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An asymptotic study of the joint maximum likelihood estimation of the regularity and the amplitude parameters of a periodized Mat{é}rn model

Sébastien J Petit

TL;DR

This paper analyzes the joint maximum likelihood estimation of the regularity $\nu$ and amplitude $\phi$ parameters for a periodized Matérn Gaussian process on the circle, with the often-ignored shape parameter $\alpha$ non-identifiable in the circular setting. By exploiting a discrete Fourier transform framework and a spectral linkage between the kernel on the circle and its circulant covariance matrix, the authors establish strong consistency for $\hat{\nu}_n$ and a joint asymptotic normality for $(\hat{\nu}_n, \hat{\phi}_n)$, showing that $\alpha$ does not affect the limit and that estimation yields asymptotically optimal prediction errors. They derive refined asymptotics for the profile likelihood via symmetrized Hurwitz zeta-type functions, enabling precise characterization of spectral contributions to estimation and prediction. In the deterministic Sobolev-case, they show that joint estimation does not necessarily recover the Sobolev smoothness of the target function, though under additional spectral assumptions some functions do satisfy $\hat{\nu}_n \to \nu_0(f)$ with probability one. Overall, the work provides a rigorous finite- to asymptotic-parameter translation for GP interpolation with a Matérn kernel on the circle and highlights when parameter estimation preserves predictive efficiency.

Abstract

This work considers parameter estimation for Gaussian process interpolation with a periodized version of the Mat{é}rn covariance function introduced by Stein. Convergence rates are studied for the joint maximum likelihood estimation of the regularity and the amplitude parameters when the data are sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a fixed deterministic element of a Sobolev space of continuous functions is also considered, suggesting that a joint estimation does not select the regularity parameter as if the amplitude were fixed.

An asymptotic study of the joint maximum likelihood estimation of the regularity and the amplitude parameters of a periodized Mat{é}rn model

TL;DR

This paper analyzes the joint maximum likelihood estimation of the regularity and amplitude parameters for a periodized Matérn Gaussian process on the circle, with the often-ignored shape parameter non-identifiable in the circular setting. By exploiting a discrete Fourier transform framework and a spectral linkage between the kernel on the circle and its circulant covariance matrix, the authors establish strong consistency for and a joint asymptotic normality for , showing that does not affect the limit and that estimation yields asymptotically optimal prediction errors. They derive refined asymptotics for the profile likelihood via symmetrized Hurwitz zeta-type functions, enabling precise characterization of spectral contributions to estimation and prediction. In the deterministic Sobolev-case, they show that joint estimation does not necessarily recover the Sobolev smoothness of the target function, though under additional spectral assumptions some functions do satisfy with probability one. Overall, the work provides a rigorous finite- to asymptotic-parameter translation for GP interpolation with a Matérn kernel on the circle and highlights when parameter estimation preserves predictive efficiency.

Abstract

This work considers parameter estimation for Gaussian process interpolation with a periodized version of the Mat{é}rn covariance function introduced by Stein. Convergence rates are studied for the joint maximum likelihood estimation of the regularity and the amplitude parameters when the data are sampled according to the model. The mean integrated squared error is also analyzed with fixed and estimated parameters, showing that maximum likelihood estimation yields asymptotically the same error as if the ground truth was known. Finally, the case where the observed function is a fixed deterministic element of a Sobolev space of continuous functions is also considered, suggesting that a joint estimation does not select the regularity parameter as if the amplitude were fixed.
Paper Structure (32 sections, 39 theorems, 183 equations, 5 figures, 1 table)

This paper contains 32 sections, 39 theorems, 183 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Gaussian process interpolation on the circle
  3. Framework
  4. Maximum likelihood estimation
  5. Studying the profile likelihood using discrete Fourier transforms
  6. Linking the spectra of $k_\theta$ and $K_\theta$
  7. The consistency of $\hat{\nu}_n$ for fixed $\phi$ and $\alpha$
  8. Profiling the likelihood
  9. A symmetrized version of the Hurwitz zeta function
  10. Main results
  11. Standing assumptions
  12. Convergence rates of maximum likelihood estimation
  13. Convergence rates of the integrated squared error
  14. Numerical experiments
  15. The deterministic case
  16. ...and 17 more sections

Key Result

Proposition 2.1

Let $n \geq 1$ and $f\colon [0, \, 1] \to \RR$ be a continuous periodic function with absolutely summable Fourier coefficients $c_j(f)$. Writing $\hat{f}_n$ for the posterior mean function given $Z$ and the parameter $\theta$, we have: The convergence of eq:blup holds uniformly absolutely.

Figures (5)

  • Figure 1: Blue curve: numerical approximation of the function $\nu \mapsto \mathcal{C}_{\nu_0}(\nu)$, for $\nu_0 = 5/2$. Red vertical line: $(\nu_0 - 1)/2$.
  • Figure 2: The upper plot shows the evolution of averages of \ref{['eq:empirical_ise']}. The black dotted line stands for the profile likelihood, and the other thin solid colored lines for the likelihood with the value of $\phi_1$ indicated in the legend. The red dotted line stands for \ref{['eq:empirical_ise']} with known $\theta_0$. The thick transparent line stands for the asymptotics of $\EE ( \mathrm{ISE}_n (\nu_0, \, \alpha_0; \, \xi) )$ predicted by Theorem \ref{['thm:error_nu_fixed']}. All but three of the thin lines mentioned above closely follow this thick line. The lower plot shows means and $95\%$ bootstrap confidence intervals of ratios between averages of \ref{['eq:empirical_ise']} with estimated and known parameters. Results are reported for fewer values of $\phi_1$ for clarity.
  • Figure 3: Numerical approximation of the function $\Mset_{\infty}^f$, for $\nu_0(f) = 3/2$. A numerical approximation of the minimizer is about $1.354$.
  • Figure 4: Left: the function $f$. Right: smoothness estimates as functions of $n$. The black dotted line stands for the profile likelihood, and the other solid colored lines for the likelihood with the fixed value $\phi_1$ indicated in the legend. The lowest black horizontal line represents the approximate minimizer of $\Mset_{\infty}^f$ shown in Figure \ref{['fig:criterion']}.
  • Figure 5: Blue curve: numerical approximation of the function $\nu \mapsto \mathcal{C}_{\nu_0}^{(0)}(\nu)$, for $\nu_0 = 5/2$. Red vertical line: $(\nu_0 - 1)/2$.

Theorems & Definitions (75)

  • Proposition 2.1
  • Proposition 2.2
  • Remark 3.1
  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Theorem 4.4
  • Proposition 6.1
  • Proposition 6.2
  • Remark 6.3
  • ...and 65 more