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De.ning xt as a set of .nite-length signals, makes x a vector of signal samples and t acquisition start time. With xt acquired from a component in a stationary condition throughout the set, the output from Mc will be constant across t. Consequently, the only factor contributing to non-constant behavior in xt across t is Me. Given xt and the contextual parameters on which Me depends, the function Me can be estimated. This provided that all the necessary contextual parameters are recorded, and that su.cient relevant training data exists.
In order to cancel the e.ect of environmental changes, a reference en-vironment must be de.ned. The purpose of the reference environment, a
r
vector of contextual parameters constituting an environment of reference pe, is to correct each observation so that they appear to have been made in this
environment.
A
correction
function
(Eq.
5.3)
is
de.ned
so
that
it
for

5.3. INDICATOR CORRECTION
an observation provides the ratio between the reference environment and the environment at the time of the observation. Weighting each observation with its correction function G will thus de-correlate Me and the observations.
Me(pr)
G(pe(t)) = e(5.3)
Me(pe(t))
This methodology can be applied directly to a signal, or to each indicator derived from the signal of a given component. The former approach requires G to be a .lter, where the .lter transfer function is given by the set of relevant .ight parameters. Using the latter method makes G a simple scalar function describing the coupling between a single indicator and the set of relevant .ight parameters. A function G must thus be estimated for each indicator of the signal associated with each component on the aircraft.
This chapter makes no assumptions about the underlying physical phe-nomena responsible for the correlation between environmental context and vibrations signature. The methods developed here are purely general, and must be adapted to each component on the aircraft.

5.3 Indicator Correction
Not all indicators are sensitive to environmental changes. Others are sen-sitive, but show a change in scatter rather than localization. A signi.cant group of indicators show a substantial change in location as a function of environmental context. The relationship between indicators from this group and the applicable .ight parameters is normally possible to approximate with a
polynomial
model
(Eq.
5.4).
In
this
case
pe(t) contains not only the pa-rameter, or parameters, of interest, but also the necessary powers for each parameter.
M.e(pe(t)) = pe(t).a.(5.4)
The vector a.contains the weight of each power of each parameter, and is estimated using a set of indicator values i and their associated .ight pa-rameters pe recorded over a period where the condition of the underlying asset is stationary. As the condition is stationary, any .uctuations in the indicator value must be caused by environmental variations. By subtracting the mean value of the indicator μi, the .uctuations are isolated, and the model M.e(pe(t)) is estimated to approximate these .uctuations (Eq. 5.5).
　
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