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On a new family of weighted Gaussian processes: an application to bat telemetry data

Jose Hermenegildo Ramirez Gonzalez, Antonio Murillo Salas, Ying Sun

TL;DR

This work introduces a new nonstationary Gaussian process family defined by the covariance kernel $K_f(s,t)$, derived from limits of rescaled occupation-time fluctuations, and proves its validity under mild integrability of $f$. It shows that two subfamilies yield long-range dependence with logarithmic covariance growth and that the processes are not semimartingales, offering a flexible framework for modeling memory-rich movement data. The authors apply the model to bat telemetry, perform likelihood-based inference, and demonstrate that the logarithmic-growth kernel often provides superior fit compared to standard fractional-OU/FBM-based models, supported by AIC-based comparisons across multiple trajectories. An accompanying R Shiny app and web appendices provide simulation tools, proofs, and empirical stationarity assessments, underscoring the practical utility and theoretical robustness of the approach for ecological trajectory modeling.

Abstract

In this article we use a covariance function that arises from limit of fluctuations of the rescaled occupation time process of a branching particle system, to introduce a family of weighted long-range dependence Gaussian processes. In particular, we consider two subfamilies for which we show that the process is not a semimartingale, that the processes exhibit long-range dependence and have long-range memory of logarithmic order. Finally, we illustrate that this family of processes is useful for modeling real world data.

On a new family of weighted Gaussian processes: an application to bat telemetry data

TL;DR

This work introduces a new nonstationary Gaussian process family defined by the covariance kernel , derived from limits of rescaled occupation-time fluctuations, and proves its validity under mild integrability of . It shows that two subfamilies yield long-range dependence with logarithmic covariance growth and that the processes are not semimartingales, offering a flexible framework for modeling memory-rich movement data. The authors apply the model to bat telemetry, perform likelihood-based inference, and demonstrate that the logarithmic-growth kernel often provides superior fit compared to standard fractional-OU/FBM-based models, supported by AIC-based comparisons across multiple trajectories. An accompanying R Shiny app and web appendices provide simulation tools, proofs, and empirical stationarity assessments, underscoring the practical utility and theoretical robustness of the approach for ecological trajectory modeling.

Abstract

In this article we use a covariance function that arises from limit of fluctuations of the rescaled occupation time process of a branching particle system, to introduce a family of weighted long-range dependence Gaussian processes. In particular, we consider two subfamilies for which we show that the process is not a semimartingale, that the processes exhibit long-range dependence and have long-range memory of logarithmic order. Finally, we illustrate that this family of processes is useful for modeling real world data.
Paper Structure (14 sections, 5 theorems, 72 equations, 12 figures, 3 tables)

This paper contains 14 sections, 5 theorems, 72 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Application to telemetry data
  3. Preliminaries and previous work
  4. Inference and application to telemetry data
  5. On the R shiny interactive app
  6. Some theoretical results
  7. The new family of Gaussian processes
  8. Sample path properties
  9. Discussion
  10. Web Appendix A: Simulation study
  11. Simulations
  12. Inference on simulated data.
  13. Web Appendix B: Proofs of theoretical results.
  14. Web Appendix C: Empirical Evidence on the (Non-)Stationarity of the Application Data

Key Result

Theorem 1

Let $f:\mathbb{R}_{+}\to \mathbb{R}_{+}$ be a measurable function such that, for any $\delta>0$ Then, $K_{f}(s,t)$ given by (critical-wegithed-covariance) is a covariance function.

Figures (12)

  • Figure 1: Trajectory of bat ID $\#4$ showing longitude and latitude. The red point indicates the initial position, and the blue point indicates the final position.
  • Figure 2: Gaussian approximation (GA) applied to the likelihood of parameters in the $(\zeta_{t,f_{\sigma,\beta}})_{t\geq 0}$ model, fitted to latitude telemetry data of bat ID $\#4$.
  • Figure 3: (a) AIC comparison of the models $\zeta_{t,f_{\sigma,\beta}}$ and $\zeta_{t,f_{\alpha,0}}$ on longitude telemetry data of bat ID $\#4$. (b)-(c) GA of the likelihood profile of parameters to the application of the $(\zeta_{t,f_{\alpha,0}})_{t\geq 0}$ model on longitude telemetry data of bat ID $\#4$.
  • Figure 4: Output of the Telemetry Simulation module. The map displays a synthetic animal trajectory generated using two independent Gaussian processes with covariance kernels (\ref{['critical-wegithed-covariance']}). The plots on the side show the respective kernel functions $f_{\sigma_1,\beta_1}(u)$ and $f_{\sigma_2,\beta_2}(u)$ used for the longitude and latitude components.
  • Figure 5: Output of the Simulated trajectory $\zeta_{t,f}$ module. The top panel shows a Gaussian process trajectory generated using a user-defined kernel function $f(x)$. The bottom panel displays the corresponding function $f(x)$, which satisfies the integrability condition (\ref{['Integrability-condition2']}) and is used to construct the covariance matrix defined in (\ref{['critical-wegithed-covariance']}).
  • ...and 7 more figures

Theorems & Definitions (5)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Proposition 1
  • Proposition 2