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r.(t)= i(t) . d.(t) (6.17)
Once r.is obtained, its gain g.r is estimated (Eq. 6.18)
using
a
sliding
window
rms
(Sec.
A.2)
of
length
Lr.
g.r(t)= wrms(.r, t, Lr) (6.18)
The components w.and s.are then separated by comparing each value in r.to its gain estimate g.r(t). Any points being larger than Ts standard deviations of r.are considered to be part of s.:
|r.(t)|
s.(t)= r.(t).(>Ts) (6.19)
g.r(t)
w.(t)= r.(t) . s.(t) (6.20)
Once w.is obtained, its gain g.w is estimated using a sliding window rms (Eq. 6.21). A parametric model g.
w of the noise gain estimate is then obtained by adjusting the sigmoid parameters (Eq. 6.22)
so that the sum square di.erence between g.w and g.
w is minimized.
g.w(t)= wrms(.w, t, Lw) (6.21)
g.
w(t)= dcw + aw (6.22)
.qw(t.pw)
1+ e
6.4.1 Sigmoid Series
This method can only model an indicator time series consisting of a single transition, i.e. a single trend or a single step change, and a single change in noise gain. A useful feature extraction algorithm must be su.ciently robust to be able to analyze an indicator time series consisting of several transitions. A solution is to model each transition in the indicator series with a separate sigmoid. This can be achieved by using a model with an arbitrary number of sigmoids (Eq. 6.23)
and (Eq. 6.24).

(k)
a
d.(t)= dcd + d (k)(k) (6.23)
.q(t.p)
k∈Kd dd
1+ e
(k)aw
g.
w = dcw + (6.24)
(k)(k)
.qw (t.pw )
1+ e
k∈Kw
The choice of model order Kd and Kw is discussed in the following.

6.4. SIGMOID PROGRESSION ANALYSIS
6.4.2 Estimation Methods
The sigmoid sum model proposed above is underdetermined and non-linear, and cannot be estimated by matrix inversion like polynomial models. The problem of .nding the set of model parameters which causes the model d.to obtain a the best possible approximate of d is a non-linear optimization problem. A solution to this problem is .nding the set of model parameters which generates the most balanced power spectrum for r.. Provided that the model order is .xed, this can be simpli.ed to the problem of .nding the set of model parameters which minimizes the r.sum of squares [48].
However, if model order is itself a model parameter, minimizing square sum r.will estimate a model d.generating an r.equal to zero. I.e. d.models both d and
r.
For the noise gain parametrization, it is su.cient to minimize the sum square di.erence between g.w and g.
w. If model order is itself a parameter, it is however not possible to validate the model by evaluating the residual power spectrum. This because the sum square di.erence between g.w and g.
w is expected to assume a power spectrum of mainly low frequency, even for a correct model. Consequently, a traditional model validation technique like r2 or adjusted r2 must be used.
　
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