6.4. SIGMOID PROGRESSION ANALYSIS
and .rst of the sigmoids are frozen while the model is re-estimated using an additional two sigmoids. This process, estimating each sigmoid two times, continues until the length of the estimation window reaches the length of i.
This method was tested with the same con.guration as the previous evo-lutionary method, but using 50 individual for 50 generations at each iteration. Final
results
are
shown
in
.gure
6.20.
This method should be seen as a failure. It is exceptionally slow, and produces erratic results. A main reason for its lack of precision is that it does not solve the global optimization problem. This is visible in the output, like in the example dataset, where output behavior is abruptly changing between segments.
Trust Region using Band-Limited Di.erentiator Pre-Processing
The success or failure of a gradient search optimization is to a large extent given by the initial guess, i.e. the initial current position in parameter space. Consequently, if a more accurate initial guess can be made, robustness and convergence speed will increase.
If the position and amplitude of each signi.cant transition can be approx-imated in a linear manner, then only transition slope need to be estimated through non-linear optimization. A common approach to trend detection is using
band-limited
di.erentiators
[9].
Due to the high levels of noise, the derivation .lter is applied to a de-noised version of i,
using
noise
removal
methods
developed
in
[12]
and
further
adapted to health indicators. This de-noising method has the advantage, compared to simply altering the bandwith of the derivation .lter, of pre-serving edges while retaining a good damping of white noise. By detecting zero-crossing of the .rst derivative, local optimums along the time-series are discovered. Transition points can then be set in between each optimum. The corresponding transition amplitudes are given by the amplitude di.erence in the de-noised i between the leading and trailing optimum to each transition point. Initial approximate for dc is the mean value from the start of the series
to
the
.rst
zero-crossing.
Final
results
are
shown
in
.gure
6.21.
This method produces repeated good results, and has the lowest execution time of all the models. Approximation is very close on the example dataset, as well as all other datasets on which it has been tested.
Comparison
Although execution speed and model error for the di.erent methods can easily be compared, which algorithm best model the features of operational importance remains a subjective opinion. Execution time is of course of in-terest, but as the analysis of HUMS data is done o.-line, this point is not crucial. The main point of interest is in each algorithm’s robustness in mod-eling the operationally important features correctly for any realistic input series. For such a study, the test-sets used here are not su.cient. Con-sequently, identifying the optimal method becomes a somewhat subjective choice.
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