约束相关函数的自然坐标：偏自相关与正功率谱的几何
Natural coordinates for constrained correlation functions: Partial autocorrelations and the geometry of positive power spectra
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中文总结 AI 辅助
本文证明约束相关函数的仿射变量等价于偏自相关系数，提供自然坐标，简化区间计算并赋予准高斯似然变换经典统计解释。
中文摘要 AI 辅助
两点相关函数是宇宙切变和大尺度结构分析中的标准汇总统计量。然而，其取值受到约束：底层功率谱的非负性将任何可接受的相关性系数序列$(r_1,\ldots,r_N)$限制在一个有界凸区域内，在一维情形下由Schneider & Hartlap (2009)的递归区间几何描述。他们的形式引入了仿射变量$x_n$，将给定$r_1,\ldots,r_{n-1}$时$r_n$的可接受区间映射到$[-1,+1]$。我们将$x_n$识别为相关正Toeplitz相关矩阵的偏自相关系数$\alpha_n$，这是时间序列分析中的经典量。因此，偏自相关为可接受区域提供了自然坐标，每个$\alpha_n$在$[-1,+1]$内独立变化。这一识别使得标准的偏自相关工具箱可直接应用。$r_n$的可接受区间在原始基于行列式的形式中在高阶时难以获得，现在可通过任意阶的$\mathcal{O}(N^2)$递归得到。Wilking & Schneider (2013)的准高斯似然构造中使用的反双曲正切变换变为偏自相关的Fisher $z$-变换，为其经验高斯化效应提供了经典统计解释。数值实验展示了这些自然坐标在测试的一维设置中的实际用途。高维各向同性约束需要额外的几何输入，不在本识别的直接范围内。
英文摘要
Two-point correlation functions are a standard summary statistic in cosmic shear and large-scale-structure analyses. Their values are, however, constrained: non-negativity of the underlying power spectrum restricts any admissible sequence of correlation coefficients $(r_1,\ldots,r_N)$ to a bounded convex region, described in the one-dimensional case by the recursive interval geometry of Schneider & Hartlap (2009). Their formalism introduces an affine variable $x_n$ that maps the admissible interval for $r_n$, given $r_1,\ldots,r_{n-1}$, to $[-1,+1]$. We identify $x_n$ with the partial autocorrelation coefficient $α_n$ of the associated positive Toeplitz correlation matrix, a classical quantity in time-series analysis. The partial autocorrelations thus provide natural coordinates on the admissible region, each $α_n$ varying independently in $[-1,+1]$. This identification makes the standard partial-autocorrelation toolbox directly applicable. The admissible intervals for $r_n$, cumbersome to obtain at higher order in the original determinant-based formalism, now follow from an $\mathcal{O}(N^2)$ recursion at arbitrary order. The inverse hyperbolic tangent used in the quasi-Gaussian likelihood construction of Wilking & Schneider (2013) becomes Fisher's $z$-transformation of partial autocorrelations, providing a classical statistical interpretation of its empirical Gaussianising effect. Numerical experiments illustrate the practical use of these natural coordinates in the tested one-dimensional settings. Higher-dimensional isotropic constraints require additional geometric input and lie outside the direct scope of this identification.
发表机构
- Argelander Institut für Astronomie（阿尔兰格天文学研究所）
- Cluster of Excellence “Our Dynamic Universe” (DYNAVERSE)（“我们的动态宇宙”卓越集群）
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