Tsallis $q$-高斯的特征函数及其在测量与计量学中的应用
Characteristic Function of the Tsallis $q$-Gaussian and Its Applications in Measurement and Metrology
AI总结：
本文提出了独立 Tsallis $q$-高斯随机变量线性组合的特征函数及其数值求逆方法，为线性测量模型中输入量服从 $q$-高斯分布时的输出量精确概率分布确定及不确定度分析提供了 Monte Carlo 方法的替代计算程序。
摘要：
Tsallis $q$-高斯分布是标准高斯分布的强大推广，常用于非广延统计力学、金融市场和图像处理等多个领域。它属于以非加性熵为特征的 $q$-分布族。由于 $q$-高斯具有通用性和实用性，它是测量模型中输入量建模的自然选择。本文提出了独立 $q$-高斯随机变量线性组合的特征函数，并提出了一种用于其求逆的数值方法。所提出的技术使得在线性测量模型中确定输出量的精确概率分布成为可能，其中输入量被建模为独立的 $q$-高斯随机变量。它为通过分布传播进行不确定度分析提供了 Monte Carlo 方法的替代计算程序。
英文摘要：
The Tsallis $q$-Gaussian distribution is a powerful generalization of the standard Gaussian distribution and is commonly used in various fields, including non-extensive statistical mechanics, financial markets and image processing. It belongs to the $q$-distribution family, which is characterized by a non-additive entropy. Due to their versatility and practicality, $q$-Gaussians are a natural choice for modeling input quantities in measurement models. This paper presents the characteristic function of a linear combination of independent $q$-Gaussian random variables and proposes a numerical method for its inversion. The proposed technique makes it possible to determine the exact probability distribution of the output quantity in linear measurement models, with the input quantities modeled as independent $q$-Gaussian random variables. It provides an alternative computational procedure to the Monte Carlo method for uncertainty analysis through the propagation of distributions.
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