Figure 6.31: Decomposition of synthetically generated dataset.

6.6.2 Sigmoid Progression Analysis
The sigmoid feature extractor requires 3 con.guration parameters to be set; Lr, Ts, Tq. The former parameter in N , while the two latter are in R.
To reduce the number of dimensions in the search space, the parameters controlling the splitting of r.and d.are estimated separately. This generates two optimization problems; {Lr,Ts} and {Tq}. The .rst problem is solved by minimizing square sum error for s. The second optimization problem is solved by minimizing the number of miss-placed edges.
6.6. CALIBRATION

Figure 6.32: Decomposition of synthetically generated dataset.

6.6.3 Non-Parametric Progression Analysis
The non-parametric feature extractor requires 10 con.guration parameters to be set; Ls.mm, Ls.wrms, Lb, Lw, Jb, Jw, Ts, Tp, Tm and Tw. The six former parameters are in N , while the four latter are in R.
To reduce the number of dimensions in the search space, the parameters controlling the estimation of s, b and w are estimated separately. This gener-ates three optimization problems; {Ls.mm,Ls.wrms,Ts}, {Lb,Jb,Tp,Tm} and {Lw,Jw,Tw}. The .rst and last problem, i.e. the estimation of the parame-ters controlling s and w, are solved by minimizing square sum error for s and w respectively. The second optimization problem, i.e. estimating the param-eters for identifying edges, is solved by minimizing the number of miss-placed edges.
Figure
6.33
shows
the
synthetic
components
as
well
as
the
estimated
ones
using the con.guration parameters obtained by the calibration procedure.

6.7 Conclusion
Using synthetic datasets generated by the progression model, the sigmoid analysis methods outperforms easily the non-parametric analysis. This can however to a large extent be contributed to the fact that the parametric analysis method and the progression model uses the same progression primi-tive; the sigmoid. As there is no physical evidence that indicator progressions assume sigmoid shapes, other than that observable indicator trends look "sig-moid like", this might give the parametric method undeserved good results when looking only at approximation error. Two other points of interest is computing time and robustness. As the parametric method uses non-linear optimization to estimate its progression model, it is signi.cantly slower than the line-based and non-parametric once and is not guaranteed to .nd the optimal solution. Although Trust Region with di.erentiator pre-processing signi.cantly obtains both convergence speed and robustness, using non-linear optimization still poses a certain risk.
The linear method has the advantage of not requiring non-linear optimiza-tion. Looking exclusively estimation error, this method will in some cases produces large deviations between algorithm output d.and the pre-de.ned target d. This can to some extent be contributed to the fact that this is a real-time algorithm. As the method must detect a signi.cant change in indi-cator slope before it can adjust its output, it sometimes "overshoots" sudden changes by a few samples. Another reason for large estimation error on sim-ulated data, compared to the sigmoid model, is that the sigmoid model and the simulated data uses the same primitive, giving undeserved good results. As far as the non-parametric method is is concerned, it will for the task of separating d and r largely emulate a lowpass .lter, for whose performance is di.cult to challenge.
　
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