Probability Distributions of Positioning Errors for Some Forms of Center-of-Gravity Algorithms. Part II

TL;DR

This paper derives probability distributions for complex center-of-gravity algorithms used in particle track fitting, highlighting their tail behaviors and regions of reduced probability to improve outlier handling.

Contribution

It provides new analytical probability density functions for specific center-of-gravity formulas involving complex combinations of independent Gaussian variables.

Findings

Probability distributions exhibit Cauchy-(Agnesi) tails beneficial for outlier attenuation.

Complex structures with reduced probability regions identified to prevent false likelihood maxima.

Analytical expressions derived for distributions of novel center-of-gravity combinations.

Abstract

The center of gravity is one of the most frequently used algorithm for position reconstruction with different analytical forms for the noise optimization. The error distributions of the different forms are essential instruments to improve the track fitting in particle physics. Their Cauchy-(Agnesi) tails have a beneficial effects to attenuate the outliers disturbance in the maximum likelihood search. The probability distributions are calculated for some combinations of random variables, impossible to find in literature, but relevant for track fitting: $x_{g3}=\theta(x_2-x_1)[ (x_1-x_3)/(x_1+x_2+x_3)] + \theta(x_1-x_2)[(x_1+2x_4)/(x_1+x_2+x_4)]$ and $x_{g4}=\theta(x_4-x_5)[(2x_4+x_1-x_3)/(x_1+x_2+x_3+x_4)]+ \theta(x_5-x_4)[(x_1-x_3-2x_5)/(x_1+x_2+x_3+x_5)]$ and $x_{g5}=(2x_4+x_1-x_3-2x_5)/(x_1+x_2+x_3+x_4+x_5)$. The probability density functions of $x_{g3}$, $x_{g4}$ and $x_{g5}$ have complex structures with regions of reduced probability. These regions must be handled with care to avoid false maximums in the likelihood function. General integral equations and detailed analytical expressions are calculated assuming the set $\{x_i\}$ as independent random variables with Gaussian probability distributions. %