A methodological approach of predicting threshold channel bank profile by multi-objective evolutionary optimization of ANFIS

Accepted Manuscript A methodological approach of predicting threshold channel bank profile by multi-objective evolutionary optimization of ANFIS

Azadeh Gholami, Hossein Bonakdari, Isa Ebtehaj, Bahram Gharabaghi, Saeed Reza Khodashenas, Seyed Hamed Ashraf Talesh, Ali Jamali PII: DOI: Reference:

S0013-7952(17)30841-4 doi:10.1016/j.enggeo.2018.03.030 ENGEO 4808

To appear in:

Engineering Geology

Received date: Revised date: Accepted date:

31 May 2017 27 March 2018 28 March 2018

Please cite this article as: Azadeh Gholami, Hossein Bonakdari, Isa Ebtehaj, Bahram Gharabaghi, Saeed Reza Khodashenas, Seyed Hamed Ashraf Talesh, Ali Jamali , A methodological approach of predicting threshold channel bank profile by multi-objective evolutionary optimization of ANFIS. The address for the corresponding author was captured as affiliation for all authors. Please check if appropriate. Engeo(2018), doi:10.1016/j.enggeo.2018.03.030

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ACCEPTED MANUSCRIPT A Methodological Approach of Predicting Threshold Channel Bank Profile by Multi-objective Evolutionary Optimization of ANFIS

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Reza Khodashenas 4, Seyed Hamed Ashraf Talesh5, Ali Jamali5

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Azadeh Gholami 1, Hossein Bonakdari 1, 2,*, Isa Ebtehaj 1, 2, Bahram Gharabaghi 3, Saeed

Department of Civil Engineering, Razi University, Kermanshah, Iran.

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Environmental Research Center, Razi University, Kermanshah, Iran.

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School of Engineering, University of Guelph, Guelph, Ontario, N1G 2W1, Canada

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Department of Water Engineering, Ferdowsi University of Mashhad, Mashhad, Iran.

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Department of Mechanical Engineering, University of Guilan, Rasht, Iran.

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Corresponding author: Hossein Bonakdari, Phone: +98 833 427 4537, Fax: +98 833 428 3264,

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E-mail: [email protected]

ABSTRACT

More accurate stable channel design methods are necessary for analyzing the complex bank profile cross sections of alluvial channels that achieve equilibrium state. This study introduces a new hybrid method that combines an adaptive neuro-fuzzy inference system (ANFIS), Differential Evolution (DE) algorithm and Singular Value Decomposition (SVD) to predict the

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ACCEPTED MANUSCRIPT bank profile of a threshold channel. SVD and DE serve to optimally determine the consequent linear parameters and antecedent nonlinear parameters of the TSK-type fuzzy rules in ANFIS. Moreover, by defining two objective functions and using the Pareto curve, the tradeoff of function is selected as the optimal modeling point. The authors carried out laboratory

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experiments at four discharge rates of 1.16, 2.18, 2.57 and 6.20 L/s to measure the coordinates of

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the points in a stable channel boundary profile. The ANFIS-DE/SVD results are compared with

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the results of a simple ANFIS model and 7 previous research works (based on numerical and experimental models and mathematical principles). The RMSE error index (0.019) of the ANFIS-

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DE/SVD model is lower than the ANFIS model (0.027), but both models outperform the best

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available model (CKM, Cao and Knight, 1998) (RMSE = 0.120). The ANFIS-DE/SVE model is more accurate for larger y (water surface level) values than the simple ANFIS model. Moreover,

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the superior performance of the hybrid ANFIS-DE/SVD over the simple ANFIS model is more

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pronounced at greater discharge rates. ANFIS-DE/SVD estimates the bank profile shape of a stable channel as a third-degree polynomial equation, which can be used to design and

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implement stable alluvial channels.

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Keywords: Bank profile shape, Discharge, Hybrid method, Pareto curve, Threshold channel

1. Introduction

Natural channels and streams tend to reach stability at the banks (Huang et al., 2014), while there is ongoing sediment transfer in the central region of the channel bed. Most researchers define a stable channel as a channel whose bank profile is in a state where the particles at the wetted perimeter are in incipient motion (Dey, 2001). Hence, the term ‘threshold channel’ is used in

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ACCEPTED MANUSCRIPT conditions where channel widening is the ultimate state (Yu and Knight, 1998). Stable channel also refers to a state when the lateral diffusion of momentum due to turbulence from the channel centre towards the banks leads to shear stress redistribution to the channel boundaries (Vigilar and Diplas, 1997). The above definition is one of the most descriptive of a channel that will

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achieve equilibrium state (Diplas, 1990). The cross-sectional shape of a channel in stable state is

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very important for the bank profile’s geometric formation. Fig. 1 represents the characteristics of

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a bank profile. In the figure, x is the transverse distance from the channel centre, y is the vertical boundary point level in stable channel form, h is the flow depth at the channel centre in stable

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state and T is the water surface width.

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Fig. 1. Bank profile characteristics

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A few researchers have noted the importance of channel bank profile shape in stable state and

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suggested various optimized shapes. Glover and Florey (1951) presented the primary assumptions for stable channels. They defined equilibrium as the lack of particle motion at all

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channel points, in which case the bank profile is referred to as cosine. The movement of particles

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on the channel bed is incompatible with the notion of a stable channel and the stable channel paradox is derived (Ikeda et al., 1988). Parker (1978) resolved particle movement on the channel bed by using lateral momentum transfer from areas with high momentum towards areas with low momentum and non-uniform distribution of shear stress, where the stress on the channel bed and banks is more and less than the critical stress allowed, respectively. Therefore, particle movement on the channel bed is justified. Ikeda (1981) was the first to suggest an exponential channel shape. Pizzuto (1990) modeled the final widening process in a channel and introduced

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ACCEPTED MANUSCRIPT an exponential equation for stable channel shape. According to Pizzuto (1990), stable channel shape changes over time to a cosine bank profile shape. Diplas and Vigilar (1992) suggested the fifth-degree polynomial function for a bank profile with a numerical solution to the momentum equilibrium and force equilibrium equations for sediment particles in impending motion

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(threshold channel). Vigilar and Diplas (1997) provided a numerical model and considered the

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lateral diffusion of momentum induced by turbulence to investigate a channel with "stable bank

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and mobile bed" that consists of a flat bed and two curving bank regions. Then Vigilar and Diplas (1998) verified their numerical model using experimental data and suggested a three-

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degree polynomial bank profile. Yu and Knight (1998) presented a numerical model by

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combining the depth averaged momentum equation with the resistance laws and threshold conditions. This combination yielded a geometric model based on physics for threshold channel

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design. Cao and Knight (1998) provided a numerical model to solve the stable channel paradox

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in terms of the secondary flow portion in the lateral diffusion of momentum. Moreover, combining the bank profile equations based on Cao and Knight’s (1997) entropy concept, flow

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continuity condition and frictional resistance with sediment transport relations produces a

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geometric model for stable alluvial channels. Babaeyan-Koopaei and Valentine (1998) used an experimental model and studied the bank profile of a straight stable channel. They suggested

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hyperbolic functions for the bank profile shape of stable channels. Dey (2001) proposed a polynomial shape using the power law and the model is highly compliant with Parker (1978), Diplas and Vigilar (1992) and Yu and Knight’s (1998) models. Khodashenas (2016) compared the performance of 13 previous models using experimental sets. Their comparison results indicate that all prior models produced significant error in predicting the bank profile of a threshold channel and thus required further study. Preceding studies therefore denote the

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ACCEPTED MANUSCRIPT importance of predicting stable channel bank profile shape. Most studies reported above are based on either experimental and numerical models or mathematical principles. Today, hydrology and hydraulics scientists widely use artificial intelligence methods as powerful timeand cost-saving tools (Emiroglu et al., 2010, 2011; Kisi et al., 2013; Zhang et al., 2015; Ebtehaj

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et al., 2016; Gholami et al., 2017a; Huang et al., 2017). Madvar et al. (2011), Taher-Shamsi et al.

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(2013) and Bonakdari and Gholami (2016) used artificial neural networks (ANN), while

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Gholami et al. (2017b) and Shaghaghi et al. (2017) used a group method of data handling (GMDH) neural model to predict stable channel geometry. All models estimate the geometry and

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hydraulic parameters, such water surface width (w), depth at the centreline (h) and longitudinal

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water surface slope (s) for stable channel design. However, the mentioned researchers did not report any bank profile results nor proposed shapes. More recently, combining fuzzy logic with

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neural networks has led to a new neuro-fuzzy system called the Adaptive Neural Fuzzy Inference

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System (ANFIS) (Khanlari et al., 2012; Mishra and Basu, 2013; Basarir et al., 2014; Kayabasi et al., 2015; Gholami et al., 2017c). The ANFIS model is one of the most common and powerful

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techniques for modeling complicated, nonlinear problems (Kulatilake et al., 2010; Azimi et al.,

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2017; Öge, 2017). However, the number of IF-THEN fuzzy rules in the ANFIS network structure becomes very large with the increment in membership functions (MF) in the system’s

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input apace. In this case, a large number of IF-THEN fuzzy rules is not desirable to map inputs to output parameters due to the overfitting problem that reduces the ANFIS model’s generalization ability for predicting unforeseen data. To achieve highly precise models for solving nonlinear problems, all inconsistent objective functions should be considered, which have a remarkable effect on modeling (Coello et al., 2007). In multiobjective optimization the existence of different objective functions leads to a set of Pareto optimal solutions (Shaghaghi et al., 2017). Thus, a

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ACCEPTED MANUSCRIPT program was coded in MATLAB to design a hybrid ANFIS model associated with two evolutionary algorithms. The authors found no study related to the use of these algorithms to predict stable channel bank profile shape. Therefore, the main objective of this study is to estimate the cross-sectional profile shape of a

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threshold channel bank profiles using a new hybrid multiobjective method. In this method, the

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multiobjective Differential Evolution (DE) algorithm and Singular Value Decomposition (SVD) are applied in combination with the ANFIS network (ANFIS-DE/SVD). DE is employed to

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optimally select the variables of the Gaussian membership function’s nonlinear coefficients in the antecedent part of ANFIS. Moreover, the SVD technique is utilized to compute the linear

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coefficients of the consequent part of the ANFIS network. Training error (TE) and prediction

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error (PE) are considered significant conflicting objectives in the optimal multiobjective ANFIS network design. Experiments were conducted to gain a wide range of data for different discharge

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rates. The bank profile shape result is compared with existing models (based on analytical and

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numerical investigations) and a simple ANFIS model.

2. Hybrid ANFIS with DE and SVD (ANFIS-DE/SVD)

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2.1. Overview of ANFIS

An adaptive neuro-fuzzy inference system (ANFIS) includes a set of Takagi-Sugeno-Kang (TSK) fuzzy rules to map the input onto the output space (Sugeno and Kang, 1988). The main purpose of an ANFIS model is to find a function (f) using n inputs, one output and a number of M different observational samples: n

Rulel : If x1 is Al( j1) , x2 is Al( j2 ) , ...AND xn is Al( jn ) Then y   wil xi  w0l i 1

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(1)

ACCEPTED MANUSCRIPT Where l  1,2,..., n, j i  1,2,..., r, r is the number of fuzzy sets in each rule,



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W l  w0l , w1l , w2l ,..., wnl is the parameter set of the consequent part and Al j is the ith membership function (MF) for the lth rule. The entire fuzzy set in space xi can be expressed as follows:

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Ai   A1 , A2 ,..., Ar 



(2)

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The Gaussian membership function (MF) is utilized in this study because it is used more often

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than other MFs, such as triangular, bell shaped, etc. (Ali et al., 2015; Alemdag et al., 2016; Liao, 2017; Bouarroudj et al., 2017; Ebtehaj et al., 2018; Gholami et al., 2017c). The MF value range

A( j )

xi   0 . Each fuzzy set





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defined as follows:

 A j  xi , c j ,  j   exp xi  c j 2 2 2j 

A j   j  1,2,..., r  of the Gaussian MF is

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degree would not be zero 

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is  i ,i i  1,2,..., n . For each xi   i ,i  there is A(j) in Eq. (2), otherwise the MF

(3)

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where σj and cj are the tunable variance and centre in the antecedent part, respectively.

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The number of parameters related to the antecedent part of the fuzzy rules is NP = n × r, where r is the number of fuzzy sets in each antecedent part and n is related to the input vector dimension.

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The fuzzy rule given as Eq. (1) is a fuzzy relationship in an U × R space, where A(j) is a fuzzy set

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in Ui U  U1  U 2  ...  U n  . The Mamdani algebraic product concept is used to calculate the order of each TSK-type IF-THEN fuzzy rule. Next, a singleton fuzzifier is used with a product inference engine to sum the portion of each rule in a fuzzy system that contains N fuzzy rules as in Eq. (1). The following linear regression is expressed: f X  

n

 px X yl  D

(4)

l 1

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ACCEPTED MANUSCRIPT where D is the difference between the actual value (y) and estimated value f(X), and p is defined as:

i 1 μ A n

(ji) l

f X   N    l 1 

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 μ (ji) ( xi )  i 1 A l  n

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

( xi ) is the membership degree of input xi in the relationship with the lth linguistic

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( j) A l

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where 

( xi )

value related to fuzzy rule Al ji  . The above equation can be rewritten in matrix form for a given

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m input-output data pair  X i , yi , i  1,2,..., m as follows:

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Y  P.W  D

(6)

where P   p1 , p2 ,..., ps   R M  S , S  N n  1 and W  w1 , w2 ,..., wS   R S . The governing T

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 

1 T

P Y

(7)

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W  PT P

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equation for estimating the bank profile shape of a stable channel is:

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Correcting the consequent part of the TSK-type fuzzy rule coefficients using the above method results in more accurately approximated data along with a minimized vector D. Direct solutions

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to normal equations are susceptible to error rounding and highly susceptible to singularity. As a

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result, a method is required to prevent the error problem and optimize the estimation errors. In the present study, the Singular Value Decomposition (SVD) method is used as a powerful computational technique to optimally estimate the linear coefficients in the conclusion part of the fuzzy rules and to prevent singularity. In addition, to optimize the nonlinear coefficients in the Gaussian MF related to the antecedent part of the TSK-type fuzzy rules, an evolutionary algorithm, i.e. Differential Evolution (DE) is used. Fig. 2 represents the flowchart of the two-

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ACCEPTED MANUSCRIPT objective ANFIS design using the DE and SVD algorithms as part of the hybrid ANFIS-DE/SVD model. The following section explains ANFIS-DE/SVD in detail.

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Fig. 2. Hybrid ANFIS-DE/SVD model flowchart

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2.2. Application of DE in the ANFIS design

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2.2.1. DE algorithm

If the number of rules in the system is considered n-input-single-output = Rʹ, the actual Rʹ(n + 1)

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value of the Gaussian MF parameters {σ, c} are produced randomly at the beginning as a series

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of conjunct substrings of binary numbers in the antecedent part of ANFIS. σ and c are the standard deviation and centre (respectively) of the Gaussian MF, which are optimized through

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training. In this study, the values of {σ, c} in different substrings are optimized for threshold

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channel bank profile shape modeling using the DE algorithm. The DE algorithm was introduced by Storn and Price (1997). DE is a population-based algorithm that employs the natural evolution

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process to find optimum solutions. The main strategy of this algorithm is to generate new

Mutation:

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and selection.

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populations using different operators, the most frequent of which include mutation, crossover

In addition to increasing solution vector diversity, this function boosts the exploration ability of the solving space in the DE algorithm. For each parent xik (i  1,2,..., M ) (where M and k are the maximum numbers of parents and generations) related to generation k (k  1,2,..., K ) , a trial

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ACCEPTED MANUSCRIPT vector is created with the mutation target vector. The trial vector (vik 1 ) is calculated using the mutation operator as follows:



vik 1  xik  F  xik1  xik2

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(8)

where vik+1 is the trial vector of the ith factor in the k+1th generation, xik is the ith parent in the kth

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generation and F is the mutation scale factor.

Crossover:

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To create new offspring, uik, the DE algorithm uses a discrete recombination method. The components of the parent vector, vik, are combined with the trial vector, forming the offspring

r  CRor  j  r1~ D 

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k 1  vi , j  k   xi , j

Otherwise

(9)

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u

k 1 i, j

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vector as follows:

where CR is the crossover rate that is in the [0, 1] range, r1~ D is a random integer in [1, D] and D

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is the dimension of a vector.

Selection:

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The generated offspring is evaluated after the mutation and crossover operations. If the fitness of the parent is worse than that of the generated offspring, parent xik is replaced by the generated offspring uik+1.

uik 1 if f (uik 1 )  f ( xik ) xi, G 1    xik Otherwise

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2.2.2. Distribution control of the mutation scale factor (F)

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ACCEPTED MANUSCRIPT In this study, the F value is computed as the parent vector’s position based on its difference from the optimal solution. A schematic diagram of the Pareto front to determine the optimum F value is presented in Fig. 3. The F value determined based on the population distribution for different Pareto front values is inversely related to the generation value, and in each population different

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values are allocated to F according to the Pareto front. Due to the population growth in the lower

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front during the gradual evolution of higher generations, the values assigned to F approach zero.

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This means the search space exploration is limited to all possible spaces around the optimal

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value.

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Fig. 3. Mutation scale factor (F) based on the optimal solution proximity in the Pareto front

2.3. Application of SVD in the ANFIS design

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As a direct solution to Eq. (6) may lead to singularity, SVD is utilized in the current study to

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estimate the optimal parameters in the consequent part of the TSK-type fuzzy rules (Golub and Reinsch, 1970). With this method, matrix P  R M S is transformed into three different matrices,

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including an orthogonal matrix (V  R S S ) , a diagonal matrix with non-negative components

Therefore,

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(Q  R S S ) and a column-orthogonal matrix (U  R M  S ) .

P  UQV T

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First, the problem of selecting parameter W is alleviated in Eq. (6) to find the retrieved inversion of matrix Q, which is expressed as q and defines the equation for calculating W as follows:   1 W  V diag  qj  

 T U Y  

(12)

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3. Experimental model For this study, the authors obtained four datasets from the hydraulic laboratory of the Department of Civil and Geological Engineering, University of Saskatchewan, Canada. Two

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parameters, namely the depth or vertical alignment of points located at the channel boundary (y)

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and the transverse distance of points on the channel axis (x) were measured in balanced channel mode. The flume used for the channel was 20 × 1.22 × 0.6 m (length, width and height) with a

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longitudinal channel slope (S) of 0.0023. The channel bottom was filled with uniform sand (d50 = 0.53 mm). The cross-section shapes in the experiments were triangular and trapezoidal. The

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experiments were done at four discharge rates of 1.157, 2.18, 2.57 and 6.2 l/s. The laboratory

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specifications for the four experimental sets are given in Table 1. A magnetic flowmeter was used to measure the discharge. The experiments were carried out such that in each run, the

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desired discharge passed through the channel and the rate was kept constant until balanced state

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was reached in the cross section and channel plan. Then the flowing water was withdrawn from the channel and the channel geometry and coordinates of the different points on the stable

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channel boundary were measured in different cross sections. The water surface and channel bed

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levels were measured using a point gage at transverse distances of 8, 9 and 11 m from the channel entrance. The point gauge measurements were checked with a laser gauge. In the experiments, the flow parameters measured in stable state at two cross sections were: 1. The end section upstream of the channel with zero bed load 2. The end section downstream of the channel with maximum bed load at the channel bottom (wider section).

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ACCEPTED MANUSCRIPT The dimensions and shape of the stable channel measured in the first run (upstream section) were used in this study. A schematic of the laboratory flume is shown in Fig. 4.

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Table 1 Experimental characteristics

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Fig. 4. Schematic of flume used in the experiments

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4. Parameter definitions and overview of existing models

The various parameters used in this study are defined in this section. Fig. 1 displays the bank

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profile shape. In this figure, the parameters are made dimensionless as follows:

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y*=y/h and x*=x/h, T*=T/h.

where x* = non-dimensional transverse distance, y* = non-dimensional balance-border and T* is

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the non-dimensional water surface width.

respectively: (S s  1)d50

T 

  cr

 c

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h

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 Vigilar and Diplas (1997) presented the following relationships to determine h, T*and  cr ,

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T  16.1814  3  44.3206  2  43.5548   21.1496 h

  cr  0.0223  3  .01481  2  0.31403   1.031

(14) (15)

where Ss =ρs, ρ = relative density, ρs = mass density of the sediment, ρ = mass density of the water, d50 = mean sediment size (m) and  cr = the critical bed stress depth that can be obtained

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ACCEPTED MANUSCRIPT with Eq. (15), where μ = tan(θ) is the submerged coefficient of the sediment particles’ static friction and θ is the angle of repose of the sediment. Yu and Knight (1998) presented an empirical relationship for μ in uniform sediment:





  tan 0.302log 100d 50 5  0.126log 100d 50 4  1.811log 100d 50 3  0.57log 100d 50 2  5.952log 100d 50   37.52

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(16)

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Here τ*c is a dimensionless value that can be obtained from Van Rijn’s (1984) equations, where  S  1g  D*  d 50  s 2  is the dimensionless particle size and ν is the kinematic viscosity of water.   

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Table 2 presents the relationships examined in this study, which are Glover and Florey’s (1951)

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model (GFM), Ikeda’s (1981) model (IKM), Pizzuto’s (1990) model (PIM), Vigilar and Diplas’ (1998) model (VDM), Cao and Knight’s (1997) model (CKM), Babaeyan-Koopaei and

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Valentine’s (1998) model (BVM) and Dey’s (2001) model (DEM).

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Table 2 Models studied in this research

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5. Model performance evaluation using statistical indices In the present study, artificial intelligence models are compared with earlier models and

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experimental values from different statistical equations based on regression. The deviation of the predicted values from the experimental data is measured with the error indices Mean Absolute Relative Error (MARE), Mean Absolute Percentage Error (MAPE), Root Mean Squared Error (RMSE), Mean Absolute Error (MAE), Mean Error (ME), determination coefficient (R2), BIAS and ρ. Error indices RMSE, MARE, MAPE, MAE and ME include units and scales similar to the predicted and experimental values, hence they are appropriate for comparison. Index values closer to zero represent lower error between the predicted values and experimental data. R2 is

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ACCEPTED MANUSCRIPT used to find the correlation between the predicted and experimental values, with values closer to 1 signifying greater model accuracy. The BIAS index is an indicator of model overestimation or underestimation, and in the present study, positive and negative BIAS values indicate model underestimation and overestimation respectively. The ρ index is very suitable for model

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performance evaluation due to the combination of composite error and correlation coefficient

 Oi  Pi   Oi  i 1

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 Pi  Oi

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M

1

i 1

M

1

ME 

i 1

M

(Oi  Pi ) i 1

R

i 1

M

M

i 1

i 1

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 (Oi  Oi ).( Pi  Pi )

(16)

(17)

(18)

(19)

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MAE 

(Oi  Pi ) 2



100 M

MAPE 

M

CR

1 M

RMSE 

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(Gholami et al., 2017d). Eqs. (16-23) represent the calculations of these indices:

(20)

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SI 

(21)

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RMSE 1 M



i 1

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BIAS 

 (Pi  Oi )

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 (Oi  Oi ) 2  (Pi  Pi ) 2

(22)

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 Oi i 1

SI 1 R

(23)

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ACCEPTED MANUSCRIPT where Oi is the observational output parameter, Pi is the parameter predicted by the models, Pi and Oi are the parameter means and M is the number of parameters.

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6. Threshold channel bank profile modeling using ANFIS-DE/SVD

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This section presents and explains threshold channel bank profile modeling using the hybrid

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ANFIS-DE/SVD. In the current study, two variables, i.e. transverse distance from the center line (x) and the discharge (Q) were employed as input parameters to estimate the vertical level (y) as

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the output parameter. There were 241 samples in total, of which 50% were used for training

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(121) and the rest for model validation in the testing stage. Other parameters to be set were related to the DE algorithm, including population size (NP), crossover ratio (CR) and mutation

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scale factor (F). The trial and error process yielded optimum NP and CR values of 80 and 0.5

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respectively. Furthermore, the F parameter was determined using the Pareto curve according to

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the schematic in Fig. 3.

Moreover, for optimized threshold channel bank profile modeling using ANFIS-DE/SVD, two

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objective functions called Training Error (TE) and Prediction Error (PE) were defined and the

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Pareto curve (Fig. 5) served to select the optimal point. In evaluating the multiobjective optimization, there is a probability of different optimum points at all target points in addition to the possible optimum points according to each target. This enables designers to choose an optimum point. The TE and PE objective functions are calculated as follows:

TE 

1 MT

MT

 (OT  PT ) i 1

i

2

(24)

i

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ACCEPTED MANUSCRIPT 1 PE  MT

MT

 (OP  PP ) i 1

i

2

(25)

i

where TE and PE are the training and prediction errors, respectively, OT and OP are the observed values in training and testing respectively, PT and PP are the predicted values in

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training and testing respectively, and MT and MP are the numbers of samples in training and

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testing, respectively. The TE points in Fig. 5 represent maximum and minimum training and

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prediction errors; in contrast, the PE points indicate the lowest prediction error and highest training error. In this case, a point that represents good model training and testing efficiency

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must be selected in order to obtain an accurate model. Moreover, a good tradeoff between the TE

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and PE points in both modes simultaneously is considered the optimal point. Fig. 5 clearly specifies the tradeoff among training and testing error points.

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Fig. 6 indicates the Gaussian MF utilized in this study for each input parameter. The optimal

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Gaussian MF values were attained using the DE algorithm that determines the tradeoff point. In fact, an optimal Gaussian MF was attained at the tradeoff point where both objectives exhibited

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the greatest compromise and were optimized simultaneously. The x and y-axes in this figure

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denote the input parameter and membership function values, respectively. The figure indicates that the MF parameter was equal to 3 and was optimized by the DE algorithm for each input. The

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optimal Gaussian MF parameters are presented in Table 3.

Table 3 Optimal Gaussian MF parameters at the tradeoff point

Fig. 5. Pareto curve of training error (TE) and prediction error (PE)

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ACCEPTED MANUSCRIPT

Fig. 6. Optimal membership function (MF) of the tradeoff design point

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7. Results and Discussion

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7.1 Model Performance Evaluation

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In this section, the performance of the ANFIS and ANFIS-DE/SVD models in predicting stable channel shape is evaluated and the results are compared. The authors’ experimental results for

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predicting stable channel section shape at four discharge rates were used to train and test the

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models. The numbers of measured points for 1.157, 2.18, 2.57 and 6.2 l/s were 39, 127, 32 and 43 data, respectively. Therefore, there were a total of 241 x data for 4 discharge rates and a

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corresponding 241 y data. The amount of x* and y* (dimensionless quantity) was obtained by

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dividing the x and y values predicted by ANFIS-DE/SVD by the channel center depth (h). Fig. 7 represents regression plots of the values predicted by ANFIS and ANFIS-DE/SVD compared to

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the experimental values in training and testing modes for all discharge rates. It can be seen that

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with ANFIS almost all data are scattered between the -10% error line and the exact line, which represents model overestimation. The -10% and +10% error lines as well as the exact line were

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drawn by multiplying 0.9, 1.1 and 1 by the range of numbers located on the horizontal and vertical (x and y) axes in the scatter plots. Here, 0, 0.3, 0.6, 0.9 and 1.1 (located on the x and yaxes) were multiplied by 0.9, and the values obtained (0, 0.27, 0.54, 0.81 and 0.99) were used to plot the -10% error line in Fig. 7. Accordingly, 0, 0.33, 0.66, 0.99 and 1.21 were applied to plot the +10% error line and 0, 0.3, 0.6, 0.9 and 1.1 were used for the exact line in the regression plots in Fig. 7. Evolutionary optimization led to better compliance between the predicted data and experimental ANFIS-DE/SVD model values, as all data are compressed around the exact

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ACCEPTED MANUSCRIPT line. Therefore, it can be concluded that applying evolutionary optimization to the ANFIS model increases model performance. The fitted ANFIS-DE/SVD line matches the exact line, indicating high model accuracy. The R2 index values of both models are quite close, but that of ANFIS is greater than ANFIS-DE/SVD. The high R2 index value (close to 1) is a necessary condition but

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not sufficient to produce good model performance and accuracy. Here, the R2 value indicates

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high model accuracy, which is evident despite the high R2 for ANFIS (0.9988) and extreme

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scattering. Consequently, other statistical indices were used for better model evaluation. Table 4 shows the various statistical indices used to compare the two models in training and testing

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modes. Here, the indices related to the testing data are used to compare the two models, but these

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index values are given for training mode as well. It is clear from the table that the MAPE value for ANFIS-DE/SVD was lower than for ANFIS (by about 11% in testing). Fig. 8 shows the

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cumulative distribution of the relative error charts for both ANFIS and ANFIS-DE/SVD models.

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Furthermore, this graph contains a magnified view of the 0-30% error range (horizontal axis) to better compare the ANFIS and ANFIS-DE/SVD models according to the cumulative relative

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error distribution. With the ANFIS model 90% of estimated data and with the ANFIS-DE/SVD

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model 100% of data had a relative error of 20%. With ANFIS, almost 100% of data had less than 40% error. About 1.6% and 1.2% of all estimated data by ANFIS and ANFIS-DE/SVD

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respectively, had errors greater than 100%. However, the maximum relative error of ANFIS (324) was about 19% higher than ANFIS-DE/SVD (274). The magnified 0-30% error range in the bottom row graph in Fig. 8 indicates that almost 80% and 63% of the vertical boundary levels estimated by ANFIS-DE/SVD and ANFIS respectively had less than 7% error. The difference between the predicted values and experimental data is in terms of MAE error and the real difference is expressed as ME. It is obvious from the table that the MAE difference between the

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ACCEPTED MANUSCRIPT predicted values and experimental data for the ANFIS model (0.0203) was higher than the ANFIS-DE/SVD model (0.0116) (by about 75% in testing mode). The actual difference between the predicted and experimental values (ME) for the ANFIS model was greater than for ANFISDE/SVD. However, a notable point is the index value closer to 1 with ANFIS-DE/SVD

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(0.00046) compared to ANFIS (-0.018), which was about 97% lower. For ANFIS, this negative

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index value represents model overestimation, but evolutionary optimization could solve this

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problem almost perfectly. Besides, the difference between the predicted values and experimental data was greater for ANFIS, but this difference was in the large y values. The MAPE for ANFIS-

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DE/SVD was lower than for ANFIS, but not significantly (about 10.25% lower). However, the

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RMSE index for ANFIS-DE/SVD (0.016) was approximately 63% lower than for ANFIS (RMSE = 0.0261). The RMSE value shows the square difference between experimental and observed

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data, and a reduction for ANFIS-DE/SVD signifies that this model performed more accurately

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for large y values (near the water surface level) than ANFIS. Water surface width is another significant parameter in stable channel design. The value of this parameter varies with changes in

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discharge rate. With increasing discharge in stable channels, the flow depth, side bank width and

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slope, and water surface level increase. Therefore, it is vital to predict y at the water surface level accurately. In this regard, applying evolutionary optimization of the ANFIS-DE/SVD model

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mitigated the simple ANFIS model’s problem, which is one of the advantages of the ANFISDE/SVD model proposed in this study. The SI index is a non-dimensional RMSE index type whose values confirm the previous explanation. The SI index value for the ANFIS model (0.0715) was much higher than for ANFIS-DE/SVD (0.0428) (by about 67%). The positive BIAS index value for the ANFIS model indicates overestimation (according to Eq. 21) and it can be deduced from Fig. 7 that the fitted

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ACCEPTED MANUSCRIPT line is exactly at the right of the exact line. The R2 index value, with the same output value displacement, did not change significantly. RMSE and MAPE show the model’s deviation from experimental data but do not express the data relations. The ρ index is a combination of the error and correlation coefficient (R) indices (Gandomi and Roke, 2013), and the smaller the ρ the more

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accurate the model is. However, ρ for the ANFIS model (0.000075) in this study was less than

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for ANFIS-DE/SVD (0.00067), which was due to the high R value of ANFIS. This index value is

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almost identical for both models and close to zero due to the presence of the SI index in the equation numerator (ρ) (Eq. 22). A model that can predict the shape of stable threshold channel

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bank profiles is of significant importance. In this study, the ANFIS-DE/SVD model had lower

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error index values than ANFIS in predicting threshold channel shape and thus exhibited superior performance and accuracy. Therefore, the evolutionary optimized model is introduced as the

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superior model in this study. On account of the high accuracy, especially for large y values (near

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the water surface level boundary), the performance of ANFIS-DE/SVD is examined for different

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discharge rates in the following section.

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Fig. 7. Comparison of ANFIS (bottom row) and ANFIS-DE/SVD (top row) in predicting stable

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channel shape in training and testing modes

Table 4 ANFIS and ANFIS-DE/SVD models in predicting stable channel boundary compared with experimental data in training and testing modes according to different indices

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ACCEPTED MANUSCRIPT Fig. 8. Cumulative distribution of the relative error for the ANFIS and ANFIS-DE/SVD models in different error ranges (horizontal axis): (a) in all error ranges and (b) a magnified graph of errors in the 0-33% range

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7.2 ANFIS-DE/SVD compared to preceding models

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In this section, ANFIS-DE/SVD model performance is evaluated and compared to previous

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models, namely GFM, IKM, PIM, VDM, CKM, BVM and DEM (Table 2). The y* values were calculated with the equations presented in this table based on the experimental values, and the

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coordinates of the points at a stable channel boundary were achieved. Fig. 9 compares the bank

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profile shapes predicted by previous models, ANFIS and ANFIS-DE/SVD with the experimental values. According to Fig. 9, ANFIS-DE/SVD complies best with the experimental data and the

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ANFIS model exhibits a little difference. The most notable point in this graph is the clear

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difference between ANFIS at the upper water surface levels (larger y* amounts) and the experimental values with an increase in RMSE index, as explained in the previous section. The

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figure clearly indicates that the optimal evolutionary ANFIS-DE/SVD model solved this problem

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at high water levels. Table 5 contains the error index values for the ANFIS-DE/SVD model in comparison with previous models. Fig. 10 provides bar graphs of the error indices for these

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models. Evidently, ANFIS-DE/SVD had the lowest error index values and was thus the most accurate model in predicting stable threshold channel shape (MARE = 0.0973, RMSE = 0.0192, SI = 0.0496, ρ = 0.0248). In Fig. 10, the MARE, RMSE and SI bars are the shortest for ANFISDE/SVD compared to the other models. ANFIS came after ANFIS-DE/SVD but still outperformed existing models. Among prior models, CKM (Cao and Knight, 1997) performed the best. With higher MARE and RMSE errors of about 12.5% and 84% respectively compared to

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ACCEPTED MANUSCRIPT ANFIS-DE/SVD, CKM was weaker. According to the figure CKM predicted the channel shape similar to the experimental model with a small margin. CKM uses the entropic principle and applies the proportion of secondary flow in the redistribution of boundary shear stress. The calculation is such that initially, by maximizing the entropy, the shear stress distribution becomes

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uniform as materials are fixed and do not move. In this case, a parabolic distribution is proposed

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for a stable channel shape. A comparison of the proposed model with the experimental values

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confirms the CKM accuracy. ANFIS-DE/SVD was close to CKM, proving that the model presented in this study is very accurate and can predict flow hydraulics well. However, unlike

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other models, in addition to considering lateral diffusion in the shear stress distribution, CKM

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considers the proportion of secondary flow.

DEM uses the power law and ignores secondary flow in calculating the transfer of transverse

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momentum in shear stress distribution (Dey, 2001). As seen in Fig .9, DEM predicted the

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channel shape at a lower water surface level. According to Fig. 10, with MARE and RMSE error differences of 24% and 85% respectively compared to ANFIS-DE/SVD, DEM produced weaker

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results than CKM. Overall, it can be concluded that the ANFIS-DE/SVD and ANFIS models

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presented in this study perform similar to CKM and DEM. The error values of GFM, PIM and VDM are almost the same, but each suggests three different stable channel shapes. Fig. 9

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demonstrates that GFM predicted smaller water surface widths and bank slopes more appropriately (Ikeda et al., 1988). GFM ignores the lateral diffusion term in boundary shear stress distribution and assumes the stress distribution across the channel to be uniform. PIM and VDM spread the amount of transverse momentum created by turbulence, but these models ignore the contribution of secondary flow. PIM proposes the exponential shape, therefore according to Fig. 9 of water surface width prediction, PIM predicted the maximum width for a

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ACCEPTED MANUSCRIPT threshold channel. VDM introduces a polynomial stable channel shape, while IKM introduces the exponential shape. However, as seen Figs. 9 and 10, IKM exhibited the maximum difference from the experimental data (MARE= 0.89). With this model, channel widening at the water surface stops before reaching a balanced state. Therefore, channel width variations at the water

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surface are imperceptible and the greatest difference is found in the bank areas. The BVM model

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suggests the hyperbolic function for a stable channel shape with a broad difference from the

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experimental model. Ikeda (1981) was the first to present a relation in the form of an exponential function for the bank profile shape of a stable channel. This relation contains a K coefficient,

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whose computation is complex and requires trial and error. Furthermore, the PIM model

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(Pizzuto, 1990) forecasts the bank profile shape of a stable channel as an exponential function after channel widening; however, according to this model the central channel depth decreases up

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to 1.1 mm (less than a sediment particle diameter) after 15-30 hours and the channel shape

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becomes almost fixed. Subsequently, although channel widening stops and there is impalpable change in the channel shape at the bank toe, there is always a change in the channel bank and a

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stable channel is not reached. Pizzuto stated that the stable shape of a bank profile is close to a

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cosine bank profile shape over time. The PIM model error is evidently lower than IKM due to existing changes in the bank profile after channel widening according to PIM. Hence, PIM

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estimates a bank profile shape better than IKM. Despite the low capability of a cosine function to predict bank profile shape, models based on the cosine function outperform models based on the exponential function. Therefore, it can be said that exponential function-based models are generally less accurate in estimating the bank profile shape of stable channels. Although Ikeda (1981) and Diplas (1990) considered the diffusion of lateral momentum by turbulence and nonuniform shear stress distribution at the channel bed and walls, they calculated the shear stress

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ACCEPTED MANUSCRIPT distribution based on the normal-depth method. This led to greater conformity of the exponential function achieved by their model only with their experimental results, which did not fit the other experimental dataset (Khodashenas, 2016). Therefore, the IKM model is not very accurate. Furthermore, the BVM model did not conform acceptably to other experimental data because the

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hyperbolic equation proposed for bank profile shape estimation was obtained by fitting the

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hyperbolic function to the observed data the researchers measured in the experimental channel.

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According to the tables as well as Figs. 9 and 10, it can be said that ANFIS-DE/SVD is the superior model and most compliant with the experimental values. Among prior relationships,

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CKM is selected as the most accurate. Accordingly, Fig. 11 compares the experimental values

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with the bank profile shapes (y* values predicted by the models compared to the x* values on the horizontal axis) proposed by these two models. The polynomial equation of the fitted line is a

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third-order polynomial with R2 of 1 as per Eqs. (26) and (27) for CKM and ANFIS-DE/SVD

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respectively, as the most accurate models:

(26)

y* = 0.0122x*3 + 0.1324x*2 - 0.0135x* + 0.0165, R2= 1

(27)

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y* = 0.0016x*3 + 0.1801x*2 + 0.0047x* - 0.001, R2= 1

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The ANFIS-DE/SVD model equation (Eq. 27) proposed in this work is the most accurate for

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predicting, designing and implementing stable channel shapes and can be used in other studies.

Fig. 9. Bank profile shapes predicted by various models compared with experimental data

Table 5 Statistical indices for evaluating ANFIS and ANFIS-DE/SVD as well as preceding models in comparison with experimental data

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ACCEPTED MANUSCRIPT Fig. 10. Error index bar graph to evaluate and compare existing models with experimental data

Fig. 11. Bank profile shapes proposed by two models (ANFIS-DE/SVD and CKM) compared to

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experimental data

7.3 Evaluation of ANFIS-DE/SVD for different discharge rates

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Previous sections described the accuracy of ANFIS-DE/SVD for large y values (near the water

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surface level) along with the most important advantage, which is evolutionary model optimization to simplify the ANFIS model. The main factor contributing to stable channel

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boundary changes and subsequently changes in the water surface width and bank slope is the

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channel discharge. Hence, ANFIS-DE/SVD performance at different discharge rates is assessed in this section. Fig. 12 compares the stable channel bank profile shape predicted by the ANFIS

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and ANFIS-DE/SVD models with corresponding experimental data at different discharge rates

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(1.157, 2.18, 2.57 and 6.2 l/s). Table 6 also displays the different error indices for model comparison in terms of the 4 discharge rates. According to Fig. 12 and Table 6, the ANFIS-

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DE/SVD prediction error values are lower than ANFIS for all discharge rates and the data compliance of this model is much greater too. Moreover, a greater difference between ANFIS and experimental data was observed near the water surface (higher y*) for all discharge rates. According to Table 6, with increasing discharge the error values are relatively lower and at lower discharge the two models’ prediction error is higher. For instance at 1.157 l/s, ANFIS and ANFIS-DE/SVD with maximum MARE of 0.219 and 0.188 respectively exhibit the lowest performance accuracy. According to Fig. 12, the two models are not sufficiently accurate in 26

ACCEPTED MANUSCRIPT areas near the water surface at this discharge rate. At 2.57 and 6.2 l/s respectively, ANFIS and ANFIS-DE/SVD exhibit maximum accuracy with MARE of 0.085 and 0.088 (although at both discharge rates ANFIS-DE/SVD outperforms ANFIS). At 2.57 l/s, despite ANFIS-DE/SVD predicting y* the best, ANFIS predicts the channel shape almost identical to the experimental

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model. With increasing discharge, the difference between the values predicted by ANFIS and

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experimental data is greater, whereby at 1.157 and 6.2 l/s MAE reached 0.0215 and 0.022

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respectively. The actual difference between the predicted values and experimental data (ME) is always negative for this model, representing model overestimation at all discharge rates. ANFIS-

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DE/SVD almost resolves this problem by reducing the y* values and approaching the

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experimental values. The notable point in Fig. 12 and Table 6 is the increasing error margin of ANFIS and ANFIS-DE/SVD with increasing discharge, as the MAE error index differences

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between the models for 1.157, 2.18, 2.57 and 6.2 l/s, respectively, are 54%, 2%, 76.4% and

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100%. The error difference between the models at 2.18 l/s reached the lowest value, meaning the models comply (Fig. 12). With increasing discharge, the flow depth and water surface width

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increased and the bank slope reduced, meaning that at high discharge a threshold shape is

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imminent and the classic cosine shape is lost (Vigilar and Diplas, 1997). As seen in Fig. 12, at greater discharge the water surface width increases a little and ANFIS-DE/SVD is able to predict

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the water surface widening well. Accordingly, with increasing discharge the ANFIS-DE/SVD model index error reduced in this study and the difference from the simple ANFIS increased. Therefore, it is concluded that ANFIS-DE/SVD performs accurately at high discharge and can solve existing problems of simpler models. These are considered important advantages of this model. Therefore, according to this study, the optimized evolutionary ANFIS-DE/SVD is a

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ACCEPTED MANUSCRIPT model with superior performance in estimating stable threshold channel shape and it can be utilized in various fields, especially in channel design considering different flow rates.

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DE/SVD compared to experimental data at different discharge rates

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Fig. 12. Shapes of a stable threshold channel bank profile predicted by ANFIS and ANFIS-

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Table 6 Evaluation of ANFIS and ANFIS-DE/SVD models compared to experimental data at

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different discharge rates in predicting stable channel shape

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8. Conclusion

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In this study, a new hybrid model based on ANFIS combined with the DE algorithm and SVD

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(ANFIS-DE/SVD) was employed to predict stable channel profile shape. To provide a flexible algorithm for multiple datasets, two different objective functions were defined and the Pareto

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curve was used to select the optimal point, which is the tradeoff between the two objectives. The

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bank profile shape proposed by ANFIS-DE/SVD was compared with the top seven existing models in terms of prediction accuracy. ANFIS-DE/SVD with the lowest prediction error index was selected as the model with the best performance. Among existing models, CKM (Cao and Knight’s model, 1998) exhibited the highest prediction accuracy. In addition to the contribution of transverse momentum diffusion, the CKM model also considers the proportion of secondary flow in the shear stress distribution at the channel bed and banks. A polynomial equation was developed using ANFIS-DE/SVD for predicting stable channel profile shape. The main

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ACCEPTED MANUSCRIPT advantage of the hybrid ANFIS-DE/SVD over the simple ANFIS model is the significantly higher degree of prediction accuracy for greater discharge rates. The new hybrid ANFISDE/SVD model presented in this study can thus be used to design stable alluvial channels. In following up with the present ANFIS-DE/SVD model, the authors suggest using different

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evolutionary algorithms (EA) instead of DE such as the genetic algorithm (GA) and particle

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swarm optimization (PSO) in further studies.

9. Acknowledgement

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The experiments for this study were performed at the Hydraulic Laboratory of Civil and

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period of sabbatical leave of the fifth author.

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Geological Engineering Department, University of Saskatchewan, Saskatoon, Canada, in a

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methods. Engineering Geology, 131, 11-18. 31. Khodashenas, S. R., 2016. Threshold gravel channels bank profile: a comparison among 13 models. International Journal of River Basin Management, 14(3), 337-344. 32. Kisi, O., Bilhan, O., Emiroglu, M. E., 2013. Anfis to estimate discharge capacity of rectangular side weir. In Proceedings of the Institution of Civil Engineers-Water Management, 166(9), 479-487, Thomas Telford Ltd.

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ACCEPTED MANUSCRIPT 33. Kulatilake, P. H. S. W., Qiong, W., Hudaverdi, T., Kuzu, C., 2010. Mean particle size prediction in rock blast fragmentation using neural networks. Engineering Geology, 114(3-4), 298-311. 34. Liao, T. W., 2017. A procedure for the generation of interval type-2 membership

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functions from data. Applied Soft Computing, 52, 925-936.

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35. Madvar, H. R., Ayyoubzadeh, S. A., Atani, M. G. H., 2011. Developing an expert system

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for predicting alluvial channel geometry using ANN. Expert Systems with Applications, 38(1), 215-222.

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36. Mishra, D. A., Basu, A., 2013. Estimation of uniaxial compressive strength of rock

Engineering Geology, 160, 54-68.

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materials by index tests using regression analysis and fuzzy inference system.

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37. Öge, İ. F., 2017. Prediction of cementitious grout take for a mine shaft permeation by

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adaptive neuro-fuzzy inference system and multiple regression. Engineering Geology, 228, 238-248.

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38. Parker, G., 1978. Self-formed straight rivers with equilibrium banks and mobile bed, Part

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2. The gravel river. Journal of Fluid Mechanics, 89(01), 127-146. 39. Pizzuto, J. E., 1990. Numerical simulation of gravel river widening. Water Resources

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Research, 26, 1971-1980. 40. Shaghaghi, S., Bonakdari, H., Gholami, A., Ebtehaj, I., Zeinolabedini, M., 2017. Comparative analysis of GMDH neural network based on genetic algorithm and particle swarm optimization in stable channel design. Applied Mathematics and Computation, 313, 271-286.

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ACCEPTED MANUSCRIPT 41. Storn, R., Price, K.,1997. Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. Journal of global optimization, 11(4), 341359. 42. Sugeno, M., Kang, G. T., 1988. Structure identification of fuzzy model, Fuzzy Set

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Systems, 28 (1), 15-33.

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43. Taher-Shamsi, A., Tabatabai, M. R. M., Shirkhani, R., 2013. An evaluation model of

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artificial neural network to predict stable width in gravel bed rivers. International Journal of Environmental Science and Technology, 9, 333-342.

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44. Van Rijn, L. C., 1984. Sediment transport, Part I: bed load transport. Journal of Hydraulic

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Engineering-ASCE, 110, 1431-1456.

45. Vigilar, G., Diplas, P., 1997. Stable channels with mobile bed: formulation and numerical

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solution. Journal of Hydraulic Engineering-ASCE, 123(3), 189-199.

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46. Vigilar, G., Diplas, P., 1998. Stable channels with mobile bed: model verification and graphical solution. Journal of Hydraulic Engineering-ASCE, 124(11), 1097-1108.

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47. Yu, G., Knight, D. W., 1998. Geometry of self-formed straight threshold channels in

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uniform material. Proceeding of the Institute of Civil Engineering, Water Maritime and Energy, London, 130(1), 31-41.

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48. Zhang, W., Goh, A. T., Zhang, Y., Chen, Y., Xiao, Y., 2015. Assessment of soil liquefaction based on capacity energy concept and multivariate adaptive regression splines. Engineering Geology, 188, 29-37.

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Fig. 1. Bank profile characteristics

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Fig. 2. Hybrid ANFIS-DE/SVD model flowchart

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Fig. 3. Mutation scale factor (F) based on the optimal solution proximity in the Pareto front

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Fig. 4. Schematic of flume used in the experiments

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Fig. 5. Pareto curve of training error (TE) and prediction error (PE)

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Fig. 6. Optimal membership function (MF) of the trade-off design point

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0.9

+10%

y = 1.0003x - 1E-04 R² = 0.9978

0.9

+10%

y = 0.9936x + 0.0028 R² = 0.9974

-10%

y* (EXP)

0.6

0.3

0.6

0.3

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y*(Train) Exact line 10% Linear (y* (Train))

0

0

+10%

y = 0.9506x + 0.0011 R² = 0.9988

0.9

-10%

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y* (EXP)

-10%

y* (EXP)

0.3 0.6 0.9 y* (ANFIS-DE/SVD)

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+10%

y = 0.954x - 0.0011 R² = 0.9987

0.9

0

0.3 0.6 0.9 y* (ANFIS-DE/SVD)

y*(Test) Exact line 10% Linear (y* (Test))

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y* (EXP)

-10%

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y* (Train) Exact line 10% Linear (y* (Train))

0 0

0.3

0.6

0.6

0.3

y* (Test) Exact line 10% Linear (y* (Test))

0

0.9

0

0.6

0.9

y* (ANFIS)

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y* (ANFIS)

0.3

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Fig. 7. Comparison of ANFIS (bottom row) and ANFIS-DE/SVD (top row) in predicting stable

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channel shape in training and testing modes

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(a)

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120 110 100 90 80 70 60 50 40 30 20 10 0

ANFIS-DE/SVD

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Threshold of Relative Absolute Error (%)

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ANFIS 40

80

120 160 200 240 Culumative Frequency (%)

280

320

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0

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110 100 90 80 70 60 50 40 30 20 10 0

ANFIS-DE/SVD

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Threshold of Relative Absolute Error (%)

(b)

0

3

6

ANFIS 9

12 15 18 21 24 Culumative Frequency (%)

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30

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Fig. 8. Cumulative distribution of the relative error for the ANFIS and ANFIS-DE/SVD models in different error ranges (horizontal axis): (a) in all error ranges and (b) a magnified graph of errors in the 0-33% range

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1

0.6

GFM IKM VDM CKM BVM EXP Model PIM DEM ANFIS Model ANFIS-DE/SVD Model

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Y*

0.8

0.2

0 1

1.5

2 X*

2.5

3

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3.5

4

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Fig. 9. Bank profile shapes predicted by various models compared with experimental data

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MARE SI RMSE

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0.6 0.4

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Error value

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0

Models

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Fig. 10. Error index bar graph to evaluate and compare existing models with experimental data

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CKM EXP ANFIS-DE/SVD 0.8 y = 0.0016x3 + 0.1801x2 + 0.0047x - 0.001

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y* (y/h)

R² = 1

0 1

2

3

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0

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y = 0.0122x3 + 0.1324x2 - 0.0135x + 0.0165 R² = 1

x* (x/h)

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Fig. 11. Bank profile shapes proposed by two models (ANFIS-DE/SVD and CKM) compared to

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experimental data

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1.2

Q= 1.157 Lit/s

ANFIS-GA/SVD

1

Q= 2.18 Lit/s

ANFIS-GA/SVD

EXP

EXP

1

ANFIS

ANFIS

0.8

y*

y*

0.8

0.6

0.6 0.4

0.2

0.2 0

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0

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0.4

0

0.5

1

1.5

2

2.5

3

0

0.5

1

x*

2

2.5

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1.2

1.2 Q= 2.57 Lit/s

3

Q= 6.2 Lit/s

ANFIS-GA/SVD

EXP

EXP

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ANFIS-GA/SVD

1

1.5

x*

1

ANFIS

ANFIS

0.8

y*

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y*

0.8 0.6

0.6 0.4

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0.4

0 0

0.5

1

1.5

2

3

0

0.5

1

1.5

2

2.5

3

x*

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x*

2.5

0.2 0

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0.2

Fig. 12. Shapes of a stable threshold channel bank profile predicted by ANFIS and ANFIS

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DE/SVD compared to experimental data at different discharge rates

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ACCEPTED MANUSCRIPT Table 1 Experimental characteristics Cross section’s

Discharge Test No.

d50 (mm)

S

h (cm)

(Q) (l/s)

shape

1.157

0.53

0.0023

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triangular

2

2.57

0.53

0.0023

61.3

triangular

3

6.2

0.53

0.0023

80

4

2.18

0.53

0.0023

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triangular trapezoidal

ACCEPTED MANUSCRIPT Table 2 Models studied in this research Model

Equation

Glover and Florey (GFM) (1951)

y   1  Cosx * 

Ikeda (IKM) (1981)

T  x 2 y  exp   K  

Pizzuto (PIM) (1990)

  T    y   exp   x *  2   

Cao and Knight (CKM) (1997)

y   x *  / 4

     

1 T2 ydy = h 0 displacement thickness.

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K

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

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Cosine

μ

exponential

exponential

polynomial

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2

y   (a3 x   a2 x   a1 x   a0 )

BabaeyanKoopaei and Valentine (BVM) (1998)

  x *  y   3.5 tanh     4  

2

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a0, a1, a2 and a3 are a function of μ and δ*c ,dimensionless critical stress depth that can be obtained in Vigilar and Diplas (1997)

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Vigilar and Diplas (VDM) (1998)

Three-degree polynomial

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hyperbolic

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Dey (DEM) (2001)

Suggested shape

Parameters

 2 2 1







2 m 1 dy *   1  y *  Cmx* 1      1 dx *

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C =0027and m = 4.5,

polynomial

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1.67

37.42

2.37

270.76

1.32

214.92

4.32

-12.21

3.48

201.70

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303.04

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3.69

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c

x

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σ

Q

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Table 3 Optimal Gaussian MF parameters at the trade-off point

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ACCEPTED MANUSCRIPT Table 4 ANFIS and ANFIS-DE/SVD models in predicting stable channel boundary compared with experimental data in training and testing modes according to different indices Models

stage

R2

MAPE RMSE

MAE

ME

SI

BIAS

ρ

Train 0.9986 11.65

0.0267 0.0212 -0.019

0.0720 0.0190

0.00010

Test

0.0261 0.0203 -0.0180

0.0715 0.0178

0.000075

9.53

0.014

0.0111 -7.8E-8

0.0394 7.84E-8 5.61E-6

11.12

0.016

0.0116 0.00046 0.0428 0.00046 0.00067

0.997

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DE/SVD Test

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Train 0.998

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ANFIS-

0.9987 12.26

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ANFIS

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ACCEPTED MANUSCRIPT Table 5 Statistical indices for evaluating ANFIS and ANFIS-DE/SVD as well as preceding models in comparison with experimental data Models ANFISANFIS CKM

DEM

VDM

GFM

PIM

BVM

IKM

Indices

DE/SVD

MARE

0.0973

0.1063

0.1095 0.120

RMSE

0.0192

0.0272

0.119

0.1310 0.2930 0.3103 0.3370 1.033

1.070

SI

0.0496

0.0703

0.084

0.093

0.2372 0.728

0.754

ρ

0.0248

0.0350

0.042

0.046

0.1037 0.1097 0.1190 0.365

0.378

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0.2650 0.3015 0.3230 0.849

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0.2070 0.219

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0.889

ACCEPTED MANUSCRIPT Table 6 Evaluation of ANFIS and ANFIS-DE/SVD models compared to experimental data at different discharge rates in predicting stable channel shape

Models

MARE

RMSE

MAE

ME

ANFIS

0.220

0.0007

0.0215

-0.018

ANFIS-DE/SVD

0.188

0.00035

ANFIS

0.125

8.5E-5

0.0049

-0.0035

ANFIS-DE/SVD

0.115

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Discharge

0.0050

0.0001

ANFIS

0.085

0.00074

0.0217

-0.020

ANFIS-DE/SVD

0.099

0.00024

0.0123

-0.0010

0.104

0.0007

0.022

-0.018

0.088

0.00023

0.011

-0.0005

2.18

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6.5E-5

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2.57

ANFIS

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6.2

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ANFIS-DE/SVD

IP

0.0140

CR

1.157

T

(l/s)

53

-0.003

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A new approach is applied to estimate the bank profile of a threshold channel

 Most influential parameters on the bank profile of a threshold channel are studied A wide-ranging experiments are used to evaluate the models



The proposed method can be used as an alternative in practical applications.

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

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