Biomedical Engineering Reference
In-Depth Information
p(sja; ; t
0
; M
k
)
= p(fs
1
; s
2
; : : : s
N
gja; ; t
0
; M
k
)
(14)
Q
N
=
p(s
i
ja; ; t
0
; M
k
):
i=1
p(sjM
k
; t
0
) can then be computed by marginalizing the likelihood func-
tion 14 over the model parameters (a; ) using an appropriate prior prob-
ability p(a; jM
k
; t
0
):
Z
p(sjM
k
; t
0
) =
dadp (sja; ; M
k
; t
0
) p(a; jM
k
; t
0
):
(15)
For each model class M
k
, there will be a prior probability distribution
for the parameters. For example, for M
1
, i.e., there is a peak present, we
choose the prior:
p(a; jt
0
; M
1
) = p(ajt
0
; M
1
)p(jt
0
; M
1
);
(16)
where we have assumed that prior for the intensity and that for the noise
are independent. If we know nothing about the values of a and , then
we choose uniform priors, or some other priors that are very broad in the
parameter space on the grounds that when we integrate against 14 only the
neighborhood of the maximum likelihood value of (a
;
) will contribute.
Up to this point, we have set up the concept of doing a model compar-
ison for the data in a window located at t
0
, all necessary terms have been
computed and the odds is ready to be computed. The window will then
slide across the spectrum point by point. When the window is sliding, the
window width will get wider accordingly if necessary because of instrumen-
tal reasons. For each window, the odds is computed. One can easily imagine
that as the window comes across a peak, the odds will increase, and will
decrease when the window passes a peak. One can then set a threshold for
the condence we need to have to declare a peak to be detected.
Once we have detected that a peak lies within a certain region of the
time axis, we then x the position and intensity of the peak using the
maximum likelihood method, i.e., maximizing the likelihood 14 over the
parameters (a; ; t
0
). This requires solving the following equations, for a
window located at t
0
with isolated data s = (s
1
; s
2
s
N
):
@L(a;;t
0
)
@a
= 0
(17)
@L(a;;t
0
)
@
= 0;
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