[Paper Review] Probability Distributions of Positioning Errors for Some Forms of Center-of-Gravity Algorithms

Gregorio Landi, Giovanni Landi|arXiv (Cornell University)|Apr 19, 2020

Particle physics theoretical and experimental studies12 references4 citations

TL;DR

This paper derives exact and approximate probability density functions (PDFs) for positioning errors in center-of-gravity (COG) algorithms used in particle physics, particularly for two- and three-strip COG in silicon microstrip detectors. It shows that standard variance-minimization methods fail due to heavy-tailed (Cauchy-like) error distributions, and provides analytical PDFs for key COG forms—x=ξ/(ξ+η), y=(ξ−η)/(2(ξ+η)), w=ξ/η, and complete COG2 and simplified three-strip COG—enabling optimal maximum-likelihood track fitting.

ABSTRACT

The center of gravity is a widespread algorithm for position reconstruction in particle physics. For track fitting, its standard use is always accompanied by an easy guess for the probability distribution of the positioning errors. This is an incorrect assumption that degrades the results of the fit. The explicit error forms show evident Cauchy-(Agnesi) tails that render problematic the use of variance minimizations. Here, we report the probability distributions for some combinations of random variables, impossible to find in literature, but essential for track fitting: $x=ξ/{(ξ+η)}$, $y={(ξ-η)}/[2{(ξ+η)}]$, $w=ξ/η$, $x=θ(x_3-x_1) (-x_3)/(x_3+x_2) +θ(x_1-x_3)x_1/(x_1+x_2)$ and $x=(x_1-x_3)/(x_1+x_2+x_3)$. The first three are directly connected to each other and are partial forms of the two-strip center of gravity. The fourth is the complete two-strip center of gravity. For its very complex form, it allows only approximate expressions of the probability. The last expression is a simplified form of the three-strip center of gravity. General integral forms are obtained for all of them. Detailed analytical expressions are calculated assuming $ξ$, $η$, $x_1$, $x_2$ and $x_3$ independent random variables with Gaussian probability distributions (the standard assumption for the strip noise).

Motivation & Objective

• To correct the widespread but incorrect assumption that COG positioning errors follow a Gaussian distribution, which degrades track fitting performance.
• To derive exact and approximate probability density functions (PDFs) for key COG algorithms—especially two- and three-strip COG—under the standard assumption of Gaussian-distributed strip noise.
• To enable optimal track fitting by providing accurate PDFs that account for signal-to-noise ratios and systematic errors, replacing suboptimal least-squares methods.
• To address the complex statistical behavior of COG2, including its characteristic gap and non-Gaussian tails, through analytical and approximate PDF formulations.
• To complete the statistical framework for high-precision track reconstruction by supplying essential PDFs previously missing in the literature for COG-based algorithms.

Proposed method

• Derives the PDF for x = ξ/(ξ+η) using the cumulative distribution function (CDF) and transformation techniques, assuming independent Gaussian-distributed random variables ξ and η.
• Applies change-of-variables and integration over multi-dimensional regions in (ξ, η, β) space to derive the CDF and PDF for the complete two-strip COG2 algorithm.
• Uses Fubini’s theorem to reorder double integrals and apply variable transformations that decouple integrations, enabling a more accurate and analytically tractable approximation.
• Constructs a refined approximation for the COG2 PDF by incorporating Cauchy-like tail terms derived via integration by parts, improving accuracy for inclined tracks.
• Derives a simplified analytical form for the three-strip COG PDF using similar techniques, assuming independent Gaussian noise on each strip.
• Validates results through extensive numerical simulations using MATLAB and symbolic computation via Mathematica, ensuring consistency with simulated data.

Experimental results

Research questions

• RQ1 What is the true probability distribution of positioning errors in two- and three-strip center-of-gravity algorithms when strip signals are subject to Gaussian noise?
• RQ2 Why do standard least-squares fitting methods fail in track reconstruction when applied to COG data, and what statistical properties of the error distribution cause this failure?
• RQ3 How can the complex, non-Gaussian error distribution of the complete COG2 algorithm—characterized by a gap and heavy tails—be accurately modeled analytically?
• RQ4 What is the impact of signal-to-noise ratio and strip position on the shape of the COG error PDF, and how can this be incorporated into maximum-likelihood fitting?
• RQ5 Can a closed-form approximation of the COG2 PDF be derived that captures both the central peak and the Cauchy-like tails with high accuracy?

Key findings

• The positioning error distributions for COG algorithms exhibit pronounced Cauchy-(Agnesi) tails, invalidating the assumption of Gaussian errors and making variance minimization suboptimal.
• The PDF for the two-strip COG2 algorithm is analytically intractable in closed form due to a characteristic gap in the data, requiring a complex multi-integral formulation.
• A refined analytical approximation for the COG2 PDF is derived that matches numerical simulations with negligible error, incorporating both the central peak and Cauchy-like tails.
• The Cauchy-like tail terms, while small (~10⁻⁵) for typical tracks, become significant (~10⁻¹) for highly inclined tracks, where they must be included for accuracy.
• The simplified three-strip COG PDF is derived in closed form, enabling efficient maximum-likelihood fitting with improved resolution over standard methods.
• The derived PDFs allow for optimal track fitting by incorporating hit-specific signal-to-noise ratios and correcting systematic errors, significantly reducing parameter variances compared to least-squares fitting.