Biomedical Engineering Reference
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need to evaluate the odds 10 and compute the evidence in 15 by marginal-
izing the likelihood function 14 over the prior distribution p(a; jM k ; t 0 ) of
parameters (a; ). If the odds is large compare to one, it strongly suggests
that there is a peak in the window; otherwise, it is more likely that there is
no peak. Once the odds concludes that a peak is present, parameters may
be t by maximizing the likelihood function 14, via solving equation 17.
Equation 23 provides an alternative way of computing the evidence if the
likelihood is strongly peaked in parameter space.
3. Example
Here, we give an example of nding peaks in a static TOF-SIMS spectrum.
In static TOF-SIMS, many primary ion pulses are used to probe the sample
surface and for each pulse only a few secondary ions are generated and
detected. The detector usually has sub-nanosecond time resolution and thus
counts each secondary ion impact, which implies a Poisson process. The nal
spectrum is a summation of detected secondary ions from all primary ion
pulses. The peak lineshape x is derived using equation 4. The window width
N is chosen such that the window covers the region from the left half-max
to the right half-max of the peak lineshape x. This is a (rough-and-ready)
compromise between the desire to include as much data as possible in the
window to improve the sampling statistics and the realization that nearby
peaks may overlap and that our peak shape model is probably not very
good out on the tails of the peak. An example peak is shown in Fig. 1
(a), overlapped with appropriately scaled peak lineshape (black dots). The
natural log of the odds is plotted versus the window position t 0 in Fig. 1(b).
It behaves as expected: when the window encounters the peak, it goes up,
reaches its maximum when the window is right on top of the peak, and then
decreases as the window leaves the peak. This allows a threshold to be set to
identify a region where we are condent about peak presence. In Fig. 1(c),
log of the maximized likelihood p(sja ; ; M 1 ; t 0 ) for each window position
is plotted. The interpretation is the following: when the window is located
in a region with only noise, the likelihood remains high. This means it is
highly likely that only dark current is observed. As the window encounters
the peak, the data in the window begin to climb, the likelihood begins to
drop implying data in the window look like neither noise nor a centered
peak. As the window eventually overlaps the peak, the likelihood peaks
up and forms a spike. When the window leaves the peak, the likelihood
decreases again and then recovers as the window totally leaves the peak.
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