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Fig. 7. Proposed methodology for accurate long-term prediction.

Figure 7 Proposed methodology for accurate long-term prediction.

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Fig. 1. Generalized real-time recurrent network architecture. used to train recurrent networks. The algorithm that uses this approximation is called the RTRL [25] and is used for training recurrent networks that run continuously in real time [26]. Since real-time operation is required in machine condition monitoring applications, real-time recurrent networks are a good choice for serving the purpose. An example of the network considered with the RTRL training algorithm is shown in Fig. 1, where it contains NV external input connections (e.g., N = 2 in Fig. 1) and M outputs (e.g., !/ = 4 in Fig. 1).
Fig. 1. Generalized real-time recurrent network architecture. used to train recurrent networks. The algorithm that uses this approximation is called the RTRL [25] and is used for training recurrent networks that run continuously in real time [26]. Since real-time operation is required in machine condition monitoring applications, real-time recurrent networks are a good choice for serving the purpose. An example of the network considered with the RTRL training algorithm is shown in Fig. 1, where it contains NV external input connections (e.g., N = 2 in Fig. 1) and M outputs (e.g., !/ = 4 in Fig. 1).
PERFORMANCE COMPARISON FOR SHORT-TERM PREDICTION shown that the weight change based on (6) and subsequent analysis using the RTRL algorithm can be represented as [26]
PERFORMANCE COMPARISON FOR SHORT-TERM PREDICTION shown that the weight change based on (6) and subsequent analysis using the RTRL algorithm can be represented as [26]
Fig. 3. Long-term prediction errors for simulated data using teacher-forcer modification. Fig. 2. Long-term prediction using a trained RNN structure. recurrent networks for this prediction paradigm. The differ- ence in long-term prediction as opposed to one-step prediction (short-term) is that once the network is trained by the RTRL algorithm, the outputs for the trained network are fed back to the input neurons recursively. This means that the predicted output x,(n) in Fig. 2 is recursively fed back to Z~! and used in the subsequent time-step for prediction. This method was applied to the simulated data, and the total prediction error (Eprediction,) increased as the number of time-steps in- creased. For a representative example at the 250th time-step, the maximum error exceeded 0.35 for the 32-step prediction. As an alternative, the teacher-forcer training algorithm was investigated wherein the desired output, instead of the actual output obtained, was fed back through the recurrent connection while training. At each presentation of an input, the weights were updated, with the maximum training error below 0.35. To compare the performance of this technique with conventional incremental training, result from the 32-step prediction was investigated at various time-steps. For a representative example at the 250th time-step, the maximum error was less than 0.25 (Fig. 3). This represents an improvement over the conventional incremental training.
Fig. 3. Long-term prediction errors for simulated data using teacher-forcer modification. Fig. 2. Long-term prediction using a trained RNN structure. recurrent networks for this prediction paradigm. The differ- ence in long-term prediction as opposed to one-step prediction (short-term) is that once the network is trained by the RTRL algorithm, the outputs for the trained network are fed back to the input neurons recursively. This means that the predicted output x,(n) in Fig. 2 is recursively fed back to Z~! and used in the subsequent time-step for prediction. This method was applied to the simulated data, and the total prediction error (Eprediction,) increased as the number of time-steps in- creased. For a representative example at the 250th time-step, the maximum error exceeded 0.35 for the 32-step prediction. As an alternative, the teacher-forcer training algorithm was investigated wherein the desired output, instead of the actual output obtained, was fed back through the recurrent connection while training. At each presentation of an input, the weights were updated, with the maximum training error below 0.35. To compare the performance of this technique with conventional incremental training, result from the 32-step prediction was investigated at various time-steps. For a representative example at the 250th time-step, the maximum error was less than 0.25 (Fig. 3). This represents an improvement over the conventional incremental training.
Fig. 4. Predicted values for simulated data using windowed inputs.
Fig. 4. Predicted values for simulated data using windowed inputs.
Fig. 5. Test bearing with an initial 0.27-mm-wide scratch on the outer ring. Fig. 6. Variation of peak value for wavelet prefiltered data.
Fig. 5. Test bearing with an initial 0.27-mm-wide scratch on the outer ring. Fig. 6. Variation of peak value for wavelet prefiltered data.
Fig. 8. Network performance with incremental training for one-step prediction.
Fig. 8. Network performance with incremental training for one-step prediction.
EFFECT OF MOMENTUM AND LEARNING RATE ON MSE based on the data from the previous time-steps, with which it was trained.
EFFECT OF MOMENTUM AND LEARNING RATE ON MSE based on the data from the previous time-steps, with which it was trained.
Fig.9. Weight variation with time-steps in incremental training.
Fig.9. Weight variation with time-steps in incremental training.
Fig. 10. RNN long-term prediction using weights trained by incremental training.
Fig. 10. RNN long-term prediction using weights trained by incremental training.
Fig. 11. Typical result of competitive training for n = 10 clusters at four different time-steps. (The cluster centers are shown in solid black markers Ml.
Fig. 11. Typical result of competitive training for n = 10 clusters at four different time-steps. (The cluster centers are shown in solid black markers Ml.
Fig. 12. RNN long-term prediction using weights trained by the proposed methodology. Fig. 13. Comparison of MSE in incremental and the proposed methods at different time-steps.
Fig. 12. RNN long-term prediction using weights trained by the proposed methodology. Fig. 13. Comparison of MSE in incremental and the proposed methods at different time-steps.
Related topics:
Instrumentation and Measurement ScienceRecurrent Neural NetworkWavelet TransformsRecurrent Neural NetworksCompetitive LearningElectrical and Electronic EngineeringContinuous wavelet transform
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