H. Liu et al. : Effluent Quality Prediction of Papermaking WWTPs Using SEL
By adding Lagrangian coefficients β i and β ∗ i , the weight vector w can be expressed as follows : w = n i = 1 ( β i − β ∗ i ) ϕ ( x ) (5) Finally, the SVR can be defined as the following regression function : f ( x ) = n i = 1 ( β i − β ∗ i ) K ( x , x ) + b (6) where K ( x , x ) corresponds to the kernel function. The radial basis function (RBF) was used in this work, which can be expressed as follows : K ( x , x ) = exp( − γ x − x 2 ) (7) where γ is kernel parameter which is used to control the radial range of the kernel function. 3) ARTIFICIAL NEURAL NETWORKS Artificial neural networks algorithm has a powerful self-learning and self-adaptive ability to reduce prediction error. Figure 3 shows the classic topology structure of arti- ficial neural networks with three layers including input layer, hidden layer, and output layer.
The output value of the k -th neuron in the output layer is written as y k , which can be defined as follows : y k = f ( M j = 1 w jk x j − θ k ) , k = 1 , 2 , · · · Q (9) The network weights and the thresholds need to be adjusted depending on the minimum mean square error (MSE) which is defined as follows : MSE = 1 l l n = 1 ( t n − y n ) 2 (10) where l is the number of training samples, t n corresponds to the expected value of the neural node and y n denotes the predicted value. The MSE will be reduced to certain extent by repeating the back-propagation mechanism. Another stop- ping criterion of the algorithm is that the training time reaches its maximum. C. META-LEARNING ALGORITHM As the most efficient meta-learning algorithm, MLR com- bines weight values with output values coming from base-learning algorithms to obtain a further generalization result. The specific form can be expressed as follows : y = w 0 + w 1 x 1 + w 2 x 2 +· · ·+ w k x k (11) where y denotes the final prediction result, w k corresponds to the weight value, and x k corresponds to the output value from the base-learning algorithm. All of the weight values are obtained in the training process. By choosing appropriate weight values, MLR makes the square sum of the difference between the predicted values and the real values as small as possible. The formula is shown as follows : e = n i = 1 ( y ( i ) − k j = 1 w j x ( i ) j ) 2 (12) where n is the number of samples, k is the number of base-learning algorithms, y ( i ) represents the real value of the i -th sample, and k j = 1 w j x ( i ) j represents the predicted value of the i -th sample. D. MODELING PERFORMANCE INDICES To determine whether the final results have a better predic- tion accuracy, three evaluation indices including determinate coefficient (R 2 ), mean absolute percentage error (MAPE), and root mean square error (RMSE) are used in this work, which is calculated from Equations (13), (14), and (15), respectively.
FIGURE 3. Topology structure of artificial neural networks.
Firstly, it is assumed that the number of input layer nodes, hidden layer nodes, and output layer nodes is N , M , and Q , respectively. Moreover, x i represents the i -th input value in the input layer, w ij represents the weight value from the input layer to the hidden layer, w jk represents the weight value from the hidden layer to the output layer and θ j corresponds to the threshold value. The input information is firstly propagated forward from the input layer to the hidden layer then the error is propagated backward according to weight values and transfer function f ( x ). Through repeated correction of weight values, the predicted values are gradually closer to the actual values. During the whole process, x j represents the output value of the j -th neuron in the hidden layer which can be defined as follows : x j = f ( N i = 1 w ij x i − θ j ) , j = 1 , 2 , · · · M (8)
N
2
( ˆ y i − y i )
i = 1
R 2 = 1 −
(13)
N
2
( y i −¯ y i )
i = 1
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VOLUME 8, 2020
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