## Regularization methods for unfolding of discrete distributions

Paper: 285
Session: A (poster)
Presenter: Anykeyev, Vladimir, Institute of High Energy Physics, Protvino
Keywords: algorithms, analysis, application programming, neural networks, programming style

Regularization methods for unfolding of
discrete distributions

Vl. B. Anykeyev
anikeev@mx.ihep.su

Institute for High Energy Physics
Protvino, Moscow region, RUSSIA, RU-142284

One of the goals of modern experiments in high energy physics is the
investigation of discrete distributions, p.e. multiplicity distributions.
As the result of finite detector resolution and
ineffectiveness of reconstruction algorithms
the migration of events from one class to another occurs.
Formally this
process is decsribed by the following mathematical model
N^{obs}_n = sum(A_{ni}N^{tr}_i + eps_n), n = 1..N
where N^{tr}_i is the true number of events from class i,
N^{obs}_n is the measured number of events in class n,
A_{ni} is the probabilty of migration of event from class
i into class n, eps_n is the
fluctuation of number of events N^{obs}_n.
The problem is to estimate N^{tr}_i. The solution can be
unstable with respect to noises eps_n.

Nowadays to solve this problem they use the correcting factors method
and the method based on the Bayes theorem.
It was shown [1], that estimates
obtained on the basis of these methods are model dependent,
while the goal is to obtain model and detector independent estimates.
To obtain estimates with such properties in the class of linear estimates it
is natural to use the idea of pseudoinverse operator for the system of
equations (1), which is defined as [2]
A^+ =
(A^*(AA^*)^{-1} N<=I)
((A^*A)^{-1}A^* N>I)
Explanation of the origin of possible instability of the solution
with respect to noises is given. Several regularization methods
were proposed and tested on data from the DELPHI detector.

[1] V.B.Anykeyev, A.A.Spiridonov and V.P.Zhigunov,
Nucl. Instr. and Meth. A322 (1992) 280.

[2] V.V.Vojevodin, Ju.A.Kuznetsov,
Matrices and computations, Moscow, "Nauka", 1984.