Feature request: LMS-based calibration routine for ECAL
Preface
A calibration procedure for ECAL based on pseudo-inverse LMS has been discussed many times. The idea is based on the fact that (neglecting preshower and main part segmentation) for some E_{full}
the whole energy deposition must be distributed in some 3 \times 3
stencil. Defining energy deposition in the cell i, j
as
\begin{equation}
E_{i,j} = k_{i,j} \cdot A_{i,j}
\end{equation}
and having some amount N_{evt}
of events hitting the cell i, j
one can write 9 equations for every event N
:
\begin{equation}
E_{\Sigma, N} = \sum\limits_{n=-1, m=-1}^{i\lt1, j\lt1} k_{i+n, j+m} \cdot A_{i+n, j+m}
\end{equation}
In general, this should be valid for any single of N_{evt}
events (so in principle one can just take 9 events to resolve it for 9 cells, a 9 \times 9
SLE), however due to fluctuations this constrain is not exactly true (yet, this assumption might be useful for fitted peaks data). On the other hand, we are not limited in the events and can build a redundant SLE and resolve it using least mean square method via pseudo-inverse matrix (LMS), as N_{evt} \times 9
SLE for every cell. Additionally, a strip-diagonal (banded) matrix can be built considering all the calibration runs as a whole (there are special numerically-stable methods developed for this family of SLE, see e.g. [1]). The main drawback of (2) is that it will be very mach affected by instabilities of periphery cells as subtle energy deposition will suffer from larger relative jitter than middle ones. This effect can be mitigated by using weighted version of LMS.
Also, some constrains can be applied to additionally stabilize the fluctuations caused by periphery cells (see "constraints" paragraph below).
Note, that this method of deriving calibration coefficient is reconstruction-agnostic in the sense that we do not make any assumptions on A_{i,j}
except for linear proportion.
Feature proposal
Study a possible numerical procedure in few steps:
- Build a model test source, generating random errors for
A_{i,j}^N
for certain "central cell". By random smearing we imply:- Gaussian errors in absolute terms caused by light gathering/leakage fluctuations:
\Delta A_{i,j}
- Fluctuations of the energy deposition fractions caused by the fact that event is not always centered within a stencil This test source must support fractioning between Prsh and Main parts of ECAL as external parameters
- Gaussian errors in absolute terms caused by light gathering/leakage fluctuations:
- Generated output has to be provided in ASCII form, easy for inspection
- Build a GSL or numpy-based utility to consume this output and resolve it wrt (2) and (1) as a first approach.
Constraints for further consideration
For instance, a stencil with fractions E_{i,j}/E_{full}
may be assumed constant across any 3 \times 3
group with constrains defined by axial symmetry of the shower profile. A simple stencil that does not take into account beam inclination would be that with respect to central cell E_{i,j}
:
-
E_{neighb} = E_{i,j}/E_{full}
:E_{i-1,j} = E_{i+1,j} = E_{i, j-1} = E_{i, j+1}
. E_{diag} = E_{i-1,j-1} = E_{i-1,j+1} = E_{i+1, j-1} = E_{i+1, j+1}
Note: it is possible to develop a stencil which takes into account beam inclination as well. In that case there should be 4 constrains, but weaker.
[1] https://www.sciencedirect.com/science/article/pii/S0377042705001020