propagation network (FFBPN) with six neurons in the hidden layer was used for the MLP network. The optimum number of neurons in the hidden layer was determined through trial and error by comparing the accuracy of models with different numbers of hidden layer neurons using the statistical indices of mean absolute relative error (MARE) and regression coefficient (R2 ). For each neuron number, a loop was developed to consider the effect of random initial weights and

Fig. 5. A three-layer MLP neural network model.

biases. A hyperbolic tangent sigmoid function and a linear function were used as the transfer functions in the second and third layers, respectively, and the Levenberg–Marquardt optimization method was used for network training. 2.2.2. Radial basis function network (RBF) Always having three neuron layers (e.g., input, hidden and output), RBF networks are a type of ANN which restricts the inputs taken into account by each node to values within a certain “radius” of a central value. Their general structure is illustrated in Fig. 6. The number of neurons in the hidden layer was set to the number of observation data inputs (p), and a Gaussian function was used as the transfer function for this layer. The output of the jth hidden node (yj ) was calculated as [25]: yj = f

  X − Uj  2j2

(6)

where ∗ and Uj are the Euclidian norm and center of the jth radial basis function f, respectively, while ␴j represents the size of the

Fig. 6. Structure of a typical RBF neural network model.

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B. Balouchi et al. / Applied Soft Computing 34 (2015) 51–59

radius around the RBF center of the jth hidden node (Uj ) within which the value of the Gaussian function will differ significantly from zero (i.e., the model will take the input into account) [25]. Thus the Gaussian RBF center of the jth hidden node can be specified by the mean Uj and the standard deviation ␴j . The network output (zm ) is calculated through a linear weighted summation of the outputs of each node in the hidden layer: zm = f (X) =

j=p 

yj wmj

(7)

j=1

where wmj is the weight of the connection between the hidden and output nodes. Training the RBF models includes two stages: (i) determining the basis functions of the hidden layer nodes and (ii) determining the weights of the connections to the output layer. The former involves finding optimal values of Uj and ␴j . The optimal values for the spread (␴j = 10), mean, and number of neurons (16) were determined by trial and error (in the same manner as for the MLP model). A Gaussian transfer function was utilized. A more detailed description of RBF models and their applications can be found elsewhere [31,45,46,48]. 2.2.3. M5P model tree A decision tree is a tree in which each branch node represents a choice between a number of alternatives and each leaf node represents a classification or decision [36]. M5P model trees split the input progressively. The set T of examples is either associated with a leaf, or some test is chosen that splits T into subsets corresponding to the test outcomes [37]. The same process is applied recursively to the subsets [38]. Splits are based on minimizing the intra-subset variation in the output values down each branch. In each node, the standard deviation of the output values for the examples reaching a node is taken as a measure of the error of this node and calculating the expected reduction in error as a result of testing each attribute and all possible split values. The attribute that maximizes the expected error reduction is chosen. The standard deviation of the reduction (␴red ) is calculated as [35]: red = (T ) −

i=n  i=1

  Ti  (Ti ) ·   T 

(8)

where T is the set of examples that reach the node and T1 , T2 , . . ., Tn are the sets that result from splitting the node according to the chosen attribute (in the case of multiple split). Splitting in M5P ceases when the class values of all the examples that reach a node only vary slightly, or only a few examples remain [35]. This very active division process often produces overly elaborate structures that must be pruned back, for example, by replacing a sub-tree with a leaf. In the final stage, a smoothing process is performed to compensate for the sharp discontinuities that will inevitably occur between adjacent linear models at the leaves of the pruned tree, particularly for some models constructed from a smaller number of training examples. In the smoothing process, the adjacent linear equations are adjusted in such a way that the estimated outputs for similar input vectors that correspond to different equations remain close nonetheless. It should be noted that based on its algorithm the M5P model in the present study formed only one branch. More details about the M5P model and its applications can be found elsewhere [43,44]. 2.3. Analysis of model performance The validation data set was divided into three different ranges ∗ (0 ≤ d∗ < 0.24, 0.24 ≤ d∗ < 0.34, and 0.34 ≤ d∗ < 0.44), of dsm sm sm sm

Fig. 7. Assessing the performance of MLP model in predicting the maximum scour depth ratio in the training and validation stages.

and the performance of the models was analyzed separately for each data subset. The accuracy of the three models’ predictions were analyzed using several standard statistical measures, including the mean absolute relative error (MARE), root mean square error (RMSE), mean absolute error (MAE) and the linear correlation coefficient between measured and predicted values (R2 ) (Eqs. (9)–(11)):

  i=n 1 RMSE =  (Oi − Pi )2 n

(9)

i=1

 1   Oi − Pi  n i=n

MAE =

(10)

i=1





100   Oi − Pi   P  n i i=n

MARE =

(11)

i=1

where n is the number of observations in the data set and Oi and Pi are, respectively, the ith observed and predicted values of dsm . 3. Results and discussion Based on river confluence data obtained from the physical model, the three soft computing techniques (MLP, RBF and M5P) were trained and validated. In the present study where the input data were not arbitrarily large, experimental results showed the training and testing phases to be sufficient. Therefore, as crossvalidation would simply have added complexity to the proposed methodology without significantly increasing its accuracy it was omitted [47]. In addition, in order to compare the MLP, RBF and M5P models’ accuracy on an even footing, it was best to use the same train-test approach for all models. The inputs for all three soft computing models were the flow ratio, the sediment (live-bed) load ratio and the densimetric Froude number, while the output was the maximum scour depth ratio. The bounds of these three input parameters (Qtd , Qbd and Fd ) were ∗ ) is mentioned in Table 2; the bound of the output parameter (dsm between 0 and 0.44. The training and validation stage performance of the MLP, RBF and M5P models in the prediction of the maximum scour depth ratio is presented in Figs. 7–9. As can be seen from the figures, all three models were fairly successful at predicting the maximum scour depth ratio in the validation stage. For both training and validation, the MLP and RBF models R2 > 0.981 (Figs. 7 and 8), while for the M5P model R2 < 0.906 (Fig. 9),

B. Balouchi et al. / Applied Soft Computing 34 (2015) 51–59

57

Table 3 Comparing the performance of all soft computing models using statistical indices. Statistical index

RMSE

Model

Training Validation Full set

MAE

Training Validation Full set

MARE

Training

Fig. 8. Assessing the performance of RBF model in predicting the maximum scour depth ratio in the training and validation stages.

Validation Full set

∗ well, the M5P was indicating that while all models predicted dsm not as accurate as the MLP or RBF models. Values of RMSE, MAE, MARE and R2 for all three models for the training, validation and complete dataset (Table 3), were used to rank models according to accuracy (1–3, from best to worst score for each statistic), and the ranks were summed to obtain an overall accuracy score for each model. In general, the best values of RMSE, MAE, MARE and R2 were found for the MLP model (total score of 17), followed by the RBF model (19) and M5P model (34). The RBF model showed the most accurate performance of all three models in the calibration phase but fell behind the MLP model in the validation phase. Since validation phase performance is of greater importance than performance during the calibration phase, it can be concluded that, in the present study the MLP model demonstrated the best prediction ability of the three models. When the models were tested with data from each of three dif∗ ranges combined, the MLP model showed the best overall ferent dsm performance in the validation phase (total score of 16), followed ∗ by the RBF (22) and M5P (34) models; however, when each dsm range was analyzed independently, the RBF model performed best ∗ < 0.24), while the MLP model perat low scour depths (0 ≤ dsm formed better at the middle and higher end of the range of scour ∗ < 0.44) (Table 4). Thus, while overall d∗ shows depths (0.24 ≤ dsm sm a direct relationship with the Fd , Qtd , and an inverse relationship with Qbd , for high values of Fd and Qtd , the MLP model should be chosen, while for high Qbd values the RBF model should be chosen.

R2

Training Validation Full set

RBF

M5P

0.011 1 0.007 1 0.01 1

0.011 1 0.012 2 0.037 3

0.024 2 0.025 3 0.025 2

Value Rank Value Rank Value Rank

0.0081 2 0.0064 1 0.0076 1

0.0058 1 0.0071 2 0.013 2

0.019 3 0.018 3 0.019 3

Value Rank Value Rank Value Rank

2.64 2 2.38 1 2.57 2

1.9 1 2.45 2 2.1 1

6.6 3 6.03 3 6.41 3

Value Rank Value Rank Value Rank

0.9817 2 0.9939 1 0.9843 2

0.9897 1 0.9858 2 0.9886 1

0.9054 3 0.9003 3 0.9035 3

Total score

17

19

34

The optimal training times for the MLP, RBF and M5P models were 35, 235, and 0.14 s, respectively. Therefore, the optimal training time for the MLP model was almost 7-fold shorter than that of the RBF model. Although the M5P model had the lowest training Table 4 Comparison of soft computing models performance in the validation stage. Subset ∗ ∗ ∗ < 0.24); subset 2: (0.24 ≤ dsm < 0.34); subset 3: (0.34 ≤ dsm < 0.44). 1: (0 ≤ dsm Statistical index

Model MLP

RBF

M5P

Value Rank Value Rank Value Rank

0.0094 2 0.0076 1 0.0033 1

0.0072 1 0.011 2 0.007 2

0.018 3 0.033 3 0.015 3

Value Rank Value Rank Value Rank

0.0084 2 0.007 1 0.0026 1

0.006 1 0.01 2 0.0046 2

0.015 3 0.022 3 0.013 3

Value Rank Value Rank Value Rank

3.65 1 2.51 1 0.69 1

27.33 3 3.56 2 1.12 2

6.45 2 6.99 3 3.25 3

Value Rank Value Rank Value Rank

0.0835 3 0.9818 1 0.9896 1

0.157 1 0.94 2 0.98 2

0.019 2 0.54 3 0.84 3

Total score (entire dataset)

16

22

34

Score (subset 1)

8

6

10

Score (subset 2)

4

8

12

Score (subset 3)

4

8

12

RMSE

Subset 1 Subset 2 Subset 3

MAE

Subset 1 Subset 2 Subset 3

MARE

Subset 1 Subset 2 Subset 3

R2

Subset 1 Subset 2 Subset 3

Fig. 9. Assessing the performance of M5P model tree in predicting the maximum scour depth ratio in the training and validation stages.

MLP Value Rank Value Rank Value Rank

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time, its accuracy did not meet that of the MLP and RBF models (Tables 3 and 4). This suggests that the MLP model should be used ∗ at river confluences under live-bed conditions. It in predicting dsm should be noted that the time analyses provided in this paper were carried out using a processor with Intel® CoreTM i7-4500U CPU @ 2.40 GHz and 6.00 GB RAM. 4. Summary and conclusions River confluences are one of the most important zones in river networks, and predicting the dsm in these zones is of important practical interest in the fields of river engineering, fluvial geomorphology, sedimentology and navigation. However, expert systems to estimate dsm at river confluences are generally lacking. In this paper, three soft computing models—multi-layer perceptron (MLP), the radial basis function (RBF) and the M5P tree models—were used to predict the dsm at river confluences under live-bed conditions. The simulation models were trained and validated using experimental laboratory data. The following can be concluded: ∗ with fair accuracy in the • While all three models predicted dsm validation phase, the ANN models (in particular the MLP model) were the most precise. • The MLP model was the most precise model overall the full ∗ values, as well as specifically in the higher range range of dsm ∗ < 0.44); however, the RBF model of scour depths (0.24 ≤ dsm ∗ at the lower end of the proved most accurate at predicting dsm ∗ range of values tested (0 ≤ dsm < 0.24). • Error and time analyses demonstrate that the best soft computing model investigated was the very accurate and time-efficient MLP model.

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