AIEC FCAL Simulation Q-Learning

Overview

It has been difficult to find correlations between the LED amplitudes for FCAL blocks and existing pi0-based calibrations. We are limited by the amount and type of information in the experimental data. One possible way to gain insight is to simulate the model we assume for the FCAL LED responses and see if a model can be trained to extract the configuration parameters.

Here is the currently assumed model:

$A(i,t)=g_{i,t}^{-1}\omega _{i}\alpha _{t}R_{i,t}$

i	block number
$t$	time (could be be actual time or run number)
$A_{i,t}$	Amplitude of ADC readout for block i at time t
$g_{i,t}$	PMT Gain of block i at time t
$\omega _{i}$	Optical coupling of block i (no time dependence)
$\alpha _{t}$	Amplitude of LED pulser (no block dependence)
$R_{i,t}$	Radiation damage (integrated) to block i at time t. (assume to be "1" for now)

One thing we see in the data is that the LED amplitudes drift over time such that the amplitudes for one time step tend to be close to what they were for the previous time step. It is believed that this is partly (maybe largely) due to the LED pulser drifting. What we are interested in though are the PMT gains which are also drifting. The two signals are convoluted. The above model assumes PMT gain drifts are independent from each other while the LED pulser drift is common to all PMTs.

The simulation will insert a time drift of the LED pulser amplitude and train an ML model with time-dependence (e.g. LSTM) to return the LED pulser amplitude for the step. The hypothesis is that it may use the knowledge of the prior time steps to help estimate the pulser amplitude for the current time step. The ultimate goal is to estimate the PMT gains, but starting with the simpler problem of just estimating the pulser amplitude may give an indication of how much this tactic is worth pursuing.

Setup

1. Simulation should start with small, but configurable sample of blocks. Say, 50.
2. Simulation should sample a set of initial gains from distribution of gains extracted from calibration DB
   • Initially, do not adjust the gains with time
3. Simulation should vary the LED pulser amplitude using a sawtooth pattern in time similar to what was observed in the LED data.
   • Amplitude should vary by around ~5% (see block 707 in ring 9 plot)
   • Time period should be about 50 time steps
   • Simulation should add systematic entropy to sawtooth pattern so it is not perfectly regular. This should *not* be a random value added to the amplitude. Rather, the saw tooth drift should switch directions at a random point in time (or whenever it reaches +/- 5%.
4. Model should take N (e.g. 50) inputs and a single output. The output should be trained to be the LED pulser amplitude.

Ideally, the model would perform well even if one started introducing data with a different set of gains. Granted, it may take some number of time steps to readjust.