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 $(r_1,\ldots,r_N)$ 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.