6.5. NON-PARAMETRIC PROGRESSION ANALYSIS

6.5 Non-Parametric Progression Analysis
The trend analysis model developed in the previous section was based on the model believed to generate the observable time series. Although choosing such an analysis model permits estimating an accurate approximate of the original data, model estimation is di.cult due to the non-linear nature of the model. It is thus desirable to .nd a trend analysis method not in need of non-linear optimization techniques. This is achievable through the use of
band-limited
di.erentiators
[9].
The
method
developed
in
this
section
is
based on the same principle as band-limited di.erentiators, but is further specialized
to
.t
the
characteristics
of
condition
indicator
time
series
[52].

In the previous section, it was assumed that the observed indicator time series are the sum of a deterministic process and a random noise process. Like in the previous section, i(n) is split in four components; an outlier process s(n), a random noise process w(n), an edge process b(n), and a trend process c(n).
These components correspond exactly to the component used in the syn-thetic indicator progression model. The traversal through states associated with mechanical degradation is as already explained manifested as changes in the value of c and the gain of w. A .rst step in the fault detection process is thus to separate these four components.
6.5.1 Outlier Separation
The dataset i is
de-trended
(Eq.
6.26)
by
having
its
moving
median
(Sec.
A.1)
at
window
size
Ls.mm removed. This .lter is an e.ective form of de-noising, and will remove all of s and some of w, while keeping most of c and
.intact.Themodi.eddataset bi1
w and very little of c and b.
will consequently contain all of s, some of

1
i.
As the modi.ed dataset has very little trend or edge contribution, it will have zero mean. An outlier is de.ned as a point of value Ts standard deviation outside the mean of the dataset (Eq. 6.27).
Windowed rms (Eq. A.2)
at
window
size
Ls.wrms is used as signal scatter might vary along the time line.
(n)= i(n) . mm(i, n, Ls.mm) (6.26)
.|i1
Before proceeding with the separation of w, b and c, the outlier component is
removed
from
the
dataset
(Eq.
6.28).

(n)| wrms[i, n, Ls.wrms
..()= isn1
(n).(

>Ts)
(6.27)

]

i1(n)= i(n) . s.(n) (6.28)

6.5.2 Edge Separation
Wavelet expansions allow a signal x to be represented as a weighted sum of scalings and dilatations of a wavelet function ψ(t). The Continuous Wavelet Transform (CWT) (Sec. A.3.1)
　
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