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Predicting discharge coefficient of triangular labyrinth weir using Support Vector Regression, Support Vector Regression-firefly, Response Surface Methodology and Principal Component Analysis
Author’s Accepted Manuscript Predicting Discharge Coefficient of Triangular Labyrinth Weir Using Support Vector Regression, Support Vector Regression-firefly, Response Surface Methodology and Principal Component Analysis Hojat Karami, Sohrab Karimi, Rahmanimanesh, Saeed Farzin
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S0955-5986(16)30263-1 http://dx.doi.org/10.1016/j.flowmeasinst.2016.11.010 JFMI1294
To appear in: Flow Measurement and Instrumentation Received date: 25 April 2016 Revised date: 19 November 2016 Accepted date: 30 November 2016 Cite this article as: Hojat Karami, Sohrab Karimi, Mohammad Rahmanimanesh and Saeed Farzin, Predicting Discharge Coefficient of Triangular Labyrinth Weir Using Support Vector Regression, Support Vector Regression-firefly, Response Surface Methodology and Principal Component Analysis, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2016.11.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Predicting Discharge Coefficient of Triangular Labyrinth Weir Using Support Vector Regression, Support Vector Regression-firefly, Response Surface Methodology and Principal Component Analysis
a
b
Hojat Karami , Sohrab Karimi *, Mohammad Rahmanimaneshc, Saeed Farzind a, b, d
Department of Civil Engineering, Semnan University, Semnan, Iran
c
Faculty of Electrical and Computer Engineering, Semnan University, Semnan, Iran
*
Corresponding author: e-mail: [email protected],Phone: (+98) 918-333-5727
Abstract Weirs are hydraulic structures which conduct the most powerful flow with large overflow. Discharge flow predication is based on capacity discharge designation by designer. In this paper, the discharge capacity in triangular labyrinth side-weirs is computed by using new techniques with high precision. The four employed techniques for computation of discharge capacity are: Support Vector Regression (SVR), Support Vector Regression– Firefly (SVR- Firefly), Response Surface Methodology (RSM) and Principal Component Analysis (PCA).A comparison between the computed discharge capacity and empirical results is considered in this paper. Determination coefficient (R2), Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), SI and δ are five statistical indicators which help us to measure the precision of the designed models. The statistical indices indicated that the SVR-Firefly model has the highest ability among the models for simulation, with average MAPE=0.49%, R2=0.991 and RMSE=0.0035. Like the results achieved by the SVR-Firefly, comparatively good results were obtained by both PCA and SVR models. The SVR model suggested the average MAPE value near 1.073 in the training mode under the most unfavorable conditions. The MAPE value equal to 1.23 was also obtained in the test mode. This proves that the value of error rate is tolerable. Keywords: Weir, Discharge capacity, Support Vector Regression (SVR), Support Vector Regression–Firefly (SVR-Firefly), Response Surface Methodology (RSM), Principal Component Analysis (PCA).
1. Introduction A weir is a structure over which passes the extra water in upstream and downstream of dams. Weirs are also considered as the most vital hydraulic structures in water resources projects. Capacity of weirs needs to be sufficient to protect dams and ensure the safety of channels constructed for the transfer of water. Insufficiency of the weir capacity is the most important cause of damages to the channels and even to dams [1, 2]. To keep the safety of water conveyance structures, the most important issue is proper functioning of weirs under the most abnormal conditions. Water flow passes over the weir when its level is raised at the back of the latter and exceeds the height of the weir’s crest. When this happens, the velocity profile becomes non-linear and curved. In case the discharge falls behind the level of channel and river, the flow decreases over the weir [3]. The designed structure must be safe and powerful for operation. Also, high efficiency is an important factor for designing of structures. The danger arising from inundation can be reduced by selecting an appropriate weir’s discharge capacity which is, in general, one of the most important elements in this respect. It is necessary that weirs act based on a capacity enough to unload the basic inundation of the designed model such that financial losses and death rate does not go beyond the expected level. It is necessary to have a good understanding of the purpose for which weirs are constructed in order to lower considerably the costs of construction and deal with the problems of inundation. There are many kinds of weirs including shafts, side weirs, stepped, sharp-crested, ogee, labyrinth, normal, siphon, and broad-crested weirs [4-6]. Soft computing techniques have been employed by various researchers in the last decade in order to solve sophisticated problems. These techniques are new and quite accurate, that can help to calculate the discharge capacity based on hydraulic parameters and geometry of the weirs. Among these parameters, one can name the Froude number of the weir’s upstream flow, crest height, flow depth, and weir height. Subramanya [7], Bagheri and Heidarpour [8], Karimi et al. [9], Kisi [10], Bonakdari et al. [11], Karami et al. [12], Bihan et al. [13], and Kisi et al. [14] used these computing techniques to calculate the discharge capacity of labyrinth side weirs. Using the new methods of support vector regression (SVR), support vector regression-firefly (SVRfirefly), response surface methods (RSM) and principal component analysis (PCA), which solve complicated engineering problems efficiently and desirably accurate, and increase the accuracy of the work of researchers, has become popular in the last decade because most of the engineering problems are complex [15- 22]. This research attempts to calculate the discharge capacity (Cd) of a triangle labyrinth. In this respect, SVR, SVR-Firefly, RSM and PCA were the four techniques used for computation of the discharge capacity. Afterwards, a comparison was made between the latter and results of the conducted experiments. The developed networks were trained and tested by considering the dimensionless parameters which affect the process of calculating the discharge capacity. Finally, statistical parameters of RMSE, MAPE, R2, and MAE were employed to select the most accurate model to predict the discharge model.
2. Methods
The development of five methods (Regression, SVR, SVR-Firefly, RSM and PCA) was carried out by means of specified dimensionless parameters (input data= 95 data in training mode and 26 data in the test mode). These parameters include: (l/h), (l/w), (h/b), (sin θ *w/l), and (y/(sin *w)). The experimental model was used to determine the accuracy of
the resulted obtained from the above-mentioned developed methods in train and test modes. These parameters are shown in Table 1.
Table 1. Parameters employed to approximate the average discharge coefficient (in this research)
2.1. Experimental Model
The experimental model designed by Kumar et al. [4] was used to estimate the discharge coefficient of the studied weirs. This model was actually a rectangular channel of 12 m in length, 0.28 m in width, and 0.41 m in depth. In these experiments, triangular weirs were used (Figure 1). The latter were placed at a distance of 11 m from the entrance of the channel. The water height was measured by using point gages with ±0.1 measurement accuracy above the weirs. A nape flow was produced by making pores in the channel wall and also in the weirs. The upstream portion of the channel was selected to install grid walls and preventive flows so that vortexes and the disturbance of the watersurface are not formed and diminished. The parameters which were employed in this research can be found in Table 2.
Fig 1. The plan of the experimental channel used in this study (Kumar et al, [4])
Table 2. Parameters applied to approximate the average discharge coefficient (Kumar et al, [4])
2.2. Support Vector Regression and Support Vector Regression – Firefly
Support Vector Regression (SVR) [15, 16] map the input data to a high dimensional feature space via a nonlinear function (i.e., the kernel function) and then compute a linear regression function on data in the mapped feature space. Suppose there is ( ) a training data set ( ) . In ε-SV regression [16], the goal is to find a function that has at most ε deviation from the actually obtained targets for all the training data, and at the same time is as flat as possible[16].In the case of linear function, f is taking the form ( ) (1) Where denotes the space of the input patterns, and denotes the dot product in X. In this paper, we employ Radial Basis kernel function (RBF) and utilize a kind of firefly algorithm to tune SVR hyper-parameters. The firefly algorithm, inspired by the flashing behavior of fireflies, is a kind of meta-heuristic algorithms used in optimization problem
[23], and along with SVR (i.e., SVR-Firefly algorithm) are widely used for regression problem [17, 18, 24, 25]. 2.3. Response Surface Methodology
Response surface methodology (RSM) [19, 20] consists of a group of techniques used in finding a relationship between a response y, and several associated input variables denoted by x1, x2, . . . , xP. In general, such a relationship can be approximated by a lowdegree polynomial model of the form: ( )
(2)
where X = (x1, x2, . . . , xP), f (x) is a vector function of k elements that consists of powers and cross-products of powers of x1, x2, . . . , xP up to a certain degree denoted by d (≥ 1), β is a vector of k constant coefficients referred to as parameters, and ϵ is a random experimental error [26]. Two important models are commonly used in RSM, i.e., the firstdegree model (d = 1), and the second-degree model (d = 2). In this paper, we use seconddegree RSM, where ∑
∑∑
∑
(3)
RSM is widely used for optimization problem in many research area, e.g. collagen extraction from eggshell membrane [21], thermodynamic optimization of the recuperative heat exchanger [27], and extraction of bioactive compounds from fruit [28].
2.4. Principal Component Analysis
The idea of principal component analysis (PCA) [22] is to reduce the dimensionality of a data set consisting of a large number of interrelated variables, while retaining as much as possible of the variation present in the data set. This is achieved by transforming to a new set of variables, the principal components (PCs). This transformation is defined in such a way that the first principal component has the largest possible variance, and each succeeding component in turn has the highest possible variance under the constraint that it is orthogonal to the preceding components. Principal components are used in conjunction with a variety of other statistical techniques such as regression analysis [22], and used in many optimization problems such as [29, 30].
3. Result 3.1. Regression Model
The discharge coefficient of triangular labyrinth weir can be written as a function of non-dimensional parameters of (l/h), (l/w), (h/b), (sin θ *w/l), and (y/(sin *w)). Different
non-linear regression models were developed. Finally, it led to the following equation for estimation of discharge coefficient of triangular labyrinth weir. C d 1.48 l h
1.34
l
w
1.05
h b sin w l y 1.11
0.25
sin w
0.25
(4)
Figure 2 shows not bad agreement between the measured and developed regression model in training and testing state.
Fig 2. Comparing estimated discharge coefficient with experimental result in training state and testing state for regression model
3.2. The results of analysis obtained through using SVR, SVR-Firefly, RSM and PCA techniques
The results of analysis obtained through using SVR, and SVR-Firefly techniques were compared to evaluate the preciseness of the approximated model at each and every step of designing the model. Each model was designed on the basis of the criteria of coefficient of determination (R2), Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), MAE, as can be seen in the following forms:
MAPE
1 n xi yi n i 1 xi
1 n xi yi 2 n i 1
RMSE
n R x i x y i y i 1
2
MAE =
(5)
1 n ∑x _ y i n i =1 i
(6)
x n
i 1
i
x
2
n
i 1
yi y
2
2
(7)
(8)
Where yi and xi are prediction and observation Cd values, respectively, and y and x are the mean prediction and observation Cd values, respectively. The approximated values relating to the discharge coefficient (Cd) are shown in Figures 3 and 4 according to Support Vector Regression (SVR), the Support Vector Regression – Firefly (SVR-Firefly), the Response Surface Methods (RSM)and the Principal Component Analysis (PCA) methods used in the present research. Then the approximated values were compared with the experimental values. As it can be seen in these Figures, these values are relatively well consistent with each other in nearly all the four models presented above, because all the models show a relative error average of lower than 2%. While approximating the discharge coefficient in the best part of the values, the SVR-Firefly model showed the largest relative error of around 1%. Like in the latter model, the Cd estimated in the PCA model shows a
similar value, however, the difference is that the approximated values in the SVR-Firefly model are not completely close to actual values and are typically estimated as overdesign. The same process, though, is not followed in the PCA model; in fact, one cannot say for sure that the approximated values are more or less than the actual ones especially at points in which the emergence of an error is not impossible.
Fig 3. Comparing estimated discharge coefficient with experimental result in training state
Fig 4. Comparing estimated discharge coefficient with experimental result in testing state
90% of the approximated discharge coefficient values, indicated in this study through using models SVR-Firefly and PCA, show an error rate of lower than 2.5%, as can be seen in Figures5 and 6. According to what was said above, it is concluded that the results of model SVR- Firefly comply with those obtained by model PCA. The discharge coefficient values approximated under testing condition in the four models of SVR-Firefly, PCA, SVR and RSM are, by and large, similar to experimental ones; however, the maximum relative error percentage made in using RSM model is around 4.5%. According to Figure 5 and compared to the experimental data, 90% of the approximated data indicate a relative error of 2.5%. The accuracy of estimating the Cd by using the models SVRFirefly, PCA, SVR and RSM and the preciseness of the data which were not applied during the construction of the above models are shown in Figure 6. It can be seen, in RSM model and given the above Figure, that the latter approximates relatively precisely the Cdas compared with the experimental data; also according to Figure 6, the approximation of the Cd through models SVR-Firefly, PCA, SVR and RSM indicates the preciseness with a relative effort lower than 2.5% in 90% of the data under test conditions.
Fig 5. Error distribution for all models (Train)
Fig 6. Error distribution for all models (Test)
As it can be observed from Tables 3 and 4, the preciseness of approximating the discharge coefficient by using various statistical indicators under 2 conditions, namely, using the train and test data, respectively. One can see from these two tables that the values of Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), R2 and MAE indicators are relatively accurate in all the four models. Furthermore, the relative error is, on average, around 1% for all the above models. The first indicator is R2 which was applied to test the accuracy of the models shown above. It could be observed regarding the table that with an R2= 0.991 in the train mode, and an R2 of 0.987 in the test model, the best result than that of the rest of the models is presented in SVR-Firefly model. The
MAPE value is about 1.073 in the train mode and 1.23 in the test mode. This value was obtained under the worst conditions and is similar to RSM. One can observe that the value resulted in this model is not large for any of the above models, but as previously noted, the R2 criterion was taken into consideration to locate the outliners and only the MAPE value, being positive, does not say that the approximated results are also good and positive. As reported by the Table, with a minimum MAPE value of 0.49% under train mode and 0.80% under test mode, the SVR-Firefly model represents more positive results than the other models with respect to this indicator.
The RMSE indicates the second root of the squared error and it’s taking into account for analyzing the models in a computable manner. This indicator exponentiates the difference existing between the anticipated values and the actual values, and establishes very big errors. The above Table shows that the value of RMSE in the SVR-Firefly model is lower than that of all the other models: 0.0035% in the train mode and 0.0057% in the test mode. The values shown by this indicator represents an identical process to those presented by the R2 indicator. Similar to the MAPE and RMSE indicators. In addition and based on the Tables 3 and 4, the results produced by this indicator is more positive and better for the SVR-Firefly model as compared with the other models and the results of the preceding criteria. The SVR-Firefly method exhibits better results than traditional SVR because of the usage of a meta-heuristic algorithm for parameter optimization and tuning. By using firefly algorithm, the SVR hyper parameters are optimally tuned and consequently the SVR model is best fitted to data distribution, resulting more accurate output estimation. Table 3. Statistics Indexes - (Train) Table 4. Statistics Indexes - (Test)
The values of the discharge capacity anticipated by the SVR-Firefly model by applying the given parameters are shown in Table 5. The latter suggests that the anticipated values are not in line with a particular process. This issue leads to this conclusion that these values are always anticipated larger or lower than the actual values. It should be noted, though, that this model can anticipate relatively well in different hydraulic conditions so that the greatest relative error identified by this model is around 2.07%. Table 5. The values of the discharge capacity predicted by using the SVR-Firefly model
4. Conclusion
One of the techniques to control inundations and redirect and calculate the flow in channels is the use of weirs. One of the most significant hydraulic parameters in the weirs is the discharge capacity over the latter. It is anticipated by a calculation performed through using the four models of Support Vector Regression (SVR), the Support Vector Regression – Firefly (SVR-Firefly), the Response Surface Methods (RSM) and the
Principal Component Analysis (PCA). Therefore, dimensionless parameters such as (l/h), (l/w), (h/b), (sin *w/l), and (y/ (sin *w) were used in this paper to train and test the designed models. The results generated by the SVR, SVR-Firefly, RSM and PCA models were then compared with the experimental results. As it can be observed by the experiments, with an R2 value equivalent to 0.991 in the training mode, an R2 value equal to 0.987 in the test model, and the minimum MAPE value of roughly 0.49% in train model and 0.80% in the test mode, the SVR-Firefly model generates the most positive and the best results in contrast to the rest of the models. The PCA and SVR models also present comparatively good results resembling to those of the SVR-Firefly model. The MAPE value obtained under the worst condition is consistent with that of the RSM model which is, on average, near 1.073 in the train model and 1.23 in the test mode; and it can be concluded that the error percentage is an acceptable value.
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Fig 1. the plan of the experimental channel used in this study (Kumar et al, [4])
Fig 2. Comparing estimated discharge coefficient with experimental result in training state and testing state for regression model
Fig 3. Comparing estimated discharge coefficient with experimental result in training statefor all models
Fig 4. Comparing estimated discharge coefficient with experimental result in testing state for all models
Fig 5. Error distribution for all models (Train)
Fig 6. Error distribution for all models (Test)
Table 1. Parameters employed to approximate the average discharge coefficient (in this research)
L
Crest length of water (m)
F
Froud number
b
Channel width (m)
Vertex angle (rad)
w
Crest height (m)
h
head over the crest of the weir(m)
y = (h+w)
head behind the weir(m)
Table 2. Parameters applied to approximate the average discharge coefficient (Kumar et al, [4])
F
l/h
l/w
h/b
min 0.608 3.88 2.68 0.028 max 3.261 135.25 11.76 0.260
(degree) 30 180
(sin *w)/l 4.37E-17 0.28
y/(sin *w) 1.13 1.40E+16
Cd 0.54 0.906
Table 3.Results of Statistics Indexes for four models- (Train)
PCA
RSM
SVR
SVRFirefly
R2
0.99
0.98
0.98
0.99
MAPE (%)
0.63
1.07
0.90
0.49
RMSE
0.004
0.007
0.006
0.003
MAE
0.002
0.005
0.004
0.001
Table 4.Results of Statistics Indexes for four models - (Test)
PCA
RSM
SVR
SVR-Firefly
R2
0.98
0.97
0.96
0.98
MAPE (%)
0.83
1.23
1.31
0.80
RMSE
0.005
0.008
0.009
0.005
MAE
0.003
0.005
0.006
0.003
Table 5. The values of the discharge capacity predicted by using the SVR-Firefly model
θ (degree)
L (m)
w (m)
y (m)
Q (m3/s)
30 30 30 30 60 60 60 60 90 90 90 90 120 120 120 120 150 150 150 150 180 180 180 180
1.082 1.082 1.082 1.082 0.560 0.560 0.560 0.560 0.396 0.396 0.396 0.396 0.323 0.323 0.323 0.323 0.290 0.290 0.290 0.290 0.280 0.280 0.280 0.280
0.092 0.092 0.092 0.092 0.101 0.101 0.101 0.101 0.103 0.103 0.103 0.103 0.106 0.106 0.106 0.106 0.108 0.108 0.108 0.108 0.100 0.100 0.100 0.100
0.011 0.017 0.026 0.032 0.013 0.031 0.051 0.029 0.014 0.047 0.069 0.058 0.027 0.044 0.073 0.060 0.014 0.071 0.034 0.052 0.055 0.072 0.045 0.061
0.003 0.006 0.009 0.012 0.002 0.006 0.011 0.006 0.002 0.008 0.012 0.010 0.003 0.007 0.012 0.010 0.001 0.011 0.004 0.008 0.007 0.011 0.005 0.008
Cd (Exp) 0.860 0.760 0.684 0.625 0.872 0.705 0.573 0.713 0.789 0.702 0.572 0.626 0.791 0.740 0.665 0.697 0.797 0.698 0.796 0.736 0.656 0.675 0.660 0.680
Cd (SVR-Firefly) 0.8414 0.760 0.683 0.635 0.871 0.704 0.575 0.714 0.789 0.693 0.572 0.631 0.791 0.742 0.666 0.695 0.796 0.691 0.763 0.735 0.655 0.672 0.673 0.679
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S0955-5986(16)30263-1 http://dx.doi.org/10.1016/j.flowmeasinst.2016.11.010 JFMI1294
To appear in: Flow Measurement and Instrumentation Received date: 25 April 2016 Revised date: 19 November 2016 Accepted date: 30 November 2016 Cite this article as: Hojat Karami, Sohrab Karimi, Mohammad Rahmanimanesh and Saeed Farzin, Predicting Discharge Coefficient of Triangular Labyrinth Weir Using Support Vector Regression, Support Vector Regression-firefly, Response Surface Methodology and Principal Component Analysis, Flow Measurement and Instrumentation, http://dx.doi.org/10.1016/j.flowmeasinst.2016.11.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Predicting Discharge Coefficient of Triangular Labyrinth Weir Using Support Vector Regression, Support Vector Regression-firefly, Response Surface Methodology and Principal Component Analysis
a
b
Hojat Karami , Sohrab Karimi *, Mohammad Rahmanimaneshc, Saeed Farzind a, b, d
Department of Civil Engineering, Semnan University, Semnan, Iran
c
Faculty of Electrical and Computer Engineering, Semnan University, Semnan, Iran
*
Corresponding author: e-mail: [email protected],Phone: (+98) 918-333-5727
Abstract Weirs are hydraulic structures which conduct the most powerful flow with large overflow. Discharge flow predication is based on capacity discharge designation by designer. In this paper, the discharge capacity in triangular labyrinth side-weirs is computed by using new techniques with high precision. The four employed techniques for computation of discharge capacity are: Support Vector Regression (SVR), Support Vector Regression– Firefly (SVR- Firefly), Response Surface Methodology (RSM) and Principal Component Analysis (PCA).A comparison between the computed discharge capacity and empirical results is considered in this paper. Determination coefficient (R2), Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), SI and δ are five statistical indicators which help us to measure the precision of the designed models. The statistical indices indicated that the SVR-Firefly model has the highest ability among the models for simulation, with average MAPE=0.49%, R2=0.991 and RMSE=0.0035. Like the results achieved by the SVR-Firefly, comparatively good results were obtained by both PCA and SVR models. The SVR model suggested the average MAPE value near 1.073 in the training mode under the most unfavorable conditions. The MAPE value equal to 1.23 was also obtained in the test mode. This proves that the value of error rate is tolerable. Keywords: Weir, Discharge capacity, Support Vector Regression (SVR), Support Vector Regression–Firefly (SVR-Firefly), Response Surface Methodology (RSM), Principal Component Analysis (PCA).
1. Introduction A weir is a structure over which passes the extra water in upstream and downstream of dams. Weirs are also considered as the most vital hydraulic structures in water resources projects. Capacity of weirs needs to be sufficient to protect dams and ensure the safety of channels constructed for the transfer of water. Insufficiency of the weir capacity is the most important cause of damages to the channels and even to dams [1, 2]. To keep the safety of water conveyance structures, the most important issue is proper functioning of weirs under the most abnormal conditions. Water flow passes over the weir when its level is raised at the back of the latter and exceeds the height of the weir’s crest. When this happens, the velocity profile becomes non-linear and curved. In case the discharge falls behind the level of channel and river, the flow decreases over the weir [3]. The designed structure must be safe and powerful for operation. Also, high efficiency is an important factor for designing of structures. The danger arising from inundation can be reduced by selecting an appropriate weir’s discharge capacity which is, in general, one of the most important elements in this respect. It is necessary that weirs act based on a capacity enough to unload the basic inundation of the designed model such that financial losses and death rate does not go beyond the expected level. It is necessary to have a good understanding of the purpose for which weirs are constructed in order to lower considerably the costs of construction and deal with the problems of inundation. There are many kinds of weirs including shafts, side weirs, stepped, sharp-crested, ogee, labyrinth, normal, siphon, and broad-crested weirs [4-6]. Soft computing techniques have been employed by various researchers in the last decade in order to solve sophisticated problems. These techniques are new and quite accurate, that can help to calculate the discharge capacity based on hydraulic parameters and geometry of the weirs. Among these parameters, one can name the Froude number of the weir’s upstream flow, crest height, flow depth, and weir height. Subramanya [7], Bagheri and Heidarpour [8], Karimi et al. [9], Kisi [10], Bonakdari et al. [11], Karami et al. [12], Bihan et al. [13], and Kisi et al. [14] used these computing techniques to calculate the discharge capacity of labyrinth side weirs. Using the new methods of support vector regression (SVR), support vector regression-firefly (SVRfirefly), response surface methods (RSM) and principal component analysis (PCA), which solve complicated engineering problems efficiently and desirably accurate, and increase the accuracy of the work of researchers, has become popular in the last decade because most of the engineering problems are complex [15- 22]. This research attempts to calculate the discharge capacity (Cd) of a triangle labyrinth. In this respect, SVR, SVR-Firefly, RSM and PCA were the four techniques used for computation of the discharge capacity. Afterwards, a comparison was made between the latter and results of the conducted experiments. The developed networks were trained and tested by considering the dimensionless parameters which affect the process of calculating the discharge capacity. Finally, statistical parameters of RMSE, MAPE, R2, and MAE were employed to select the most accurate model to predict the discharge model.
2. Methods
The development of five methods (Regression, SVR, SVR-Firefly, RSM and PCA) was carried out by means of specified dimensionless parameters (input data= 95 data in training mode and 26 data in the test mode). These parameters include: (l/h), (l/w), (h/b), (sin θ *w/l), and (y/(sin *w)). The experimental model was used to determine the accuracy of
the resulted obtained from the above-mentioned developed methods in train and test modes. These parameters are shown in Table 1.
Table 1. Parameters employed to approximate the average discharge coefficient (in this research)
2.1. Experimental Model
The experimental model designed by Kumar et al. [4] was used to estimate the discharge coefficient of the studied weirs. This model was actually a rectangular channel of 12 m in length, 0.28 m in width, and 0.41 m in depth. In these experiments, triangular weirs were used (Figure 1). The latter were placed at a distance of 11 m from the entrance of the channel. The water height was measured by using point gages with ±0.1 measurement accuracy above the weirs. A nape flow was produced by making pores in the channel wall and also in the weirs. The upstream portion of the channel was selected to install grid walls and preventive flows so that vortexes and the disturbance of the watersurface are not formed and diminished. The parameters which were employed in this research can be found in Table 2.
Fig 1. The plan of the experimental channel used in this study (Kumar et al, [4])
Table 2. Parameters applied to approximate the average discharge coefficient (Kumar et al, [4])
2.2. Support Vector Regression and Support Vector Regression – Firefly
Support Vector Regression (SVR) [15, 16] map the input data to a high dimensional feature space via a nonlinear function (i.e., the kernel function) and then compute a linear regression function on data in the mapped feature space. Suppose there is ( ) a training data set ( ) . In ε-SV regression [16], the goal is to find a function that has at most ε deviation from the actually obtained targets for all the training data, and at the same time is as flat as possible[16].In the case of linear function, f is taking the form ( ) (1) Where denotes the space of the input patterns, and denotes the dot product in X. In this paper, we employ Radial Basis kernel function (RBF) and utilize a kind of firefly algorithm to tune SVR hyper-parameters. The firefly algorithm, inspired by the flashing behavior of fireflies, is a kind of meta-heuristic algorithms used in optimization problem
[23], and along with SVR (i.e., SVR-Firefly algorithm) are widely used for regression problem [17, 18, 24, 25]. 2.3. Response Surface Methodology
Response surface methodology (RSM) [19, 20] consists of a group of techniques used in finding a relationship between a response y, and several associated input variables denoted by x1, x2, . . . , xP. In general, such a relationship can be approximated by a lowdegree polynomial model of the form: ( )
(2)
where X = (x1, x2, . . . , xP), f (x) is a vector function of k elements that consists of powers and cross-products of powers of x1, x2, . . . , xP up to a certain degree denoted by d (≥ 1), β is a vector of k constant coefficients referred to as parameters, and ϵ is a random experimental error [26]. Two important models are commonly used in RSM, i.e., the firstdegree model (d = 1), and the second-degree model (d = 2). In this paper, we use seconddegree RSM, where ∑
∑∑
∑
(3)
RSM is widely used for optimization problem in many research area, e.g. collagen extraction from eggshell membrane [21], thermodynamic optimization of the recuperative heat exchanger [27], and extraction of bioactive compounds from fruit [28].
2.4. Principal Component Analysis
The idea of principal component analysis (PCA) [22] is to reduce the dimensionality of a data set consisting of a large number of interrelated variables, while retaining as much as possible of the variation present in the data set. This is achieved by transforming to a new set of variables, the principal components (PCs). This transformation is defined in such a way that the first principal component has the largest possible variance, and each succeeding component in turn has the highest possible variance under the constraint that it is orthogonal to the preceding components. Principal components are used in conjunction with a variety of other statistical techniques such as regression analysis [22], and used in many optimization problems such as [29, 30].
3. Result 3.1. Regression Model
The discharge coefficient of triangular labyrinth weir can be written as a function of non-dimensional parameters of (l/h), (l/w), (h/b), (sin θ *w/l), and (y/(sin *w)). Different
non-linear regression models were developed. Finally, it led to the following equation for estimation of discharge coefficient of triangular labyrinth weir. C d 1.48 l h
1.34
l
w
1.05
h b sin w l y 1.11
0.25
sin w
0.25
(4)
Figure 2 shows not bad agreement between the measured and developed regression model in training and testing state.
Fig 2. Comparing estimated discharge coefficient with experimental result in training state and testing state for regression model
3.2. The results of analysis obtained through using SVR, SVR-Firefly, RSM and PCA techniques
The results of analysis obtained through using SVR, and SVR-Firefly techniques were compared to evaluate the preciseness of the approximated model at each and every step of designing the model. Each model was designed on the basis of the criteria of coefficient of determination (R2), Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), MAE, as can be seen in the following forms:
MAPE
1 n xi yi n i 1 xi
1 n xi yi 2 n i 1
RMSE
n R x i x y i y i 1
2
MAE =
(5)
1 n ∑x _ y i n i =1 i
(6)
x n
i 1
i
x
2
n
i 1
yi y
2
2
(7)
(8)
Where yi and xi are prediction and observation Cd values, respectively, and y and x are the mean prediction and observation Cd values, respectively. The approximated values relating to the discharge coefficient (Cd) are shown in Figures 3 and 4 according to Support Vector Regression (SVR), the Support Vector Regression – Firefly (SVR-Firefly), the Response Surface Methods (RSM)and the Principal Component Analysis (PCA) methods used in the present research. Then the approximated values were compared with the experimental values. As it can be seen in these Figures, these values are relatively well consistent with each other in nearly all the four models presented above, because all the models show a relative error average of lower than 2%. While approximating the discharge coefficient in the best part of the values, the SVR-Firefly model showed the largest relative error of around 1%. Like in the latter model, the Cd estimated in the PCA model shows a
similar value, however, the difference is that the approximated values in the SVR-Firefly model are not completely close to actual values and are typically estimated as overdesign. The same process, though, is not followed in the PCA model; in fact, one cannot say for sure that the approximated values are more or less than the actual ones especially at points in which the emergence of an error is not impossible.
Fig 3. Comparing estimated discharge coefficient with experimental result in training state
Fig 4. Comparing estimated discharge coefficient with experimental result in testing state
90% of the approximated discharge coefficient values, indicated in this study through using models SVR-Firefly and PCA, show an error rate of lower than 2.5%, as can be seen in Figures5 and 6. According to what was said above, it is concluded that the results of model SVR- Firefly comply with those obtained by model PCA. The discharge coefficient values approximated under testing condition in the four models of SVR-Firefly, PCA, SVR and RSM are, by and large, similar to experimental ones; however, the maximum relative error percentage made in using RSM model is around 4.5%. According to Figure 5 and compared to the experimental data, 90% of the approximated data indicate a relative error of 2.5%. The accuracy of estimating the Cd by using the models SVRFirefly, PCA, SVR and RSM and the preciseness of the data which were not applied during the construction of the above models are shown in Figure 6. It can be seen, in RSM model and given the above Figure, that the latter approximates relatively precisely the Cdas compared with the experimental data; also according to Figure 6, the approximation of the Cd through models SVR-Firefly, PCA, SVR and RSM indicates the preciseness with a relative effort lower than 2.5% in 90% of the data under test conditions.
Fig 5. Error distribution for all models (Train)
Fig 6. Error distribution for all models (Test)
As it can be observed from Tables 3 and 4, the preciseness of approximating the discharge coefficient by using various statistical indicators under 2 conditions, namely, using the train and test data, respectively. One can see from these two tables that the values of Root Mean Square Error (RMSE), Mean Absolute Percentage Error (MAPE), R2 and MAE indicators are relatively accurate in all the four models. Furthermore, the relative error is, on average, around 1% for all the above models. The first indicator is R2 which was applied to test the accuracy of the models shown above. It could be observed regarding the table that with an R2= 0.991 in the train mode, and an R2 of 0.987 in the test model, the best result than that of the rest of the models is presented in SVR-Firefly model. The
MAPE value is about 1.073 in the train mode and 1.23 in the test mode. This value was obtained under the worst conditions and is similar to RSM. One can observe that the value resulted in this model is not large for any of the above models, but as previously noted, the R2 criterion was taken into consideration to locate the outliners and only the MAPE value, being positive, does not say that the approximated results are also good and positive. As reported by the Table, with a minimum MAPE value of 0.49% under train mode and 0.80% under test mode, the SVR-Firefly model represents more positive results than the other models with respect to this indicator.
The RMSE indicates the second root of the squared error and it’s taking into account for analyzing the models in a computable manner. This indicator exponentiates the difference existing between the anticipated values and the actual values, and establishes very big errors. The above Table shows that the value of RMSE in the SVR-Firefly model is lower than that of all the other models: 0.0035% in the train mode and 0.0057% in the test mode. The values shown by this indicator represents an identical process to those presented by the R2 indicator. Similar to the MAPE and RMSE indicators. In addition and based on the Tables 3 and 4, the results produced by this indicator is more positive and better for the SVR-Firefly model as compared with the other models and the results of the preceding criteria. The SVR-Firefly method exhibits better results than traditional SVR because of the usage of a meta-heuristic algorithm for parameter optimization and tuning. By using firefly algorithm, the SVR hyper parameters are optimally tuned and consequently the SVR model is best fitted to data distribution, resulting more accurate output estimation. Table 3. Statistics Indexes - (Train) Table 4. Statistics Indexes - (Test)
The values of the discharge capacity anticipated by the SVR-Firefly model by applying the given parameters are shown in Table 5. The latter suggests that the anticipated values are not in line with a particular process. This issue leads to this conclusion that these values are always anticipated larger or lower than the actual values. It should be noted, though, that this model can anticipate relatively well in different hydraulic conditions so that the greatest relative error identified by this model is around 2.07%. Table 5. The values of the discharge capacity predicted by using the SVR-Firefly model
4. Conclusion
One of the techniques to control inundations and redirect and calculate the flow in channels is the use of weirs. One of the most significant hydraulic parameters in the weirs is the discharge capacity over the latter. It is anticipated by a calculation performed through using the four models of Support Vector Regression (SVR), the Support Vector Regression – Firefly (SVR-Firefly), the Response Surface Methods (RSM) and the
Principal Component Analysis (PCA). Therefore, dimensionless parameters such as (l/h), (l/w), (h/b), (sin *w/l), and (y/ (sin *w) were used in this paper to train and test the designed models. The results generated by the SVR, SVR-Firefly, RSM and PCA models were then compared with the experimental results. As it can be observed by the experiments, with an R2 value equivalent to 0.991 in the training mode, an R2 value equal to 0.987 in the test model, and the minimum MAPE value of roughly 0.49% in train model and 0.80% in the test mode, the SVR-Firefly model generates the most positive and the best results in contrast to the rest of the models. The PCA and SVR models also present comparatively good results resembling to those of the SVR-Firefly model. The MAPE value obtained under the worst condition is consistent with that of the RSM model which is, on average, near 1.073 in the train model and 1.23 in the test mode; and it can be concluded that the error percentage is an acceptable value.
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Fig 1. the plan of the experimental channel used in this study (Kumar et al, [4])
Fig 2. Comparing estimated discharge coefficient with experimental result in training state and testing state for regression model
Fig 3. Comparing estimated discharge coefficient with experimental result in training statefor all models
Fig 4. Comparing estimated discharge coefficient with experimental result in testing state for all models
Fig 5. Error distribution for all models (Train)
Fig 6. Error distribution for all models (Test)
Table 1. Parameters employed to approximate the average discharge coefficient (in this research)
L
Crest length of water (m)
F
Froud number
b
Channel width (m)
Vertex angle (rad)
w
Crest height (m)
h
head over the crest of the weir(m)
y = (h+w)
head behind the weir(m)
Table 2. Parameters applied to approximate the average discharge coefficient (Kumar et al, [4])
F
l/h
l/w
h/b
min 0.608 3.88 2.68 0.028 max 3.261 135.25 11.76 0.260
(degree) 30 180
(sin *w)/l 4.37E-17 0.28
y/(sin *w) 1.13 1.40E+16
Cd 0.54 0.906
Table 3.Results of Statistics Indexes for four models- (Train)
PCA
RSM
SVR
SVRFirefly
R2
0.99
0.98
0.98
0.99
MAPE (%)
0.63
1.07
0.90
0.49
RMSE
0.004
0.007
0.006
0.003
MAE
0.002
0.005
0.004
0.001
Table 4.Results of Statistics Indexes for four models - (Test)
PCA
RSM
SVR
SVR-Firefly
R2
0.98
0.97
0.96
0.98
MAPE (%)
0.83
1.23
1.31
0.80
RMSE
0.005
0.008
0.009
0.005
MAE
0.003
0.005
0.006
0.003
Table 5. The values of the discharge capacity predicted by using the SVR-Firefly model
θ (degree)
L (m)
w (m)
y (m)
Q (m3/s)
30 30 30 30 60 60 60 60 90 90 90 90 120 120 120 120 150 150 150 150 180 180 180 180
1.082 1.082 1.082 1.082 0.560 0.560 0.560 0.560 0.396 0.396 0.396 0.396 0.323 0.323 0.323 0.323 0.290 0.290 0.290 0.290 0.280 0.280 0.280 0.280
0.092 0.092 0.092 0.092 0.101 0.101 0.101 0.101 0.103 0.103 0.103 0.103 0.106 0.106 0.106 0.106 0.108 0.108 0.108 0.108 0.100 0.100 0.100 0.100
0.011 0.017 0.026 0.032 0.013 0.031 0.051 0.029 0.014 0.047 0.069 0.058 0.027 0.044 0.073 0.060 0.014 0.071 0.034 0.052 0.055 0.072 0.045 0.061
0.003 0.006 0.009 0.012 0.002 0.006 0.011 0.006 0.002 0.008 0.012 0.010 0.003 0.007 0.012 0.010 0.001 0.011 0.004 0.008 0.007 0.011 0.005 0.008
Cd (Exp) 0.860 0.760 0.684 0.625 0.872 0.705 0.573 0.713 0.789 0.702 0.572 0.626 0.791 0.740 0.665 0.697 0.797 0.698 0.796 0.736 0.656 0.675 0.660 0.680
Cd (SVR-Firefly) 0.8414 0.760 0.683 0.635 0.871 0.704 0.575 0.714 0.789 0.693 0.572 0.631 0.791 0.742 0.666 0.695 0.796 0.691 0.763 0.735 0.655 0.672 0.673 0.679