Design of a support vector machine with different

Nat Hazards DOI 10.1007/s11069-016-2540-5 ORIGINAL PAPER

Design of a support vector machine with different kernel functions to predict scour depth around bridge piers Hassan Sharafi1 • Isa Ebtehaj1 • Hossein Bonakdari1 Amir Hossein Zaji1

•

Received: 14 July 2015 / Accepted: 19 August 2016  Springer Science+Business Media Dordrecht 2016

Abstract Scour depth is a vital subject in bridge pier design. The exact estimation of scour depth can prevent damage caused by bridge failure and facilitate optimal bridge pier design. In this article, the support vector machine (SVM) method is applied to predict scour depth around bridge piers. The SVM technique is developed using six kernel functions, including polynomial, sigmoid, exponential, Gaussian, Laplacian and rational quadratic. Scour depth is modeled as a function of three dimensionless variables, namely geometric characteristics, flow and bed materials. The performance of SVM (in training and testing) is evaluated using dimensionless variables gathered from a wide range of field datasets. The SVM designed using the polynomial kernel function produced the most accurate results compared with the other kernel functions (RMSE = 0.078, MRE = -0.181, MARE = 0.332, MSRE = 0.025). Sensitivity analysis is performed to identify the effect of each dimensionless parameter on predicting scour depth around bridge piers. The testing results of SVM-polynomial are compared with that of the artificial neural networks (ANN), adaptive neuro-fuzzy inference systems (ANFIS) and nonlinear regression-based methods presented in this study and in the literature. Evidently, SVM-polynomial predicted scour depth with higher accuracy and lower error than when using ANN, ANFIS and nonlinear regression-based equations. Moreover, as an alternative method, a simple program is presented by the SVM-polynomial to calculate scour depth around bridge piers. Keywords Bridge pier  Scour depth  Support vector machine (SVM)  Sensitivity analysis  Traditional equation

& Hossein Bonakdari [email protected] 1

Department of Civil Engineering, Razi University, Kermanshah, Iran

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1 Introduction Attributable to three-dimensional flow patterns that come in contact with bed materials, stream and river channel scour is considered a complex phenomenon that is challenging for bridge engineers worldwide (Chiew 1992). Due to an extensive assortment of non-physical variables, predicting complex problems regarding local scour around bridge piers is intricate. Many significant variables, viz. pier and bed material characteristics, have an effect on prediction, with pooled consequences of the turbulent boundary layer, time-dependent flow patterns, sediment transport mechanism, the physical parameters of flow and sediment characteristics as well as pier geometry. For this reason, the scour process is deemed a significant factor in the design of bridge foundation depth in a channel bed and is the best strategy to protect foundations as well. Scour depth overestimation and underestimation may result in significant redundant expenditures and costly bridge failures, respectively; therefore, bridge construction is viewed as a costly enterprise. Due to such influential variables, determining and formulating mathematical models for the scour process is complicated. As a result, to attain safe and cost-effective bridge structures and to prevent devastating bridge failures that can cause loss of life, the accurate estimation of scour depth around piers and abutments is indispensable. In the past, to estimate equilibrium scour depth at bridge piers several experimental formulae have been projected, but they are affected by related uncertainties (Melville and Sutherland 1988; Melville and Chiew 1999). Existing methods’ results differ extremely from one another, thus causing great disagreement concerning the design, pier foundation costs and protection methods against scour (Ettema et al. 1998). Thus far, many studies have been carried out, but a general theory has still not been attained owing to the complex problem structures. For this reason, improvements in current research in this area, including tools and techniques to develop reliable and effective methods for scour depth prediction around bridge piers, is required. Therefore, enhancement in these areas is projected to lead to safe, economical and effective bridge pier design (Azamathullah et al. 2010). As an alternative to equations identified in the literature, artificial intelligence (AI) has recently been recognized as a professional tool for modeling complex hydraulic and hydrologic systems. AI methods serve as valuable tools in many hydraulic and environmental engineering applications owing to the very good accuracy (Bonakdari et al. 2011; Ebtehaj and Bonakdari 2013, 2014; Sattar 2014a, b; Zaji and Bonakdari 2015; Sattar and Gharabaghi 2015; Ebtehaj et al. 2015; Thompson et al. 2016). One of the most popular applications of AI is for predicting local scour at bridge piers. For instance, artificial neural networks (ANNs) (Bateni et al. 2007), genetic programming (GP) (Guven et al. 2009), gene expression programming (GEP) (Khan et al. 2012), group method of data handling (GMDH) (Najafzadeh et al. 2013, 2014; Najafzadeh and Sattar 2015) and the adaptive neuro-fuzzy inference system (ANFIS) (Azamathulla and Ghani 2010) are often used to predict local scour depth around hydraulic structures. Zounemat-Kermani et al. (2009) predicted the depth of scour holes around pile groups using ANFIS and two types of ANN, namely feedforward backpropagation (FFBP) and radial basis function (RBF). The results demonstrated the superior performance of FFBPNN over RBF-NN, ANFIS and traditional equations. Kazeminezhad et al. (2010) presented a new approach based on artificial neural networks using tenfold and holdout cross-validation to develop a more accurate model for predicting scour depth around submarine

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pipelines. They indicated that existing methods produce high errors in scour depth estimation. Moreover, the authors demonstrated that ANNs were more accurate than existing approaches in scour depth prediction. Samadi et al. (2014) employed two different decision tree techniques based on classification and regression trees as model trees to estimate scour depth downstream of overfall spillways. A comparison of these methods indicated that the model trees outperformed classification and regression trees in terms of prediction capability. Najafzadeh and Lim (2015) predicted scour depth downstream of a sluice gate with an improved neuro-fuzzy method based on GMDH using a particle swarm optimization algorithm. The authors compared the testing results of NF-GMDH-PSO with existing methods. They found the NF-GMDH-PSO provided better results than existing methods. Mohammadpour et al. (2016) predicted temporal scour depth at abutments using ANFIS [grid partitioning (GP) and subtractive clustering (SC)] and ANN [feedforward backpropagation (FFBP) and radial basis function (RBF)] and with a wide range of experimental datasets to overcome the limitations of conventional methods. The authors found that ANFIS-GP and ANN-FFBP performed well in predicting scour depth at abutments. The support vector machine is a powerful AI technique with several applications in different fields, such as classification (Tang et al. 2009; Rahman et al. 2011), regression (Liao et al. 2011; Lv et al. 2014), pattern recognition (Sathiya et al. 2014; Tsai et al. 2015), water resource management (Ch et al. 2013; Mohammadpour et al. 2014) and time series (Sˇteˇpnicˇka et al. 2013; Grigorievskiy et al. 2014). According to these studies, SVM can be used successfully in various science fields. The aim of this study is to develop SVM using six different kernel functions for scour depth prediction around bridge piers by applying real field data. Dimensional analysis is employed to estimate the effective parameters on determining scour depth, after which different kernel functions are applied to predict these parameters. The performance of each kernel function is evaluated using statistical indices. Upon selecting the best kernel function and by using dimensional analysis, the effect of each parameter on scour depth can be evaluated. Moreover, the developed models’ performance in predicting scour depth using SVM is compared with ANN, ANFIS, and traditional and nonlinear regression (NLR) equations.

2 Materials and methods 2.1 Data presentation To investigate scour depth around bridge piers, actual field data from 14 different sites (Landers and Mueller 1999; Mohammed et al. 2005) are used in this study. The total number of measurements is 476 samples for different pier forms, including cylindrical (43 samples), sharp (95 samples), square (107 samples) and round (231 samples). Table 1 presents the statistical indices of the different variables applied in this study. The Pearson correlation coefficients matrix (Pearson 1895) between the considered input variables is presented in Table 2. The correlation coefficients are given values in the -1 to 1 interval, where 1 and -1 represent the total positive and negative correlations, respectively, and 0 represents no correlation between the considered variables. According to Table 2, except for the L/y and D/y input variables that have the same structure, there is no strong correlation between the other input variables. The input–output surface for the input variables is presented in Fig. 1.

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Nat Hazards Table 1 Statistical indexes of different variables Statistical

Variable

Index

D (m)

L (m)

V (m/s)

y (m)

d50 (m)

r

ds (m)

Min

0.30

0.00

0.00

0.10

0.00

1.20

Max

22.90

27.40

4.50

22.50

0.11

20.30

7.70

Mean

1.40

10.74

1.32

4.48

0.01

3.33

1.02

SD

1.43

4.08

0.80

3.54

0.02

2.47

1.20

CV

1.02

0.38

0.61

0.79

1.79

0.74

1.18

Table 2 Correlation coefficient matrix between the input variables

0.00

r

Corr

Fr

d50/y

D/y

L/y

Fr

1

0.51

0.43

0.45

0.06

1

0.37

0.45

-0.03

d50/y D/y L/y r

1

0.85

0.25

1

0.2 1

Fig. 1 Input–output surface for input parameters

The intricate scour mechanism around bridge piers depends on a number of factors, including geometric cross-bridge characteristics, flow, sediments around the pier, and bed materials. Therefore, a functional relationship to determine scour depth is presented as follows:

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ds ¼ f ðU; y; D; L; d50 ; r; g; qw ; qs ; mÞ

ð1Þ

where ds, U, y, D, L, d50, r, g, qw, qs and t are local scour depth, average velocity, flow depth, pier width, pier length, median diameter of particles, standard deviation of bed grain size, gravitational acceleration, water density, sediment density and kinematic viscosity, respectively. By using dimensional analysis, the relative scour depth can be expressed as the following relationship:   ds L d50 D ; ð2Þ ¼f ; ; Fr; r; Re y y y y  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where Fr and Re are the Froude number U= gðs  1Þd50 and Reynolds number (U  D/t). Here, s is the sediment relative density, which is equal to the ratio of the sediment density (qs) to the water density (qw). According to s, the effect of the Reynolds number is insignificant (Melville and Sutherland 1988) and the above-mentioned equation is rewritten as follows:   ds L d 50 D ; ð3Þ ¼f ; ; Fr; r y y y y To determine the significance of each dimensionless variable on scour depth around bridge piers presented in Eq. (3), six different models are introduced as follows:   ds d 50 D L Model 1: ¼ U1 Fr; ; ; ;r y y y y   ds d 50 D L ¼ U2 ; ; ;r Model 2: y y y y   ds D L Model 3: ¼ U3 Fr; ; ; r y y y   ds d 50 L ¼ U4 Fr; ; ;r Model 4: y y y   ds d 50 D Model 5: ¼ U5 Fr; ; ;r y y y   ds d 50 D L ¼ U6 Fr; ; ; Model 6: y y y y After determining the scour depth estimation parameters for bridge piers and following a dimensional analysis to determine the dimensionless parameters, the field dataset collected is divided into two categories: training and validation. SVM is trained using Eq. (3) and training data. In training and SVM modeling to predict scour depth, six different kernel functions are applied, i.e., sigmoid, polynomial, Gaussian, exponential, rational quadratic and Laplacian. Subsequent to SVM modeling and applying the four statistical indices and validation data, each kernel function’s performance is evaluated to select the best one. Once the best kernel function is selected, the SVM designed with the best kernel function is used to perform sensitivity analysis and effectually determine each parameter given in Eq. (3). The presented model’s accuracy is compared with traditional regression-based equations (Table 3) and the results of a nonlinear regression equation as well. The flowchart of the methodology employed in this study is shown in Fig. 2.

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2.2 Support vector machines Vapnik (2000) introduced SVM as a powerful soft computing method for solving classification and regression problems. The SVM method used for regression problems is called support vector regression (SVR). The goal of SVR is to develop a model that can predict the experimental, measured or observed output dataset, namely the target vector x1 ; ~ x2 ; . . .; ~ xn g. SVR output vector T = {t1, t2, …, tn} using input variables X ¼ f~ Y = {y1, y2, …, yn} amounts closer to T signify higher SVR model performance. Equation (8) represents the linear regression model that establishes a relation between input and output variables. yi ¼ wTi xi þ b

ð8Þ

where xi and yi are the ith input and output samples, respectively, and w and b are the weights and biases of the considered model, respectively. In the SVR procedure, the loss function Le is utilized to penalize the model. Le is defined as:  0 jti  yi j  e ð9Þ L e ð ti ; y i Þ ¼ fi jti  yi j [ e where fi is a nonnegative slag variable. According to Eq. (9), the model is penalized when the difference between the target and output vectors exceeds a predefined constant, e. From þ Eq. (9), it can be concluded that e  f i  ti  yi  þ e þ fi . Thus, the loss function can be rewritten as: þ Le ðti ; yi Þ ¼ f i þ fi

ð10Þ

In the training procedure, SVR attempts to minimize Le by minimizing the empirical risk function, Remp, which is defined as: Remp ½yi  ¼

n 1X Le ðti ; yi Þ n i¼1

ð11Þ

where n is the number of samples. In the minimization procedure of Remp, unfavorable model expansion may occur. As such, a complexity term should be added to the objective function. The regularized risk function Rreg is defined as the sum of the Remp defined in Eq. (11) and the norm of the weight matrix w. Therefore, by minimizing Rreg, the smallest model with the highest accuracy is found. Rreg is defined as follows (Smola 1996; Cimen 2008): 1 Rreg ½yi  ¼ Remp ½yi  þ wT w 2

ð12Þ

Table 3 Traditional equations for predicting scour depth around bridge piers Authors

Equations

Richardson and Davis (2001)

ds/y = 2.6(D/y)0.65(Fr0.43)

(4)

Johnson (1992)

ds/y = 2.02(Fr0.21)(D/y)0.98(r-0.98)

(5)

Shen et al. (1969)

ds/y = 3.4(Fr0.67)(D/y)0.67

(6)

Laursen and Toch (1956)

ds/y = 1.35(D/y)0.7

(7)

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Fig. 2 Methodology flowchart

Equations (7) and (5) are used to obtain the following form of Rreg: 8 n  < ti þ yi þ e þ fþ X i 0  1  þ T Rreg ½yi  ¼ C fi þ fi þ w w such that ti  yi þ e þ f  0 i :  2 i¼1 fi and fþ i 0

ð13Þ

In Eq. (13), C serves as a trade-off parameter to find the degree of Remp. After minimization, the following linear regression equation is obtained. jSj X  þ    ai þ a for S ¼ ij0\aþ i þ ai \C yj ¼ i xi xj

ð14Þ

i¼1  where aþ i , ai and S are the Lagrange multipliers and support vector, respectively. According to Fig. 3, to transfer the linear regression SVR from Eq. (14) to the nonlinear future, kernel functions K(xi, xj) are employed. Then, Eq. (14) can be rewritten as:

yj ¼

jSj X 

    aþ i þ ai K xi ; xj

ð15Þ

i¼1

Various kernel functions can be utilized in the SVR procedure. Choosing the appropriate kernel function directly affects SVR regression performance. In the present study,

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Fig. 3 SVM method structure

six types of SVR models with different kernel functions are developed, and their performance is compared. Details of the considered kernel functions are given in Table 4. To train the SVR model, the C and e constants must be predefined. Selecting appropriate values for these constants significantly affects SVR prediction performance. Moreover, according to Table 4, each kernel function needs to predefine the kernel constant. Trial and error is employed to determine the three parameters: C, e and the kernel constant. Therefore, some additional loops were added to the main SVR program, and nearly 1000 runs were done to find the optimum trial and error constants.

2.3 Artificial neural networks Artificial neural networks are inspired by the biological system of the human brain that contains series of connected components called neurons. This system processes the data input to the system in order to find the nonlinear relationship between input vectors and target parameters. Different neurons are associated with each other throughout the various networks via a set of weights, on which network performance is very much dependent. One of the most common neural networks and which is used in this study is the multilayer perceptron (MLP). This network contains an input layer, an output layer and Table 4 Kernel functions utilized

Kernel name

Kernel equation

Kernel constant

Polynomial

k(x, x0 ) = (xTx0 ? 1)d

 0 2 k kðx; x0 Þ ¼ exp  kxx 2r2

 0 k kðx; x0 Þ ¼ exp  kxx 2r2

 0 k kðx; x0 Þ ¼ exp  kxx r

d

Gaussian Exponential Laplacian Sigmoid Rational quadratic

123

r r r

0

T 0

d

0

kxx0 k2 kxx0 k2 þd

d

k(x, x ) = tanh (x x ? d) kðx; x Þ ¼ 1 

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also a hidden layer. The number of variables in the input and output layers varies from problem to problem. Considering that the use of a hidden layer in the majority of studies presents acceptable results (You and Seo 2009; Kamatchi et al. 2012; Ebtehaj and Bonakdari 2016), this study also employs a hidden layer in the neural network. In ANN modeling, the input variables are first received. Then, with respect to the weight of each input, they are gathered and a bias is added. The obtained result is given to the transfer function as an argument. After sending the results to the output, the predicted results are compared with the present values and the estimation error is calculated. To minimize the error value, the weight and bias values are manipulated in the learning process in order to achieve an acceptable result. The output signal value corresponding to k neurons can be calculated using the following equation: ! m X ð16Þ wkj xj þ bk yk ¼ u j¼1

where yk is the output signal of neuron k; u is the transfer function; wkj is the weight of neuron k; xj is the input signal; and bk is the bias. The Levenberg–Marquardt algorithm (LM) is the best gradient algorithm used in different studies to train an MLP network. Moreover, a sigmoid-type transfer function is used in the current study, which is expressed as follows: u ð xÞ ¼

1 1 þ ex

ð17Þ

2.4 Adaptive neuro-fuzzy inference systems (ANFIS) ANFIS is an artificial intelligence method that is a combination of neural networks and fuzzy logic. In fact, this method benefits from fuzzy logic and neural networks to overcome the limitations of each of the two methods. The input space is divided into different local regions. A local region is extended based on adjustable coefficients or linear functions. Then the dimension of each input is divided using membership functions. Different regions are activated simultaneously, whereas the input space is covered by membership functions (Aminossadati et al. 2012). A fuzzy system based on if–then rules is presented as: Rule1: If x1 is A1 and x2 is B1 and . . .; then f1 ¼ p1 x1 þ q1 x2 þ    þ r1

ð18Þ

Rule1: If x1 is A2 and x2 is B2 and . . .; then f2 ¼ p2 x1 þ q21 x2 þ    þ r2

ð19Þ

where Ai and Bi are the fuzzy sets, fi is the output set, and pi, qi, ri are the design parameters that are adjusted through the learning process. Generally, an ANFIS network has five layers that are expressed as follows. In the first layer, each node consists of a fuzzy set. The output of each is proportional to the grade of the fuzzy set’s membership function. The nodes found in this layer are adaptive and calculated as follows: ð20Þ O1;i ð xÞ ¼ lAi ðxÞ where O1,i is the membership function (MF) related to the input of fuzzy set Ai. Due to the good performance of Gaussian MF for different engineering problems (Nguyen et al. 2014; Sun et al. 2016; Mohammadpour et al. 2016), this MF in the range of [0 1] is used in this study and is expressed as:

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jx  ci j2 lAi ð xÞ ¼ exp 2r2i

! ð21Þ

where x is the input parameter, and c and r are the premise parameters that are adapted using a hybrid algorithm (combining backpropagation and least squares). In the second layer, the firing strength of each rule is calculated. The output of this layer is calculated as follows: O2;i ð xÞ ¼ wi ¼ lAi ðx1 Þ      lCi ðxn Þ

ð22Þ

In the third layer, the ratio of the firing strength of the ith rule to all rules’ firing strengths is evaluated. The relative firing strength is calculated as follows: ¼ O3;i ð xÞ ¼ x

xi ðx1 þ    þ xn Þ

ð23Þ

After computing the weight of each rule, the output of each node is calculated in the fourth layer as follows: O4;i ¼

n X

 i fi x

ð24Þ

i¼1

In the fifth layer, the overall output, which is equal to the sum of the input signals to this layer, is calculated as: Pn i f x O5;i ¼ Pi¼1 ð25Þ n x i¼1  ii

2.5 Statistical indices In this study, to evaluate the performance of the presented models and compare them with existing methods, one statistical index is used to assess the absolute model performance, i.e., root-mean-square error (RMSE), as well as three relative indices, i.e., mean relative error (MRE), mean absolute relative error (MARE) and mean squared relative error (MSRE). The RMSE index refers to the square difference between the observed and predicted values and includes the lower limit. A value of zero for this index represents the best model performance. s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  n

2 1 X ð26Þ ðds =yÞobsi ðds =yÞSVMi RMSE ¼ n i¼1 The MRE index presents the relative prediction error value regardless of scour depth over- or underestimation. It does not include upper and lower limits, and negative or positive values indicate underestimation or overestimation, respectively. The MARE index is a nonnegative index with a zero lower limit. MSRE represents the square root of the mean relative error ds/y estimation by SVM. !  X 1 n ðds =yÞobsi ðds =yÞSVMi ð27Þ MRE ¼ n i¼1 ðds =yÞobsi

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!  X 1 n ðds =yÞobsi ðds =yÞSVMi MARE ¼ n i¼1 ðds =yÞobsi

ð28Þ

!2  X 1 n ðds =yÞobsi ðds =yÞSVMi MSRE ¼ n i¼1 ðds =yÞobsi

ð29Þ

3 Results and discussion Figure 4 presents the scatter plot of six different kernel functions in designing SVM for scour depth prediction around bridge piers. For ds/y values greater than 0.5, the rational quadratic, exponential, Laplacian and Gaussian kernel functions exhibit overestimation with high relative error. At these scour depth range values, the Gaussian and sigmoid kernel functions display underestimation. The polynomial kernel function outperforms other kernel functions, because for all ds/y ranges, the values provided by the kernel function lead to superior performance over other models. However, for assessing model superiority over other models, a quantitative model study is required as well. Figure 5 shows the anticipated scour depth results using SVM with different kernel functions. MRE determines the average estimated error and average relative error with regard to underestimation or overestimation, and it does not include upper and lower limits. Concerning Fig. 5, the value of this index is negative for all kernel functions. Consequently, all predictions made by the six kernel functions follow the same process and, on average, exhibit underestimation. A higher MRE index value (by taking the absolute value) is related to the sigmoid kernel function whose value reaches 1.7. The best performance is attributed to the polynomial kernel function (MRE = -0.18). The Gaussian kernel function (MRE = -1.57) performs almost the same as the sigmoid. The exponential

Fig. 4 Scatter plot of observed and predicted scour depth for six kernel functions

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Fig. 5 Performance evaluation for six kernel functions using different statistical indexes

(MRE = -0.672), Laplacian (MRE = -0.732) and rational quadratic (MRE = -0.780) kernel functions exhibit similar performance. The values provided by these indices are about 3–4 times the minimum MRE value (polynomial kernel function). In case of predicting the relative scour depth (ds/y) using SVM, the polynomial kernel function can be used (SVM-polynomial). The MARE index shows the average relative error is about 0.33 for the polynomial, which is almost half the index for the exponential (MARE = 0.75). The MSRE and RMSE indices represent the root-mean-square error for both relative and absolute errors (respectively), and they also use the polynomial kernel function (RMSE = 0.078; MSRE = 0.025), resulting in the best performance for all kernel functions. Therefore, using a polynomial kernel function in SVM design to predict scour depth leads to the best performance. Table 5 presents the sensitivity analysis results for the SVM-polynomial models. Besides Model 1, only Model 2 does not use the Froude number (Fr) parameter in scour depth estimation, resulting in a significant decrease in SVM-polynomial performance (the relative error increased about sixfold, MARE = 1.808). Other statistical indices Table 5 Sensitivity analysis results for the SVM-polynomial model Sensitivity analysis

MRE

MARE

MSRE

RMSE

Model 1: ds/y = f (Fr, d50/y, D/y, L/y, r)

-0.180

0.332

0.025

0.078

Model 2: ds/y = f (d50/y, D/y, L/y, r)

-1.801

1.808

0.461

0.218

Model 3: ds/y = f (Fr, D/y, L/y, r)

-1.192

1.221

0.237

0.164

Model 4: ds/y = f (Fr, d50/y, L/y, r)

-0.618

0.722

0.211

0.220

Model 5: ds/y = f (Fr, d50/y, D/y, r)

-1.245

1.316

0.254

0.173

Model 6: ds/y = f (Fr, d50/y, D/y, L/y)

-0.914

0.949

0.159

0.139

Italic values present the best model

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demonstrate an increasing trend, but Model 2 also exhibits underestimation. However, concerning the large index value (MRE = -1.8), using Model 2 leads to uncertainty in scour depth estimation. In fact, Fr is an important parameter in predicting scour depth around bridge piers, because this parameter considers the average flow velocity, which has high impact on flow around bridge piers. Not using parameter d50/y among all parameters proposed in Model 1 and in scour estimation (Model 3), similar to Model 2, is associated with significant performance degradation. This is because this parameter considers the effect of sediment on predicting scour depth and the other dimensionless parameters in Eq. (3) do not consider the effect of sediment except for Fr that the used of this parameter with average velocity. According to the statistical indices presented in Table 5, it can be seen that the Fr (Model 2) (MARE = 1.808; MSRE = 0.461; RMSE = 0.218) parameter has greater impact than d50/y (Model 3) (MARE = 1.221; MSRE = 0.237; RMSE = 0.164) on SVM-polynomial performance. Not utilizing D/y as an effective variable in scour depth prediction results in decreased performance of the SVM model. The statistical indices of Model 4 increased much more compared with Model 1, whereby the MARE, MSRE and RMSE values for Model 4 are almost 4, 10 and 2 times greater than Model 1. Thus, pier width (D) is an important parameter in scour depth prediction. In contrast to Model 1, Model 5 (MRE = -1.245, MARE = 1.316, MSRE = 0.254, RMSE = 0.173) does not only consider the relative pier length-to-flow depth (L/y) ratio in scour depth estimation, and it therefore presents a process similar to Models 2 and 3. Therefore, the two different variables related to pier characteristics (D/y and L/y) should be considered as an input combination to reach accurate results. Similar to other dimensionless parameters introduced as effective variable in scour depth prediction, not using the standard deviation of bed grain size (r) parameter (Model 6) leads to reduced SVM model accuracy. D/y (Model 4) and r (Model 6) have the least impact on scour depth estimation. However, not using either of these two parameters leads to reduced SVM-polynomial performance, which produces two and three times the relative error value. Therefore, among two dimensionless parameters that consider the pier characteristics (D/y and L/y), pier length (L) is more effective than pier width (D). However, both dimensionless parameters related to pier characteristics (L/y and D/y) should be used to make accurate predictions. Moreover, not using the standard deviation of grain size (r), which considers the effect of bed material and is not considered in the other dimensionless parameters (Eq. 3), results in decreased prediction accuracy of scour depth around bridge piers. In general, the sensitivity analysis results show that the ratio of pier width to flow depth (D/y) and Froude number (Fr) have the lowest and highest effects (respectively) on scour depth prediction. The other effective variables on scour depth around bridge piers (ds/y) in SVM include the standard deviation of bed grain size (r), ratio of median diameter of particles to flow depth (d50/y) and ratio of pier length to flow depth (L/y), which were ranked from lowest to highest, respectively. According to the provided explanations, not using any of the following parameters in estimating scour depth (Eq. 3) reduces SVMpolynomial accuracy. Consequently, if SVM-polynomial considers the effective dimensionless parameters in scour depth estimation in the form of Eq. (3), it performs the best. Figure 6 evaluates the ability of SVM-polynomial to estimate scour depth in comparison with regression (Fig. 6a) and artificial intelligence (Fig. 6b) techniques. It can be seen that none of the regressions presented in Table 3 consider all five parameters proposed in Eq. (3) to estimate ds/y. The nonlinear regression (NLR) analysis applied in MINITAB software together with the field data employed in this study yields the following relationship:

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Fig. 6 Comparison of SVM-polynomial with regression-based equations, ANN and ANFIS

    ds =y ¼ 0:28 Fr0:47 ðd50 =yÞ0:1 ðD=yÞ0:44 ðL=yÞ0:23 r0:13

ð30Þ

Figure 6a shows that Laursen and Toch’s (1956) relationship yields high estimation relative error values, as the values presented in this relationship predict about three times the actual values. The statistical indices demonstrate the underestimation performance of this model (MRE = -2.63, MARE = 2.63, MSRE = 1.034, RMSE = 0.396). This relationship includes the dimensionless parameter D/y as an effective parameter in scour depth estimation. According to the sensitivity analysis results from this study, using this parameter to estimate scour depth has the least impact on the dimensionless parameters provided in Eq. (3), and it can thus be considered one of the most important factors of poor model performance. Shen et al. (1969), Richardson and Davis (2001), and Laursen and Toch’s (1956) relationships present overestimations with large relative error, but the average performance signifies underestimation. However, the two relationships of Shen et al. (1969) and Richardson and Davis (2001) include the relative pier width-to-flow depth ratio from Laursen and Toch’s (1956) relationship in addition to the Froude number (Fr) parameter to estimate scour depth. According to the sensitivity analysis, the relative pier width-to-flow depth ratio has the greatest effect on scour depth estimation. But according to Table 6, it does not present good performance, and Richardson and Davis’ (2001)

Table 6 Comparison of SVMpolynomial, ANFIS, ANN and regression-based equations

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Methods

Statistical indices MRE

MARE

MSRE

RMSE

SVM-polynomial

-0.18

0.332

0.025

0.078

ANFIS

-0.2

0.42

0.03

0.078

ANN

-0.29

0.47

0.05

0.08

Laursen and Toch (1956)

-2.63

2.63

1.034

0.396

Shen et al. (1969)

-1.949

1.979

0.97

0.419

Johnson (1992)

-0.007

0.448

0.051

0.101

Richardson and Davis (2001)

-3.523

3.523

2.266

0.623

NLR

-0.413

0.602

0.075

0.116

Nat Hazards Box 1 SVM-polynomial model output for predicting relative scour depth (ds/y)

InVar1=input('Input the Fr'); InVar2=input('Input the d50/y'); InVar3=input('Input the D/y'); InVar4=input('Input the L/y'); InVar5=input('Input the Sigma'); x=[InVar1;InVar2;InVar3;InVar4;InVar5]; Kernel=@(xi,xj) (xi'*xj+1)^d; y=0; fori=1:Number of samples y=y+etaS(i)*Kernel(xS(:,i),x); end y=y+b; disp('ds/y = '); disp(y);

relation is weaker than Laursen and Toch’s (1956). Table 6 shows Johnson’s (1992) relation (MRE = -0.007, MARE = 0.448, MSRE = 0.051, RMSE = 0.101), which performs better than the NLR equation (MRE = -0.413, MARE = 0.602, MSRE = 0.075, RMSE = 0.116). The differences between Johnson’s (1992) relation and NLR are the data ranges used and the fact that NLR employs the L/y parameter. A comparison between Johnson’s (1992) relation and the SVM-polynomial results indicates that these relations estimate ds/y with better accuracy than other relations, although Johnson’s (1992) relationship produces greater relative error with underestimated estimation compared to SVM-polynomial (Fig. 6). In addition, the accuracy of SVM-polynomial in predicting scour depth is evaluated using two artificial intelligence methods (ANN and ANFIS). Figure 6b shows the ds/ y values predicted by three techniques (SVM, ANN and ANFIS). According to this figure, ANN shows the weakest performance compared to the two other methods for a lower range of relative scour depth (ds/y \ 0.3). For some of the ds/y samples predicted by ANN, the predicted values have high relative error in the form of overestimation and underestimation. With increasing scour depth, the accuracy of ANN increases, but in some samples, the ANN underestimates with high relative error. To select the best artificial intelligence method, the statistical indices presented in Table 6 are utilized. Among the indices presented in this table, the SVM indicates the best performance. Concerning the relative indices presented in this table, the values of MRE, MARE and MSRE for SVM represent a significant difference from the other two methods. According to the SVM-polynomial predictions compared with existing regression-based equations and two artificial intelligence techniques (ANFIS and ANN), this method can serve as an alternative as it uses a simple program (Box 1) to calculate scour depth.

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Nat Hazards

4 Conclusion In this study, scour depth around piers was predicted using SVM. The effective input variables on scour depth were first recognized. These variable consist of ratio of pier length to flow depth (L/y), ratio of pier width to flow depth (D/y), ratio of median diameter of particles to flow depth (d50/y), Froude number (Fr) and standard deviation of bed grain size (r). The results achieved in this study are summarized as follows: • To design the SVM network, six different kernel functions were utilized. The results demonstrated that the polynomial kernel function leads to superior SVM performance (MRE = -0.118, MARE = 0.332, MSRE = 0.025, RMSE = 0.078). • The sensitivity analysis applied in this study indicates that the Froude number (Fr) and ratio of pier width to flow depth (D/y) have the highest and lowest effects (respectively) on scour depth around bridge piers. Moreover, the most accurate prediction was obtained when using all dimensionless parameters presented in Eq. (3) [ds/y = f(Fr, D/ y, L/y, d50/y, r)]. • Concerning the important effect of all dimensionless parameters in Eq. (3) on predicting scour depth, a nonlinear regression-based equation was presented as Eq. (30). • The comparison of SVM-polynomial with nonlinear regression-based equations (Eq. 30) and traditional equations (Table 3) showed that using SVM-polynomial confirms the significant increase in scour depth prediction accuracy over regressionbased equations. • The proposed SVM-polynomial for predicting scour depth was compared with two artificial intelligence (AI) methods, namely ANN and ANFIS. The results signified that SVM-polynomial is more accurate than the two AI techniques and overcomes the weakness of high relative error of these approaches in scour depth prediction. • Owing to the superior performance of SVM-polynomial compared to traditional methods, the authors recommend it to design engineers to use as an alternative means of predicting scour depth using a simple program presented in this study.

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