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A new family of Gaussian processes for modeling animal movement: application to bat telemetry data

This paper introduces a new family of Gaussian processes derived from branching particle systems, characterized by long-range dependence and non-stationarity, which are shown to effectively model animal movement with strong memory through an analysis of bat telemetry data.

Original authors: Jose Hermenegildo Ramirez Gonzalez, Antonio Murillo Salas, Ying Sun

Published 2026-07-17

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CC BY 4.0

Original authors: Jose Hermenegildo Ramirez Gonzalez, Antonio Murillo Salas, Ying Sun

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the path of a wandering explorer. In the world of science, this explorer is an animal, and the map is a computer model called a "Gaussian process." Think of these models as mathematical crystal balls. They don't just guess where an animal will go next; they look at where it has been to figure out how it thinks. Some animals are like forgetful tourists who only care about the step they are taking right now. Others are like deep thinkers with long memories, where a decision made hours or even days ago still influences their current direction. This paper lives in the field of statistics and ecology, a corner of science dedicated to understanding how living things move through space and time. The key idea here is "long-range dependence," which is just a fancy way of saying that the past has a heavy, lingering weight on the future. Scientists care about this because if we want to protect endangered species or understand how they migrate, we need to know if they are reacting to the wind blowing right now or remembering a storm from three days ago.

The authors of this paper, a team of statisticians, decided to build a brand-new type of crystal ball specifically for animals with very strong memories. They noticed that the old tools used to track animal movement were like trying to measure the ocean with a ruler; they worked okay for short trips, but they struggled when the animal's path showed signs of remembering things from far back in time. The researchers created a new family of mathematical models derived from some very complex physics involving branching particles (think of a tree where every branch splits into more branches, and we are tracking how long the "branches" stay occupied). They found that their new model has a special superpower: it grows its memory in a "logarithmic" way.

To understand the difference, imagine two ways a memory could fade. The old models often assumed that if you look far enough back in time, the memory fades away quickly, like a shout that gets quieter and quieter until it's gone (exponential decay). Or, they assumed the memory grew so fast that it became overwhelming (polynomial growth). The new model, however, suggests the memory grows slowly, like a whisper that never quite disappears but gets louder very gradually. It's a "Goldilocks" memory: not too fast, not too slow, just right for animals that seem to carry their history with them without being crushed by it.

The team tested this new model using real-life data from five bats flying around in Germany. They tracked the bats' longitude and latitude (their east-west and north-south positions) over time. When they fed this data into their new "logarithmic memory" model, it fit the bats' movements surprisingly well. In fact, for two out of the ten different flight paths they analyzed, their new model was the best fit of all, beating out the popular "fractional Ornstein-Uhlenbeck" models that scientists usually use. The authors suggest that for these specific bats, the idea that their memory grows slowly (logarithmically) describes their behavior better than the idea that it grows explosively fast.

However, the paper is careful not to declare a total victory. The new model didn't win every single race; in seven out of the ten cases, the older, more complex models still did a slightly better job. The authors explicitly argue against the idea that all animal movement can be described by simple, stationary models (models that assume the animal's behavior doesn't change over time). Their data showed that the bats' movements were "non-stationary," meaning their behavior changed as the day went on, and their memory wasn't just a simple repetition of the past. They also ruled out the idea that a standard "integrated Ornstein-Uhlenbeck" process (a model based on standard Brownian motion) could explain the long-term patterns they saw, because that model's memory fades too quickly to match the bats' actual behavior.

To help others use their discovery, the team didn't just write equations; they built a fun, interactive app. Imagine a video game where you can tweak the "memory settings" of a virtual bat and watch it fly across a map. You can see how changing the math changes the flight path, making the abstract numbers feel real. They also provided a way for other scientists to plug in their own data to see if this new "slow-growing memory" model fits their animals too.

In the end, this paper suggests that when we look at animals with strong memories, like bats, chimpanzees, or elephants, we might need to stop using tools that assume memory fades too fast or grows too wild. Instead, we should consider tools that let the past influence the future in a steady, slow, logarithmic whisper. While the new model isn't a magic bullet that solves every movement puzzle, it offers a fresh, more accurate lens for understanding the deep, lingering connections between an animal's past and its future.

Technical Summary: A New Family of Gaussian Processes for Modeling Animal Movement

Problem Statement
Modeling animal movement is critical for ecological and biological research, particularly for understanding migratory species threatened by habitat loss and climate change. A significant challenge in this domain is developing predictive models that accurately capture trajectories exhibiting strong memory-dependent movement. While Gaussian processes (GPs) are established tools for describing animal trajectories, existing models often struggle to represent long-range dependence (LRD) where remote past positions continue to influence future movement without overstating that influence. Furthermore, empirical diagnostics on bat telemetry data suggest that observed trajectories are often non-stationary and not intrinsically stationary, necessitating models that can accommodate these properties while exhibiting specific asymptotic covariance behaviors.

Methodology
The authors introduce a new family of centered, non-stationary Gaussian processes, denoted as $\zeta = (\zeta_t)_{t \geq 0}$ , derived from the limiting fluctuations of the rescaled occupation-time process of certain branching particle systems. The covariance function is defined as:
$K_f(s, t) := 2 \int_0^{s \wedge t} f(r) \left[ (s+t-2r)\log(s+t-2r) - (s-r)\log(s-r) - (t-r)\log(t-r) \right] dr$
where $f: \mathbb{R}^+ \to \mathbb{R}^+$ is a measurable, locally integrable weighting function.

The study focuses on two specific subfamilies:

Polynomial weights ( $C_1$ ): $f_\alpha(u) = u^\alpha$ for $\alpha > -1$ .
Exponential weights ( $C_2$ ): $f_{\sigma^2, \beta}(u) = \sigma^2 e^{-\beta u}$ for $\sigma^2 > 0, \beta \geq 0$ .

The exponential subfamily ( $C_2$ ) is selected for applied analysis due to its structural similarity to the covariance of the integrated fractional Ornstein–Uhlenbeck process ( $\mu_H$ ), a common benchmark in telemetry modeling. The authors derive the likelihood function for finite-dimensional distributions of $\zeta$ under the $C_2$ specification, enabling maximum likelihood estimation (MLE) of parameters $\sigma^2$ and $\beta$ .

Key Theoretical Results

Positive Definiteness: Theorem 1 establishes sufficient conditions on $f$ (local integrability) for $K_f$ to be a valid covariance function.
Long-Range Dependence (LRD): Proposition 1 demonstrates that the process exhibits LRD with index $\kappa = 1$ . Specifically, the covariance between a fixed past position and a remote future position grows at a logarithmic order:
$\lim_{T \to \infty} \frac{E(\zeta_r \zeta_{s+T})}{\log(T)} = 2 \int_0^r f(u)(r-u) du$
This contrasts with models exhibiting polynomial covariance growth, offering a "milder" retention of remote past influence.
Stationarity Properties: The process is shown to be non-stationary and, for weights satisfying $1 \asymp f$ on compact intervals, not intrinsically stationary on compact time intervals.
Path Properties: The process is proven to have infinite total variation and zero quadratic variation on $[0, 1]$ , implying it is not a semimartingale. It possesses Hölder continuous paths with specific regularity indices depending on the weight function.

Application and Empirical Results
The proposed model was applied to three-dimensional migratory movement data from five bats in Germany (Telemetry data from Teague O'Mara et al., 2019).

Data Diagnostics: Empirical analysis (rolling means, variances, ADF, KPSS, and DFA tests) indicated that raw telemetry series are non-stationary. First differencing did not uniformly induce stationarity, and DFA exponents exceeded 1 for all series, suggesting strong large-scale dependence.
Model Fitting: The $\zeta$ process with exponential weights ( $C_2$ ) was fitted separately to longitude and latitude coordinates.
Comparative Performance: The proposed model was compared against several competing models, including the fractional Ornstein–Uhlenbeck position model ( $\mu_H$ $μ_{H}$ ), integrated Ornstein–Uhlenbeck ( $\mu_{1/2}$ $μ_{1/2}$ ), fractional Brownian motion ( $\sigma W_H$ $σ W_{H}$ ), and stationary confluent hypergeometric and Stein-type models.
- The proposed $\zeta$ -type model achieved the smallest Akaike Information Criterion (AIC) in two of the ten coordinate series.
- The fractional Ornstein–Uhlenbeck model ( $\mu_H$ ) achieved the best fit in seven series.
- In nine of the ten datasets, the preferred model was one of these two non-stationary, non-intrinsically stationary LRD models.
- Notably, for Bat #4 longitude, the boundary model with $\beta=0$ (constant weight) provided the best fit, suggesting that for some trajectories, the logarithmic growth regime may be more appropriate than the polynomial growth induced by $\mu_H$ when $H > 1/2$ .

Significance and Claims
The paper claims that the proposed family of Gaussian processes offers a useful alternative for modeling animal movement in settings where trajectories exhibit strong memory but where polynomial covariance growth (as seen in fractional Brownian motion or integrated fractional Ornstein–Uhlenbeck models) may overstate the influence of distant past positions. The logarithmic asymptotic growth of the covariance provides a distinct LRD structure that retains remote past contributions in a "milder" way.

The authors emphasize that the model is particularly relevant for species with robust memory capabilities, such as bats, chimpanzees, and elephants. To facilitate practical application, the authors have released the inference code and developed an interactive R Shiny application for simulation, estimation, and prediction.

Limitations and Future Directions
The study acknowledges that the assumption of uncorrelated coordinate processes (longitude and latitude) simplifies analysis but may be inadequate if cross-coordinate dependence is non-negligible. A natural extension proposed is the development of a multidimensional version of the model. Additionally, while maximum likelihood estimators were derived, asymptotic results for parameter estimation within the $C_1$ and $C_2$ families are currently unavailable, representing a direction for future research.

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