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Gaussian Processes Generated By Monotonically Modulated Stationary Kernels

Stephen Crowley

TL;DR

Problem addressed: understanding Gaussian processes generated by monotonically modulating stationary kernels and how their spectral properties and zero statistics transform. Approach: establish an explicit isometry between the original and modulated RKHS via a conjugation operator $M_\theta$, derive the eigenfunction mapping $\phi_n(t)=\psi_n(\theta(t))\sqrt{\dot{\theta}(t)}$, and prove eigenvalue invariance; compute the zero-crossing expectation using Kac-Rice. Key results: eigenfunction form and eigenvalue invariance, isometric embedding, and a closed-form zero-counting formula $E[N([0, T])] = \sqrt{-\ddot{K}(0)} (\theta(T)-\theta(0))$. Significance: enables principled time-warping of stationary Gaussian processes with preserved spectral structure and tractable zero statistics.

Abstract

This article examines Gaussian processes generated by monotonically modulating stationary kernels. An explicit isometry between the original and the modulated reproducing kernel Hilbert spaces is established, preserving eigenvalues and normalization. The expected number of zeros over the interval $[0,T]$ is shown to be exactly $\sqrt{-\ddot{K}(0)}(θ(T)-θ(0))$, where $\ddot{K}(0)$ is the second derivative of the kernel at zero and $θ$ is the modulating function.

Gaussian Processes Generated By Monotonically Modulated Stationary Kernels

TL;DR

Problem addressed: understanding Gaussian processes generated by monotonically modulating stationary kernels and how their spectral properties and zero statistics transform. Approach: establish an explicit isometry between the original and modulated RKHS via a conjugation operator , derive the eigenfunction mapping , and prove eigenvalue invariance; compute the zero-crossing expectation using Kac-Rice. Key results: eigenfunction form and eigenvalue invariance, isometric embedding, and a closed-form zero-counting formula . Significance: enables principled time-warping of stationary Gaussian processes with preserved spectral structure and tractable zero statistics.

Abstract

This article examines Gaussian processes generated by monotonically modulating stationary kernels. An explicit isometry between the original and the modulated reproducing kernel Hilbert spaces is established, preserving eigenvalues and normalization. The expected number of zeros over the interval is shown to be exactly , where is the second derivative of the kernel at zero and is the modulating function.
Paper Structure (3 sections, 4 theorems, 24 equations)

This paper contains 3 sections, 4 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Conclusion

Key Result

Theorem 1

For any stationary kernel $K (t, s) = K (|t - s|)$, the eigenfunctions of the integral covariance operator defined by the $\theta$-modulated kernel are given $\forall \theta \in \mathcal{F}$ by which satisfies the eigenfunction equation where $\psi_n$ are the normalized eigenfunctions of the covariance operator defined by the original unmodulated kernel $K (|t - s|)$ which satisfy

Theorems & Definitions (10)

  • Definition 1
  • Remark 1
  • Theorem 1: Eigenfunctions
  • proof
  • Corollary 1: Eigenvalue Invariance
  • proof
  • Theorem 2: Operator Conjugation
  • proof
  • Theorem 3: Expected Zero-Counting Function
  • proof