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Microergodicity implies orthogonality of Matérn fields on bounded domains in $\mathbb{R}^4$

Natesh S. Pillai

Abstract

Matérn random fields are one of the most widely used classes of models in spatial statistics. The fixed-domain identifiability of covariance parameters for stationary Matérn Gaussian random fields exhibits a dimension-dependent phase transition. For known smoothness $ν$, Zhang \cite{Zhang2004} showed that when $d\le3$, two Matérn models with the same microergodic parameter $m=σ^2α^{2ν}$ induce equivalent Gaussian measures on bounded domains, while Anderes \cite{Anderes2010} proved that when $d>4$, the corresponding measures are mutually singular whenever the parameters differ. The critical case $d=4$ for stationary Matérn models has remained open. We resolve this case. Let $d=4$ and consider two stationary Matérn models on $\mathbb R^4$ with parameters $(σ_1,α_1)$ and $(σ_2,α_2)$ satisfying \[ σ_1^2α_1^{2ν}=σ_2^2α_2^{2ν}, \qquad α_1\neq α_2. \] We prove that the corresponding Gaussian measures on any bounded observation domain are mutually singular on every countable dense observation set, and on the associated path space of continuous functions. Our approach can be viewed as a spectral analogue of the higher-order increment method of Anderes \cite{Anderes2010}. Whereas Anderes isolates the second irregular covariance coefficient through renormalized quadratic variations in physical space, we detect the first nonvanishing high-frequency spectral mismatch via localized Fourier coefficients and use a normalized Whittle score to identify parameters. More broadly, the localized spectral probing framework used here for detecting subtle covariance differences in Gaussian random fields may be useful for studying identifiability and estimation in other spatial models.

Microergodicity implies orthogonality of Matérn fields on bounded domains in $\mathbb{R}^4$

Abstract

Matérn random fields are one of the most widely used classes of models in spatial statistics. The fixed-domain identifiability of covariance parameters for stationary Matérn Gaussian random fields exhibits a dimension-dependent phase transition. For known smoothness , Zhang \cite{Zhang2004} showed that when , two Matérn models with the same microergodic parameter induce equivalent Gaussian measures on bounded domains, while Anderes \cite{Anderes2010} proved that when , the corresponding measures are mutually singular whenever the parameters differ. The critical case for stationary Matérn models has remained open. We resolve this case. Let and consider two stationary Matérn models on with parameters and satisfying We prove that the corresponding Gaussian measures on any bounded observation domain are mutually singular on every countable dense observation set, and on the associated path space of continuous functions. Our approach can be viewed as a spectral analogue of the higher-order increment method of Anderes \cite{Anderes2010}. Whereas Anderes isolates the second irregular covariance coefficient through renormalized quadratic variations in physical space, we detect the first nonvanishing high-frequency spectral mismatch via localized Fourier coefficients and use a normalized Whittle score to identify parameters. More broadly, the localized spectral probing framework used here for detecting subtle covariance differences in Gaussian random fields may be useful for studying identifiability and estimation in other spatial models.
Paper Structure (19 sections, 14 theorems, 266 equations, 3 figures)

This paper contains 19 sections, 14 theorems, 266 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Fixed-domain asymptotics and microergodicity
  3. Our contribution
  4. Idea of the proof
  5. Related Work
  6. Organization of the paper
  7. Main Result
  8. The critical dimension, Anderes' method, and our approach
  9. Why $d=4$ is the critical threshold
  10. Score interpretation and likelihood-ratio expansion
  11. Simulation study
  12. Simulation results
  13. Continuous modifications and dense observation sigma-fields
  14. Localized Fourier coefficients
  15. Diagonal asymptotics
  16. ...and 4 more sections

Key Result

Theorem 1

Let $D \subset \mathbb{R}^4$ be a bounded domain with nonempty interior and $S \subset D$ be countable and dense. Denote by $\mathbf{P}_j^S$ the law of the coordinate process $\{Y_j(t):t\in S\}$ on $\mathbb{R}^S$. Assume eq:micro-match. Then In particular, for the dense grid the laws of the coordinate processes $\{Y_j(t):t\in D_\infty\}$ are mutually singular.

Figures (3)

  • Figure 1: Empirical distribution of $T_N$ in Experiment 1 under the two microergodically matched models with $\alpha_1=1$ and $\alpha_2=2$. The statistic concentrates near $0$ under model 1 and near $1$ under model 2.
  • Figure 2: Empirical distribution of $T_N$ in Experiment 2 under the two microergodically matched models with $\alpha_1=1$ and $\alpha_2=1.2$. Separation persists but with increased variance due to weaker spectral separation.
  • Figure 3: Monte Carlo distribution of the Whittle estimator $\widehat{\alpha}$ based on the (exact) grid Fourier coefficients in dimension $d=4$. The true parameter value $\alpha = 3$ is indicated by the dashed line.

Theorems & Definitions (28)

  • Theorem 1
  • Lemma 1: Diagonal likelihood expansion for the localized Matérn coefficients
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Corollary 1
  • proof
  • Lemma 4
  • ...and 18 more