Estimating migration proportions from discretely observed continuous diffusion processes

We model two time and space scales discrete observations by using a unique continuous diffusion process with time dependent coefficient. We define new parameters for the large scale model as functions of the small scale distribution cumulants. We use the non - uniform distribution of the observation time intervals to obtain consistent and unbiased estimators for these parameters. Closed form expressions for migration proportions between spatial domains are derived as functions of these parameters. The models are applied to estimate migration patterns from satellite tag data.

💡 Research Summary

The paper tackles the problem of estimating large‑scale migration proportions when the underlying movement process is observed only at discrete, irregular time points. The authors propose a unified continuous‑diffusion framework in which the diffusion coefficient D(t) is allowed to vary with time, thereby capturing both short‑term fine‑scale movements and long‑term cumulative displacement within a single stochastic model.

At the core of the methodology is the decomposition of the observed data into “small‑scale” increments ΔX_i measured over non‑uniform intervals Δt_i. Each increment is assumed to follow a Gaussian distribution with mean v_iΔt_i (where v_i is a possibly time‑varying drift) and covariance 2D_iΔt_i. Traditional maximum‑likelihood estimators become biased when Δt_i are heterogeneous, because they implicitly treat each observation as equally informative. To overcome this, the authors derive unbiased, consistent estimators by weighting each increment according to its interval length. The resulting estimators are:

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