6.30
shows how the di.erent wavelet scales reacts to fast and slow movement in c.(Fig.
6.27)
and
g.w (Fig.
6.28).
ac (j)(n)= cwt(c, ψ) (6.42)
a(g.jw )(n)= cwt(.gw,ψ) (6.43)
6.6 Calibration
Both the parametric and the non-parametric feature extractor depends on a number of tuning parameters to perform a correct decomposition of the input data. An optimal con.guration of these parameters depends on the type of input, i.e. the energy distribution between s, w, b and c, and is essential for the performance of the algorithms.
The fundamental function of the feature extractor is to map an input vector to a set of output vectors. In the most general of terms, this is the same functionality provided by other non-linear mapping tools, like fuzzy logic classi.ers and arti.cial neural networks. There are two main approaches for calibrating non-linear mapping tools; through expert knowledge, as normally applied to fuzzy logic systems, or through the use of training data.
Traditional training methods using marked training sets are inapplicable in this scenario, as the correct decomposition, i.e. the desired output, for a real time series cannot be known. Using expert knowledge, it is possible to derive the con.guration parameters from a set of speci.cations describ-ing behavior of the di.erent signal components. Any such speci.cation will however mostly be guesswork based on user experience. As an alternative approach, a synthetic dataset is generated using the model developed in the previous section. The correct decomposition of this dataset is known from the de.nition of the dataset, and can be used as a target for automatic training.
The four arti.cially generated components are added together before the source splitter algorithm is applied. The output from the source splitter, an estimate of the four components, is compared to the original components and an error metric is produced. This procedure is used as object function for an evolutionary algorithm, set to .nd the optimal value for the con.guration parameters.
An evolutionary algorithm runs 5 individuals per dimension for 20 gen-erations with an elite ratio of 0.1 and a crossover fraction of 0.8. As the dataset contains stochastic elements, the optimization procedures maximize performance across 10 di.erent datasets generated with di.erent seed for the random number generator.
6.6.1 Linear Progression Analysis
The line based feature extractor requires three parameters; Th1, Th2 and Thc [5].
In
addition
to
this,
it
requires
the
same
parameters
as
the
sigmoid
model
to separate s.and w.. This produces two optimization sets; {Th1,Th2,Thc}and {Lr,Ts}. The .rst problem is solved by .nding the parameters that minimizes the square sum error for s. The second problem is solved by .nding the parameters that minimizes the square sum error for d.
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