Aart - hit time info in PDFs

Aashowerfit uses a likelihood based entirely on the poissonian hit stochastics, i.e. it is computed as the product of the probability of observing a hit and the probability of observing no hit for each PMT. On the other hand, track reconstruction uses only the hit time information. Aart presented a couple of slides which explain how these two methods can be combined.

A key ingredient is the probability density for observing a first hit at time t1 (explained on slide 5). An associated 'first hit time likelihood' can be defined as the product of these probability densities over all PMTs in your detector.

Maarten: This is indeed exactly what is done in the track reconstruction!

If we incorporate a constant background rate and define a critical time, tc, after which all signal hits must have been observed, then it turns out (see slides 7 through 9)  that the corresponding log-likelihood can be written down as the sum of the individual log-likelihoods for the PMTs observing a (signal or background) hit before time tc, for the PMTs not to observe any signal hits after tc, and for the PMTs observing a first hit before tc.

If we associate any registered hit times greater than tc with non-hit PMTs, it turns out that this is nothing other than the sum of the hit/no-hit likelihood and the first hit time likelihood.

There was some discussion surrounding the multiplication of the background rate with the hit times on slide 5 and 6.

Maarten & Ronald: The term Rt in the exponent probably should be R(t - t0), since you have implicitly used a reference start time here.

Additionally, there was a discussion about the physical dimension of the quantities incorporated in the log-likelihood on slide 6.

Ronald: A likelihood is always defined as some product of the probabilities of a model giving rise to certain observations. Hence the argument in your (log-)likelihood needs to be dimensionless.

Maarten & Aart: You can also maximize probability densities.

Some discussion also arose concerning the infinitely long track hypothesis, which is currently assumed in track reconstruction:

Maarten: Taking into account first hit time will be essential for moving away from the infinitely long track hypothesis in the track reconstruction in Jpp.

Brían: I suppose this will not be as important for ARCA as it is for ORCA?

Maarten: True. The muon tracks in ARCA will pass through the entire detector, unless you focus on the 100 GeV muons. For ORCA it will be quintessential to include the hit time informaiton, since the muons will oftentimes end before the edge of the detector.

Some final comments:

Maarten, Ronald: We might be able to compete with the machine learning algorithms, since this includes almost all info.

Alfonso: Can we not use ToTs as well?

Ronald: That would be the penultimate goal!

Brían

Showed a slide regarding the definition of the hit/no-hit likelihood.

Aart: Some of the confusion here might be alleviated after reading the LATEX document I linked in my slides.

Otherwise working on documenting some of the steps involved in the combined (track+shower) likelihood and reviewing them.

Jordan

Ran into some memory issues at Lyon. Setting RSS memory to 7 GB works alleviates problems, but it's not understood why.

Ronald: Try running it on a node, so you can track live what happens.

Maarten: Memory ought not be a problem. Loading all PDFs amounts to 1 GB at most. Also note that sampling a 1D-PDF for each event and each PMT should be extremely fast.

Thijs

Shared some slides regarding discrepancies between the MLE event starting times computed via a toy MC and via JSirene. In JSirene negative starting times yield maximum likelihood, whilst the toy MC prefers a starting time of 0.0 ns.

The discrepancy seems to be caused by the fact that JSirene incorporates the light generated by all tracks individually, each with their own EM-equivalent energy and direction, etc., whilst the toy MC assumes a single shower, with a single direction and energy.  

In particular, if you resolve all tracks individually, there are a couple of low energy mesons, which create cascades below the neutrino vertex in the direction opposite that of the neutrino. Incorporating these effects in the toy MC brings the MLE event starting time to negative values and resolves the discrepancy w.r.t. JSirene.