论文标题
某些形式的重心算法的定位错误的概率分布
Probability Distributions of Positioning Errors for Some Forms of Center-of-Gravity Algorithms
论文作者
论文摘要
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.在这里，我们报告了一些随机变量组合的概率分布，在文献中找不到，但对于轨道拟合至关重要：$ x =ξ/{（ξ+η）} $，$ y = {（ξ-η）}/[2 {2 {（ξ+η）} $，$ w = em ph =ξ/η$，$ x = (-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.假设$ξ$，$η$，$ x_1 $，$ x_2 $和$ x_3 $独立的随机变量（带有高斯概率分布）（带状噪声的标准假设），则计算详细的分析表达式。
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).
