The Annals of Applied Statistics

Modelling ocean temperatures from bio-probes under preferential sampling

Daniel Dinsdale and Matias Salibian-Barrera

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


In the last 25 years there has been an important increase in the amount of data collected from animal-mounted sensors (bio-probes) which are often used to study the animals’ behaviour or environment. We focus here on an example of the latter, where the interest is in sea surface temperature (SST), and measurements are taken from sensors mounted on elephant seals in the southern Indian Ocean. We show that standard geostatistical models may not be reliable for this type of data, due to the possibility that the regions visited by the animals may depend on the SST. This phenomenon is know in the literature as preferential sampling, and, if ignored, it may affect the resulting spatial predictions and parameter estimates. Research on this topic has been mostly restricted to stationary sampling locations such as monitoring sites. The main contribution of this manuscript is to extend this methodology to observations obtained by devices that move through the region of interest, as is the case with the tagged seals. More specifically, we propose a flexible framework for inference on preferentially sampled fields where the process that generates the sampling locations is stochastic and moving over time through a two-dimensional space. Our simulation studies confirm that predictions obtained from the preferential sampling model are more reliable when this phenomenon is present, and they compare very well to the standard ones when there is no preferential sampling. Finally, we note that the conclusions of our analysis of the SST data can change considerably when we incorporate preferential sampling in the model.

Article information

Ann. Appl. Stat., Volume 13, Number 2 (2019), 713-745.

Received: April 2018
Revised: October 2018
First available in Project Euclid: 17 June 2019

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Animal movement models bio-probes Laplace approximation preferential sampling template model builder


Dinsdale, Daniel; Salibian-Barrera, Matias. Modelling ocean temperatures from bio-probes under preferential sampling. Ann. Appl. Stat. 13 (2019), no. 2, 713--745. doi:10.1214/18-AOAS1217.

Export citation


  • Albertsen, C. M., Whoriskey, K., Yurkowski, D., Nielsen, A. and Flemming, J. M. (2015). Fast fitting of non-Gaussian state-space models to animal movement data via template model builder. Ecology 96 2598–2604.
  • Auger-Méthé, M., Field, C., Albertsen, C. M., Derocher, A. E., Lewis, M. A., Jonsen, I. D. and Flemming, J. M. (2016). State-space models’ dirty little secrets: Even simple linear Gaussian models can have estimation problems. Sci. Rep. 6 26677.
  • Auger-Méthé, M., Albertsen, C. M., Jonsen, I. D., Derocher, A. E., Lidgard, D. C., Studholme, K. R., Bowen, W. D., Crossin, G. T. and Flemming, J. M. (2017). Spatiotemporal modelling of marine movement data using Template Model Builder (TMB). Mar. Ecol. Prog. Ser. 565 237–249.
  • Banerjee, S. and Gelfand, A. E. (2003). On smoothness properties of spatial processes. J. Multivariate Anal. 84 85–100.
  • Banerjee, S., Gelfand, A. E. and Sirmans, C. F. (2003). Directional rates of change under spatial process models. J. Amer. Statist. Assoc. 98 946–954.
  • Bolker, B. M., Gardner, B., Maunder, M., Berg, C. W., Brooks, M., Comita, L., Crone, E., Cubaynes, S., Davies, T., Valpine, P. et al. (2013). Strategies for fitting nonlinear ecological models in R, AD model builder, and BUGS. Methods Ecol. Evol. 4 501–512.
  • Breed, G. A., Jonsen, I. D., Myers, R. A., Bowen, W. D. and Leonard, M. L. (2009). Sex-specific, seasonal foraging tactics of adult grey seals (Halichoerus grypus) revealed by state–space analysis. Ecology 90 3209–3221.
  • Breed, G. A., Costa, D. P., Jonsen, I. D., Robinson, P. W. and Mills-Flemming, J. (2012). State-space methods for more completely capturing behavioral dynamics from animal tracks. Ecol. Model. 235–236 49–58.
  • Brillinger, D. R., Stewart, B. S. and Littnan, C. L. (2008). Three months journeying of a Hawaiian monk seal. In Probability and Statistics: Essays in Honor of David A. Freedman. Inst. Math. Stat. (IMS) Collect. 2 246–264. IMS, Beachwood, OH.
  • Brillinger, D. R., Preisler, H. K., Ager, A. A., Kie, J. G. and Stewart, B. S. (2002). Employing stochastic differential equations to model wildlife motion. Bull. Braz. Math. Soc. (N.S.) 33 385–408.
  • Buderman, F. E., Hooten, M. B., Ivan, J. S. and Shenk, T. M. (2016). A functional model for characterizing long-distance movement behaviour. Methods Ecol. Evol. 7 264–273.
  • Carton, J. A. and Giese, B. S. (2008). A reanalysis of ocean climate using simple ocean data assimilation (SODA). Mon. Weather Rev. 136 2999–3017.
  • Carton, J. A., Chepurin, G. A., Chen, L. and Grodsky, S. A. (2018). Improved global net surface heat flux. J. Geophys. Res. 123 3144–3163.
  • Conn, P. B., Thorson, J. T. and Johnson, D. S. (2017). Confronting preferential sampling when analysing population distributions: Diagnosis and model-based triage. Methods Ecol. Evol. 8 1535–1546.
  • da Silva Ferreira, G. and Gamerman, D. (2015). Optimal design in geostatistics under preferential sampling. Bayesian Anal. 10 711–735.
  • Diggle, P. J., Menezes, R. and Su, T. (2010). Geostatistical inference under preferential sampling. J. R. Stat. Soc. Ser. C. Appl. Stat. 59 191–232.
  • Diggle, P. J. and Ribeiro, P. J. Jr. (2007). Model-Based Geostatistics. Springer Series in Statistics. Springer, New York.
  • Dinsdale, D. and Salibian-Barrera, M. (2019a). Methods for preferential sampling in geostatistics. J. R. Stat. Soc. Ser. C. Appl. Stat. 68 181–198.
  • Dinsdale, D. and Salibian-Barrera, M. (2019b). Supplement to “Modelling ocean temperatures from bio-probes under preferential sampling.” DOI:10.1214/18-AOAS1217SUPP.
  • Dujon, A. M., Lindstrom, R. T. and Hays, G. C. (2014). The accuracy of fastloc-GPS locations and implications for animal tracking. Methods Ecol. Evol. 5 1162–1169.
  • Evans, K., Lea, M.-A. and Patterson, T. (2013). Recent advances in bio-logging science: Technologies and methods for understanding animal behaviour and physiology and their environments. Deep-Sea Res., Part 2, Top. Stud. Oceanogr. 88 1–6.
  • Fedak, M. (2004). Marine animals as platforms for oceanographic sampling: A situation for biology and operational oceanography. Mem. Natl. Inst. Polar Res., Spec. Issue 58 133–147.
  • Fedak, M. A. (2013). The impact of animal platforms on polar ocean observation. Deep-Sea Res., Part 2, Top. Stud. Oceanogr. 88–89 7–13.
  • Gelfand, A. E., Sahu, S. K. and Holland, D. M. (2012). On the effect of preferential sampling in spatial prediction. Environmetrics 23 565–578.
  • Gloaguen, P., Etienne, M.-P. and Le Corff, S. (2018). Stochastic differential equation based on a multimodal potential to model movement data in ecology. J. R. Stat. Soc. Ser. C. Appl. Stat. 67 599–619.
  • Gneiting, T. (2013). Strictly and non-strictly positive definite functions on spheres. Bernoulli 19 1327–1349.
  • Gneiting, T. and Raftery, A. E. (2007). Strictly proper scoring rules, prediction, and estimation. J. Amer. Statist. Assoc. 102 359–378.
  • Gould, J., Roemmich, D., Wijffels, S., Freeland, H., Ignaszewsky, M., Jianping, X., Pouliquen, S., Desaubies, Y., Send, U., Radhakrishnan, K. et al. (2004). Argo profiling floats bring new era of in situ ocean observations. Eos Trans. AGU 85 185–191.
  • Griewank, A. and Walther, A. (2008). Evaluating Derivatives: Principles and Techniques of Algorithmic Differentiation, 2nd ed. SIAM, Philadelphia, PA.
  • Guinet, C., Vacquié-Garcia, J., Picard, B., Bessigneul, G., Lebras, Y., Dragon, A. C., Viviant, M., Arnould, J. P. and Bailleul, F. (2014). Southern elephant seal foraging success in relation to temperature and light conditions: Insight into prey distribution. Mar. Ecol. Prog. Ser. 499 285–301.
  • Gurarie, E. and Ovaskainen, O. (2011). Characteristic spatial and temporal scales unify models of animal movement. Amer. Nat. 178 113–123.
  • Gurarie, E., Fleming, C. H., Fagan, W. F., Laidre, K. L., Hernández-Pliego, J. and Ovaskainen, O. (2017). Correlated velocity models as a fundamental unit of animal movement: Synthesis and applications. Mov. Eco. 5 13.
  • Hooten, M. B. and Johnson, D. S. (2017a). Basis function models for animal movement. J. Amer. Statist. Assoc. 112 578–589.
  • Hooten, M. B., Johnson, D. S., McClintock, B. T. and Morales, J. M. (2017b). Animal Movement: Statistical Models for Telemetry Data. CRC Press, Boca Raton, FL.
  • Jacobs, S. (2006). Observations of change in the Southern Ocean. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 364 1657–1681.
  • Jeong, J. and Jun, M. (2015). A class of Matérn-like covariance functions for smooth processes on a sphere. Spat. Stat. 11 1–18.
  • Johnson, D. S., London, J. M., Lea, M.-A. and Durban, J. W. (2008). Continuous-time correlated random walk model for animal telemetry data. Ecology 89 1208–1215.
  • Jonsen, I. D., Flemming, J. M. and Myers, R. A. (2005). Robust state–space modeling of animal movement data. Ecology 86 2874–2880.
  • Kristensen, K., Nielsen, A., Berg, C. W., Skaug, H. and Bell, B. M. (2016). TMB: Automatic differentiation and Laplace approximation. J. Stat. Softw. 70 1–21. DOI:10.18637/jss.v070.i05.
  • Lindgren, F. and Rue, H. (2015). Bayesian spatial modelling with R-INLA. J. Stat. Softw. 63 1-25.
  • Lindgren, F., Rue, H. and Lindström, J. (2011). An explicit link between Gaussian fields and Gaussian Markov random fields: The stochastic partial differential equation approach. J. R. Stat. Soc. Ser. B. Stat. Methodol. 73 423–498.
  • Liu, Y., Battaile, B. C., Trites, A. W. and Zidek, J. V. (2015). Bias correction and uncertainty characterization of dead-reckoned paths of marine mammals. Animal Biotelemetry 3 51-61.
  • McClintock, B. T., King, R., Thomas, L., Matthiopoulos, J., McConnell, B. J. and Morales, J. M. (2012). A general discrete-time modeling framework for animal movement using multistate random walks. Ecol. Monogr. 82 335–349.
  • McIntyre, T., Ansorge, I. J., Bornemann, H., Plötz, J., Tosh, C. A. and Bester, M. N. (2011). Elephant seal dive behaviour is influenced by ocean temperature: Implications for climate change impacts on an ocean predator. Mar. Ecol. Prog. Ser. 441 257–272.
  • Morales, J. M., Haydon, D. T., Frair, J., Holsinger, K. E. and Fryxell, J. M. (2004). Extracting more out of relocation data: Building movement models as mixtures of random walks. Ecology 85 2436–2445.
  • Pati, D., Reich, B. J. and Dunson, D. B. (2011). Bayesian geostatistical modelling with informative sampling locations. Biometrika 98 35–48.
  • Preisler, H. K., Ager, A. A. and Wisdom, M. J. (2013). Analyzing animal movement patterns using potential functions. Ecosphere 4 1–13.
  • Robusto, C. C. (1957). Classroom Notes: The Cosine-Haversine Formula. Amer. Math. Monthly 64 38–40.
  • Roquet, F., Wunsch, C., Forget, G., Heimbach, P., Guinet, C., Reverdin, G., Charrassin, J.-B., Bailleul, F., Costa, D. P., Huckstadt, L. A. et al. (2013). Estimates of the Southern Ocean general circulation improved by animal-borne instruments. Geophys. Res. Lett. 40 6176–6180.
  • Roquet, F., Williams, G., Hindell, M. A., Harcourt, R., McMahon, C., Guinet, C., Charrassin, J.-B., Reverdin, G., Boehme, L., Lovell, P. et al. (2014). A southern Indian Ocean database of hydrographic profiles obtained with instrumented elephant seals. Sci. Data 1 140028.
  • Roulston, M. S. and Smith, L. A. (2002). Evaluating probabilistic forecasts using information theory. Mon. Weather Rev. 130 1653–1660.
  • Rue, H., Martino, S. and Chopin, N. (2009). Approximate Bayesian inference for latent Gaussian models by using integrated nested Laplace approximations. J. R. Stat. Soc. Ser. B. Stat. Methodol. 71 319–392.
  • Russell, J. C., Hanks, E. M., Haran, M. and Hughes, D. (2018). A spatially varying stochastic differential equation model for animal movement. Ann. Appl. Stat. 12 1312–1331.
  • Shaddick, G. and Zidek, J. V. (2014). A case study in preferential sampling: Long term monitoring of air pollution in the UK. Spat. Stat. 9 51–65.
  • Siegert, S., Ferro, C. A. and Stephenson, D. B. (2014). Evaluating ensemble forecasts by the ignorance score—correcting the finite-ensemble bias. Preprint. Available at arXiv:1410.8249.
  • Simpson, D., Lindgren, F. and Rue, H. (2012). In order to make spatial statistics computationally feasible, we need to forget about the covariance function. Environmetrics 23 65–74.
  • Ungar, E. D., Henkin, Z., Gutman, M., Dolev, A., Genizi, A. and Ganskopp, D. (2005). Inference of animal activity from GPS collar data on free-ranging cattle. Rangeland Ecology and Management 58 256–266.
  • Votier, S. C., Bearhop, S., Witt, M. J., Inger, R., Thompson, D. and Newton, J. (2010). Individual responses of seabirds to commercial fisheries revealed using GPS tracking, stable isotopes and vessel monitoring systems. J. Appl. Ecol. 47 487–497.
  • Weimerskirch, H., Bonadonna, F., Bailleul, F., Mabille, G., Dell’Omo, G. and Lipp, H.-P. (2002). GPS tracking of foraging albatrosses. Science 295 1259.
  • Whoriskey, K., Auger-Méthé, M., Albertsen, C. M., Whoriskey, F. G., Binder, T. R., Krueger, C. C. and Flemming, J. M. (2017). A hidden Markov movement model for rapidly identifying behavioral states from animal tracks. Ecol. Evol. 7 2112–2121.
  • Wilson, R. and Wilson, M.-P. (1988). Dead reckoning: A new technique for determining penguin movements at sea. Meeresforschung 32 155–158.

Supplemental materials

  • Simulation Code. Code used for the simulations in this paper, with an example shown in the README.