Engineering Science and Technology, an International Journal xxx (2014) 1—10

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Engineering Science and Technology, an International Journal

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Full length article

Neuro-fuzzy GMDH based particle swarm optimization for prediction of scour depth at downstream of grade control structures

Mohammad Najafzadeh*

Department of Civil Engineering, Graduate University of Advanced Technology-Kerman, P.O. Box 76315-116, Kerman, Iran

ARTICLE INFO

Article history: Received 13 May 2014 Received in revised form 15 September 2014 Accepted 15 September 2014 Available online xxx

Keywords: Neuro-fuzzy GMDH Particle swarm optimization Grade-control structures Maximum scour depth

ABSTRACTS

In the present study, neuro-fuzzy based-group method of data handling (NF-GMDH) as an adaptive learning network was utilized to predict the maximum scour depth at the downstream of grade-control structures. The NF-GMDH network was developed using particle swarm optimization (PSO). Effective parameters on the scour depth include sediment size, geometry of weir, and flow characteristics in the upstream and downstream of structure. Training and testing of performances were carried out using non-dimensional variables. Datasets were divided into three series of dataset (DS). The testing results of performances were compared with the gene-expression programming (GEP), evolutionary polynomial regression (EPR) model, and conventional techniques. The NF-GMDH-PSO network produced lower error of the scour depth prediction than those obtained using the other models. Also, the effective input parameter on the maximum scour depth was determined through a sensitivity analysis.

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1. Introduction

Grade-control structures were basically made of gravel, stone, wood, concrete, or other materials to control bed erosion of rivers and downstream of dam spillway. These structures were utilized to convey water from higher to lower elevation with regarding to the controlling the energy and flow velocity of the water. The grade-control structures can control both erosion and scour phenomena by enhancement of stabilizing the banks and bed of rivers (or channels). In addition, they can be applied for enhancement of environmental water quality, reduction of hazardous pollution. The grade-control structures can be considered as a mean for operations of fish passage, water table control, and reduced turbidity.

The grade-control structures are commonly used to prevent excessive bed channel degradation in alluvial materials. This action can destroy the downstream of the grade-control structures. Hence, an accurate prediction of the maximum scour depth in alluvial beds has provided remarkably attentions due to the high complexity of problem. In the past few decades, field and experimental investigations have been widely carried out to identify the behavior of the scour process at the downstream of the grade-control structures (e.g., Refs. [24,7,8]). For instance, Bormann and

* Tel.: +98 9169159830.

E-mail address: moha_najafzadeh@yahoo.com. Peer review under responsibility of Karabuk University.

Julien [7] proposed an empirical equation for large-scale model of grade-control structures based on the physical properties of jet diffusion and bed material stability. Beside, D'Agostino and Ferro [8] utilized the incomplete self-similarity (ISS) theory to represent effective parameters on the scour depth at the downstream of grade-control structures. Through their investigations, it was found that the ISS method provides more accurate prediction than empirical equations.

Traditional methods based regression models are restricted to the conditions of laboratory and field studies. Occasionally, this issue produced lack of validation for empirical equations in the scour depth prediction. Recently, different artificial intelligence approaches such as artificial neural network (ANN), adaptive neuro-fuzzy inference system (ANFIS), genetic programming (GP), linear genetic programming (LGP), gene-expression programming (GEP), evolutionary polynomial regression (EPR) were carried out to predict the local scour depth at the downstream of hydraulic structures [3,11,4,5,12,18].

Among the various intelligence methods, the GMDH network is known as self-organization approach which can be employed to solve various problems in non-linear systems with large degree of complexity [2]. Recently, the GMDH networks were utilized to predict the scour depth around bridge piers and abutments (e.g., Refs. [27—29]. In case of GMDH applications, results provided relatively more accurate prediction than those yielded using empirical equations and other soft computing tools. The GMDH approach has been used to identify the behavior of non-linear

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Nomenclature P mass density of water

Ps mass density of bed material

s maximum scour depth R correlation coefficient

b width of weir RMSE root mean square error

z height of weir MAPE mean absolute percentage of error

g acceleration due to gravity BIAS difference between averages of predicted and observed

B width of channel values

h0 flow depth over weir SI scatter index

Q water discharge a, b, w weighting coefficient of Gaussian membership

d50 bed grain-size for which 50% of sampled particles are function

finer x input variable

d90 bed grain-size for which 90% of sampled particles are y output variable

finer K number of fuzzy rules

h tail water depth F value of Gaussian membership function

H difference in height from water level upstream of the weir to the tail water level E error values obtained in each partial description

systems such as explosive cutting process, tool life testing in gun drilling, constructing optimal educational test, control engineering, and engineering geology [2,6,15,33,36,1,16]. In the past two decades, Nagasaka et al. [26] used multi-stage fuzzy decision rule as neuro-fuzzy GMDH to model grinding characteristics. Also, [37] proposed the orthogonal and successive projection approach for learning of the NF-GMDH. Hwang [13] applied to forecast the unreliable mobile communication. The NF-GMDH model was configured through least square training method. From his study, it was found to be excellent for the solution of the forecasting problems. The neuro-fuzzy GMDH networks were used to identify the physical meaning of complicated problems in computer sciences. Mitrakis et al. [20] developed self-organizing neuro-fuzzy multilayered classifiers (SONeFMUC) model which was composed of small-scale interconnected fuzzy neuron classifiers (FNCs) arranged in layers. Results of performances indicated that the proposed neuro-fuzzy classifier was compromisingly compared with other well-known classification techniques. Mitrakis and Theo-charis [21] proposed an efficient structure learning algorithm to develop the SONeFMUC model. Mozaffari et al. [22] applied an evolvable SONeFMUC network with GMDH algorithm to fault diagnosis of hydraulic systems. They found that an efficient bio-inspired method can effectively determine the constructive parameters of the multi-layered neuro-fuzzy classifier. In case of the scour depth prediction, a few contributions were conducted by the NF-GMDH to characterize the scour process at downstream of sluice gates and pile groups due to waves [30,31].

In this research, the neuro-fuzzy GMDH network was utilized for predicting the scour problem due to considerable complication of scour process. In this way, the structure of neuro-fuzzy GMDH as a self-organized method was developed using the PSO algorithm for the scour depth prediction at grade-control structures. Also, the performance of the NF-GMDH-PSO algorithm is compared with gene-expression programming, evolutionary polynomial regression, and empirical equations.

2. Data presentation

Previous investigations of the scour process at the downstream of the grade-control structures demonstrated that the maximum scour depth depends on the geometric properties of weir, upstream flow condition, and physical properties of bed materials [8,11,18]. Therefore, the following equation can be used for the scour depth prediction:

s = f (b, z, B, h0; p, Ps, g, Q, d50, d90, h, H)

where s, b, z, B, h0, r, rs, g, Q, d50, d90, h, and H are the scour depth, width of weir, height of weir, width of channel, flow depth over weir, mass density of water, mass density of bed material, acceleration due to gravity, water discharge, bed grain-size for which 50% of sampled particles are finer, bed grain-size for which 90% of sampled particles are finer, tail water depth, and difference in height from water level upstream of the weir to the tail water level, respectively. To perceive the effective parameters on the scour depth, general feature of the scour phenomenon at downstream of the grade-control structures was illustrated in Fig. 1. From Fig. 1, b and l parameters are known as jet deflection angle (jet angle near bed hole) and downstream face angle of the grade-control structure, respectively.

The following equation was resulted using dimensional analysis in form of [18]:

s/z = f {H/b, b/z, h0/(h0 + z), A5q, Ago, h0/z, d9a/d5a, b/B) (2) where A50 is calculated as follows:

g( P - 1 )d50

and A90 is obtained by substituting d90 with d50 in Eq. (3).

It was proven that the use of non-dimensional parameters produced better predictions of the maximum scour depth than that of dimensional parameters (e.g., Refs. [3,11,27]. In this study, Eq. (2) was utilized to develop the NF-GMDH-PSO model for predicting the maximum scour depth at the downstream of the grade-control structures.

The datasets were divided into three groups. The first one consisted of 114 series (DS-1) from D'Agostino and Ferro [8] experiments. The second one included 88 series experiments (DS-2) from Bormann and Julien [7]. The third one contained 110 series dataset (DS-3) that 97 series of them are collected from Veronese [35], Mossa [25], Falciai and Giacomin [10], Lenzi et al. [19]. Reminding 13 series data were reported by D'Agostino and Ferro [8].

In the case of datasets conditions, Veronese [35] carried out experimental study in a flume with rectangular cross section and width of 0.5 m. Weir width of grade-control structure was 0.5 m. Also, difference between water levels in the upstream and

Fig. 1. Configuration of scour profile down stream of grade-control structure [18].

downstream of grade-control structure was fixed as 1 m within all experiments. In his study, four types of sediment with median diameters of 9.1,14.2, 21, and 36.2 mm were utilized. In addition, Q/b values between 0.001 and 0.083 m2/s produced local scour depth at the downstream of experimental models in the range of 0.055—0.22 m. Bormann and Julien [7] conducted local scour experiments with the Q/b between 0.3 and 2.5 m2/s and maximum scour depth was reported in the range of 0.1 —1.4 m. Ratio of B/b was considered as 1 in all experiments and vertical distance from the grade-control structure to flume bed was 2.13 m. They utilized two types of non-cohesive sediments with d50 of 0.3 and 0.45 mm as erodible beds at the downstream of physical model. Mossa [25] performed laboratory works for the scour depth prediction at the downstream of grade-control structures in coarse bed sediment with d50 of 2 mm. The Q/b values were regulated between 0.0045 and 0.0148 m2/s. Furthermore, weir width of structure was 0.3 m and ratio of B/b was constant value of 1 during the experiments. In his research, maximum scour depth was reported between 0.0352 and 0.145 m.

D'Agostino and Ferro [8] carried out local scour experiments at the downstream of grade-control structure. They designed two weirs for structure with height of 0.41 and 0.71. Weir width was 0.5 m and ratios of b/B were considered as 0.3 and 0.6 m. Also, they used coarse bed sediments with d50 values of 9.1 and 11.5 mm at the downstream of grade-control structure.

The Q/b values were regulated between 0.0167 and 0.167 m2/s, ds values were observed between 0.045 and 0.285 m. Falciai and Giacomin [10] measured the scour depth at the downstream of grade-control structure by using a very large-scale physical model. Water discharge per unit weir width was regulated between 1.2 and 10.2 m2/s. Width of rectangular flume, B, were in the range of 0.4—3.5 m. Different coarse bed sediments with d50 values of 0.019—0.1 m were localized at the downstream of grade-control structure. The ds valued were measured between 0.4 and 3.5 m, respectively.

Lenzi et al. [19] conducted an experimental investigation to observe the scour depth at the downstream of the grade-control structures. Width of rectangular flume, B, were between 3.3 and 25 m. They utilized a coarse bed sediment with d50 value of 8.5 mm at downstream of grade-control structure. The Q/b parameter was fixed within the range of 0.0067—0.0292 m2/s and the ds and tail water depth were measured between 0.032 and 0.53 m, 0.033—0.077 m, respectively.

Field datasets reported from Missiaga river in Italy. Ratio of b/B was 1. Coarse sediment with d50 of 60 mm was in river bed. The Q/b parameter was varied within the range of 0.369—0.531 m2/s and the maximum scour depth values were measured between 0.15 and 0.65 m.

The datasets were obtained from different conditions of investigations including small-scale experiments, field scale, large

and very large-laboratory scale. Effects of different scales on performing the predictive data-exiting approaches and regression based techniques are neglected. It may be expected to reduce capacity generalization of models. Meantime, this feature was issued in previous works [11,18]. Table 1 presents the ranges of input and output variables to develop the models. Out of the each dataset, about 75% and 25% were selected randomly to perform training and testing stages, respectively.

3. Principle of neuro-fuzzy GMDH network

The GMDH model is one of the learning machine approaches based on the polynomial theory of complex systems, designed by Ivakhnenko [14]. From this network, the most significant input parameters, number of layers, number neurons of middle layers, and optimal topology design of the network are defined automatically. Therefore, the GMDH network is included those of active neurons known as a self-organized model. The structure of the GMDH network is configured thorough the training stage with polynomial model which produces the minimum error between the predicted value and observed output. The other definitions related to the mechanism of learning method were given in literature (e.g., Refs. [23,2]). In this paper, a neuro-fuzzy GMDH model based PSO algorithm is proposed for the scour depth prediction. The structure of neuro-fuzzy GMDH is constructed automatically using a heuristic self-organized algorithm [13]. The neuro-fuzzy GMDH network is a very flexible algorithm, can be hybridized easily by other iterative and evolutionary algorithms.

Furthermore, a simplified fuzzy reasoning rule is utilized to improve the GMDH network as follows (Takashi et al., 1998):

If x1 is Fk1 and x2 is Fk2, then, output y is wk.

Gaussian membership function is used in term of Fkj which is related to the kth fuzzy rules in the domain of the jth input values xj.

Fkj (xj) = exp( - (xj - akj f/bj (4)

Table 1

Ranges of input—output parameters for the scour depth prediction.

Parameter Range

b/z 0.0322-33.33

H/b 0.0316-4.63

h0/(hg +z) 0.0343-0.884

d90/ds0 1-5.267

b/B 0.3-1

h/H 0.05-3.96

A50 0.00298-27.33

A90 0.00228-27.33

h0/z 0.0355-7.64

s/z 0.048-4

Fig. 2. A case for structure of the neuro-fuzzy GMDH.

which akj and bkj are constant values for each rules. Also, y parameter is defined as output that was expressed as follows:

J2Ukwk k=1

uk = 11 iFkj (-xj )

which wk is real value for kth fuzzy rules [13].

The neuro-fuzzy GMDH model is one of the adaptive learning networks that has hierarchical structure. In this model, each neuron has two input variables and one output. General structure of the NF-GMDH network was indicated in Fig. 2. Through Fig. 2, output of each neuron in a layer is considered as the input variable in the next layer. The final output is calculated using average of the outputs from the last layer. From Fig. 2, it can be said that the inputs from the mth model and pth layer are the output variables of the (m - 1) th and mth model in the (p - 1)th layer. Mathematical function for calculating the ypm is that,

z f I yP-1,m-1 ; yp-1,m

Fig. 3. Hybridization of the NF-GMDH network and PSO algorithm as parallel process.

mim = exp -

yp-l,m-l - aP™ "k,i

yp-1m - apm

which mpm and wpm are the kth Gaussian function and its corresponding weight parameter, respectively, that are related to the mth model in the pth layer. In addition, ap,m and bpm are the Gaussian parameters that utilized for the ith input variable from the mth model and pth layer. Also, final output is expressed using the following function:

The learning process of feed forward neuro-fuzzy GMDH is known as an iterative method to solve the complicated systems. A sample for the structure of the neuro-fuzzy GMDH technique was sketched in Fig. 2. In each iteration, error parameter for network can be obtained as follows:

parallel action in each PD. According to the Eq. (2), the NF-GMDH-PSO has eight input variables and one output. Each PD was generated in the structure of the NF-GMDH model with two fuzzy rules and six weighting coefficients fa1, a2, b1, b2, w1, w2 }.

Interaction between the NF-GMDH network and the PSO algorithm as a parallel process was illustrated in Fig. 3.

For the DS-1, 28 neurons were produced in the first layer. After that the second layer was generated using 28 neurons from the first layer. This process could be continued until minimum error of training network was obtained. The error values related to the PDs for the three datasets were shown in Fig.s 4—6. The NF-GMDH-PSO model composed of three layers was generated through optimization process. Meantime, the proposed NF-GMDH-PSO network was considered for 2nd and 3rd data sets. The PSO parameters utilized to predict the maximum scour depth were presented in Table 2. From performing the training stage, many partial descriptions for the first layer are given as follows:

for DS-1:

E = Ь* - У)'

which y*is the predicted value.

4. Development of neuro-fuzzy GMDH using PSO algorithm

The PSO algorithm based evolutionary approach has been fundamentally designed by Eberhart and Kennedy [17] which solves several of continuous and binary problems with large domains. In addition, it can easily be developed for combinatorial problems for which appropriate specialized algorithms do not exist. The original PSO formulas determine each particle as potential solution to a problem in D-dimensional space [9,34]. In this study, neuro-fuzzy GMDH model is developed using the PSO algorithm.

The basic structure of the NF-GMDH model has been consisted of partial descriptions (neurons). As mentioned in previous section, grouped parameters in form of Gaussian variables and weight related to the fuzzy rule are unknown in each partial description (PD).

The PSO algorithm was applied to optimize grouped-unknown parameters in PDs. Performing the NF and PSO algorithms is a

1.0288

f -1.0288

+1.1489 exp

b -1.489 0.5

f -1.489

0.5 exp

+ 0.5 exp

(A50 - 1.5)2

0.7387

(A50 - 0.7387)2

0.6411

0.6411

0---------------------------

1 2 3 4 5 6 7 8 9 10 11 12 1314151617 18 19 20 21 22 23 24 25 26 27 28

Number of Neuron

Fig. 4. Values of training error related to the DS-1 for three layers of proposed NF-GMDH-PSO.

=0.9729 exp

+ 0.5 exp

for DS-2:

(A50 - 1.5) 1.5

(A50 - 1.5)2 0.7661

M. Najafzadeh / Engineering Science 2

I dar. t r \

daa - 1.5

tF - 1.5

0.7661

and Technology, an International Journal xxx (2014) 1-10 and for DS-3:

0.5001 exp

+ 1.4993 exp

b- 0.8329

0.5024

b - 1.1213

- 0.8329

0.5024

0.5024

h0 - 1.1213

0.5024

0.9846 exp

+ 0.2697 exp

b- 0.568

0.3441

b - 0.25

h0 - 0.568

0.3441

- 0.25

0.3643

0.3643

0.5 exp

ht - 0.734

0.5288

(A50 - 0.734)2

0.5288

+ 0.5001 exp

(A50 - 1.395)2

0.4721 exp

+ 0.25 exp

0.8725

0.8746

(A50 - 0.8725)2

0.8746

* - 0 3859 j (A50 - 0.3859)2

0.5786

= 0.2522 exp

+ 0.4974 exp

(A50 - 0.7395)2 0.9499

0.5786

dF - 0.7395

(A50 -1)2 Vd=0

0.4722

0.9499

0.4722

0.5 exp

(A50 - 1.4466)2 0.6435

> - 1.4466

0.6435

+ 0.7908 exp

(A50 - 1.4336)2 1.0144

> - 1.4336

1.0144

The superscript and subscript of each parameter represent the number of pertaining layer and partial description, respectively. In addition, fuzzy rules used to build these PDs were given in Table 3.

5. Results and discussion

The results of the NF-GMDH-PSO networks, GEP, EPR, and regression models were presented in this section. In this way,

Fig. 5. Values of training error related to the DS-2 for three layers of proposed NF-GMDH-PSO.

Fig. 6. Values of training error related to the DS-3 for three layers of proposed NF-GMDH-PSO.

correlation coefficient (R), root mean square error (RMSE), scatter index (SI), BIAS, and mean absolute percentage of error (MAPE) can be defined to evaluate error indicators in the training and testing stages (e.g., Refs. [3,27]):

Pill ( Yi(Actual) - Y(Actual) )(Yi(Model) - Y(Model)

^jPMU(Yi,

(Actual) - Y (Actual) ) ' P ill ( Yi(Model) - Y (Model)

RMSE =

Pill (Yi(Model) - Yi(Actual))

MAPE = —

Pill I Yi(Model) - Yi(Actual) I

srM Y Z^i=1 Yi(Actual)

BIAe ^ (/i(Model) - Yi(Actual)

= 4-i M

(1 /M ) Pill yi(Actual)

The statistical results of the NF-GMDH-PSO networks for training and testing stages were presented in Table 4. Trough the training stages, it can be found that the NF-GMDH-PSO network developed by the DS-1 produced more accurate performance (R = 0.95, RMSE = 0.049, MAPE = 0.154, BIAS = 0.0003, and SI = 0.168), compared to the other NF-GMDH-PSO models. For the DS-2, the training result of the NF-GMDH-PSO algorithm has produced lower error parameter (RMSE = 0.216 and MAPE = 0.45) than that NF-GMDH-PSO developed by the DS-3 (RMSE = 0.39 and MAPE = 0.77). In contrast, the NF-GMDH-PSO network developed by the DS-2 has lower correlation coefficient (R = 0.83) and higher scatter index (SI = 0.76) than those NF-GMDH-PSO predicted using the DS-3 (R = 0.66 and SI = 0.4).

In the testing stages, the NF-GMDH-PSO network developed by the DS-1 predicted the maximum scour depth with more accurate performance (R = 0.95, RMSE = 0.0494, MAPE = 0.47, BIAS = -0.00041, and SI = 0.169), compared to the NF-GMDH-PSO models given by the DS-2 and DS-3. For the DS-3, the performance of the NF-GMDH-PSO produced the maximum scour depth with higher correlation coefficient (R = 0.84) and relatively higher error (RMSE = 0.307 and MAPE = 1.27) than the NF-GMDH-PSO model proposed by the DS-2 (RMSE = 0.214 and MAPE = 1.62). The NF-GMDH-PSO model for DS-3 provided the maximum scour depth with quite lower of BIAS (0.0086) than that obtained NF-GMDH-PSO using the DS-2 (BIAS = 0.234). Scatter plots between predicted and observed scour depth values for both training and testing stages by the NF-GMDH-PSO models were indicated in Figs. 7 and 8, respectively. Furthermore, a traditional equation given by D'Agostino and Ferro [8] was used to evaluate the maximum scour depth prediction:

where Y,(Moael) is the predicted values (network output), Victual) is the observed values (target), and M is the total of events.

Table 2

Values of the PSO properties for predicting the scour depth at downstream of grade-control structures.

Parameter Range (DS-1) Range (DS-2) Range (DS-3)

Omega 0.04—0.09 0.04—0.09 0.04—0.09

Number of particles 40 40 40

Number of variables 6 6 6

Maximum iteration 100 100 100

Error 0.00001 0.00001 0.00001

C1 and C2 1.5 and 2.5 1.5 and 2.5 1.5 and 2.5

a,, b,, and w, 0.5—1.5 0.25—1 0.5—1.5

s ¡z =0.54(b /z)0'593 (h/H)-0126 (A50)

(d90/d50)

Eq. (25) results in a relatively high error (RMSE = 0.134 and MAPE = 1.123) and lower accuracy (R = 0.66) for the maximum scour depth compared to those obtained by NF-GMDH-PSO for DS-1. Originally, Eq. (25) was validated for [38] experimental conditions and this lack of validation indicated higher sensitivity to the range of datasets in comparison with the NF-GMDH-PSO model.

Moreover, the lack of validation for traditional methods is related to the limitation of effective parameters range, which

Table 3

Fuzzy rules for the three PDs.

The Ist neuron of the Ist layer The 2nd neuron of the Ist layer The 3rd neuron of the Ist layer

Fuzzy rules 1. If (b/z is b/z mfl) and (h0/z is h0/z mfl) 1. If (h0/z is h0 /z mfl) and (A50 is A50 mfl) 1. If (A50 is A50 mfl) and (d90/d50 is d90/d50 mfl)

then ((S/z)l is (S/z)l mfl) then ((S/z)] is (S/z)] mfl) then ((S/z)3 is (S/z)] mfl)

2. If (b/z is b/z mf2) and (h0/z is h0/z mf2) 2. If (h0/z is h0/z mf2) and (A50 is A50 mf2) 2. If (A50 is A50 mf2) and (d90/d50 is d90 /d50 mf2)

then ((S/z)l is (S/z)l mf2) then ((S/z)] is (S/z)] mf2) then ((S/z)] is (S/z)] mf2)

Table 4

Results of performances for training and testing stages of NF-GMDH-PSO.

Training stage R RMSE MAPE BIAS SI

DS-1 0.95 0.047 0.154 0.0003 0.168

DS-2 0.83 0.39 0.77 0.041 0.76

DS-3 0.66 0.216 0.45 -0.0065 0.40

Testing stage R RMSE MAPE BIAS SI

DS-1 0.95 0.0494 0.47 -0.00041 0.169

DS-2 0.84 0.307 1.22 0.234 0.4

DS-3 0.66 0.214 1.62 0.0086 0.46

s/z(Observed)

Fig. 7. Scatter plot of observed and predicted scour depth for training stages.

s/z(Observed)

Fig. 8. Scatter plot of observed and predicted scour depth for testing stages.

commonly are failed to present the accurate feature for physical behavior of the scour process [11,12].

Guven and Gunal [11] proposed a model based GEP approach for predicting the maximum scour depth at the downstream of the grade-control structures. The GEP model was utilized to evaluate the maximum scour depth. The performance of the GEP model indicated higher error (RMSE = 0.45, MAPE = 4.37, BIAS = 0.358) than the NF-GMDH-PSO model given by the DS-1. In addition, Laucelli and Giustolisi [18] presented an equation for predicting the maximum scour depth at downstream of the grade-control structures using the EPR model. The EPR model yielded quite remarkable over-predictions of the scour depth (RMSE = 0.888 and MAPE = 10.8), compared to the NF-GMDH-PSO models. BIAS and SI parameters as metric values for the EPR model were resulted 0.883 and 3.041, respectively. From the statistical error parameters, the EPR model demonstrated significant over-predictions of the maximum scour depth in comparison with the GEP method. The results of performances for comparisons were given in Table 5. Also, scatter plots between predicted and observed scour depth values for evaluating the models given by GEP, EPR, and NF-GMDH-PSO models from DS-1, was illustrated in Fig. 9. In this study, the

Table 5

Evaluation of different models to predict the scour depth at downstream of grade-control structures.

Methods Source R RMSE MAPE BIAS SI

NF-GMDH-PSO Obtained 0.95 0.0494 0.47 -0.00041 0.169

using DS-1

GEP [11] 0.925 0.45 4.37 0.358 1.55

EPR [18] 0.88 0.888 10.8 0.883 3.042

Eq. (25) [8] 0.66 0.134 1.123 0.0743 0.458

Fig. 9. Scatter plot of observed and predicted scour depth for evaluating the different models.

Table 6

Results of sensitivity analysis for NF-GMDH-PSO model given using DS-1.

Functions R RMSE MAPE BIAS SI

s/z = f (b/z, ho/(ho + z), A50, A90, h0/z, dgo/dso, b/B) 0.955 0.044 0.418 -0.0016 0.151

s/z = f (H/b, ho/(ho + z), A50, A90, ho/z, d9o/d5o, b/B) 0.854 0.078 0.771 -0.021 0.267

s/z = f (H/b, b/z, A5o, A9o, ho /z, d9o/d5o, b/B) 0.924 0.0634 0.652 0.0317 0.217

s/z = f (H/b, b/z, ho / (ho + z), A», d9o /d5o, b/B) 0.85 0.07 0.61 -0.027 0.254

s/z = f (H/b, b/z, ho / (ho + z), A;o, ho/z, d9o /d5o, b/B) 0.948 0.0486 0.492 0.0151 0.166

s/z = f (H/b, b/z, ho / (ho + z), A;o, A9o, d9o /d5o, b/B) 0.938 0.0542 0.523 0.0062 0.179

s/z = f (H/b, b/z, ho / (ho + z), A5o, A», ho /z, b/B) 0.918 0.0613 0.55 0.21 -0.015

s/z = f (H/b, b/z, ho / (ho + z), A;o, Ao, ho /z, d9o /d5o) 0.911 0.065 0.605 -0.012 0.222

physical reasons for the failure of the traditional method were investigated. Efficiency of Eq. (25) was evaluated by the DS-1. Correlation coefficient given by Eq. (25) is quite low value whereas values of other statistical error parameters approved that Eq. (25) can be considered as more efficient approach in comparison with the GEP and EPR models. Another interesting point is that, ranges of d90/d50, b/B, and b/z for testing stages have remarkable restrictions. For the DS-1, ratio of d90/d50 are constant values (1.53). Values for ratio of b/B are 0.3 and 0.6. Also, b/z values are 0.365 and 0.7317. Exponents of d90/d50, b/B, and b/z are -0.856, -0.751, and 0.593, respectively. These values indicated that non-dimensional parameters affected considerably on the s/z predicted by Eq. (25). Therefore, it is evident that Eq. (25) has quite lower generalization capacity for the maximum scour depth prediction. Performing the traditional approaches is encountered with major limitations related to the datasets range. Also, different scales of experimental and field investigations affected on the efficiency of the models. Datasets (observed data in laboratory and filed studies) used in this study were validated in different scales and inherently produced errors in the predictions. Guven and Gunal [11], Guven and Gunal [39], Laucelli and Giustolisi [18] have not considered these issues in their studies. In the present research, the DS-1 and DS-2 were observed with small and large scale of experiments, respectively. The third one was collected for both conditions of small and field scales. In addition, the NF-GMDH-PSO model demonstrated high sensitivity to the scales of experimental studies.

6. Sensitivity analysis

To determine the importance of each input variable on the maximum scour depth, the NF-GMDH-PSO model developed by the DS-1 was applied to perform sensitivity analysis technique. The analysis was conducted such that, one parameter of Eq. (2) was eliminated each time to evaluate the effect of that input on output. Results of the analysis demonstrated that b/z (R = 0.854, RMSE = 0.078, MAPE = 0.771, BIAS = -0.021, and SI = 0.267) is the most effective parameter on the maximum scour depth whereas the H/b (R = 0.955, RMSE = 0.044, MAPE = 0.418, BIAS = -0.0016, and SI = 0.151) has the least influence on the s/z for the NF-GMDH-PSO model, respectively. The other effective parameters on the s/z parameter includeA50, b/B, d90/d50, h0/z, h0/h0 + z, and A90 which were ranked from higher to lower values, respectively (Table 6).

7. Conclusion

In this study, neuro-fuzzy GMDH network was presented for a new application, namely predicting the maximum scour depth downstream of the grade-control structures. The neuro-fuzzy GMDH network was developed using the PSO algorithm for training stage. Eight effective parameters on the maximum scour depth were defined using the dimensional analysis. The NF-GMDH-PSO model was designed using three layers. Each layer included 28

PDs. Also, GEP, EPR, and traditional equation models were considered for comparison with performances of the NF-GMDH-PSO networks. Performing the training and testing of the NF-GMDH-PSO network were carried out using three datasets (DS-1, DS-2, and DS-3). Performances of the NF-GMDH-PSO model for the DS-1 indicated more accurate prediction (R = 0.95 and RMSE = 0.0468) than those developed using the DS-2 and DS-3. Statistical error parameters indicated that the NF-GMDH-PSO model produced quite better scour depth prediction (BIAS = -0.00041 and SI = 0.169) compared to those obtained by the GEP and EPR models. Furthermore, statistical results demonstrated that the GEP model presented by [11] yielded more accurate scour depth prediction (RMSE = 0.45 and MAPE = 4.37), compared to the EPR approach. From the performances, it is concluded that Eq. (25) provided relatively higher error (RMSE = 0.134 and MAPE = 1.123) and lower accuracy (R = 0.66), compared to the NF-GMDH-PSO models. Also, the results of sensitivity analysis indicated that b/z is the most important parameter in the modeling of the maximum scour depth by the NF-GMDH-PSO network. This study was proven that the NF-GMDH-PSO model as an adaptive learning network can be used as a powerful soft computing tool for predicting the scour depth downstream of the grade-control structures as well as the other artificial intelligence methods.

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