ORIGINAL PAPER
Estimation of scour depth at bridges with complex pier foundations
using support vector regression integrated with feature selection
Nhat-Duc Hoang1• Kuo-Wei Liao2•Xuan-Linh Tran1
Received: 28 December 2017 / Accepted: 23 May 2018
Springer-Verlag GmbH Germany, part of Springer Nature 2018
Abstract
This study aims at establishing machine learning models based on the support vector regression (SVR) for estimating local scour around complex piers under steady clear-water condition. A data set consisting of scour depth measurement cases has been collected to construct the prediction models. The data set includes eight influencing factors that consider aspects of pier geometry, flow property, and river bed material. Moreover, to enhance the performance of the SVR model, filter and wrapper feature selection strategies are used. The research finding is that all feature selection approaches can help to improve the prediction accuracy compared with the SVR model that uses all available features. Notably, the feature selection method based on the variable neighborhood search (VNS) algorithm achieves the best performance (MAPE = 21.65%, R2= 0.85). Accordingly, the prediction model produced by SVR and VNS can be useful for assisting decision makers in the task of structural health monitoring as well as the design phase of bridges.
Keywords Scour depth prediction Bridge Scour Complex pier foundations Support vector regression Feature selection Variable neighborhood search
1 Introduction
Bridge scour is generally defined as the removal of sedi-ment (e.g., sand and gravel) from around bridge abutsedi-ments or piers [1]. Scour which is caused by swiftly moving water can scoop out scour holes; this leads to the deterioration of the integrity of a bridge structure [2, 3]. About 60% of
bridge failures in the United States are related to scour [4]. More importantly, scour failures have the tendency to happen quickly without any prior warning and it is very difficult to monitor them during flood events [4,5].
River bed scouring can be basically categorized into the three types: general scour, contraction scour, and local scour [6]. In recent history, it has been observed that the majority of bridge failures has been caused by local scour of the streambed [7–9]; therefore, this particular type of scour is considered to be the most important part for bridge safety analysis. As pointed out by [9], besides bridge safety, failures caused by scour often lead to considerable costs, including direct expenditures for repairing damaged bridges and indirect costs due to the impact on trans-portation (e.g., maintaining traffic flow without the bridge and of the cost for the time lost utilizing alternate routes) and on the economy of local communities. Accordingly, structural health monitoring of bridges regarding to scour is a crucial problem and has attracted an increasing attention of many scholars and hydraulic engineers [10–13].
In addition, most of the pier scour research works has focused on dealing with the scour with uniform piers [12,14]; the impacts of the pile caps and pile groups on the scour depth had not been taken into account. In real-world Electronic supplementary material The online version of this
article (https://doi.org/10.1007/s13349-018-0287-2) contains supplementary material, which is available to authorized users.
& Xuan-Linh Tran tranxuanlinh@dtu.edu.vn Nhat-Duc Hoang hoangnhatduc@dtu.edu.vn Kuo-Wei Liao
kliao@ntu.edu.tw
1 Faculty of Civil Engineering, Institute of Research and Development, Duy Tan University, R. 809 – No. 03 Quang Trung, Da Nang 550000, Vietnam
2 Department of Bioenvironmental Systems Engineering, National Taiwan University, No.1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan
circumstance, due to geotechnical and economical reasons, most of the bridges do not feature the property of uniform piers; their cross-sectional dimension varies over the length of the pier. The complex piers are composed of several components, i.e., column, pile cap, and pile group. This imposes a significant challenge for predicting the bridge scour.
Needless to say, models that can accurately estimate the scour depth at bridge piers are highly desirable. The reason is that under-predicting the scour depths can lead to catastrophic consequences, because the bridge foundation design is not sufficient for ensuring the required support to the structure [9]. On the contrary, overestimating the scour depth results in uneconomical designs of the bridge foundations.
Both theoretical and experimental studies have pointed out that bridge scour is affected by various influencing factors (i.e., pier geometry, property of water flow, and characteristics of the river bed) [15]. Moreover, the underlying functions that relate to the influencing factors and the scour depth are highly non-linear and difficult to express them explicitly [16]. These facts lead to the sophisticated procedures for estimating the scour depth using empirical formulae [15,17–19] and the inaccuracy in scour depth prediction of those approaches [9,20].
Within this context, various scholars have relied on machine learning to establish scour depth prediction models from experimental data sets. Muzzammil [21] carried out a comparative study on the performance of artificial neural network (ANN), adaptive network-based fuzzy inference system, and multiple linear regression used for estimating scour depth at uniform bridge abutments. Scour depth prediction models based on ANN and fuzzy ANN have been extensively utilized [16, 22–26]. Najaf-zadeh and Azamathulla [27] proposed a quadratic poly-nomial of group method of data handling network for estimating scour depth around bridge piers. Genetic Pro-gramming and Gene-Expression ProPro-gramming for model-ing of local scours have been proposed in [28–30].
As can be seen from the literature, studies on scour depth prediction at bridges with complex piers are still very limited. Our current research aims at contributing to the body of knowledge by proposing a scour depth estimation model for bridge structures with complex piers that employs the Support Vector Regression (SVR). SVR is widely known as a powerful and reliable tool for non-linear modeling [31,32]; however, this machine learning method has been rarely applied in the problem of interest.
In addition, feature or influencing factor selection on experimental/real-world data sets has also been left unex-plored in scour depth modeling. Feature selection is indeed very crucial; as summarized by [33], the procedure of feature selection can help to enhance the prediction
performance of the machine learning models, construct more cost-effective models, and build a more comprehen-sible model with fewer variables involved. Therefore, this study equips the SVR with the feature selection algorithms. Both the widely known filter and wrapper strategies of feature selection are employed. Notably, a new wrapper method based on the stochastic search of variable neigh-borhood search (VNS) is proposed in this study. The research finding is that the VNS-based wrapper method is useful in identifying a highly relevant subset of scour depth influencing factors and enhancing the SVR prediction accuracy.
The rest of this paper is organized in the following manner. The second section describes notable factors that influence the scour depth at bridges with complex pier foundations. Research material and method is stated in the third section, followed by the proposed framework description. Experimental result and comparison are reported in the next section. Several conclusions of this study are provided in the final part.
2 Influencing factors and the collected data
set of scour depth at bridges
with complex pier foundations
Notably, for predicting scour depth at non-uniform piers, many important impact factors should be considered. In this study, those factors are categorized into three groups: the pier geometry, flow property, and material character-istic at the riverbed.
2.1 Pier geometry
In this category of factors, the pier width perpendicular to the flow direction (bc), pile-cap width (bpc), and soil
cov-ering height (level of the top surface of the pile cap below the surrounding bed level (Y) are generally recognized as influential variables [18]. As a common perception, a larger pier width (bc) leads to an increase of the scour depth (ds).
[34] simulated scouring of non-uniform piers and found that when the non-uniform ratio of bc/bpcis greater, the pier
scour depth is smaller.
Melville and Raudkivi [35] divided non-uniform piers into three separated sections based on the soil covering height (Y): (Zone 1) the section in which the pile cap is below the bottom of the scour hole (Y/bc[ 2.4), (Zone 2)
the section in which the pile-cap top is within the scour hole (2.4 C Y/bcC 0), and (Zone 3) the section in which
the pile-cap top is above the bed level (Y/bc\ 2.4). The
authors experimentally found that compared to the scour results of a uniform pier, Zone 1 does not show influence on the scour, Zone 2 reduces the scour, and Zone 3
increases the scour depth. Moreover, to characterize the pile-cap feature, the longitudinal extension of pile-cap face out from pier face (Lu) can be employed [18].
2.2 Flow property
Inevitably, this group of variables has to consider the mean velocity of the approach flow (V). Depending on the magnitude of velocity (V), the scour can be further divided into two types, namely, clear-water and live-bed scour [35]. In the first type, the scour depth increases as a function of the flow velocity without sediment movement. In the sec-ond type, the flow velocity surpasses the critical mean velocity for particle motion (Vc); this leads to the transports
of sediment across the bed surface and complicates the scour status.
As shown in the previous works of Melville [15, 19], when the flow velocity (V) exceeds the threshold velocity (Vc), the scour depth first decreases and then increases to
another peak. As a consequence, the average scour depth of the live-bed scour is smaller than that of the clear-water scour depth. Therefore, for safety purpose, scour depth in the clear-water condition is often considered in bridge safety evaluation. Furthermore, the flow depth (y), which is the distance from the water surface to the river bed, should be employed to estimate the scour depth [36]. It is gener-ally known that when the ratio of y/bcis greater, the impact
of the water flow on the scour depth is greater and vice versa. [36] showed that if the ratio of y/bc[ 3 or 4 (i.e.,
deep water), the impact of the change of flow depth on the scour depth can be ignored.
2.3 Characteristics of river bed material
The third group of variables, including the median grain size (d50), river bed material geometric standard deviation
(rg), and the critical velocity of sediment movement (Vc),
characterizes the feature of the river bed material [27,36]. In general, the greater the size of the river better material, the smaller the local scour depth and vice versa. The reason is that large size of the river bed material provides better resistance against the scour effect. Besides the size of the river bed material, the roughness and the size distribution of the material may also impose influence on the local scour depth; these two factors can be inferred via the critical velocity of sediment movement (Vc) and the
geo-metric standard deviation (rg).
2.4 The collected data set
This study has collected 170 data samples of clear-water scour from existing experimental works and four more data samples conducted in the Hydrotech Research Institute of
the National Taiwan University [37]. For more details regarding the data measurement and collection, the readers are guided to the previous works [18,35,38,39].
In total, 174 data instances can be used for model con-struction and verification. The eight influencing factors, including the flow depth y, the pier width perpendicular to the flow direction bc, the pile-cap width bpc, the
longitu-dinal extension of pile-cap face out from pier face Lu, the
soil covering height Y, the ratio of the mean velocity to the critical velocity of sediment movement V/Vc, the median
grain size d50, and the river bed material geometric
stan-dard deviation rg, are employed to estimate the scour depth
dsof complex pier foundations.
It is noted that the bed material used in the experiment is sand. Moreover, the properties of the river bed, including the erodibility, are characterized by the median grain size (d50), river bed material geometric standard deviation (rg),
and the critical velocity of sediment movement (Vc). The
statistical descriptions of all variables are provided in Table1. Scatter plots of the variables are illustrated in Fig.1. The collected data set used in this study is provided in the Appendix.
3 Research material and method
3.1 Support vector regression (SVR)
Support vector regression (SVR), introduced by Vapnik [40], is a competent tool for non-linear function approxi-mation. The learning process of SVR guarantees to convert to a single a global optimum. Moreover, this algorithm often demonstrates outstanding prediction performance due to its implementation of the structural risk minimization principle which considers both the training error and the generalization of the model [41,42, 43]. SVR is notably characterized by the usage of kernels for dealing with non-linear data, absence of local minima in model learning, and
Table 1 Data description
Variable Notation Unit Min Average Std. Max
IF1 y m 0.13 0.22 0.12 0.60 IF2 bc m 0.01 0.04 0.03 0.15 IF3 bpc m 0.05 0.10 0.06 0.37 IF4 Lu m 0.01 0.13 0.41 2.78 IF5 Y m - 0.67 - 0.02 0.12 0.21 IF6 V/Vc – 0.53 0.89 0.13 1.18 IF7 d50 mm 0.06 0.62 0.22 1.00 IF8 rg – 1.00 1.12 0.12 1.30 Y ds m 0.01 0.09 0.05 0.34
sparseness of the solution [42]. A non-linear function is learned by the algorithm via a kernel function K which maps the data into high-dimensional kernel-induced feature space (see Fig.2).
In SVR, the goal is to learn a function f(x) that best describes the mapping between a set of L training samples (x1, y1),…., (xL, yL). Similar to the conventional linear
regression, SVR algorithm attempts to locate a hyperplane f(x) with the smallest structural risk in the high-dimen-sional feature space [31]:
fðxÞ ¼ wT/ðxÞ þ b ð1Þ
where /(x) is a non-linear mapping from the original input space to the feature space. w and b are model parameters and estimated via the following minimization problem:
Min: 1 2jjwjj 2 þ CX L i¼1 niþ ni ð2Þ subjected to yi hw; /ðxð iÞi þ bÞ e þ ni hw; /ðxiÞi þ b ð Þ yi e þ ni ni;ni 0 8 > < > :
where C is the the penalty factor that determines the trade of between model accuracy and model complexity; niand
ni*denote the slack variables. i is the index of training data
sample. e is the error toleration threshold meaning that the data sample xiis not penalized as long as its error does not
exceed this threshold [44]. Fig. 1 Scatter plots of scour depth influencing factors
X (Influencing Factors) Y (Scour Depth) Kernel Mapping Φ(xj) Φ(xi)
Φ(x)
Input Space Feature space Y (Scour Depth)Fig. 2 Illustration of the SVR learning process
After solving the above optimization problem with the Lagrangian and Karush–Kuhn–Tucker conditions for optimality, the final SVR model can be expressed as fðxÞi¼ XL i¼1 ai ai Kðxi; xÞ þ b ð3Þ
where aiand ai*are nonzero Lagrangian multipliers. K(xi,x)
denotes a kernel function. The commonly used kernel function is the radial basis function (RBF) [37,45,46] is shown as follows:
Kðxi; xjÞ ¼ exp cjjxi xjjj 2
ð4Þ where c is a free parameter of the RBF.
3.2 Feature selection
In general, feature selection (or input variable selection) methods can be divided into two types: filter and wrapper algorithms [47]. Filter algorithms evaluate the relevance of each feature before the utilization of any machine learning algorithm; the relevance of each feature is quantified using statistical properties of features. On the other hand, wrap-per methods determine the relevance of features according to the accuracy of the prediction model that employs such features on the training data.
ReliefF [48] is a popular filter method for feature analysis. Based on probability and information theories, ReliefF is able to detect conditional dependencies between features and provide a unified view on the relevance of features in function approximation tasks [49]. This method assigns a weight value for each input feature that indicates its importance; the higher the weight, the more relevant the input feature. Due to its efficiency, this algorithm is selected to be used in this study.
In the case of the wrapper strategy, these methods are implemented by first determining the machine learning algorithm (SVR in this study), the performance criterion (e.g., root mean square error), and the search strategy. Based on the defined search strategy, an iterative process is performed; the machine learning algorithm is trained at each iteration with the available data set and the perfor-mances of the model according to different subsets of features are calculated. After the searching process, a desirable subset of feature is determined according to the pre-defined performance criterion.
Sequential Forward Selection (SFS), Sequential Back-ward Selection (SBS), and Metaheuristic-Based Feature Selection (MFS) are the three commonly used wrapper algorithms due to their ease of implementation [50, 51]. SFS starts from the empty set of feature and sequentially add a certain feature that can reduce the prediction error of the machine learning model. Meanwhile, SBS operates in
the opposite direction of SFS. SBS begins with the full set of features and sequentially remove a feature that helps to improve the model prediction accuracy.
The main drawback of SFS is that this algorithm cannot cast out features that become redundant after the advent of other features. On the other hand, the main disadvantage of SBS is its inability to consider the relevance of a feature after it has been rejected. MFS formulates the feature selection as an optimization problem in which the decision variables are the selected subset of features. MFS tends to identify a better subset of features due to its ability to avoid the aforementioned drawbacks of SFS and SBS. Never-theless, MFS is often criticized for their computational burden due to the entwinement of the learning and feature selection tasks [52,53]. Therefore, this study employs the Variable Neighborhood Search (VNS) as a single solution-based optimizer instead of other population-solution-based mizers (e.g., genetic algorithm and particle swarm opti-mization) for performing feature selection with the SVR model. The aim of this algorithm selection is to seek for quality solutions of feature subsets with reasonable com-putational cost.
3.3 Variable neighborhood search (VNS)
Variable Neighborhood Search (VNS), proposed by Mladenovic´, Hansen [54], is a single solution-based metaheuristic method for solving a set of combinatorial optimization and global optimization problems. VNS ini-tially searches within small neighborhoods until a local optimum is encountered, at which point the search process switches to a larger neighborhood, which might help to escape from the local optimum [55].
In VNS, k neighborhood relation N1, N2,…,Nk is
employed, which are ordered according to increasing size. The algorithm begins with the neighborhood N1 and
per-forms neighborhood descend-based movements until it reaches a local minimum. If no further improvement is found using a neighborhood Ni, VNS continues the search
in an enlarged neighborhood Ni?1. If an improvement is
achieved, VNS returns to the neighborhood N1. The VNS
algorithm is demonstrated in Fig.3.
4 The proposed SVR-based model
for estimating scour depth at bridges
with complex pier foundations
This section of the study describes the structure of the SVR-based scour depth prediction at bridges with complex pier foundations. An overview of the prediction model is shown in Fig.4. The proposed model can be divided into three consecutive steps: feature selection, model training,
and model prediction. It is noted that the model is pro-grammed and operates in MATLAB environment.
It is noted that the original data set has been normalized using the Z-score method. Accordingly, in the first step, the relevance of scour depth influencing factors is evaluated via one among four approaches: ReliefF, SFS, SBS, and VNS. It is noted that to reliably assess the relevance of input factors and avoid the randomness due to data selec-tion, the feature selection process is carried out on the basis of a tenfold cross-validation process. Herein, the original data set is separated into ten mutually exclusive folds in which each fold in turn is utilized as testing data and the other nine folds are utilized for model construction.
Hence, the following cost function is employed to evaluate the quality of a feature subset:
f ¼ P10 i¼1RMSE Testing i 10 ð5Þ
where RMSEiTesting= the model error measured in terms of
Root Mean Square Error (RMSE) for the ith testing data fold.
It is worth noticing that the implementation of ReliefF requires the setting of the number of nearest neighbor (Kn)
[48]; in this study, Knis set to be 5% of the total number of
the data samples. Furthermore, for the case of VNS-based feature selection, a general form of a solution candidate is X = [x1,…xi,…,xD], where xiis an integer of the range [1,
D]; D = 8 is the number of available influencing factors in the current data set. The valid subset of features is formed by selecting unique values of X. For instance, a solution candidate X = [1, 2,…,8] indicates that all influencing factors are selected; a solution candidate X = [1, 2, 3, 5, 3, 6, 7, 8] indicates means that the factor 4 is not included. In addition, the neighborhood set N of the VNS algorithm is selected to be [0.001, 0.005, 0.01, 0.05, 0.1, 0.5, 1, 4, 8].
When the most appropriate subset of scour depth influ-encing factors has been identified by a method of feature selection, the data set with selected features is established by excluding rejected features from the original data set. Based on the newly formed data set, the training and pre-dicting phases of SVR are consecutively carried out. It is noted that the training phase of the SVR mode requires the specification of the penalty parameter C and the RBF parameter c; in this study, these two parameters are selected via a tenfold cross-validation-based grid search procedure with the parameter set of [0.01, 0.05, 0.1, 0.5, 1, 5, 10, 50, 100]. In addition, the error toleration threshold (e) is fixed to be 0.1.
5 Experimental result and comparison
This section reports the proposed SVR-based scour depth prediction model accompanied with the four feature selection algorithms (ReliefF, SFS, SBS, and VNS). It is noted that the Root Mean Squared Error (RMSE), the Mean Absolute Percentage Error (MAPE), and the coefficient of determination (R2) are used to quantify the prediction accuracy of the prediction model. RMSE indicates the deviation between the output values actually observed and the output values computed from a trained model. MAPE is the deviation between actual and prediction values divided by the actual value, and is presented in terms of a per-centage. In addition, R2shows the proportion of the vari-ability in the output variable explained by the model.
The feature selection outcome produced by ReliefF is shown in Fig.5. As can be seen from the result, ReliefF rejects IF7(d50) and IF8(rg) from the final feature subset.
___________________________________
Variable Neighborhood Search:
Define objective function (f)
Define solution boundary (B)
Define a neighborhood set N
1, N
2,…,N
kDefine the maximum number of iteration (Iter)
Randomly initial a solution x
∈
B
g = 1 // Iteration counter
While g < Iter
Choose the most improving neighbor x* of x in N
iIf f(x*) < f(x)
x = x*
i = 1
Else
i = i + 1
End if
End while
Return x
___________________________________
Fig. 3 VNS algorithm Feature Selection ReliefF SFS SBS VNS SVR Model Training Ten Fold SeparationOriginal Data Set
Data Set with Selected Features SVR Model Prediction SVR Model Training SVR Model Prediction Scour Depth Prediction Result
IF3 (bpc), IF4 (Lu), IF5 (Y), and IF6 (V/Vc) receive
com-paratively high weighting values; meanwhile, the weights of IF1(y) and IF2 (bc) are relatively low. Based on such
outcome, the SVM coupled with ReliefF method is run using two scenarios: one with a feature subset of (1, 2, 3, 4, 5, 6) and one a feature subset of (3, 4, 5, 6). Tenfold cross-validation processes point out that the first scenario (RMSE = 0.021; MAPE = 24.559%; R2= 0.822) is better than the second scenario (RMSE = 0.021; MAPE = 27.095%; R2= 0.801). Therefore, the first scenario of feature subset is selected for the SVR model supported by ReliefF; this model is denoted as SVR–ReliefF.
Based on experiments, it is interesting to find that the two methods of SFS and SBS yield the same result of feature selection (see Tables2, 3). These two methods reach a consensus that IF2, IF4, IF5, IF7, and IF8are useful
for scour depth prediction; IF1, IF3, and IF6are recognized
as not useful features. These results of the two wrapper methods are quiet contradictory to the feature subset found by the filter approach of ReliefF. The reason is that ReliefF suggested that IF7 and IF8 are redundant. Notably, the
model performance using SFS/SBS, denoted as SVR–SFS/ SBS (RMSE = 0.021; MAPE = 22.743%; R2= 0.828),
demonstrates certain improvement compared with SVR– ReliefF (RMSE = 0.021; MAPE = 24.559%; R2= 0.822). The feature selection outcome performed by VNS optimization algorithm is reported in Table4. Since VNS is a stochastic search, the VNS-based feature selection is performed in 20 runs to reliably evaluate the feature importance. The maximum number of iteration is set to be 300. A typical run of the VNS-based feature selection is shown in Fig.6. Moreover, the average selection score (ASC) of a feature (see Fig.7) can be computed as the average time that the feature is selected by VNS.
As can be seen from the result, IF1, IF2, IF4, IF5, IF7,
and IF8receive high ASC; notably, the ASC values of IF2,
IF4, and IF5 are 1 meaning that those features are
abso-lutely relevant for scour depth prediction. It is suspected that IF6and IF3are redundant, since their ASC values are
relatively small (0.05 and 0.15). Therefore, two scenarios of feature subsets are employed for the SVR model sup-ported by VNS (denoted that SVR–VNS): subset 1 which includes the features of (1, 2, 3, 4, 5, 7, 8) with IF6being
excluded and subset 2 which includes the features of (1, 2, 4, 5, 7, 8) with IF6and IF3being excluded. Experiments
point out that SVR–VNS with the subset 1 (RMSE = 0.018; MAPE = 21.653%; R2= 0.852) is better than that with the subset 2 (RMSE = 0.020; MAPE = 23.933%; R2= 0.845). Therefore, SVR–VNS with the subset 1 is selected to be used for result comparison with other models.
It is noted that from the preliminary inspection shown in Fig.1, the factors of X4(the longitudinal extension of
pile-cap face out from pier face) and X5 (the soil covering
height) have low linear correlations with the scour depth. The correlation coefficients for X4 and X5 are 0.13 and
0.02, respectively. However, analysis results using the feature selection methods of SBS and SFS show that these two factors are useful for scour depth prediction (as illus-trated in Table2, 3). In addition, the VNS optimization process has revealed that X4and X5are highly relevant for
scour depth estimation; their average selection scores are both 1 (see Fig.7). These facts imply that the relation between these two factors with the scour depth is non-Fig. 5 Feature evaluation using ReliefF method
Table 2 Feature selection using SFS method
Iteration Influencing factors RMSE
1 2 3 4 5 6 7 8 1 x 0.034 2 x x 0.030 3 x x x 0.024 4 x x x x 0.022 5 x x x x x 0.021
Note ‘x’ = a selected feature
Table 3 Feature selection using SBS method
Iteration Influencing factors RMSE
1 2 3 4 5 6 7 8
1 x x x x x x x x 0.0245
2 x x x x x x x 0.0229
3 x x x x x x 0.0226
4 x x x x x 0.0214
linear. Therefore, the application of SVR in analyzing the functional mapping between the influencing factors and the output of scour depth is very suitable. It is because SVR can effectively model non-linear relations between input and output variables ([56]; [57]).
Table5summarizes the experimental result comparison that reports the performance of the four model used in this study. SVR denotes the model that employs all available eight features for scour depth prediction. As can be observed, all models equipped with feature selection methods (SVR– ReliefF, SVR–SFS/SBS, and SVR–VNS) show improve-ment compared with SVR. It can be seen that wrapper approaches (SVR–SFS/SBS and SVR–VNS) are better than the filter approach of ReliefF. Among them, SVR–VNS achieves the most desirable outcome in terms of all perfor-mance measurements (RMSE, MAPE, and R2) (see Fig.8). Illustrations of the scour depth prediction result yielded by the most desirable SVR–VNS model are shown in Fig. 9. Table 4 Feature selection using VNS algorithm
IF Run ASC 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 1 x x x x x x x x x x x x x 0.65 2 x x x x x x x x x x x x x x x x x x x x 1.00 3 x x x 0.15 4 x x x x x x x x x x x x x x x x x x x x 1.00 5 x x x x x x x x x x x x x x x x x x x x 1.00 6 x 0.05 7 x x x x x x x x x x 0.50 8 x x x x x x x x x x x x x x x 0.75
Note ‘x’ = a selected feature
ASC average selection score, IF influencing factors
Fig. 6 Typical run of VNS-based feature selection
In addition, the performance of the proposed SVR–VNS is compared with the conventional formula-based approa-ches including the HEC-18 Richardson and Davis [6],
Melville and Coleman [19], and Ataie-Ashtiani et al. 18] approaches. The result comparison is reported in Table6. It is clearly shown that the newly proposed method Table 5 Prediction result
summary Phase Performance SVR SVR–ReliefF SVR–SFS/SBS SVR–VNS
Mean Std. Mean Std. Mean Std. Mean Std.
Training RMSE 0.009 0.000 0.009 0.000 0.013 0.001 0.012 0.001 MAPE (%) 14.203 1.223 14.420 0.861 16.896 1.033 16.655 0.725 R2 0.971 0.003 0.971 0.003 0.941 0.007 0.948 0.004 Testing RMSE 0.021 0.009 0.021 0.009 0.021 0.008 0.018 0.010 MAPE (%) 25.475 13.140 24.559 9.634 22.744 8.197 21.653 6.978 R2 0.814 0.085 0.822 0.087 0.828 0.119 0.852 0.152
Fig. 8 Model comparison in terms of RMSE, MAPE, and R2. Note Prediction models 1, 2, 3, and 4 denote SVR, SVR–ReliefF, SVR–SFS/SBS, and SVR–VNS, respectively
(MAPE = 21.65%, RMSE = 0.02, R2= 0.85) is superior to other formula-based models in all performance measure-ment metrics.
6 Conclusion
Prediction models for scour depth prediction at complex pier foundations based on SVR and feature selection methods are investigated in this study. The SVR, the machine learning algorithm based on the statistical learning theory, is employed or constructing mapping functions used for the prediction of scour depth. To construct and verify these machine learning based approaches, a data set containing 174 records of scour depth measurement has been collected for this study.
Moreover, both filter and wrapper feature selection strategies have been integrated into the proposed SVR prediction model. The filter method is based on the ReliefF algorithm; meanwhile, the wrapper feature selection strat-egy employs the SFS, SBS, and VNS algorithms. Experi-mental results have shown that the SVR models integrated with feature selection methods are better than the SVR that uses all available input factors. Notably, the SVR–VNS demonstrates the best outcome; this method achieves an outstanding results with MAPE = 21.65% and R2= 0.85, which are considered to very desirable, since the problem of scour depth estimation at complex pier foundations has been proved to be very challenging.
Accordingly, the prediction model produced by SVR and VNS can be helpful to support decision makers in design phase of bridge which is susceptible to scouring. Based on the estimated results obtained from the proposed SVR–VNS, if the predicted scour depth surpasses the pre-specified critical value, scour prevention measures must be performed for the monitored bridge. The future improve-ments of the current study may include: (1) collecting more data samples to enhance the applicability of current pre-diction model; (2) investigating the performance of the SVR models used with other types of kernel function (e.g.,
hybrid kernel functions) in scour depth estimation; and (3) utilizing other advance machine learning approaches to establish more accurate modeling tools.
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