Natural coordinates for constrained correlation functions: Partial autocorrelations and the geometry of positive power spectra
Abstract
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 (r1,…,rN) 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 xn that maps the admissible interval for rn, given r1,…,rn−1, to [−1,+1]. We identify xn with the partial autocorrelation coefficient αn 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 αn varying independently in [−1,+1]. This identification makes the standard partial-autocorrelation toolbox directly applicable. The admissible intervals for rn, cumbersome to obtain at higher order in the original determinant-based formalism, now follow from an 𝒪(N2) 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.
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BibTeX
@misc{erben2026natural,
title = {{Natural coordinates for constrained correlation functions: Partial autocorrelations and the geometry of positive power spectra}},
author = {Thomas Erben},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.24500},
eprint = {2609.24500},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.24500},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Natural coordinates for constrained correlation functions: Partial autocorrelations and the geometry of positive power spectra
AU - Thomas Erben
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.24500
UR - https://arxiv.org/abs/2609.24500
AB - 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,…,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,…,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 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.
ER -