The Annals of Applied Statistics

Bayesian data fusion approaches to predicting spatial tracks: Application to marine mammals

Yang Liu, James V. Zidek, Andrew W. Trites, and Brian C. Battaile

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Bayesian Melding (BM) and downscaling are two Bayesian approaches commonly used to combine data from different sources for statistical inference. We extend these two approaches to combine accurate but sparse direct observations with another set of high-resolution but biased calculated observations. We use our methods to estimate the path of a moving or evolving object and apply them in a case study of tracking northern fur seals. To make the BM approach computationally feasible for high-dimensional (big) data, we exploit the properties of the processes along with approximations to the likelihood to break the high-dimensional problem into a series of lower dimensional problems. To implement the alternative, downscaling approach, we use R-INLA to connect the two sources of observations via a linear mixed effect model. We compare the predictions of the two approaches by cross-validation as well as simulations. Our results show that both approaches yield similar results—both provide accurate, high resolution estimates of the at-sea locations of the northern fur seals, as well as Bayesian credible intervals to characterize the uncertainty about the estimated movement paths.

Article information

Ann. Appl. Stat., Volume 10, Number 3 (2016), 1517-1546.

Received: January 2016
Revised: May 2016
First available in Project Euclid: 28 September 2016

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Bayesian melding downscaling bio-logging conditional independence INLA Dead-Reckoning tracking marine mammals Northern fur seal


Liu, Yang; Zidek, James V.; Trites, Andrew W.; Battaile, Brian C. Bayesian data fusion approaches to predicting spatial tracks: Application to marine mammals. Ann. Appl. Stat. 10 (2016), no. 3, 1517--1546. doi:10.1214/16-AOAS945.

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Supplemental materials

  • Supplement to “Bayesian data fusion approaches to predicting spatial tracks: Application to marine mammals”. Supplement A: Details of the dead-reckoning algorithm. We provide additional details of the Dead-Reckoning Algorithm to help understand how it works and why it is biased. Supplement B: Details of Bayesian melding. This supplement includes the detailed derivations of the inferential methods needed for our Bayesian Melding approach. It includes the following subsections: 1. Explicit form of $\langle\boldsymbol{\beta} ,\boldsymbol{\eta} |\mathbf{X},\mathbf{Y}\rangle$. 2. Derivation of (3.10) and (3.11). 3. Explicit expression for (3.11). 4. Explicit expression of $[\boldsymbol{\phi} ,\boldsymbol{\beta} ,\boldsymbol{\eta}_{G}|\mathbf{X}_{G},\mathbf{Y}]$. 5. Marginal distribution of $\boldsymbol{\eta}$ at the non-GPS points. 6. Integration over the variance parameters $\boldsymbol{\phi}$. 7. The goodness of normal approximation credible intervals. Supplement C: Figures in color. The colored version of Figures 1, 4, 5 and 6.