Fig. 1. Multi-static location model
Fig. 2. Target location model implementation flow chart of multi-static sky-wave over-the-horizon radar based on LACMODA-ELM
Fig. 3. Influence of the number of hidden layer nodes on the locating error of ELM
Fig. 4. Prediction error of neural network. (a) Prediction error of three neural networks; (b) prediction error from ELM neural network optimized by different algorithms
Fig. 5. Longitude and latitude predicted by ELM optimized by different algorithms. (a) Longitude prediction; (b) latitude prediction
Fig. 6. Location predicted by ELM optimized by different algorithms
No. | Function name | Dimension | Search scope | Optimal value | Peak |
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f1 | Step | 10 | [-100,100] | 0 | Unimodal | f2 | Quartic | 10 | [-1.28,1.28] | 0 | Unimodal | f3 | Rastrigin | 10 | [-5.12,5.12] | 0 | Multimodal | f4 | Ackley | 10 | [-32,32] | 0 | Multimodal |
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Table 1. Benchmark function
Function | Algorithm | Optimal value | Worst value | Average value | Standard deviation |
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f1 | PSO | 1.0115 | 1.2092×104 | 4.1767×103 | 4.5314×103 | DA | 0.0309 | 1.4434×104 | 347.4895 | 1.6229×103 | LACMODA | 2.0781×10-26 | 0.1107 | 0.0064 | 0.0237 | f2 | PSO | 0.0084 | 1.1167 | 0.0229 | 0.0856 | DA | 0.0380 | 8.1144 | 0.2746 | 1.2505 | LACMODA | 1.1679×10-4 | 0.0182 | 0.0015 | 0.0022 | f3 | PSO | 19.8995 | 107.2528 | 23.6501 | 13.8974 | DA | 7.2221 | 91.2835 | 14.3237 | 18.0229 | LACMODA | 0 | 3.1857 | 0.0599 | 0.2826 | f4 | PSO | 4.9902 | 19.4590 | 7.4746 | 4.6434 | DA | 2.5799 | 19.7779 | 3.9696 | 3.7016 | LACMODA | 0.0027 | 1.2011 | 0.0663 | 0.1356 |
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Table 2. Simulation results of improved DA
NeuralNetwork | LongitudeAE /(°) | LatitudeAE /(°) | MAPE /(°) | Distanceerror /km |
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BP | 0.788 | 0.424 | 0.224 | 99.91 | RBF | 1.469 | 0.364 | 0.271 | 167.77 | ELM | 0.176 | 0.107 | 0.040 | 22.53 |
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Table 3. Error comparison based on different neural network prediction methods
Optimization algorithm | Longitude AE /(°) | Latitude AE /(°) | MAPE /(°) | Distance error /km |
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PSO-ELM | 0.134 | 0.031 | 0.023 | 15.20 | DA-ELM | 0.125 | 0.021 | 0.018 | 13.90 | LACMODA-ELM | 0.018 | 0.017 | 0.008 | 2.71 |
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Table 4. Error comparison of ELM prediction methods based on different optimization algorithms
Positioning method | Distance error /km |
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Ray tracing technology | 17.2 | Hybrid sky-surfacewave propagation mode | 9 | Coordinate registration andmultipath data correction | 2.75 | LACMODA-ELMpositioning model | 2.71 |
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Table 5. Comparison of errors in different positioning methods