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For the BP neural networks, the learning rate is fixed as 0.01, and the neuron number and activation functions are designated respectively. Generally, we set the max iteration as 1,000 for the tansig and logsig functions (the definitions of the functions are formulated as Equations (9) and (10)). The neuron number is assigned from 6 to 20, and the networks are likely over-fitting when the neuron number becomes large. Because BPNNs consume a lot of time in training the networks, each set-up is trained ten times and the average performances are presented in Table 1.
formula
9
formula
10
Table 1

The performance of BPNNs with tansig or logsig as the activation function

Activation functionNeuron numberAverage iteration numberAverage training time (%)Average training error (%)Average testing error (%)Smallest testing error (%)
Tansig 591 51.4039 7.09 7.21 6.63 
708 57.7563 6.60 6.88 6.28 
10 844 75.7478 5.98 6.29 5.86 
12 951 92.8658 5.81 6.19 5.57 
14 988 98.0263 5.42 5.72 5.32 
16 1,000 111.6640 5.09 5.54 5.20 
18 1,000 117.8120 4.85 5.23 4.84 
20 1,000 127.2032 4.84 5.69 4.73 
Logsig 534 42.7224 7.16 7.22 6.84 
806 66.8199 6.50 6.70 6.12 
10 897 81.2812 6.27 6.52 5.90 
12 986 97.8064 5.71 5.96 5.62 
14 982 97.2682 5.47 5.88 5.44 
16 1,000 106.4332 5.18 5.80 5.37 
18 1,000 113.0742 5.07 5.46 5.07 
20 1,000 128.5261 4.89 5.38 5.03 
Activation functionNeuron numberAverage iteration numberAverage training time (%)Average training error (%)Average testing error (%)Smallest testing error (%)
Tansig 591 51.4039 7.09 7.21 6.63 
708 57.7563 6.60 6.88 6.28 
10 844 75.7478 5.98 6.29 5.86 
12 951 92.8658 5.81 6.19 5.57 
14 988 98.0263 5.42 5.72 5.32 
16 1,000 111.6640 5.09 5.54 5.20 
18 1,000 117.8120 4.85 5.23 4.84 
20 1,000 127.2032 4.84 5.69 4.73 
Logsig 534 42.7224 7.16 7.22 6.84 
806 66.8199 6.50 6.70 6.12 
10 897 81.2812 6.27 6.52 5.90 
12 986 97.8064 5.71 5.96 5.62 
14 982 97.2682 5.47 5.88 5.44 
16 1,000 106.4332 5.18 5.80 5.37 
18 1,000 113.0742 5.07 5.46 5.07 
20 1,000 128.5261 4.89 5.38 5.03 
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