Prediction of bed load sediments using different artificial neural network models

Abstract

Modeling and prediction of bed loads is an important but difficult issue in river engineering. The introduced empirical equations due to restricted applicability even in similar conditions provide different accuracies with each other and measured data. In this paper, three different artificial neural networks (ANNs) including multilayer percepterons, radial based function (RBF), and generalized feed forward neural network using five dominant parameters of bed load transport formulas for the Main Fork Red River in Idaho-USA were developed. The optimum models were found through 102 data sets of flow discharge, flow velocity, water surface slopes, flow depth, and mean grain size. The deficiency of empirical equations for this river by conducted comparison between measured and predicted values was approved where the ANN models presented more consistence and closer estimation to observed data. The coefficient of determination between measured and predicted values for empirical equations varied from 0.10 to 0.21 against the 0.93 to 0.98 in ANN models. The accuracy performance of all models was evaluated and interpreted using different statistical error criteria, analytical graphs and confusion matrixes. Although the ANN models predicted compatible outputs but the RBF with 79% correct classification rate corresponding to 0.191 network error was outperform than others.

Keywords

bed load prediction / artificial neural network / modeling / empirical equations

Introduction

Prediction of sediment transported loads due to nonlinear relationships and complicated interaction of involved parameters is a very difficult task. Moreover, the importance of accurate prediction and corresponding extracted information of sediment transport in providing more concise interpretation in different objectives such as morphological and sedimentological features [¹,²], water engineering purposes [³–⁵], maintaining the channel geometry [⁶], and economic damage of sediment loads [⁷], as well as river basin management [⁸,⁹], has been approved.

Direct measurements or using the proposed empirical relations are the common methods in calculating the transported sediment loads. Due to possible unfeasible and uneconomical aspects in equipped all desired locations for long time direct measurement [⁴,¹⁰], the simple but accurate enough approaches for modeling of sediment loads is preferred. It is approved that the information on the channel, flow and sediment characteristics are essential elements on development of sediment load relations [¹¹]. However, because of wide range of flow conditions as well as dependency of involved parameters to river and sediment characteristics the adopted equations are not unique [¹²–¹⁴]. Moreover, the sophisticated nonlinear relation between flow and sediment loads due to lack of accepted fundamental principles and increased demand of practical application as well as considering the effect of different auxiliary factors cannot be properly modeled by statistical and regression techniques [¹⁵–¹⁷]. Therefore, it can be expectable that the same formulation not only provide dissimilar scores of accuracy but also usually do not fit with the observed data. This reason implies why significant attempts for improvement of the existing equations should be made.

Due to approved difficulties in direct bed load measurements and involvement of several factors (e.g., change the flow properties, modifying the bed geometry, friction, roughness, producing bed forms) integrating reliable bed load prediction into hydraulic calculations is a challenging task [¹⁸–²⁰]. Furthermore, the associating of bed load measurements with slopes [²¹–²³], make it not only more complicated and expensive than suspended load but also more interest in developing equations for the predictive models [¹⁶,²⁴].

In such situations, by applying the soft computing approach and in particular artificial neural network (ANNs) techniques the complex nonlinear behavior can be captured [²⁵,²⁶]. The literatures show that the ANN techniques due to producing the results faster than most conventional methods have successfully been employed in sediment load predictions in hydrological sciences [¹⁴,²⁷–³⁷].

In this paper, different optimum multilayer percepterons (MLP), generalized feed forward neural network (GFFN), and radial based function (RBF) models subjected to various training algorithms and activation functions with the aim of bed load sediment prediction in the Main Fork Red River-Idaho, USA are introduced. Among the acquired data by United State Geological Survey (USGS) and United States Department of Agriculture (USDA), the five predominant parameters in empirical bed load transport formulas including flow discharge (Q), flow velocity (V), water surface slopes (S), flow depth (d), and mean grain size (D₅₀) were used. The results of developed models then were compared with previous proposed equations and evaluated using error criteria. The models then were ranked and the best accuracy in fitting with measured data was observed for RBF and GFFN, respectively.

Compiled datacenter of study area

The Main Fork Red River as part of the Spokane River Basin watershed is located in the Panhandle National Forest of northern Idaho (Fig. 1(a)). The records of stream flow for 1986 to 2000 and transported sediments for 1986 to 1999 including 200 and 136 measurements of transported bed and suspended loads has been extracted and compiled from USGS and USDA. The recorded data sets as well as information on channel profile, cross section and transported material can be found in URL link of US Forest service of USDA. The variation of river cross section at the measurement station for 3 years (1995–1998) is presented in Fig. 1(b). According to previous studies [²,⁴,¹⁴,¹⁹]; Q, V, S, d, and D₅₀ due highest influence on bed load sediment transport were selected and analyzed. These factors can be categorized into hydrological, channel geometry, geomorphological and hydraulic characteristics. By sorting the time series of recorded data, a uniform processed data sets including 102 sets of five mentioned parameters were provided and statistically analyzed (Table 1). The wide range of observed variation in selected parameters can be referred to amount of precipitations in each water year during the measurements. The used data sets due to different units using {\displaystyle \frac{x-x_{\min }}{x_{\max }-x_{\min }}} were normalized within the range of [0, 1] to provide dimensionless input data which are necessary to improve the learning speed and model stability. The created data sets was then randomized into 55%, 25%, and 20% to organize training, testing and validation of ANN-based models. Therefore training procedure implies that the models are run with similar data sets which provide more facilities in evaluating and comparison of the performance criteria.

ANN predictive models

ANNs as an information processing paradigm is one of the main tools used in machine learning and connectionist computing systems which aim to simulate brain structure and mimic the way that humans learn. Typically, the embedded neurons (processing units) in ANNs are arranged in a series of layers including input, hidden, and output. Input (x_i), weights (w_i.j), bias (b_i), activation function (f_act), and output (O_i,j) are the components of an artificial neuron. In human brain, the received inputs in each level of neurons provide insight and then the information gets passed on to the next, more senior level. This is precisely the mechanism that ANNs as fully connected and weighted network layers try to replicate. The input layer receives various forms of data which network aims to process or learn about. The data according to weighted connections then are passed through one or more hidden layers. The hidden layer is responsible to data processing and then transfer to output unit. The feed forward ANNs are those type of neural nets in which information travels in only one direction from input to output. The MLPs as typical feed forward networks are trained slowly but easy to use and can approximate any input/output map. The jth network output (net_j) contain a set of adaptive weight of w_i,j and is expressed as:

(1)ne{t}_j={\sum }_{x\in X}(O_i\cdot w_{i,j})+b_i,

where b_i denotes the bias as a type of connection weight with a constant nonzero value which is set up into the all neurons in the back-propagation and transfer functions except for the input layer. The bias is much like a weight but with a constant input of 1, while the transfer function shifts the summed signals received from this neuron. The assigned activation state then by use of threshold value (q_j) transforms the net_j from initial activation state a_j(t−1) into new a_j(t) by:

(2)a_j\left(t\right)=f_{\mathrm{a}\mathrm{c}\mathrm{t}}\left(ne{t}_j\right(t),a_j(t-\mathrm{1}),{\theta }_j).

The output value O_j of the neuron j is then calculated from corresponding activation state a_j as:

(3)f_{\mathrm{out}}\left(a_j\right)=\mathrm{0.}

The error of each sample (E_p) and root mean square error (RMSE) between the input (x) and the actual output (y) for the kth output neuron is defined by:

(4)E_{\mathrm{p}}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}{\sum }_{k\in O}{(x_k-y_k)}^{\mathrm{2}},

(5)RMSE=\sqrt{{\displaystyle \frac{{\sum }_{k\in O}{(x_k-y_k)}^{\mathrm{2}}}{\left|O\right|}}}.

The optimum weights then can be found using and updating procedure for (n+ 1)th pattern subjected to:

(6)\mathrm{\Delta }{w}_{i,k}=-\eta {\displaystyle \frac{\partial Err\left(W\right)}{\partial w_{i,k}}},

(7)w_{i,k}(n+\mathrm{1})=w_{i,k}\left(n\right)+\nabla w_{i,k}\left(n\right),

where h is the learning rate.

The GFFN due to ability in jumping over one or more layers showed more facilities in both selecting the optimum topology and increasing the computational potency [³⁸–⁴⁰]. Moreover, it was found that the in the same number of neurons the performance of GFFN is more efficient than MLPs [⁴⁰]. The GFFN classifier uses a generalized shunting neuron (GSN) model which allows neurons to operate as adaptive nonlinear filters [³⁸,⁴⁰]. To produce the output in GFFN, all input is summed and passed through an activation function similar to a perceptron neuron (Eq. (8)).

(8)x_j={\displaystyle \frac{b_j+f({\sum }_i{w}_{ji}{I}_j+w_{jo})}{a_j+g({\sum }_i{c}_{ji}{I}_i+c_{jo})}},

where x_j: output (activity) of the jth neuron; I_j and I_i: inputs to the ith and jth neurons; a_j: passive decay rate of the neuron (positive constant); w_ji and c_ji: connection weight from the ith inputs to the jth neuron; a_j and b_j: constant biases; g and f: activation functions.

The RBFs is a hybrid network with three layers (input layer, a hidden layer, and a linear output layer) in which uses nonlinear Gaussian activation transfer functions rather than the standard sigmoidal functions and tend to learn much faster than MLPs.

(9)h_j\left(x\right)={\exp }^{-\frac{{(x-c_j)}^{\mathrm{T}}(x-c_j)}{r^{\mathrm{2}}}},

where c_j denotes the center, r is the width and (·)^T expresses the transpose of the argument.

Each input x_i at each hidden neuron j is weighted by w^h as:

(10)S_j=[x_{\mathrm{1}}{w}_{\mathrm{1},j}^h,x_{\mathrm{2}}{w}_{\mathrm{2},j}^h,\mathrm{...},x_n{w}_{n,j}^h,\mathrm{...},x_N{w}_{N,j}^h],

where x_n is the nth input; w^h_n,j is the input weight between input n and hidden neuron j. Accordingly the output of hidden neuron (f_j(s_j)) and the network output (O_m) is calculated by:

(11){\phi }_j\left(S_j\right)=\exp \left({\displaystyle \frac{\left|\right|S_j-c_j|{|}^{\mathrm{2}}}{{\sigma }_j}}\right),

(12)O_m={\sum }_{i=\mathrm{1}}^J{\phi }_j\left(S_j\right)w_{j,m}^o+w_{\mathrm{0},m}^o,

where the activation function ϕ_j(·) for hidden neuron j is normally chosen as Gaussian function; c_j and s_j denote the center and width of hidden neuron j, respectively. w^o_j,m expresses the output weight between hidden neuron j and output neuron m and w^o_0,m is the bias weight of output neuron m.

The approximating function expresses a sum of N radial basis functions associated with a different center x_i and appropriate weighted coefficients. The approximating function is linear and thus can be estimated using the matrix methods of linear least squares.

Therefore, producing optimum ANNs structure due to prevent the over-fitting problem and dependency to internal characteristics (e.g., training algorithm, number of neurons, learning rate, activation function, architecture, regularization) is a difficult but important task. Despite of different proposed relations for number of hidden neurons [⁴¹], no unique method for network configuration is accepted [⁴²,⁴³]. In this paper the optimum models and corresponding adjusted internal characteristics were found through an iterative procedure integrated with a developed code based on constructive technique (Fig. 2).

Results of modeling process

The data were randomized as 55%, 25%, and 20% for training, testing and validation. In optimizing process not only different training algorithms and activation functions but also various arrangements of neurons in hidden layers were examined. To find the optimum model, seven training algorithms (quick propagation, QP; conjugate gradient descent, CGD; step, S; momentum, MO; quasi-Newton, QN; limited memory quasi-Newton, LMQN; Levenberg-Marquardt, LM) and six activation function (logistic, Log; hyperbolic tangent, HyT; linear, Lin; softmax axon, SoA; bias axon, BiA; squash, Sq) were examined. The sum of squares and cross-entropy were utilized as output error function, respectively.

To decrease the network variables, the learning rate of used algorithms and the domain of changes in step size for hidden layers was set 0.7 and [1.0–0.001], respectively. The process is terminated using two different termination criteria. The first priority is to satisfy the target root mean square error (RMSE) and if not achieved the number of iterations will use. The maximum iteration number in this study was set for 1000 which means that if cannot attain the desired RMSE then the minimum observed network error of each examined structure among 1000 repeated epochs will be considered. With this condition even the structures which couldn’t show the target error but provide lower RMSE than previous examined model were also captured. According to embedded loop in defined procedure (Fig. 2) numerous structures even with similar architecture but different internal characteristics were generated. As a description of executed effort (Fig. 2), the process with one neuron in hidden layer (J) is begun with one of the training algorithms (T) and activation functions (F). In the loop, T and F are fixed and only J is varying and while t<7, the loop for 6 different F (Log, Hyt, Lin, SoA, BiA, Sq) is repeated. Therefore the process is repeated up to F = 6 and thus the condition F≥6 will break the internal iteration and goes to changing the J and the process with new number of neurons again is repeated. During this process, by achieving the minimum RMSE, the loop is terminated and the obtained number of neurons needs to be managed in appropriate layers to find the optimum network structure (e.g., topology characteristics, model regularization and new topology as presented in Fig. 2). The maximum number of neurons as a user option can be set by after neuron increment in which in this study using while command was managed for 17. Obviously, by changing the condition different number of neurons can be checked. In this step different arrangement of obtained neurons subjected to learning rate from 1 to 0.01 for each generated structure is examined and those topologies with the minimum observed RMSE as candidate of optimum models were considered. To have a similar condition in model evaluation, all models were trained subjected to initial randomized data sets.

Accordingly, the calculated minimum RMSE and the maximum network coefficient of determination (R²) of each structure after 3 runs were considered. Structures refer to numerous generated topologies which have been obtained from different neurons arrangement in layers. Using described procedure more than 1200 different models even in similar structure but different internal characteristics were examined. Diverse training algorithms and activation functions under different step sizes for learning rate were used to avoid the over fitting problem and escape from local minima. For example replacing the gradient descent by the momentum optimizer with step size 0.001 minimizes the chances that it gets stuck in a local minimum. Furthermore, the results of 3 runs for each structure and investigation of error improvements were also monitored to be ensured that the examined models were not over fitted and trapped in local minima. The results of selected optimum models as well as example of tested network errors against the number of neurons for MLP, GFFN, and RBF subjected to different training algorithms are given in Table 2 and Fig. 3, respectively.

Discussion and validation

The confusion matrix is a specific table layout to analyze and visualize the performance of an algorithm in classification purposes of ANN models [⁴⁴]. The rows (i) and columns (j) of this square matrix represent the used data sets in predicted and actual classes to display if the system is confusing two classes [⁴⁵]. The value in the (i,j) position is the number of records for the target column in the ith category and corresponding network output in the jth category. The perfect classification would have 0 everywhere except on the diagonal entries. The confusion matrixes of all obtained optimum models with 10 rows and target value step of 1.04 were calculated. As an example, the results of RBF (Table 3) showed that the system have trouble in distinguishing the real values in the range of 1.08–2.11, 7.29–8.33, and 9.37–10.4, but can make proper distinction between other defined classes. The rows with 0 means that there is no input in this range and also the network perform appropriate prediction which there is no data to confuse with other categories. Similar interpretation for GFFN and MLP was also carried out. Then correct classification rate (CCR) as a qualitative characteristic in classification tasks (Eq. (13)) is employed to show the portion (in %) of data correctly recognized within the general data set [⁴⁶]. The CCR and classification error (CE) were calculated for all optimized models in which for RBF can be found in Table 3. Furthermore, a comparison between the averages of CCR for all models were carried out and reflected in Table 4, respectively.

(13)CCR={\displaystyle \frac{correctly\mathrm{\hspace{0.17em}}predicted\mathrm{\hspace{0.17em}}class}{total\mathrm{\hspace{0.17em}}testing\mathrm{\hspace{0.17em}}class}}\to CE=\mathrm{1}-CCR.

In the literatures, several empirical equations subjected to different parameters for determining the bed load have been proposed. In this paper the developed equations by [⁴⁷–⁵⁶], were investigated. The dimensionless bed load rate was calculated by the empirical equations in the spreadsheet software with the replacement of the flow and geometric characteristics of the river and the characteristics of the river bed materials. This replacement in empirical relationships in spreadsheet software can easily be done. The performance of these equations can be compared with ANN models using validation data sets which randomly selected and didn’t previously have been fed to network in training and testing stages [⁴²]. Due to specified conditions in development of each equation as well as inherent approved limitation of implemented statistical techniques not only high scattering in produced outputs can be observed but also some of them even couldn’t provide estimation for all fed validation sets (Table 5). Therefore the comparisons between those equations without misestimated outputs were conducted and reflected in Fig. 4. The imperfect predictability of empirical equations showed that in similar condition the accuracy and fitting to actual data are not uniform and thus appropriate results cannot be expected. As presented in Table 5 the coefficient of determination (R²) of estimated bed loads using empirical predictors vary from 0.08 [⁴⁸] to 0.21 [⁵⁶], whereas in ANN models more consistency and closer values from 0.93 in MLP to 0.98 for RBF model were obtained. It was observed that proposed equation by Ref. [⁵⁴] provide under estimate values whereas other formulas exhibited over estimate in predicted values.

Despite of approved applicability of statistical error criteria in evaluating the performance of predictive models, the associated interpretation in assessment of low-volume data across multiple items should be considered [⁴⁰,⁴³]. Here, the accuracy performance of models were evaluated suing the mean absolute percentage error (MAPE), variance account for (VAF), RMSE, mean squared deviation (MSD), mean absolute deviation (MAD), R², calculated residuals (CR) and absolute error (AE) criteria. MAPE as the most common used index of forecast error represents the accuracy using the average of the unsigned percentage error. MAD is the average distance between each data point and the mean. This statistic gives an idea about the variability in a data set for size of the error in units and calculates the average of the unsigned errors. MSD as a nonnegative index measures the quality and accuracy of a predictor in fitted values and also express a risk function, corresponding to the expected values of the squared error loss. AE as an indicator of physical error and the uncertainty in a measurement expresses the quality of model and defines the deviation between predicted and measured values. CR is the differences between the measured and predicted values. In performance analyzing, higher values of VAF and R² as well as smaller and lower values of MAPE, CR, AE, MAD, MSD, and RMSE is of interested. According to reflected results in Table 6 and Fig. 5, in the studied area the equation proposed by Ref. [⁵⁶] was ranked as the most appropriate predictors whereas Ref. [⁴⁸], represented the most improper result.

The applicability of sensitivity analysis methods in determining the influence of input parameters on predicted outputs has been approved [⁵⁷–⁵⁹]. Using sensitivity analysis methods not only simplified but robust calibrated model from large number of parameters can be utilized but also important connections between observations and model output as well as evaluating the impacts of the uncertainties in the output can be determined [⁵⁸–⁶³]. In partial derivative algorithm (PaD) as one of the most known ANN-based sensitivity analysis technique [⁶⁴,⁶⁵], the contribution of inputs is analyzed using Jacobian matrix of the partial derivatives of outputs (Eq. (14)) with respect to inputs [⁶⁶].

\mathrm{contribution}\mathrm{\hspace{0.17em}}\mathrm{of}\mathrm{\hspace{0.17em}}i\mathrm{th}\mathrm{\hspace{0.17em}}\mathrm{variable}={\displaystyle \frac{SS{D}_i}{{\sum }_{i}SS{D}_i}}\mathrm{,}

(14)SS{D}_i={\sum }_p{\left({\displaystyle \frac{\partial o_k^p}{\partial x_i^p}}\right)}^{\mathrm{2}},

where O_k^p and x_i^p are output and input values for pattern P the SSD_i is sum of the squares of the partial derivatives, respectively.

Therefore, PaD is able to assess the sensitivity of the output against slight changes in inputs. The results of calculated influence of fed inputs on predicted output using PaD method is reflected in Fig. 6.

It can be seen that the Q and V as tow of the main hydraulic parameters are ranked as the most effective parameters whereas the d which also can be categorized into hydraulic factors is placed in lowest rank. However the values of Q and V are approximately similar and their differences are not significant. The contribution of D₅₀ as one of the main important sediment characteristics in predicted bed loads is ranked in the third position. The influence of S which is a geomorphological related factor is scored in fourth place but both D₅₀ and S are recognized with moderate influences more than d.

Conclusions

ANN models are able to reveal hidden laws of natural phenomena such as sediment transport process. In the current study, different MLP, GFFN, and RBF models for the purpose of bed load prediction in the Main Fork Red River in Idaho, USA were successfully developed and examined. The models were optimized based on the variability of internal characteristics through an organized iterative procedure using 102 sets of five dominant implemented hydraulic variables and sediment data (Q, V, S, d, and D₅₀). The calculated network RMSE and R² revealed that the optimized RBF (RMSE = 0.191) due to more compatible results between measured and predicted bed load transport slightly outperforms than GFFN. It was also observed that in similar number of neurons the GFFN (RMSE = 0.201) reflect much more robust results than MLP (RMSE = 0.289). The summary of predicted results for ANN models using validation and test data sets were presented in separated confusion matrixes which gave insight into the errors by classifier and showed the CCR probability. The corresponding level of CCR for RBF, GFFN, and MLP was 79%, 74% and 67%, respectively. Furthermore the predictability of several empirical bed load equations respect to ANN models for the study area was analyzed. In compare to ANN models which their corresponding R² values varied from 0.93 to 0.98, the empirical equations using similar fed data provided diverse predicted scatters with R² values between 0.10 and 0.21. Comparison between the empirical formulas also demonstrated an overall unsatisfactory predictions of the bed loads for a wide range of hydrodynamic and sediment conditions. This issue can be considered as one of the main shortcomings of the empirical equations that in a same condition exhibit different accuracy scores and thus the performance of such relations were found comparatively insufficient. The accuracy performance of the ANN models and empirical equations were then assessed and ranked using MAPE, RMSE, MAD, VAF, R², CR, and AE error criteria. It was observed that RBF due to satisfied more criteria (RMSE, 0.191; MAD, 0.77; MSD, 0.018; R², 0.98) than GFFN (MAPE, 5.23; VAF, 96.49) demonstrated more accurate performance. Moreover, the comparative analytical graphs of variations of CR and AE showed lower values in ANN models than empirical equations in which the lowest variation was tracked in RBF. The Q, V, and d using the PaD method were recognized as the most and least effective factors on predicted bed loads.

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