r.(t)= i(t) . d.(t) (6.11)
Once r.is obtained, its gain g.r is estimated using a sliding window rms (Eq.
6.12).

g.r(t)= wrms(.r, t, Lr) (6.12)
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.(Eq.
6.13
and
6.14).

|r.(t)|
s.(t)= r.(t).(>Ts) (6.13)
g.r(t)

6.3. LINEAR PROGRESSION ANALYSIS
w.(t)= r.(t) . s.(t) (6.14)
Once w.is obtained, its gain g.w is estimated using a sliding window rms (Eq. 6.15). A parametric model g.
w of the noise gain is obtained using the same parameterization methods that was used to identify d..
g.w(t)= wrms(.w, t, Lw) (6.15)
6.3.3 Trend Analysis
An estimate for d is generated by the parameterization process. The compo-nent .b is identi.ed from the segmentation algorithm. Any two segments left discontinuous constitutes a change in .b, with position and amplitude given by the position and amplitude of the discontinuity. The component c.is iden-ti.ed by subtracting .b from d.. In order to detect changes in the condition of the underlying asset, .uctuations in the value of c and the gain of w must be monitored. This is done by computing the time derivative of c.(Fig. 6.12)
and g.
w (Fig. 6.13);
ac.(Fig. 6.14)
and ag.
w (Fig. 6.15).

6.4. SIGMOID PROGRESSION ANALYSIS

6.4 Sigmoid Progression Analysis
Although the line-based method performs very well, it is desirable to .nd a model which permits modeling of curved shapes. According to the pro-gression model de.ned in the previous section, d is either constant, abruptly changing or gradually changing. When estimating the model d.based on a set of observations of i, it is necessary to .nd a model prototype capable of assuming any behavior exhibited by d. A possible candidate is the sigmoid, as it is the primitive already used for generating gradual transitions in d. Further, it is also, with the correct set of con.guration parameters, capable of producing abrupt transitions and straight lines.
The shape of the sigmoid prototype is adjusted by entry level dcd, transi-tion amplitude ad, transition slope qd, and transition point pd, as illustrated by
equation
(Eq.
6.16)
and
.gure
6.16
[51].

ad
d.(t)= dcd + (6.16)
.qd(t.pd)
1+ e

Subtracting d.from the observed indicator time series i obtains a scatter and outlier estimate r.(Eq.
6.17).
As
r by de.nition is Gaussian noise, an r.with a balanced power spectrum indicates that the model r.and consequently the model d.is correct. By adjusting the sigmoid shape parameters so that r.assumes a white power spectrum, d.assumes a close approximation of d.
　
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