Biomedical Engineering Reference
In-Depth Information
is independent of (a; ), for example, a and are uniformly distributed in
some region (a
min
; a
max
) and (
min
;
max
), with substitution of 21 into 15,
the integration is readily carried out:
p(sjM
k
; t
0
)
=
R
dadp(sja; ; M
k
; t
0
)p(a; jM
k
; t
0
)
R
R
a
max
max
dade
L(a
;
;t
0
)
e
2
(XX
)
0
rrL(a
;
;t
0
)(XX
)
1
a
max
a
min
1
max
min
=
a
min
min
(2)
m=2
max
min
e
L(a
;
;t
0
)
1
a
max
a
min
1
p
=
jdet[rrL(a
;
;t
0
)]j
;
(23)
where m is the dimension of parameter space. In the last step of integra-
tion, lower and upper boundaries of integration are extended to innity.
This is valid if the likelihood function is sharply peaked around (a
;
)
and (a
min
; a
max
) and (
min
;
max
) are large enough such that contribu-
tions from outside these regions are negligible. Otherwise, the integral will
result in an error function. Notejdet [rrL(a
;
; t
0
)]jis the determinant
of the Hessian matrix evaluated at (a
;
) and 1=
p
jdet [rrL(a
;
; t
0
)]j
is proportional to the `volume' within
a
and
around (a
;
) in param-
eter space, i.e.
a
a
max
a
min
max
min
:
p(sjM
k
; t
0
)p(sja
;
; M
k
; t
0
)
(24)
Notice that solving equation 17 only maximizes the likelihood with re-
spect to (a; ), the maximizing of likelihood with respect to t
0
is done
by computing the likelihood for each window position at (a
;
), i.e.,
p(sja
;
; M
k
; t
0
) and then nd the maximum point of p(sja
;
; M
k
; t
0
)
with respect to t
0
. However, maximizing over t
0
has a dierent logical char-
acter than the other parameters, because we are comparing dierent data
sets as we slide the window across the peak. The justication is based upon
the physical reasonableness of the approach: the width of the window is
large compared to the uncertainty in the position of the peak, hence near
the maximum of the likelihood, most of the data being compared comes
from overlapping windows. An alternative way of looking at this is that by
comparing p(sja
;
; M
k
; t
0
) at dierent t
0
we are actually looking for a
window in which the data best support the model M
k
.
Before we go any further to give an example, let us summarize the pro-
cedure: to detect peaks in a TOF-MS spectrum, we need to put a window of
appropriate width on the spectrum that isolates N data points. For these
N data points, we want to compare model M
1
versus M
0
, in which we
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