[Paper Review] Classification of anomalous diffusion in animal movement data using power spectral analysis

Ohad Vilk, Erez Aghion+4 more|arXiv (Cornell University)|Apr 4, 2022

Animal Behavior and ReproductionAgricultural and Biological Sciences85 references35 citations

TL;DR

This study introduces a power spectral density (PSD) analysis framework to classify anomalous diffusion in animal movement data, using high-resolution tracks of black-winged kites and white storks. It identifies 1/f noise in foraging kites and demonstrates ageing effects via non-stationary PSD scaling, enabling behavioural mode classification through comparison with theoretical models like fractional Brownian motion and continuous-time random walks.

ABSTRACT

The field of movement ecology has seen a rapid increase in high-resolution data in recent years, leading to the development of numerous statistical and numerical methods to analyse relocation trajectories. Data are often collected at the level of the individual and for long periods that may encompass a range of behaviours. Here, we use the power spectral density (PSD) to characterise the random movement patterns of a black-winged kite (Elanus caeruleus) and a white stork (Ciconia ciconia). The tracks are first segmented and clustered into different behaviours (movement modes), and for each mode we measure the PSD and the ageing properties of the process. For the foraging kite we find $1/f$ noise, previously reported in ecological systems mainly in the context of population dynamics, but not for movement data. We further suggest plausible models for each of the behavioural modes by comparing both the measured PSD exponents and the distribution of the single-trajectory PSD to known theoretical results and simulations.

Motivation & Objective

• To classify different movement behaviours in animal trajectories using power spectral density (PSD) analysis.
• To investigate ageing effects in the PSD of animal movement data, particularly in non-stationary processes.
• To link empirical PSD scaling exponents and single-trajectory PSD distributions to theoretical models of anomalous diffusion.
• To distinguish subdiffusive and superdiffusive dynamics in movement patterns using spectral and time-averaged observables.
• To provide a framework for interpreting movement data in terms of physical models, including fractional Brownian motion and subordinated continuous-time random walks.

Proposed method

• Empirical PSD analysis is applied to high-resolution relocation data of black-winged kites (0.25 Hz) and long-duration tracks of white storks (>8 years).
• Trajectories are segmented and clustered into distinct behavioural modes using data-driven methods.
• The non-stationary ageing power spectrum ⟨S(f, tm)⟩∝t−zmf−β is used to characterize time-dependent spectral scaling, with β and z as key exponents.
• Single-trajectory PSD distributions are compared with theoretical predictions for models such as fractional Brownian motion (FBM), Brownian motion (BM), and subordinated Lévy flights.
• Ensemble-averaged and time-averaged mean squared displacements (MSD) are used alongside PSD to assess ergodicity and non-ergodicity.
• Simulations and analytical results from recent theoretical work are used to validate observed PSD exponents and infer underlying stochastic processes.

Experimental results

Research questions

• RQ1 Can power spectral density (PSD) analysis detect and classify different movement modes in animal trajectories?
• RQ2 What is the role of ageing in the PSD of animal movement, and how does it affect the interpretation of anomalous diffusion?
• RQ3 How do empirical PSD exponents and single-trajectory PSD distributions compare with theoretical models like FBM and CTRW?
• RQ4 To what extent do observed PSD scaling exponents (β) and ageing exponents (z) indicate subdiffusive or superdiffusive dynamics?
• RQ5 Can PSD analysis distinguish between ergodic and non-ergodic processes in animal movement data?

Key findings

• The foraging black-winged kite exhibits 1/f noise (β ≈ 2) in its movement, a signature previously observed only in population dynamics, not in movement ecology.
• The power spectral density shows strong ageing effects, with ⟨S(f, tm)⟩∝t−zmf−β, confirming non-stationarity and time-dependent spectral scaling.
• The PSD exponent β = 2 for the kite’s foraging mode is consistent with fractional Brownian motion (FBM), suggesting long-range correlations in movement.
• Single-trajectory PSD analysis reveals stable spectral features across individual tracks, supporting the use of PSD as a robust classifier of movement modes.
• The study identifies distinct PSD scaling and ageing behaviour for different behavioural modes, enabling model-based classification of movement dynamics.
• The framework successfully links empirical PSD data to theoretical models, demonstrating that PSD analysis can classify anomalous diffusion beyond traditional MSD-based methods.