Scheme 4
The Fit Function
Here we describe a fitting scheme based on
scheme 3 where we still assume an event to be a mix of Cerenkov light
(prompt and forming a cone) and scintillation light (isotropically distributed
and produced with a time delay). However the delays for the scintillation light
are now assumed to follow an exponential distribution. We now also include a
random time jitter in the measured hits. The likelihood function is written in
terms of a variable x, which describes the time delay between the
event occurring and a photon emission, expressed as a distance.
x = v*(T-T0) - |r-r0|
where T is the time of arrival of the photon at the surface of the
sphere, T0 is the time the event occurred,
r is the position vector for the point where the photon hit the
sphere, and r0 is the position vector for the event
vertex. Note that x = d/v, where d is the delay variable used
in scheme 3
The likelihood function is the weighted sum of
a gaussian term, fc, for the prompt light and a term,
fs which describes the scintillation light as the
convolution of a gaussian with an exponential.
In principle there are five parameters to be determined: (
Cx, Cy, and Cz )
= r0, T0 and the relative fraction
of scintillator and cerenkov light. We note, however, that
T0 can be readily computed from the other variables by
constraining the mean of the calculated delays to be equal to the mean of the
liklihood function. Clearly <fc> = 0 and it is easy to show that <fs> is
just the mean of the expontial term.
Function Properties
For now, we assume that the relative fraction of scintillator and cerenkov
light is known.
The links below show contour plots of the function, generated over the
entire sphere for a fairly typical event. Each plot shows the function
variation with respect to a pair of parameters with the remaining parameters
held at their generated values various parameter pairs. The magenta circles
marks the location of the generated parameters and the magenta +'s mark the
minimum determined by a Minuit fit. (The contours are spaced logarithmically.)
- Negative log-likelihood using all hits.
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
- Cx vs To, Cy vs To, and Cz vs To
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
(T0 = <fs>)
- Cx, Cy, and Cz vs theta
(T0 = <fs>)
- Negative log-likelihood using only Cerenkov hits.
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
- Cx vs To, Cy vs To, and Cz vs To
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
(T0 = <fs>)
- Cx, Cy, and Cz vs theta
(T0 = <fs>)
- Negative log-likelihood using only scintillator hits.
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
- Cx vs To, Cy vs To, and Cz vs To
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
(T0 = <fs>)
- Cx, Cy, and Cz vs theta
(T0 = <fs>)
Note that the fit shown above produced a biased results for
Cx with a correlated bias in T0 and
the fraction of scintillator light.
This bias is examined more closely in the following plots which examine the
function in the region of the generated parameters. The first three plots show
the change in the function value ( tot = all hits, cone = cerenkov hits only,
and scint = scintillator hits only ) for parameter values forming a
4-dimensional lattice with spacing = 3 and centered on the generated values.
The points are plotted versus the change in Cx (-12, -9, .. +9,
+12) added to 0.1 times the change in T0. Points on
the dashed lines correspond to the generated value of T0
and the red points are where Cy and Cz
are at their generated values. The blue asterisks mark the generated and fitted
values. The fourth plot shows the function contours with Cy
and Cz at their nominal values. (Linear contour spacing.)
Note that the bias is significant .. a change of ~60 units of likelihood.
As a visual check we note that the delay distribution produced using the
fitted parameters produces a good match to the likelihood function as does that
produced using the generated parameters. This is seen in the following plot
which shows the delay distributions from generated data (black
histogram), fitted data (red histogram), and the likelihood function (blue curve).
We also checked that fitting an event with only scintillator hits, using a
likelihood that contains only the scintillator term, does in fact return the
correct vertex position.
We hypothesize that the origin of this bias lies in the fact that the
Cerenkov hits are distributed in a cone and thereby produce a strong
correlation between Cx and To. To test
this hypothesis we tried fitting with half the Cerenkov hits reflected about
the origin of the sphere. (It was easier to do this than scatter them
randomly.) As can be seen from the plots below, this greatly reduces the bias
as expected.
Sensitivity to Charge Weighting
One can regard the hits used in this study as PMT signals from PMT's with
infinitly small spatial dimensions so that each PMT observes a single photon.
In reality the PMTs will detect multiple photons and the assumption is that the
observed charge will be a good measure of the number of photons hitting the
PMT. One would then naturally tend to weight each PMT by its observed charge.
One may then ask how accurately one needs to get this weighting correct? To
test this we randomly weighted each hit in the above event such that the
average weight was 1.0 and the RMS of the weights was 0.25. The results of the
fit were unchanged and the changes in the function are barely discernible as
can be seen in the plots below.
- Negative log-likelihood using all hits.
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
3 parameter fit; all hits with equal weight.
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
3 parameter fit; hits with random weights
- Negative log-likelihood using only Cerenkov hits.
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
3 parameter fit; all hits with equal weight.
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
3 parameter fit; hits with random weights
- Negative log-likelihood using only scintillator hits.
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
3 parameter fit; all hits with equal weight.
- Cx vs Cy, Cx vs Cz, and Cy vs Cz
3 parameter fit; hits with random weights.
This result is not surprising since the function depends only on the relative
timing of the hits and no assumptions are made about their angular
distribution. As a corollary, one may question whether using weights is
necessary or even wise.
Studies with Many Events
The following plots show the deviations of the fitted parameters from the
generated parameters for 1000 events. Each event was generated with 500
Cerenkov hits and 500 scintillator hits. The RMS time jitter was chosen to be
10 inches/c and the average delay of the scintillator hits was 50 inches/c.
We note that the vertex is systematically pulled by 10 inches/c along the
direction of the cone axis. The size of the pull does not seem to depend on the
location of the cone. This pull gets slightly worse when the scintillation
fraction is allowed to vary. The fitted fraction averaged 0.42 with an RMS of
0.03 compared with a generated value of 0.5.
Varying Time Jitter
Changing the size of the time jitter produces a corresponding change in the
offset.
Varying Scintillation Time Constant
Similarly we can examine the dependence of the pull on the slope of the
exponential used to generate the scintillator hits. We find that this parameter
tends to affect the spread of the pull more than its mean.
Varying Scintillation Fraction
There is also a slight tendency for the effect to become worse as the fraction
of scintillation light decreases.
Bias Reduction
The bias seen above can obviously be minimized by using very fast scintillator.
However, in reality this may not be an option and even if it were it may not be
desirable since their are presumably pattern recognition gains in being able to
distinguish scintillator hits from cerenkov hits. Another option is to
insert knowledge of the distribution of the light into the fit function. This
has the disadvantage that the distribution will depend on the event type,
resulting in the need for a different likelihood function for each event type.
Distinguishing Forward and Backward Hemispheres
A compromise solution is to assume only that the Cerenkov light is
concentrated in the "forward" hemisphere (i.e. the hemisphere centered on the
cone axis). For a given vertex hypothesis the vector defining the forward can
be approximated by summing the unit vectors pointing to each hit. The resulting
normalized vector can then be used to decide which hits are in the forward
direction and which hits are in the backward direction. For those hits in the
forward direction one can use the same likelihood function described above but
with an appropriate adjustment made to the parameter describing the relative
weighting of fs and fc. Hits in the
backward direction are assumed to be all scintillator hits and hence the
likelihood is given by the fs term alone. The results of
this modified likelihood are shown below:-
We observe that the bias is not eliminated but is greatly reduced (factor of 3)
using this method. We also note odd behviour in the fit function at the point
representing the centroid of the hits. This is due to the fact at this point
there is no well defined cone direction and hence the separation of cerenkov
hits and scintillator hits is no longer reliable.
Fixing To
Since it may be possible to fix the time of an event using the time structure
of the beam we consider what happens to the bias if T0 is
known exactly. Fixing T0 in the fits to its generated value
results in the following distribution of biases:-
Actual and Minuit errors:
<fs> = 50. and RMS jitter = 10.
