voigtinference：Voigt 轮廓的精确似然计算与条件归因
voigtinference: Exact likelihood calculus and conditional attribution for the Voigt profile
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中文总结 AI 辅助
本文提出 voigtinference 软件包，利用 Voigt 分布解析导数实现闭式似然计算、Fisher 信息及条件归因，支持无分箱共振拟合，解决传统数值近似问题。
中文摘要 AI 辅助
针对 Faddeeva 函数 w(z) 及其衍生的 Voigt 轮廓 K(x,a) 的快速精确算法已存在四十年，部分实现中提供了解析一阶导数。然而，本文所回顾的既有库并未提供归一化 Voigt 分布的完整似然计算：应用实践中仍常采用伪 Voigt 近似、有限差分导数或数值卷积进行参数推断。由于 w'(z) = -2z w(z) + 2i/sqrt(pi)，Voigt 对数似然的每一阶导数均为 K 与色散部分 L(x,a) = Im w(z) 的代数函数，且仅需一次复值求值即可获得轮廓。由此可得到闭式表达的得分函数与 Hessian 矩阵，并通过一维求积解析被积函数获得期望 Fisher 信息；对于固定的内部宽度 sigma, gamma > 0，中心位置及两个宽度的最大似然估计（MLE）具有一致性，并以 sqrt(n) 速率渐近正态，尽管该分布不存在有限均值或方差，因此常规基于似然的标准误差仍然适用。给定观测时，高斯分量的条件均值为 (y - mu) - gamma L/K：这是一个再下降函数，将中等偏差归因于高斯（多普勒/分辨率）分量，而将极端偏差归因于洛伦兹尾部。软件包 voigtinference（Python，基于 NumPy/SciPy，并配有交叉验证的 Julia 伴侣）提供了完整工具集：得分函数、完整参数 Hessian、期望信息、基于牛顿法的无分箱最大似然估计（含边界诊断）、条件分量矩，以及在极端宽度比下经高精度参考验证的评估。它可直接应用于无分箱的非相对论、恒定宽度 Breit-Wigner 与高斯卷积共振拟合，并为线形精修提供解析雅可比矩阵。配套论文：arXiv:2605.01665。
英文摘要
Fast, accurate algorithms for the Faddeeva function w(z), and hence the Voigt profile K(x,a), have existed for four decades, and analytic first derivatives are available in some implementations. What the established libraries reviewed here have not provided is the full likelihood calculus of the normalized Voigt distribution: applications still commonly resort to pseudo-Voigt approximations, finite-difference derivatives, or numerical convolution for parameter inference. Because w'(z) = -2z w(z) + 2i/sqrt(pi), every derivative of the Voigt log-likelihood is an algebraic function of K and the dispersion part L(x,a) = Im w(z), from the single complex evaluation that delivers the profile. This yields the score and Hessian in closed form, and the expected Fisher information by one-dimensional quadrature of an analytic integrand; for fixed interior widths sigma, gamma > 0, the MLE of the center and both widths is consistent and asymptotically normal at rate sqrt(n), despite the distribution having no finite mean or variance, so conventional likelihood-based standard errors apply. The conditional mean of the Gaussian component given an observation is (y - mu) - gamma L/K: a redescending function that attributes moderate deviations to the Gaussian (Doppler/resolution) component and extreme ones to the Lorentzian tail. The package voigtinference (Python, NumPy/SciPy, with a cross-validated Julia companion) supplies the toolkit: score, full parameter Hessian, expected information, Newton-based unbinned maximum likelihood with boundary diagnostics, conditional component moments, and evaluation validated against high-precision references at extreme width ratios. It applies directly to unbinned non-relativistic, constant-width Breit-Wigner x Gaussian resonance fits and supplies analytic Jacobians for line-shape refinement. Companion paper: arXiv:2605.01665.
发表机构
- University of North Carolina at Chapel Hill（北卡罗来纳大学教堂山分校）
- Xiamen University（厦门大学）
机构由 AI 辅助整理，请以论文原文为准。
