Biomedical Engineering Reference
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Figure 1(d) is an expansion of Fig. 1(c) such that Fig. 1(a), Fig. 1(b) and
Fig. 1(d) have the same y axis coverage. Notice that both Fig. 1(b) and
Fig. 1(d) are plotted in log scale while Fig. 1(a) is plotted in linear scale,
which means the odds and likelihood are very sharply peaked in the region
where there is a peak and the likelihood is even shaper, which implies
extreme sensitivity to peak position. Since Fig. 1(d) is plotted in log scale,
one can t the center part of it (inside the rectangle) to a parabola and
exponentiate it resulting in a Gaussian-like curve. Then, the center of the
parabola would be the center of the Gaussian and is the best estimation of
peak position. From the curvature of the parabola one can get the width of
the Gaussian that indicates the uncertainty of estimated the peak position.
x10
2
x10
2
6
6
(a)
(b)
4
4
2
2
0
0
8.08
8.085
8.09
8.095
x 10
4
8.08
8.085
8.09
8.095
x 10
4
Time
t
0
x10
3
x10
2
0
(c)
(d)
−1
−2
−2
−4
−3
−6
8.08
8.085
8.09
8.095
x 10
4
8.08
8.085
8.09
8.095
x 10
4
t
0
t
0
Fig. 1. Illustration of peak detection procedure: (a) an example peak overlapped with
peak lineshape (black dots); (b) log of the odds; (c) log of the maximized likelihood for
M
1
; (d) expansion of the log of the maximized likelihood.
Having detected peaks using the above method, it is found that there
are often shifts in arrival times for the same peak in dierent spectra, as
illustrated in Fig. 2(a), which shows the overlapped, but shifted parent peak
of Vasopressin in two dierent spectra, labeled spectrum 1 and spectrum 2.
The multiple peaks seen in Fig. 2(a) are the isotopic pattern. Let p
1
and p
2
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