10th International River Engineering Conference Shahid Chamran University, 19-21 Jan 2016, Ahwaz
An optimal Adaptive Neural Fuzzy Inference System (ANFIS) model and regression relations to predict stable channel geometry in rivers gravel bed Azadeh Gholami 1,*, Hossein Bonakdari 2, Isa Ebtehaj1, Saba Shaghaghi3 1
Ph.D. Candidate, Department of Civil Engineering, Razi University, Kermanshah, Iran
2
Associate. Prof, Department of Civil Engineering, Razi University, Kermanshah, Iran.
3
M.Sc. Student, Department of Civil Engineering, Razi University, Kermanshah, Iran, E-mail: *Corresponding author, e-mail:
[email protected]
Abstract
Hydraulic geometry of a river has primary importance in the design, planning, management and river training in river engineering science. In investigation of stable channels dimensions, the most presented relations are based on statistical and theoretical methods that don’t have more accuracy. In last decades, using soft computing methods or artificial neural methods because of high accuracy and fewer time and cost are interested by different science researches. In the present paper, using Adaptive Neural Fuzzy Inference System (ANFIS) model, the accuracy of regression relations to predict width, depth and slope of stable channels are improved. A set of observed data (including 85 cross section data) are used to train and test ANFIS models and also to fit regression relations. The two models efficiency are evaluated and compared with observed data. Results show that ANFIS models with R2 values of 0.9224, 0.7464 and 0.9264 show a high accuracy to predict width, depth and slope of stable channels, respectively. Also, the mean absolute relative error (MARE) values in regression relation are 73, 57 and 50 times higher than ANFIS models in predicting width, depth and slope, respectively. Therefore, using ANFIS model causes to improve regression equations performance and its results can be used in the design of Executive channels. Keywords: ANFIS model, regression relations, models efficiency, observed
data, stable channel geometry.
1- Introduction Alluvial rivers continuously adjust their hydraulic geometry to achieve a stable condition to convey imposed water and sediment discharge without net scour or deposition [1]. Being a river in stable state does not mean that the bed and banks are non-erodible with time, but the average channel dimensions remain constant over a period of time in spite of bed and bank deposition and erosion [2]. Parameters of determining channel stability are defined in terms of channel 1
width, average depth of flow, longitudinal slope and flow velocity. For any imposed water and sediment load, the channel will adjust these parameters to attain a stable condition [3]. Numerous efforts have been done to extend equations for predicting hydraulic geometry of alluvial channels in stable state. For example Lee (1995), cheema et al (1997) and yen and lee (1995) realized the velocity distribution, bed-load transport, bed form changes in alluvial gravel-bed rivers. Cao and knight (1998), julien and Anthony (2001), kassem and chaudhry (2002) and olsen (2003) determined bed-load motion, numerical modeling of bed transition and self forming modeling in rivers. Yen (2002), smart et al (2002) analyzed flow resistance and maximum velocity in open channels. yang and lim (2003), and haralampides et al (2003) studied suspended-load, total load and bed load sediment transport in alluvial channels [4-13]. In last decades, soft computing methods are used to estimate functions that depend on a large number of inputs. This method has been extensively used in various hydraulic engineering problems such as stream flow prediction (chang and Chen 2003; Gholami et al. 2014, 2015a, 2015b), rainfall-runoff modeling (lin & chen, 2004), combined open channel flow (yang & chang, 2005), sediment transport (Ebtehaj & Bonakdari 2014) [14-21]. More recently, combination of fuzzy logic and neural networks have given rise to a novel neuro-fuzzy system named Adaptive Neural Fuzzy Inference System (ANFIS) which have a much application in enginnering: Asadiani Yekta et al. (2010) examined suspended sediment estimation using ANFIS, ANN and Sediment Rating Curves (SRC) [22]. Their results showed that ANFIS is more accurate than ANN in estimating the suspended sediment load as a discharge function. Samandar (2011) presented an ANFIS model to predict the friction coefficient of open channel flow [23]. ANFIS model may be a suitable method of analysing general hydraulic problems that are mostly based on laboratory tests. Since soft computing method has been rarely used to estimate hydraulic geometry of stable channels, the objective of this study is to extend afzalimehr et al. 2009 [24] studies on width, depth and slope of stable channels by use of 85 set of data and by employing ANFIS model and eventually compare the result of these two methods.
2- Material and methods 2-1- Overview of ANFIS The adaptive Neuro-Fuzzy Inference System (ANFIS) is a universal estimator introducing by Jang (1993) [25]; Jang and Gullery (1995) [26]. ANFIS is a combination of Artificial Neural Network (ANN) and with Fuzzy Inference Systems (FIS) to facilitate the process of learning and adaptation that is the main problem in design of fuzzy systems which achieve the fuzzy “IFTHEN” rules with the effective utilize of ANN learning capability for optimal generation of these rules. ANFIS is able to approximate any continues function on a complicated set of any degree of precision. This method constructed a functional mapping approximating the estimation process of internal system parameter. In this paper, the hybrid algorithm which is a combination of back-propagation and least square methods are used as a high efficient learning algorithm (Ebtehaj and Bonakdari, 2014 [21]) to train the ANFIS network. The Takagi-Sugeno fuzzy inference system which is saves the IF-THEN rules in rules platform for two rules is presented as follows:
2
Rule 1 : If
x = A1 ,
y = B1
Rule 2 : If
x = A2 ,
y = B2
Then Then
f1 = p1 x + q1 y + r1
(1)
f1 = p2 x + q2 y + r2
(2)
where A1, A2 and B1, B2 are membership functions for x and y, respectively. The p1, p2, q1, q2, r1 and r2 are parameters of output function. The ANFIS output follows an advanced five-layer neural network arrangement: The output of the ith node in layer l is indicated as Ol,i. Five different layers for an ANFIS network with two rules are as follow: In the layer 1, membership degree of each input variables is identified with node function. The output of Ol,i is defined as following equations: Oi1 = µ Ai (x)
(3)
i = 1 ,2
Oi1 = µ Ai −2 (y)
(4)
i = 3 ,4
where x and y are inputs of ith node and Ai and Bi-2 are linguistic label (“high” or “low”) of this node. The membership function described by Gaussian functions as follow: x − ci µ A ( x) = exp − 2 2σ i
2
(5)
where σi and ci are parameters set which is changed the membership function form in [0,1]. In layer 2, each node calculates the activation degree of any rule as follows: Oi2 = wi = µ Ai (x ) × µ Bi ( y )
(6)
i = 1, 2
where µAi(x) and µBi(y) are degree of x and y membership in set of Ai and Bi, respectively. In layer 3, the firing strength of ith rule to firing strength of all rules is computes as follows: Oi3 = wi =
w1 , w1 + w2
(7)
i = 1,2
where wi is the normalized membership degree of ith rule. The output of each node is calculate in layer 4 as follows: Oi4 = wi f i =
w1 ( pi x + qi y + ri ) w1 + w2
(8)
where pi, qi and ri are the adaptive parameters of ith layer. The final output as summation of all incoming signals is calculate in layer 5 as follows: Oi5 = ∑ wi f i =
∑ wi f i ∑ wi
(9) 3
2-2- Data presentation for stable channel design and experimental design Hydraulic and geometric parameters in channel in regime state are so important parameter for designing channels. These parameters are water level width (w), average flow depth (depth) and channel slope (s). Variables affecting this parameters are two flow discharge (Q) and average particle size of sediment (d50) parameters and later Shields parameter (τ*) (sediment particle moving) are important parameter to predict w, h and s at stable channel in most studies. The hydraulic geometry relations are of great practical value in prediction of channel deformation; layout of river training works; design of stable canals and intakes, river flow control works, irrigation schemes, and river improvement works; there are much relations based on theoretical and regression methods for predicting geometry of stable channels most of them focused on the following main issues: (1) the basis of hydraulic geometry relations, (2) the tendency of equilibrium state, (3) limitations of the equilibrium assumption, (4) the effect of river channel patterns, (5) the variation of channel width with discharge, (6) the effect of stream size, (7) the variation in channel velocity, (8) boundary conditions. For example Afzalimehr et al. (2009) represented three semi- theoretical equation to predict w, h and s of stable channels that are as follow [24]: (10) (11) h = 0.226 Q 0.821 0.851 (12) s = 1.565 d 50 τ * Three ANFIS model are designed to predict width, depth and slope of stable channel crosssections. The input variables are Q, d50 and τ* parameters and the input or target variable are w, h and s of stable channel. The observational data for 85 cross section rivers with gravel-bed are related to Afzalimehr et al. 2009 that are located in stability regime condition in IRAN [24]. 70% and 30% of whole data (60 and 15 data) are used to train and test models, respectively. Then Afzalimehr’ s regression relationships [24] are used to predict geometry of stable channel and also for evaluating of ANFIS model’ s efficiency. w = 5.876 Q 0.743 0.345
2-3- Statistical Analysis The results of the ANFIS models are explained in this section along with the statistical indexes. Three indices of Mean Absolute Error (MAE) and Root Mean Squared Error (RMSE), Mean Absolute Relative Error (MARE), Correlation coefficient (R) and BIAS statistical parameters as the statistical parameters are already selected:
RMSE =
1 n (Oi − Pi ) 2 ∑ n i =1
(13)
4
1 n ∑ Oi − Pi n i =1
MAE =
(14)
1 n Oi − Pi ∑ n i =1 Oi
MARE =
(15)
n
∑ (O − O ).( P − P ) i
R=
i
i
i
i =1 n
(16)
n
∑ (O − O ) ∑ ( P − P ) 2
i
i
i
i =1
2
i
i =1
n
∑ (P − O ) i
BIAS = where
i
(17)
i =1
O i
n is the output observational parameter, Pi is the parameter predicted by the ANFIS
models, Oi and Pi is the mean ANFIS and observed model parameter and N is the number of parameters. R provides a measure of how well the observed outcomes are replicated by the model. Higher model accuracy leads to RMSE, MAE, MARE values closer to zero.
3- Results and Discussions Geometry parameters of stable channels are predicted using ANFIS models in the present study. Scatter plots of width, depth and slope prediction by ANFIS model compare with observational data are showed in Figure 1 and R2 values between the models show in the plots. The width, depth and slope values are estimated by Afzalimehr et al. (2009)’s regression equations [24]. R, MARE, RMSE, MAE, BIAS error indexes between ANFIS models and Afzalimehr’s equations with observed data are shown in Table 2. In Figure 1 more data compression has been around exact line and also R2 values in three models (value close to 1) shows high accuracy of ANFIS models in prediction of all three parameters (R2 = 0.9224, 0.7464 and 0.9264). Table 1 shows comparison between ANFIS models and Afzalimehr’ s regression equations (Equation No. 10, 11 and 12). Figure 1 and Table 1 show the high accuracy of ANFIS model in predicting of slope of stable channel with high R2 value and low error indexes (R2= 0.9264 and MARE= 0.019). In width prediction model, ANFIS model with high R value is more accurately than the Afzalimehr’ s equation (almost 20%). Also in this model, relative and absolute error of MARE and RMSE have the less value in ANFIS model (MARE =0.062 and RMSE =0.113). In depth prediction model, ANFIS model with high correlation coefficient and low error values is more accurately than Afzalimehr’ s equation (R= 0.86 (Approximate increase= 53%), MARE= 0.016, RMSE= 0.14). In slope prediction model like width and depth prediction models, relative and absolute error values in ANFIS model are more less than Afzalimehr’s equation (Approximate 5
increase in R value equal to 150% , MARE= 0.019, RMSE= 0.356). In general, MARE value of Afzalimehr’ s regression equations are more higher than ANFIS model almost 73, 57 and 50 times in prediction width, depth and slope parameters, respectively. Therefore, regression models performance improve well using ANFIS model. Performance improvement of ANFIS model in width prediction is more than another models (by MARE 73 time lesser than regression relation) because in the Afzalimehr’ s widh equation [24] are existed only discharge parameter but input variables in ANFIS model are all three Q, d50, τ* parameter. The lowest efficiency improvement is slope prediction model because the presence of all three affected parameters in both models. Specifically, presence of τ* parameter in slope prediction according on shear stress relation with zero-pressure gradient is more affected. BIAS indices show the underestimation and overestimation models. This index value in ANFIS model Afzalimehr et al. (2009) equations to predict the width, depth and slope is close to zero. It can be said that, this index with the same average value predict the estimation value as underestimation and overestimation. Negative and positive value indices show the underestimation and overestimation model, respectively. In present all ANFIS models, BIAS values are negative, indicating that the models are as underestimate. And positive values indicate an overestimation in predicting models. Table 1. the different error indexes to compare and evaluate ANFIS models and Afzalimehr et al. (2009) [24] equations in prediction geometry of stable channels. Models
Width prediction
Depth prediction
Slope prediction
index
ANFIS
Afzalimehr
ANFIS
Afzalimehr
ANFIS
Afzalimehr
R
0.96
0.802
0.86
0.562
0.962
0.383
MARE
0.062
4.520
0.016
0.902
0.019
0.952
RMSE
0.113
12.622
0.14
3.162
0.356
5.532
MAE
0.046
11.978
0.058
3.153
0.130
5.482
BIAS
-0.003
11.978
-0.022
-3.153
-0.048
0.504
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Figure 1. The scatter plot diagrams to predict the width, depth and slope by ANFIS model in comparison with observational values.
In Figure 2, width, depth and slope values predicted by ANFIS model are compared with observed data in 85 cross-section of river. ANFIS model predicts well width, depth and slope data trend like observed data and have a good agreement with observed data (especially in slope prediction). In three models, the agreement of all models are good but in some points specially in maximum and minimum points ANFIS model predicts higher and lower values than those observed data, respectively.
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Figure 2. Comparison of width, depth and slope points predicted by ANFIS model with observed values.
4- Conclusions Effieciency of ANFIS model is investigated in prediction of geometric and hydraulic variables in stable channels, in the present study. Also, using available previous regression ralations, width, depth and slope variables’ s value are predicted and compared with ANFIS model’ s results. Available observational data are used for training and testing models. The results show that ANFIS model using are caused to improve regression relations’s performance. As correlation coefficient values (R) in ANFIS models approximatly 20%, 53% and 150% increas compared to 8
the regression eouations in prediction of width, depth and slope respectively which is indicative of high accuracy of ANFIS models. But decreasing of MARE error value of 73%, 57% and 50% in ANFIS model shows the efficiency improvement of width prediction model. The noticable point in this study is that despite of the regressional relation are fitted on the same field data that uses for training and testing ANFIS model, but the proposed ANFIS models still better than the regression equations. It is suggested that the other soft computing techniques such as, group method of data handling (GMDH) neural network, Gene Expression Programming model etc in prediction of stable channel parameters be used. Refrence [1] Lee, Jong-Seok, and Pierre Y. Julien. "Downstream hydraulic geometry of alluvial channels." Journal of hydraulic engineering 132.12 (2006): 1347-1352. [2] Julien, P. Y. (2002). River mechanics. Cambridge, UK: Cambridge University Press. [3] Singh, V. P., & Zhang, L. (2008a). At-a-station hydraulic geometry relations, 1:Theoretical development. Hydrological Processes, 22, 189–215. [4] Lee, J. S. (1995). Evaluation of parameters for bed form changes in a meandering channel. Ph.D. dissertation, Dankook University, Seoul, Korea (in Korean). [5] Cheema, M. N., Marino, M. A., & DeVries, J. J. (1997). Stable width of an alluvial channel. Journal of Irrigation and. Drainage Engineering, ASCE, 123(1), 55–61. [6] Yen, C. L., & Lee, K. T. (1995). Bed topography and sediment sorting in channel bend with unsteady flow. Journal of Hydraulic Engineering, ASCE, 121(8), 591–599. [6] Cao, Shuyou, and Donald W. Knight. "Design for hydraulic geometry of alluvial channels." Journal of Hydraulic Engineering 124.5 (1998): 484-492. [7] Julien, P. Y., & Anthony, D. J. (2001). Bed load motion and grain sorting in a meandering stream. Journal of Hydraulic Research, 40(2), 125–133. [8] Kassem, A. A., & Chaudhry, M. H. (2002). Numerical modeling of bed evolution in channel bends. Journal of Hydraulic Engineering, ASCE, 128(5), 507–514. [9] Olsen, N. R. B. (2003). Three-dimensional CFD modeling of self forming meandering channel. Journal of Hydraulic Engineering, ASCE, 129(5), 366–372. [10] Yen, B. C. (2002). Open channel flow resistance. Journal of Hydraulic Engineering, ASCE, 128(1), 20–39 [11] Smart, G. M., Duncan, M. J., & Walsh, J. M. (2002). Relatively rough flow resistance equations. Journal of Hydraulic Engineering, ASCE, 128(6), 568–578. [12] Yang, S. Q., & Lim, S. Y. (2003). Total load transport formula for flow in alluvial channels. Journal of Hydraulic Engineering, ASCE, 129(1), 68–72. [13] Haralampides, K., McCorquodale, J. A., & Krishnappan, B. G. (2003). Deposition properties of fine sediment. Journal of Hydraulic Engineering, ASCE, 129(3), 230–234. [14] Chang, F. J., & Chen, Y. C. (2003). Estuary water-stage forecasting by using radial basis function neural network. Journal of Hydrology, 270(1-2), 158–166. [15] Gholami, A.; Akhtari, A.A.; Minatour, Y.; Bonakdari, H. and Javadi, A.A. Experimental and numerical study on velocity fields and water surface profile in a strongly-curved 90° open channel bend. Engineering Applications of Computational Fluid Mechanics, Vol.8,No.3,pp. 447-461.2014. [16] Gholami, A.; Bonakdari, H.; Zaji, A.H. and Akhtari, A.A. Simulation of open channel bend characteristics using computational fluid dynamics and artificial neural networks. Engineering Applications of Computational Fluid Mechanics, Vol.9,No.1,pp. 355-361.2015.
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