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In order to simulate a failure prediction system, the prediction routine must first estimate the parameters of the "unstable system". Then the prediction procedure can be iterated until the desired lead is obtained. However since the simulation of this system is not done in real time, the procedure uses an if statement to determine when to switch from estimation to prediction.
For a real-time system, the fault prediction procedure is:
1. Run the estimation procedure for at least the number of samples determined by the process time-span (N*(p/skip) in this case).
2. Run the prediction faster than real time, until the desired lead is obtained. Meanwhile, the estimation procedure is also running on new real-time data.
3. Test the predicted parameter value plus the confidence interval against a minimum value threshold. Test with minus the confidence interval against a maximum value threshold if appropriate.
4. Return to step 2 cycling the prediction procedure.
In order to obtain comparisons between predicted and actual parameter values, the simulation doesn't follow the same steps as the real-time system. Instead, the simulation:
1. Runs the estimation procedure from where enough data is available to 1/2 the number of data points available.
2. Runs the prediction procedure for the rest of the data while calculating the predicted value and the confidence interval.
When the routine switches from estimation to prediction, Y(i-1:skip:p) data is switched to the Ym(i-1:skip:p) data in the X input vector. With more code, a more gradual switch between the two sets of data is possible. Implementation of a gradual switch should further increase the accuracy of the prediction.
Appendix 1 shows a listing of the prediction procedure. Figure 12 shows the parameter estimated by the LSM routine, the predicted parameter value predicted by the prediction routine, and the confidence intervals, for an AR(200/20) process. Figure 13 shows the same results for an AR(200/40) process.
Figure 14 shows the use of the prediction routine on a sine wave to demonstrate that the routine works for more than just linear drift of a parameter. The process was an AR(200/20). Figure 15 shows the same with an AR(200/40).
Figure 12, Prediction of p1 with AR(200/20)
Figure 13, Prediction of p1 with AR(200/40)
Figure 14, Prediction of sine wave with AR(200/20)
Figure 15, Prediction of sine wave with AR(200/40)
Figure 16 shows the parameter estimated by the LSM routine, the predicted parameter value predicted by the prediction routine, and the confidence intervals, for an ARMA(1,1) process with the skip factor (lag) set to 20. Figure 17 shows the same results for a skip factor set to 40.
Figure 18 shows the use of the prediction routine on a sine wave with an ARMA(1,1) and skip=20. Figure 19 shows the same with an AR(200/40).
Figure 16, Prediction of p1 with ARMA(1,1)/20
Figure 17, Prediction of p1 with ARMA(1,1)/40
Figure 18, Prediction of Sine Wave With ARMA(1,1)/20
Figure 19, Prediction of Sine Wave With ARMA(1,1)/40
Chapter 5 Discussion
Abstract - Fault Prediction With Regression Models