加权FBET型框架的Gram-Schmidt相关记账法：分离相关测量中的短程与长程相关尺度，及其在射电光度函数中的应用
Gram-Schmidt correlation bookkeeping for weighted FBET-type frameworks: separating short- and long-range correlation scales in correlated measurements, with an application to radio luminosity functions
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中文总结 AI 辅助
针对相关测量中混合短程与长程相关结构的问题，提出一种加权Gram-Schmidt正交化相关记账构造，避免协方差分解双重计数，并在AirPassengers及射电光度函数数据上验证。
中文摘要 AI 辅助
相关测量的不确定性量化通常需要一个协方差模型，该模型混合了几种性质上不同的相关结构：局部测量噪声、短程（例如周期性或季节性）相关，以及与趋势、漂移或长记忆行为相关的长程相关。加权和广义最小二乘只有在协方差矩阵事先已知或可信地指定时才能编码这种结构，而在实践中这很少见。我们描述了一种结构化的相关记账构造，旨在用于加权FBET型（wFBET）不确定性量化框架内部：与不同滞后变量相关的候选相关基函数，在允许进入协方差或度量构造之前，通过加权Gram-Schmidt过程（在明确声明的、统计上合理的度量下）进行正交化。这避免了朴素加性协方差分解形式$\Sigma = \Sigma_{\mathrm{short}} + \Sigma_{\mathrm{long}}$中固有的双重计数问题。我们在经典的AirPassengers基准数据集上对该构造（其向量化形式、算子值核形式以及实现中使用的有限秩基函数实现）进行了基准测试，并将其应用于天体物理学中按红移分箱测量的射电光度函数案例。我们不声称具有普遍最优性；目标是在加权信息几何拟合框架内提供一种透明且可审计的方式来组织相关尺度。
英文摘要
Uncertainty quantification for correlated measurements frequently requires a covariance model that mixes several qualitatively different correlation structures: local measurement noise, short-range (e.g. cyclic or seasonal) correlations, and long-range correlations associated with trend, drift, or long-memory behavior. Weighted and generalized least squares can encode such structure only when the covariance matrix is known or credibly specified in advance, which in practice it rarely is. We describe a structured correlation bookkeeping construction intended for use inside a weighted FBET-type (wFBET) uncertainty-quantification framework: candidate correlation basis functions associated with distinct lag variables are orthogonalized with a weighted Gram-Schmidt procedure (under an explicitly declared, statistically motivated metric) before they are allowed to enter the covariance or metric construction. This avoids the double counting inherent in naive additive covariance decompositions of the form $Σ= Σ_{\mathrm{short}} + Σ_{\mathrm{long}}$. We benchmark the construction (in its vectorized form, its operator-valued kernel form, and the finite-rank basis-function realization used by the implementation) on the classic AirPassengers benchmark dataset, and we apply it to the astrophysical case of radio luminosity functions measured in redshift bins. No claim of universal optimality is made; the goal is a transparent and auditable way to organize correlation scales within weighted information-geometric fitting frameworks.
