6.3 Simulation Results
6.3.1 Training and Testing at Same Rotational Speed
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The variation of initial and final fitness values (i.e., the percentage accuracy) with the population for C-SVC at 30 Hz rotational speeds are shown in Figure 6.12(a), (c) and (e) with the GA and Figure 6.12(b), (d) and (f) with the ABCA for the CWT and WPT based features, respectively). Similarly, the variation of the initial and final fitness values with the population for the ν-SVC at 30 Hz rotational speed are shown in Figure 6.13(a), (c) and (e) using the GA and Figure 6.13(b), (d) and (f) using the ABCA with the CWT and WPT based features, respectively. It includes all the population, including solutions violating constraints. From the choice of initial solutions, it could be seen that it has quite good divergence of the possible solution domain. In the last generation also all solutions have been shown including those violating constraints (however, very few), and it could be observed that the most of solutions have converged.
In the case of GSM 9 4 270× × data sets are used for the parameter estimation. Data points used for the optimization and the final testing is tabulated in Table 4.3. The cross validation accuracy in the case of GSM at 30 Hz rotational speed for the C-SVC is shown in Figure 6.14(a), (c) and (e); and for the ν-SVC is shown in Figure 6.14(b), (d) and (f) with the CWT and WPT based features, respectively. In these the contour line for the percentage accuracy is plotted and the best CV accuracy is marked. Correspondingly, the best SVC parameters are found from the best CV accuracy and the testing accuracy is tabulated. The optimized percentage accuracy (all classes) of different SVM formulation is shown in Table 6.2.
Fault Prediction Ability: After optimization of parameters 9 4 30× × data sets are used for the final testing of the fault classification for the GA and the ABCA, and 9 4 30× × data sets are used for the final testing of the fault classification for the GSM. In many times the accuracy (all classes) is more in the GA and the ABCA as compared with the GSM, which reflect the
soundness of the GA and the ABCA. If we look upon the testing accuracy for all classes, the lowest one is equal to 88.33% and this occurs at 15 Hz rotational speeds on the C-SVC case.
(a) 30 Hz (C-SVC with Morlet wavelet family using GA optimization)
(b) 30 Hz (C-SVC with Morlet wavelet family using ABCA optimization)
(c) 30 Hz (C-SVC with Mexican hat wavelet family using GA optimization)
(d) 30 Hz (C-SVC with Mexican hat wavelet family using ABCA optimization)
(e) 30 Hz (C-SVC with wavelet packet transform using GA optimization)
(f) 30 Hz (C-SVC with wavelet packet transform using ABCA optimization) Figure 6.12 Variation of the initial and final fitness (percentage accuracy) with the population for the C-SVC (a), (c), (e) using the GA and (b), (d), (f) using the ABCA for 30 Hz training and
testing speed
0 10 20 30 40 50
20 40 60 80 100
Fitness
Population number
Initial Final
0 20 40 60 80 100
20 40 60 80 100
Fitness
Population number
Initial Final
0 10 20 30 40 50
20 40 60 80 100
Fitness
Population number
Initial Final
0 20 40 60 80 100
20 40 60 80 100
Fitness
Population number
Initial Final
0 10 20 30 40 50
20 40 60 80 100
Fitness
Population number
Initial Final
0 20 40 60 80 100
20 40 60 80 100
Fitness
Population number
Initial Final
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(a) 30 Hz (ν -SVC with Morlet wavelet family using GA optimization)
(b) 30 Hz (ν-SVC with Morlet wavelet family using ABCA optimization)
(c) 30 Hz (ν-SVC with Mexican hat wavelet family using GA optimization)
(d) 30 Hz (ν-SVC with Mexican hat wavelet family using ABCA
optimization)
(f) 30 Hz (ν -SVC with wavelet packet transform using GA optimization)
(g) 30 Hz (ν -SVC with wavelet packet transform using ABCA optimization) Figure 6.13 Variation of the initial and final fitness (percentage accuracy) with the population
for the ν -SVC (a), (c), (e) using the GA and (b), (d), (f) using the ABCA for 30 Hz training and testing speed
0 10 20 30 40 50
20 40 60 80 100
Fitness
Population number
Initial Final
0 20 40 60 80 100
20 40 60 80 100
Fitness
Population number
Initial Final
0 10 20 30 40 50
20 40 60 80 100
Fitness
Population number
Initial Final
0 20 40 60 80 100
20 40 60 80 100
Fitness
Population number
Initial Final
0 10 20 30 40 50
20 40 60 80 100
Fitness
Population number
Initial Final
0 20 40 60 80 100
20 40 60 80 100
Fitness
Population number
Initial Final
(e) 30 Hz (C-SVC with Morlet wavelet family)
(f) 30 Hz (ν-SVC with Morlet wavelet family)
(g) 30 Hz (C-SVC with Mexican hat wavelet family)
(h) 30 Hz (ν-SVC with Mexican hat wavelet family)
(i) 30 Hz (C-SVC with WPT) (j) 30 Hz (ν -SVC with WPT) Figure 6.14 Cross validation accuracy for 30 Hz training and testing speed
It is also observed that other than 15 Hz rotational speed at all speeds the fault prediction is near perfect (i.e., at 10, 20, 25 and 30 Hz). Table 6.4 illustrates the percentage fault prediction (individual classes) in various rotational speeds against the best prediction accuracy (all classes). The prediction accuracy of 96.67% is the individual lowest for the WT case at 15 Hz rotational speed.
55 55
60 60
65
65 70
70
75 75
75
80 80
80
85
85
85
90
90
90
95 95
95
95 95
log2(C) log 2(γ)
Acc = 99.63 %
-10 -5 0
-10 -8 -6 -4 -2
60 70 80 90
82 84 868 8
90 92 90 92 90
92
94 94 94
94
96 96 96
96
98 98
98
98
log2(ν) log 2(γ)
Acc = 99.91 %
-10 -8 -6 -4 -2
-10 -8 -6 -4 -2
80 85 90 95
50 50
55 60 55 60
65 65
70 70
75 75
80 80
80
85 85
85
90
90
90
90
95 95
95
log2(C) log 2(γ)
Acc = 97.50 %
-10 -5 0
-10 -8 -6 -4 -2
50 60 70 80 90
75 75
80 80
85
85 85
90 90
90
90
90
90
95 95 95
95
95 95
70 70
75 85
8585
65
75 6565
65 80
60
log2(ν) log 2(γ) Acc = 98.24 %
-10 -8 -6 -4 -2
-10 -8 -6 -4 -2
60 70 80 90
50 50
55 60 55 60
65 65
70 70
75 75
80 80
80
85
85
85
90 90
90
90
95 95
95
log2(C) log 2(γ)
Acc = 97.50 %
-10 -5 0
-10 -8 -6 -4 -2
50 60 70 80 90
75 75
80 80
85
85 85
90 90
90
90 90 90
95 95 95
95
95 95
70 70
75 85
85
85
65 75
6565 65
log2(ν) log 2(γ) Acc = 98.24 %
-10 -8 -6 -4 -2
-10 -8 -6 -4 -2
60 70 80 90
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Table 6.2 Classification of gear faults (all classes) with the training and the testing from the continuous wavelet transforms (Morlet and Mexican hat family) and wavelet packet transforms
features for same speed training and testing
Training speed
(Hz)
Testing speed
(Hz)
SVM formulation
Testing accuracy (%) for Morlet wavelet family Mexican hat wavelet
family
Wavelet packet transform
CV* GA ABCA CV* GA ABCA CV* GA ABCA
For same speed
10 10 C-SVC 95.00 95.83 96.67 90.83 91.67 92.50 90.83 97.50 98.33
ν -SVC 95.00 96.67 95.83 94.17 93.33 93.33 94.17 95.83 100.00 15 15 C-SVC 99.17 99.17 99.17 94.17 93.33 93.33 94.17 88.33 90.00
ν -SVC 92.50 99.17 99.17 95.83 96.67 95.83 95.83 89.17 90.83 20 20 C-SVC 100.00 100.00 99.17 95.83 95.83 95.00 95.83 97.50 98.33
ν -SVC 100.00 100.00 100.00 97.50 97.50 95.83 97.50 98.33 99.17 25 25 C-SVC 99.17 99.17 99.17 95.00 95.00 95.00 95.00 100.00 100.00
ν -SVC 100.00 100.00 100.00 95.83 95.00 95.83 95.83 100.00 100.00 30 30 C-SVC 99.17 99.17 98.33 91.67 90.83 90.83 91.67 100.00 100.00
ν -SVC 99.17 98.33 98.33 95.00 95.83 95.83 95.00 100.00 100.00
*CV means the cross validation accuracy of the GSM. Bold values represent prediction accuracies of the maximum accuracy.
6.3.2 Training at two Different Rotational Speeds and Testing at an Intermediate