- Email: [email protected]
ANFIS-based approach for predicting sediment transport in clean sewer
Applied Soft Computing 12 (2012) 1227–1230
Contents lists available at SciVerse ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Short communication
ANFIS-based approach for predicting sediment transport in clean sewer H. Md. Azamathulla a,∗ , Aminuddin Ab. Ghani a , Seow Yen Fei b a b
River Engineering and Urban Drainage Research Centre (REDAC), Universiti Sains Malaysia, Engineering Campus, Seri Ampangan, 14300 Nibong Tebal, Penang, Malaysia School of Civil Engineering, Universiti Sains Malaysia, Engineering Campus, Seri Ampangan, 14300 Nibong Tebal, Penang, Malaysia
a r t i c l e
i n f o
Article history: Received 27 September 2011 Received in revised form 28 November 2011 Accepted 11 December 2011 Available online 19 December 2011 Keywords: Sewer pipes Sewer sediments Sediment transport Regression analysis ANFIS
a b s t r a c t The necessity of sewers to carry sediment has been recognized for many years. Typically, old sewage systems were designated based on self-cleansing concept where there is no deposition in sewer. These codes were applicable to non-cohesive sediments (typically storm sewers). This study presents adaptive neurofuzzy inference system (ANFIS), which is a combination of neural network and fuzzy logic, as an alternative approach to predict the functional relationships of sediment transport in sewer pipe systems. The proposed relationship can be applied to different boundaries with partially full flow. The present ANFIS approach gives satisfactory results (r2 = 0.98 and RMSE = 0.002431) compared to the existing predictor. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In sewer networks, the deposition of solids occurs occasionally due to the intermittent nature of flow [1]. The longer the deposits remain in the sewer the more likely that the sediment properties change and can eventually become consolidated or cemented, especially during dry weather flows. Such permanent deposits in pipe inverts will change the nature of the velocity and boundary shear distributions, which affects the sediment carrying capacity and hydraulic resistance of sewers. Previous works on sediment transport with no deposition include those of May et al. [2], Mayerle et al. [3] and Vongvisessomjai et al. [4]. Sewer system designs must satisfy two major criteria: high flow and low flow criteria. During high flows, sewer systems must convey the design discharge. For low flows, sewers should be free from sediment deposit as much as possible. Traditionally, a fixed minimum flow velocity for non-deposition, such as 0.6 m/s [5] is used as a low flow criterion. This criterion may be inadequate because the loading and sediment characteristics vary considerably under different environmental conditions [4] with permission from ASCE. Sediment transport equations which are applicable for both conditions in sewers, namely rigid boundary (clean pipes) and loose boundary (pipes with deposited beds) are being derived to meet
the challenging condition of flow. Few previous works on sediment transport with no deposition include those by Mayerle et al. [3], Ab Ghani [6], May et al. [2], Cigizoglu [7], Dogan et al. [8]. Kisi [9] applied ANN and ANFIS approaches to estimate suspended sediment. Yang et al. [10] used ANN to evaluate total sediment load formulae and prediction of sediment load concentration in rivers using neural networks [11]. Recently Ab Ghani and Azamathulla [12] have attempted gene-expression programming (GEP) for sediment transport in sewer pipe systems. The present study investigates the hydraulic characteristics of the flow in channels with a circular cross-section with different bed roughness, and their effects on sediment transport capacity (Fig. 1) [13]. The pipe channel can be represented by a trapezoidal section, especially at low depths; at high depths, the cross-section is influenced by the ‘crowing’ effect of the pipe, through the changes in velocity and shear distributions due to changes in cross-sectional shapes [1]. The present study presents multiple regression model and adaptive neuro-fuzzy inference system (ANFIS), as an alternative approach to predict the functional relationships of sediment transport in sewer pipe systems.
2. Data collection ∗ Corresponding author. Tel.: +60 4 5995867; fax: +60 4 5941036. E-mail addresses: [email protected], [email protected] (H.Md. Azamathulla), [email protected] (A. Ab. Ghani), [email protected] (S.Y. Fei). 1568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2011.12.003
The experimental data were collected from Ab Ghani [6] and Vongvisessomjai et al. [4]. Models were generated from the collection of cumulative data. The sediment transport data for clean pipe with diameter 100–150 mm consisted of 118 sets. As for the
1228
H.Md. Azamathulla et al. / Applied Soft Computing 12 (2012) 1227–1230
4. Overview of Adaptive Neural Fuzzy Inference System (ANFIS) ANFIS is a combination of Fuzzy Inference System (FIS) and ANNs, which is a successful invention by Jang [14]. It is a multilayer feed-forward network, which uses neural network learning algorithms and fuzzy reasoning to map an input space to an output space. The fuzzy decision rules are implemented as membership functions (MFs) and the model learns the best fitting parameters of the MFs. A membership function (MF) is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1. FIS, was based on if-then rules, such that we can obtain the relation between input and output variables by these rules. Since in conditions which input and output data have high uncertainty, the classic estimating method, regression for instance, do not considered uncertainty of data well, and we use FIS as a prediction model.
Fig. 1. Q-S0-D plot: clean pipe. Reprinted from Ab Ghani [6].
5. Implementation of ANFIS model present study, 85% (98 sets) data has been using for training data and 15% for testing data. 3. Multiple linear regression The sediment transport rate in channels with a circular crosssection or pipe channels depends on many factors, such as flow depth (yo ), bed slope (So ), sediment size (d50 ), density of sediment (s ), kinematic viscosity () and density () of fluid, friction factor (), pipe diameter (D), and gravitational constant (g). But, not all the parameters will have significant effect on the result. Therefore, sensitivity test was performed to investigate the effect of each parameter on the model result, followed by dimensionless analysis to determine the functional relationship of limiting velocity. From the dimensional analysis using Buckingham Theorem, an expression was obtained considering the parameters Vs = f (g, R, d50 , s , Dgr , Cv , ) as:
V gd50
=f
d
50
R
, s , Dgr , s , Cv
(1)
The following equation was obtained on the basis of nonlinear regression to describe bed load transport with a limiting velocity for no deposition (clean pipes): Vs
gd
−0.27 0.16 = 0.22Dgr Cv
d −0.29 50
R
−0.51 s
(2)
where Vs is the limiting velocity, Dgr is dimensionless grain size,
In the ANFIS model, fuzzy subtractive clustering algorithm was used to design an initial rule base. Subtractive clustering is a fast, one-pass algorithm for estimating the number of clusters and the cluster centers in a set of data [15]. In this study, the genfis2 function was applied to generate a model using data by using clustering. The genfis2 function is a fast, one-pass method that does not perform any iterative optimization. When fuzzy systems are designed by using fuzzy clustering, each cluster corresponds to a fuzzy rule. Hence, the number of clusters determines the number of rules. A sample command-line is as follow: fismat = genfis2 (datin, datout, 0.5); In general, as the number of rules is increased, the difference between the predicted and the obtained values decreases and more complex relations can be modeled with a large number of rules. A crucial point in the rule base design is selecting the number of rules. The present study focuses on developing various models with different cluster radius, in order to determine the number of clusters by studying the rules and the results obtained from each model. For the regression analysis, after obtaining an equation, it would be verified by using the testing data by equating each data in the equation. As for ANFIS model, it would generate itself to compute the predicted value. It is vital to define the criteria by which the performance of the model and its prediction accuracy are evaluated in the model development process. The correlation coefficient (r) and the root mean square error (RMSE) were used in the current study. The root mean square error, RMSE describes the average difference between predicted values and measured values, as expressed by:
1/3
Dgr = d((Ss − 1)/2 ) , R is hydraulic radius, d50 is sediment size, is relative density of sediment in water (s /) − 1, s is over0.03 C 0.04 0.86 , is the clear water all friction factor, s = 0.851Dgr c v c friction factor of the channel. This simple regression equation could produce a good prediction of sediment transport in clean pipes as it yielded r2 = 0.94, RSME = 0.084, AAE = 0.061. However, this method was not as efficient as the new soft computing technique, such as artificial neural networks (ANNs), evolutionary computation (EC), fuzzy logic (FL) and genetic programming, which have been successfully applied in water engineering problems in the last two decades. Although many successful attempts have been made, a wider application of theoretical models is restricted by their heavy demand in terms of computing capacity and time. Thus, this study will attempt a new soft computing technique, Adaptive Neuro-Fuzzy Inference System (ANFIS) to obtain a new sediment transport equation for the bed load transport in pipes with no deposition.
RMSE =
(X − Y )2 n
1/2 (3)
The correlation coefficient, r represents the degree of association between the predicted and the measured values as depicted below:
xy r = 2 x
y2
(4)
First, an attempt was made to assess the significance or influence of each input parameter on Vs / gd. Table 1 compares the ANFIS models, with one of the independent parameters removed in each case, and deleting any independent parameter from the input set five independent yielded larger RMSE and lower r2 values. These parameters have non-negligible influence on Vs / gd and so the functional relationship given in Eq. (2) is used for ANFIS modeling in this study.
H.Md. Azamathulla et al. / Applied Soft Computing 12 (2012) 1227–1230
1229
Table 1 Models for sensitivity analysis. Model set
Dependent variables
1
√
2
√
Independent variables
Error r2
RMSE
d50 /R, s , Dgr , Cv
0.98
0.002431
s , Dgr , Cv
0.96
0.0567
3
√ V gd50
d50 /R, Dgr , Cv
0.93
0.1489
4
√
d50 /R, s , Cv
0.89
1.458
5
√
d50 /R, s , Dgr
0.85
2.3765
V gd50 V gd50
V gd50 V gd50
d50 /R sug41
Fig. 3. Observed versus predicted V/
λs
gd50 by ANFIS and regression (training).
f(u)
(sugeno)
63 rules
V/√(∆gd50)
Dgr
Cv Fig. 2. Scenario of the ANFIS model using cluster radius of 0.1 (63 rules).
6. Result and discussion Table 1 shows 5 models being set up for sensitivity analysis as shown below by using regression approach. Table 1 shows the sensitivity analysis done for the 5 models. Model with the least errors and highest r2 value were used in the present analysis. Accordingly, model ((d50 /R), s , Dgr , s , Cv ) has been selected for further analysis as the parameters give the most significant effect on determining the relation of limiting velocity (Vs ). Eq. (1) was deduced through the regression analysis, based on the 85% (98 sets) training data, a quite acceptable results with RMSE = 0.084, r2 = 0.94 was obtained. Fig. 2 shows the scenario of the ANFIS model using cluster radius of 0.1. The inputs had 4 parameters ((d50 /R), s , Dgr , Cv ), output
layer had 1 parameter V/ gd50 , and the model generated 63 rules. Table 2 shows the comparison of errors from ANFIS and regression model. The performance of the ANFIS model was compared with the regression analysis and also the proposed ANFIS approach gave satisfactory results compared to the existing predictor [4] which produced r2 = 0.75 only. Overall, particularly for Table 2 Errors measures—ANFIS and regression analyses. Model
Data sets
RMSE
r2
Regression
Training Testing Training Testing Testing
0.08393 0.37646 0.18527 0.17388 3.234
0.94 0.90 0.98 0.94 0.75
ANFIS Vongvisessomjai et al. [4]
Fig. 4. Observed versus predicted V/
gd50 by ANFIS and regression (testing).
laboratory measurements, the ANFIS models could give better predictions than the traditional regression models. The ANFIS model produced the lowest errors r2 = 0.98, and RMSE = 0.002431 for the training data (Fig. 3), and r2 = 0.94, and RMSE = 0.005678 for the testing data (Fig. 4). 7. Conclusions Sediment transport in pipes is a complex phenomenon. The nature and motivation of traditional models differ significantly. These approaches are normally capable to make predictions within about one order-of-magnitude of the actual measurements. A single bed load transport model for storm sewers is given in Eq. (2). To overcome the complexity and uncertainty associated with bed load estimation, this study demonstrated that an ANFIS model could be applied for accurate prediction. The ANFIS model successfully predicted the bed load transport in storm sewers. The high value of the coefficient of determination (r2 = 0.98), and RMSE = 0.002431, indicate that the ANFIS model is an excellent fit for the measured
1230
H.Md. Azamathulla et al. / Applied Soft Computing 12 (2012) 1227–1230
data. Thus the proposed ANFIS model is robust and useful for the practitioners. References [1] C. Nalluri, A.A. Ghani, A.K.S. El-Zaemey, Sediment transport over deposited beds in sewers, Water Science and Technology 29 (1–2) (1994) 125–133. [2] R.W.P. May, J.C. Ackers, D. Butler, S. John, Development of design methodology for self-cleansing sewers, Water Science and Technology 33 (9) (1996) 195–205. [3] R. Mayerle, C. Nalluri, P. Novak, Sediment transport in rigid bed conveyances, Journal of Hydraulic Research 29 (4) (1991) 475–496. [4] N. Vongvisessomjai, T. Tingsanchali, M.S. Babel, Non-deposition design criteria for sewers with part-full flow, Urban Water Journal 7 (1) (2010) 61–77. [5] American Society of Civil Engineers (ASCE), Water Pollution Control Federation, Design and Construction of Sanitary and Storm Sewers. American Society of Civil Engineers Manuals and Reports on Engineering Practices, No. 37, 1970. [6] A. Ab Ghani, Sediment Transport in Sewers, Ph.D. Thesis, University of Newcastle Upon Tyne, UK, 1993.
[7] H. Cigizoglu, Suspended sediment estimation for rivers using artificial neural networks and sediment rating curves, Turkish Journal of Engineering Environmental Sciences 26 (1) (2002) 27–36. [8] E. Dogan, S. Tripathi, D.A. Lyn, R.S. Govindaraju, From flumes to rivers: can sediment transport in natural alluvial channels be predicted from observations at the laboratory scale? Water Resources Research 114 (2009) W08433, doi:10.1029/2008WR007637. [9] O. Kisi, Suspended sediment estimation using neuro-fuzzy and neural network approaches? Hydrological Science 50 (4) (2005) 683–696. [10] C.T. Yang, M. Reza, M.T. Aalami, Evaluation of total load sediment transport using AAN, International Journal of Sediment Research 24 (3) (2009) 274–286. [11] H.M. Nagy, K. Watanabe, M. Hirano, Prediction of sediment load concentration in rivers using artificial neural network model, Journal of Hydraulic Engineering 128 (6) (2002) 588–595. [12] A. Ab Ghani, H.Md. Azamathulla, Gene-expression programming for sediment transport sewer pipe systems, ASCE Journal of Pipeline Systems Engineering and Practice 2 (3) (2011) 102–106. [13] C. Nalluri, A. Ab Ghani, Design options for self-cleansing storm sewers, Journal of Water Science and Technology, IWA 33 (9) (1996) 215–220. [14] J.S.R. Jang, ANFIS: adaptive-network-based fuzzy inference system, IEEE Transactions on Systems, Man, and Cybernetics 23 (3) (1993) 665–685. [15] Fuzzy Logic Toolbox 2, User’s Guide, 1995–2007, http://Mathwork.com.
Contents lists available at SciVerse ScienceDirect
Applied Soft Computing journal homepage: www.elsevier.com/locate/asoc
Short communication
ANFIS-based approach for predicting sediment transport in clean sewer H. Md. Azamathulla a,∗ , Aminuddin Ab. Ghani a , Seow Yen Fei b a b
River Engineering and Urban Drainage Research Centre (REDAC), Universiti Sains Malaysia, Engineering Campus, Seri Ampangan, 14300 Nibong Tebal, Penang, Malaysia School of Civil Engineering, Universiti Sains Malaysia, Engineering Campus, Seri Ampangan, 14300 Nibong Tebal, Penang, Malaysia
a r t i c l e
i n f o
Article history: Received 27 September 2011 Received in revised form 28 November 2011 Accepted 11 December 2011 Available online 19 December 2011 Keywords: Sewer pipes Sewer sediments Sediment transport Regression analysis ANFIS
a b s t r a c t The necessity of sewers to carry sediment has been recognized for many years. Typically, old sewage systems were designated based on self-cleansing concept where there is no deposition in sewer. These codes were applicable to non-cohesive sediments (typically storm sewers). This study presents adaptive neurofuzzy inference system (ANFIS), which is a combination of neural network and fuzzy logic, as an alternative approach to predict the functional relationships of sediment transport in sewer pipe systems. The proposed relationship can be applied to different boundaries with partially full flow. The present ANFIS approach gives satisfactory results (r2 = 0.98 and RMSE = 0.002431) compared to the existing predictor. © 2011 Elsevier B.V. All rights reserved.
1. Introduction In sewer networks, the deposition of solids occurs occasionally due to the intermittent nature of flow [1]. The longer the deposits remain in the sewer the more likely that the sediment properties change and can eventually become consolidated or cemented, especially during dry weather flows. Such permanent deposits in pipe inverts will change the nature of the velocity and boundary shear distributions, which affects the sediment carrying capacity and hydraulic resistance of sewers. Previous works on sediment transport with no deposition include those of May et al. [2], Mayerle et al. [3] and Vongvisessomjai et al. [4]. Sewer system designs must satisfy two major criteria: high flow and low flow criteria. During high flows, sewer systems must convey the design discharge. For low flows, sewers should be free from sediment deposit as much as possible. Traditionally, a fixed minimum flow velocity for non-deposition, such as 0.6 m/s [5] is used as a low flow criterion. This criterion may be inadequate because the loading and sediment characteristics vary considerably under different environmental conditions [4] with permission from ASCE. Sediment transport equations which are applicable for both conditions in sewers, namely rigid boundary (clean pipes) and loose boundary (pipes with deposited beds) are being derived to meet
the challenging condition of flow. Few previous works on sediment transport with no deposition include those by Mayerle et al. [3], Ab Ghani [6], May et al. [2], Cigizoglu [7], Dogan et al. [8]. Kisi [9] applied ANN and ANFIS approaches to estimate suspended sediment. Yang et al. [10] used ANN to evaluate total sediment load formulae and prediction of sediment load concentration in rivers using neural networks [11]. Recently Ab Ghani and Azamathulla [12] have attempted gene-expression programming (GEP) for sediment transport in sewer pipe systems. The present study investigates the hydraulic characteristics of the flow in channels with a circular cross-section with different bed roughness, and their effects on sediment transport capacity (Fig. 1) [13]. The pipe channel can be represented by a trapezoidal section, especially at low depths; at high depths, the cross-section is influenced by the ‘crowing’ effect of the pipe, through the changes in velocity and shear distributions due to changes in cross-sectional shapes [1]. The present study presents multiple regression model and adaptive neuro-fuzzy inference system (ANFIS), as an alternative approach to predict the functional relationships of sediment transport in sewer pipe systems.
2. Data collection ∗ Corresponding author. Tel.: +60 4 5995867; fax: +60 4 5941036. E-mail addresses: [email protected], [email protected] (H.Md. Azamathulla), [email protected] (A. Ab. Ghani), [email protected] (S.Y. Fei). 1568-4946/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.asoc.2011.12.003
The experimental data were collected from Ab Ghani [6] and Vongvisessomjai et al. [4]. Models were generated from the collection of cumulative data. The sediment transport data for clean pipe with diameter 100–150 mm consisted of 118 sets. As for the
1228
H.Md. Azamathulla et al. / Applied Soft Computing 12 (2012) 1227–1230
4. Overview of Adaptive Neural Fuzzy Inference System (ANFIS) ANFIS is a combination of Fuzzy Inference System (FIS) and ANNs, which is a successful invention by Jang [14]. It is a multilayer feed-forward network, which uses neural network learning algorithms and fuzzy reasoning to map an input space to an output space. The fuzzy decision rules are implemented as membership functions (MFs) and the model learns the best fitting parameters of the MFs. A membership function (MF) is a curve that defines how each point in the input space is mapped to a membership value (or degree of membership) between 0 and 1. FIS, was based on if-then rules, such that we can obtain the relation between input and output variables by these rules. Since in conditions which input and output data have high uncertainty, the classic estimating method, regression for instance, do not considered uncertainty of data well, and we use FIS as a prediction model.
Fig. 1. Q-S0-D plot: clean pipe. Reprinted from Ab Ghani [6].
5. Implementation of ANFIS model present study, 85% (98 sets) data has been using for training data and 15% for testing data. 3. Multiple linear regression The sediment transport rate in channels with a circular crosssection or pipe channels depends on many factors, such as flow depth (yo ), bed slope (So ), sediment size (d50 ), density of sediment (s ), kinematic viscosity () and density () of fluid, friction factor (), pipe diameter (D), and gravitational constant (g). But, not all the parameters will have significant effect on the result. Therefore, sensitivity test was performed to investigate the effect of each parameter on the model result, followed by dimensionless analysis to determine the functional relationship of limiting velocity. From the dimensional analysis using Buckingham Theorem, an expression was obtained considering the parameters Vs = f (g, R, d50 , s , Dgr , Cv , ) as:
V gd50
=f
d
50
R
, s , Dgr , s , Cv
(1)
The following equation was obtained on the basis of nonlinear regression to describe bed load transport with a limiting velocity for no deposition (clean pipes): Vs
gd
−0.27 0.16 = 0.22Dgr Cv
d −0.29 50
R
−0.51 s
(2)
where Vs is the limiting velocity, Dgr is dimensionless grain size,
In the ANFIS model, fuzzy subtractive clustering algorithm was used to design an initial rule base. Subtractive clustering is a fast, one-pass algorithm for estimating the number of clusters and the cluster centers in a set of data [15]. In this study, the genfis2 function was applied to generate a model using data by using clustering. The genfis2 function is a fast, one-pass method that does not perform any iterative optimization. When fuzzy systems are designed by using fuzzy clustering, each cluster corresponds to a fuzzy rule. Hence, the number of clusters determines the number of rules. A sample command-line is as follow: fismat = genfis2 (datin, datout, 0.5); In general, as the number of rules is increased, the difference between the predicted and the obtained values decreases and more complex relations can be modeled with a large number of rules. A crucial point in the rule base design is selecting the number of rules. The present study focuses on developing various models with different cluster radius, in order to determine the number of clusters by studying the rules and the results obtained from each model. For the regression analysis, after obtaining an equation, it would be verified by using the testing data by equating each data in the equation. As for ANFIS model, it would generate itself to compute the predicted value. It is vital to define the criteria by which the performance of the model and its prediction accuracy are evaluated in the model development process. The correlation coefficient (r) and the root mean square error (RMSE) were used in the current study. The root mean square error, RMSE describes the average difference between predicted values and measured values, as expressed by:
1/3
Dgr = d((Ss − 1)/2 ) , R is hydraulic radius, d50 is sediment size, is relative density of sediment in water (s /) − 1, s is over0.03 C 0.04 0.86 , is the clear water all friction factor, s = 0.851Dgr c v c friction factor of the channel. This simple regression equation could produce a good prediction of sediment transport in clean pipes as it yielded r2 = 0.94, RSME = 0.084, AAE = 0.061. However, this method was not as efficient as the new soft computing technique, such as artificial neural networks (ANNs), evolutionary computation (EC), fuzzy logic (FL) and genetic programming, which have been successfully applied in water engineering problems in the last two decades. Although many successful attempts have been made, a wider application of theoretical models is restricted by their heavy demand in terms of computing capacity and time. Thus, this study will attempt a new soft computing technique, Adaptive Neuro-Fuzzy Inference System (ANFIS) to obtain a new sediment transport equation for the bed load transport in pipes with no deposition.
RMSE =
(X − Y )2 n
1/2 (3)
The correlation coefficient, r represents the degree of association between the predicted and the measured values as depicted below:
xy r = 2 x
y2
(4)
First, an attempt was made to assess the significance or influence of each input parameter on Vs / gd. Table 1 compares the ANFIS models, with one of the independent parameters removed in each case, and deleting any independent parameter from the input set five independent yielded larger RMSE and lower r2 values. These parameters have non-negligible influence on Vs / gd and so the functional relationship given in Eq. (2) is used for ANFIS modeling in this study.
H.Md. Azamathulla et al. / Applied Soft Computing 12 (2012) 1227–1230
1229
Table 1 Models for sensitivity analysis. Model set
Dependent variables
1
√
2
√
Independent variables
Error r2
RMSE
d50 /R, s , Dgr , Cv
0.98
0.002431
s , Dgr , Cv
0.96
0.0567
3
√ V gd50
d50 /R, Dgr , Cv
0.93
0.1489
4
√
d50 /R, s , Cv
0.89
1.458
5
√
d50 /R, s , Dgr
0.85
2.3765
V gd50 V gd50
V gd50 V gd50
d50 /R sug41
Fig. 3. Observed versus predicted V/
λs
gd50 by ANFIS and regression (training).
f(u)
(sugeno)
63 rules
V/√(∆gd50)
Dgr
Cv Fig. 2. Scenario of the ANFIS model using cluster radius of 0.1 (63 rules).
6. Result and discussion Table 1 shows 5 models being set up for sensitivity analysis as shown below by using regression approach. Table 1 shows the sensitivity analysis done for the 5 models. Model with the least errors and highest r2 value were used in the present analysis. Accordingly, model ((d50 /R), s , Dgr , s , Cv ) has been selected for further analysis as the parameters give the most significant effect on determining the relation of limiting velocity (Vs ). Eq. (1) was deduced through the regression analysis, based on the 85% (98 sets) training data, a quite acceptable results with RMSE = 0.084, r2 = 0.94 was obtained. Fig. 2 shows the scenario of the ANFIS model using cluster radius of 0.1. The inputs had 4 parameters ((d50 /R), s , Dgr , Cv ), output
layer had 1 parameter V/ gd50 , and the model generated 63 rules. Table 2 shows the comparison of errors from ANFIS and regression model. The performance of the ANFIS model was compared with the regression analysis and also the proposed ANFIS approach gave satisfactory results compared to the existing predictor [4] which produced r2 = 0.75 only. Overall, particularly for Table 2 Errors measures—ANFIS and regression analyses. Model
Data sets
RMSE
r2
Regression
Training Testing Training Testing Testing
0.08393 0.37646 0.18527 0.17388 3.234
0.94 0.90 0.98 0.94 0.75
ANFIS Vongvisessomjai et al. [4]
Fig. 4. Observed versus predicted V/
gd50 by ANFIS and regression (testing).
laboratory measurements, the ANFIS models could give better predictions than the traditional regression models. The ANFIS model produced the lowest errors r2 = 0.98, and RMSE = 0.002431 for the training data (Fig. 3), and r2 = 0.94, and RMSE = 0.005678 for the testing data (Fig. 4). 7. Conclusions Sediment transport in pipes is a complex phenomenon. The nature and motivation of traditional models differ significantly. These approaches are normally capable to make predictions within about one order-of-magnitude of the actual measurements. A single bed load transport model for storm sewers is given in Eq. (2). To overcome the complexity and uncertainty associated with bed load estimation, this study demonstrated that an ANFIS model could be applied for accurate prediction. The ANFIS model successfully predicted the bed load transport in storm sewers. The high value of the coefficient of determination (r2 = 0.98), and RMSE = 0.002431, indicate that the ANFIS model is an excellent fit for the measured
1230
H.Md. Azamathulla et al. / Applied Soft Computing 12 (2012) 1227–1230
data. Thus the proposed ANFIS model is robust and useful for the practitioners. References [1] C. Nalluri, A.A. Ghani, A.K.S. El-Zaemey, Sediment transport over deposited beds in sewers, Water Science and Technology 29 (1–2) (1994) 125–133. [2] R.W.P. May, J.C. Ackers, D. Butler, S. John, Development of design methodology for self-cleansing sewers, Water Science and Technology 33 (9) (1996) 195–205. [3] R. Mayerle, C. Nalluri, P. Novak, Sediment transport in rigid bed conveyances, Journal of Hydraulic Research 29 (4) (1991) 475–496. [4] N. Vongvisessomjai, T. Tingsanchali, M.S. Babel, Non-deposition design criteria for sewers with part-full flow, Urban Water Journal 7 (1) (2010) 61–77. [5] American Society of Civil Engineers (ASCE), Water Pollution Control Federation, Design and Construction of Sanitary and Storm Sewers. American Society of Civil Engineers Manuals and Reports on Engineering Practices, No. 37, 1970. [6] A. Ab Ghani, Sediment Transport in Sewers, Ph.D. Thesis, University of Newcastle Upon Tyne, UK, 1993.
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