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

Abstract

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.