6.3. LINEAR PROGRESSION ANALYSIS
The following three sections describe three methods to "reverse engineer" the component sum generated by the indicator model into its four base com-ponent. All three methods where tested on the same ten indicator series. The series have the same progression parameters for the model (Tab. 6.4),
but with di.erent seeds for the random number generator. Each method is illustrated with one of the progression from the test set. The component sum for
this
progression
is
given
in
(Fig.
6.11).
Symbol Value Symbol Value
dcw 0.5 dcc 5
Kw run(1, 4) Kc run(1, 4)
a(k) w ru(0, 1) a(k) c ru(.5, 5)
q(k) w ru(0.1, 0.2) q(k) c ru(0.2, 0.5)
p(k) w ru(100, 200) p(k) c ru(100, 200)
gs(t) 2gw(t) ab ru(1, 4)
pb 50
Table 6.4: Test set model parameters.
6.3 Linear Progression Analysis
Before proceeding with fault detection, it is necessary to identify variations in the b, c, w and s components of which an indicator time series consist. A very
interesting
method
developed
by
Sylvie
Charbonnier
et
al.
[5]
permits
dividing a time series into linear segments without the use of non-linear optimization. This section uses a variant of this algorithm as an alternative to the sigmoid based method developed in the previous section.
6.3.1 Segmentation
The segmentation algorithm starts by .nding a .rst order polynomial ap-proximation d.0 of the .rst Linit samples of the dataset i. A cumulative error metric
(Eq.
6.10)
is
used
for
validating
d.0 as an approximation of i. The dataset is then tested sample by sample starting at the beginning of the dataset. When e breaches the threshold Th1, a marker is set in the dataset. Once e breaches a second threshold Th2, d.0 is rejected. A new .rst order polynomial approximation, d1, is then estimated using the data from the marker up to the point where d0 was rejected. This process is repeated until the entire dataset is segmented.
e(n)=
n
d0(k) . i(k)
(6.10)
k=0
6.3.2 Segment Concatenation
The above segmentation process approximates a dataset i as a set d.k of linear segments which need not be continuous. In order to reduce the number of discontinuities in the approximation, the value at the beginning of each segment d.k is compared to the value at the end of it preceding segment d.k.1. If this di.erence supersedes the threshold Thc, the segments are left discontinuous. Inversely, if the gap between the segments is less than Thc, the angle of d.k is changed so that its starting point matches the ending point of d.k.1. This process removes minor discontinuous in the approximation. The component d.is then constructed by concatenating all the d.k segments.
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