Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Fixed and Increasing Domain Asymptotics for the Roughness and Scale of Isotropic Gaussian Random Fields

Varun Kotharkar, Michael L. Stein

TL;DR

This work analyzes joint estimation of roughness and scale for two-dimensional Gaussian fields with power-law covariances, formalizing the model via generalized covariances and GC–k structure. It develops a principled, likelihood-free method of moments based on two first-order bilinear differences and a two-scale identity, proving simultaneous consistency and central limit theorems under increasing-domain asymptotics and transferring them to fixed-domain settings through exact self-similar rescaling. The results extend to higher-order differences to accommodate smoother processes, and the FD theory remains valid under modest irregular sampling and Matérn-like misspecification by using tangent power-law behavior around the origin. The framework offers a scalable, robust alternative to likelihood-based methods, with explicit asymptotic covariances, and supports practical applications where grid sampling is imperfect or incomplete, including irregular designs and Matérn-like truths.

Abstract

We establish a rigorous asymptotic theory for the joint estimation of roughness and scale parameters in two-dimensional Gaussian random fields with power-law generalized covariances \cite{Matheron1973, Stein1999, Yaglom1987}. Our main results are bivariate central limit theorems for a class of method-of-moments estimators under increasing-domain and fixed-domain asymptotics. The fixed-domain result follows immediately from the increasing-domain result from the self-similarity of Gaussian random fields with power-law generalized covariances \cite{IstasLang1997, Coeurjolly2001, ZhuStein2002}. These results provide a unified distributional framework across these two classical regimes \cite{AvramLeonenkoSakhno2010-ESAIM, BiermeBonamiLeon2011-EJP} that makes the unusual behavior of the estimates under fixed-domain asymptotics intuitively obvious. Our increasing-domain asymptotic results use spatial averages of quadratic forms of (iterated) bilinear product difference filters that yield explicit expressions for the estimates of roughness and scale to which existing theorems on such averages \cite{BreuerMajor1983,Hannan1970} can be readily applied. We further show that the asymptotics remain valid under modestly irregular sampling due to jitter or missing observations. For the fixed-domain setting, the results extend to models that behave sufficiently like the power-law model at high frequencies such as the often used Matérn model \cite{ZhuStein2006, WangLoh2011EJS, KaufmanShaby2017EJS}.

Fixed and Increasing Domain Asymptotics for the Roughness and Scale of Isotropic Gaussian Random Fields

TL;DR

This work analyzes joint estimation of roughness and scale for two-dimensional Gaussian fields with power-law covariances, formalizing the model via generalized covariances and GC–k structure. It develops a principled, likelihood-free method of moments based on two first-order bilinear differences and a two-scale identity, proving simultaneous consistency and central limit theorems under increasing-domain asymptotics and transferring them to fixed-domain settings through exact self-similar rescaling. The results extend to higher-order differences to accommodate smoother processes, and the FD theory remains valid under modest irregular sampling and Matérn-like misspecification by using tangent power-law behavior around the origin. The framework offers a scalable, robust alternative to likelihood-based methods, with explicit asymptotic covariances, and supports practical applications where grid sampling is imperfect or incomplete, including irregular designs and Matérn-like truths.

Abstract

We establish a rigorous asymptotic theory for the joint estimation of roughness and scale parameters in two-dimensional Gaussian random fields with power-law generalized covariances \cite{Matheron1973, Stein1999, Yaglom1987}. Our main results are bivariate central limit theorems for a class of method-of-moments estimators under increasing-domain and fixed-domain asymptotics. The fixed-domain result follows immediately from the increasing-domain result from the self-similarity of Gaussian random fields with power-law generalized covariances \cite{IstasLang1997, Coeurjolly2001, ZhuStein2002}. These results provide a unified distributional framework across these two classical regimes \cite{AvramLeonenkoSakhno2010-ESAIM, BiermeBonamiLeon2011-EJP} that makes the unusual behavior of the estimates under fixed-domain asymptotics intuitively obvious. Our increasing-domain asymptotic results use spatial averages of quadratic forms of (iterated) bilinear product difference filters that yield explicit expressions for the estimates of roughness and scale to which existing theorems on such averages \cite{BreuerMajor1983,Hannan1970} can be readily applied. We further show that the asymptotics remain valid under modestly irregular sampling due to jitter or missing observations. For the fixed-domain setting, the results extend to models that behave sufficiently like the power-law model at high frequencies such as the often used Matérn model \cite{ZhuStein2006, WangLoh2011EJS, KaufmanShaby2017EJS}.

Paper Structure

This paper contains 52 sections, 35 theorems, 147 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Generalized covariances (GC–$k$).
  3. Variation-based inference.
  4. Our contribution.
  5. Related Work
  6. One-dimensional models.
  7. Fractional Brownian surfaces and Gaussian fields.
  8. Likelihood-based and related approaches.
  9. Quadratic forms and CLTs.
  10. Methodology: Bilinear Differences and Moment Estimators
  11. Sampling and aliasing.
  12. Two bilinear differences (first order).
  13. Quadratic variations and two-scale identity.
  14. Notation.
  15. Integrability and the $D^{(1)}_{[2]}$–from–$D^{(1)}_{[1]}$ recovery.
  16. ...and 37 more sections

Key Result

Lemma 3.1

If $0<\phi_2<1$, then

Figures (2)

  • Figure 1: Sample vs. true variograms on a log scale (bilinear, IRF-0). For 25 simulated $60\times60$ power-law fields with $\phi_2=0.8$, the plot compares estimated and true variograms on a log--log scale. Each colored line shows the bilinear ratio slope $\hat{\phi}_2-\phi_2$; a mean slope near zero indicates unbiasedness and the spread reflects sampling variability.
  • Figure 2: Fixed–domain (FD) estimator scatter plot for IRF–0 (bilinear). Each panel shows 400 replicates of $(\log \widehat{\phi}_1,\widehat{\phi}_2)$ from the bilinear method-of-moments on an $n\times n$ grid. The near-linear ridge ($\mathrm{corr}\approx0.99$) and stable scaled s.d. ($\approx1.45$) indicate $\widehat{\phi}_2$ is unbiased and that $\operatorname{sd}(\widehat{\phi}_2)\approx c/\sqrt{M_{\mathrm{int}}}$ with $c\approx1.5$.

Theorems & Definitions (73)

  • Lemma 3.1: Expectation for $D^{(1)}_{[r]}$
  • proof
  • Lemma 3.2: Integrability of the filtered spectrum
  • Lemma 4.1: Boundary remainder is negligible at $\sqrt{N}$ scale
  • proof
  • Proposition 4.2: Quadratic–form CLT under increasing domain
  • proof
  • Remark : Normalizing by $N$ vs. $|\Lambda_{n-2r}|$
  • Theorem 4.3: Joint CLT for $(Q^{(1)}_{1},Q^{(1)}_{2})$
  • Theorem 4.4: Consistency and joint asymptotic normality
  • ...and 63 more