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Continuous Time Locally Stationary Wavelet Processes

Henry Antonio Palasciano, Marina I. Knight, Guy P. Nason

TL;DR

This work develops continuous-time locally stationary wavelet processes (CLS-WP) to model irregularly spaced time series with a spectral representation across a continuous range of scales. It formulates X(t) via a homogeneous Lévy basis and a time-varying amplitude W, defines the continuous evolutionary spectrum S(u,v) = |W(u,v)|^2 and the local autocovariance c(t,τ), and derives a practical estimation pipeline that inverts a linear integral equation using an iterative soft-thresholding approach grounded in Mercer's theorem. The authors provide rigorous theory linking the raw wavelet periodogram to the spectrum, prove asymptotic properties for sampled data, and implement a complete estimation workflow including smoothing, regularization, and computation. They demonstrate the method on infant heart-rate data and irregular heart-rate series, showing coherent spectral structure across scales and demonstrating the practical relevance for irregularly sampled physiological signals. Overall, the paper delivers a principled, nonnegative spectral estimator for irregular time series, enabling continuous-scale analysis and interpretation in real-world applications while providing a solid theoretical bridge from continuous-time wavelets to local autocovariance.

Abstract

This article introduces the class of continuous time locally stationary wavelet processes. Continuous time models enable us to properly provide scale-based time series models for irregularly-spaced observations for the first time, while also permitting a spectral representation of the process over a continuous range of scales. We derive results for both the theoretical setting, where we assume access to the entire process sample path, and a more practical one, which develops methods for estimating the quantities of interest from sampled time series. The latter estimates are accurately computable in reasonable time by solving the relevant linear integral equation using the iterative soft-thresholding algorithm due to Daubechies, Defrise and De~Mol. Appropriate smoothing techniques are also developed and applied in this new setting. Comparisons to previous methods are conducted on the heart rate time series of a sleeping infant. Additionally, we exemplify our new methods by computing spectral and autocovariance estimates on irregularly-spaced heart rate data obtained from a recent sleep-state study.

Continuous Time Locally Stationary Wavelet Processes

TL;DR

This work develops continuous-time locally stationary wavelet processes (CLS-WP) to model irregularly spaced time series with a spectral representation across a continuous range of scales. It formulates X(t) via a homogeneous Lévy basis and a time-varying amplitude W, defines the continuous evolutionary spectrum S(u,v) = |W(u,v)|^2 and the local autocovariance c(t,τ), and derives a practical estimation pipeline that inverts a linear integral equation using an iterative soft-thresholding approach grounded in Mercer's theorem. The authors provide rigorous theory linking the raw wavelet periodogram to the spectrum, prove asymptotic properties for sampled data, and implement a complete estimation workflow including smoothing, regularization, and computation. They demonstrate the method on infant heart-rate data and irregular heart-rate series, showing coherent spectral structure across scales and demonstrating the practical relevance for irregularly sampled physiological signals. Overall, the paper delivers a principled, nonnegative spectral estimator for irregular time series, enabling continuous-scale analysis and interpretation in real-world applications while providing a solid theoretical bridge from continuous-time wavelets to local autocovariance.

Abstract

This article introduces the class of continuous time locally stationary wavelet processes. Continuous time models enable us to properly provide scale-based time series models for irregularly-spaced observations for the first time, while also permitting a spectral representation of the process over a continuous range of scales. We derive results for both the theoretical setting, where we assume access to the entire process sample path, and a more practical one, which develops methods for estimating the quantities of interest from sampled time series. The latter estimates are accurately computable in reasonable time by solving the relevant linear integral equation using the iterative soft-thresholding algorithm due to Daubechies, Defrise and De~Mol. Appropriate smoothing techniques are also developed and applied in this new setting. Comparisons to previous methods are conducted on the heart rate time series of a sleeping infant. Additionally, we exemplify our new methods by computing spectral and autocovariance estimates on irregularly-spaced heart rate data obtained from a recent sleep-state study.
Paper Structure (58 sections, 24 theorems, 223 equations, 24 figures)

This paper contains 58 sections, 24 theorems, 223 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Background and Notation
  3. Locally Stationary Wavelet Processes in Continuous Time: Theory
  4. Process Definition
  5. Continuous Time Evolutionary Wavelet Spectrum
  6. Continuous Time Local Autocovariance
  7. Relationship between the Wavelet Periodogram and Spectrum
  8. Solving for the Spectrum and Related Quantities
  9. Implementation of Spectral and Autocovariance Estimators
  10. Sampled Process Definition
  11. Estimation Theory
  12. Smoothing the Discretized Raw Wavelet Periodogram Across Time
  13. Estimating the Spectrum from the Smoothed Raw Wavelet Periodogram
  14. Computational Complexity
  15. Non-Stationary Haar Moving Average Example
  16. ...and 43 more sections

Key Result

Proposition 1

Let $\{X(t)\}_{t\in\mathbb R}$ be a continuous time locally stationary wavelet process, $c_X(t,\tau)$ be its autocovariance function and $c(t,\tau)$ its local autocovariance. Then, for all scales $u\in\mathbb R^+$ and times $v\in\mathbb R$, where $\gamma$ is a constant which depends on the decay structure of the wavelet $\psi$ and $K(\,\cdot\,)$ is the Lipschitz constant function from Definition

Figures (24)

  • Figure 1: Left: $n(T) = 1500$ equally-spaced samples from a single realization of continuous time non-stationary Haar ma process. Centre: the underlying evolutionary wavelet spectrum. Right: the corresponding local autocorrelation.
  • Figure 1: Left: Haar wavelet. Centre: Haar autocorrelation wavelet. Right: Haar inner product kernel.
  • Figure 2: Haar moving average evolutionary wavelet spectral estimates using daub_inv's iterative soft-thresholding algorithm. Top row: $N=100$ iterations. Bottom row: $N=10000$ iterations. From left to right: estimates computed using $R=1, 10, 100$ and $1000$ realizations.
  • Figure 2: Left: Ricker wavelet. Centre: Ricker autocorrelation wavelet. Right: Ricker inner product kernel.
  • Figure 3: Haar moving average local autocorrelation estimates. From left to right: estimates computed using $R=1$, $10$, $100$ and $1000$ realizations and $N=10000$.
  • ...and 19 more figures

Theorems & Definitions (66)

  • Definition 1
  • Remark 1
  • Example 1
  • Definition 2
  • Remark 2
  • Example 1
  • Definition 3
  • Definition 4
  • Proposition 1
  • Definition 5
  • ...and 56 more