Skip to main content
Log in

The Conditionally Autoregressive Hidden Markov Model (CarHMM): Inferring Behavioural States from Animal Tracking Data Exhibiting Conditional Autocorrelation

  • Published:
Journal of Agricultural, Biological and Environmental Statistics Aims and scope Submit manuscript


One of the central interests of animal movement ecology is relating movement characteristics to behavioural characteristics. The traditional discrete-time statistical tool for inferring unobserved behaviours from movement data is the hidden Markov model (HMM). While the HMM is an important and powerful tool, sometimes it is not flexible enough to appropriately fit the data. Data for marine animals often exhibit conditional autocorrelation, self-dependence of the step length process that cannot be explained solely by the behavioural state, which violates one of the main assumptions of the HMM. Using a grey seal track as an example we motivate and develop the conditionally autoregressive hidden Markov model (CarHMM), a generalization of the HMM designed specifically to handle conditional autocorrelation. In addition to introducing and examining the new CarHMM with numerous simulation studies, we provide guidelines for all stages of an analysis using either an HMM or CarHMM. These include guidelines for pre-processing location data to obtain deflection angles and step lengths, model selection, and model checking. In addition to these practical guidelines, we link estimated model parameters to biologically relevant quantities such as activity budget and residency time. We also provide interpretations of traditional “foraging” and “transiting” behaviours in the context of the new CarHMM parameters.

Supplementary materials accompanying this paper appear online.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others


  • Ailliot, P. (2006). Some theoretical results on Markov-switching autoregressive models with gamma innovations. Comptes Rendus Mathematique, 343(4):271–274.

    Article  MathSciNet  Google Scholar 

  • 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. Ecological Modelling, 235-236:49 – 58.

    Article  Google Scholar 

  • Curry, D. M. (2014). An algorithm for clustering animals by species based upon daily movement. Procedia Computer Science, 36:629 – 636.

    Article  Google Scholar 

  • Douc, R., Moulines, E., Rydén, T., et al. (2004). Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. The Annals of Statistics, 32(5):2254–2304.

    Article  MathSciNet  Google Scholar 

  • Forester, J. D., Ives, A. R., Turner, M. G., Anderson, D. P., Fortin, D., Beyer, H. L., Smith, D. W., and Boyce, M. S. (2007). State-space models link elk movement patterns to landscape characteristics in Yellowstone National Park. Ecological Monographs, 77(2):285–299.

    Article  Google Scholar 

  • Hooten, M., Johnson, D., McClintock, B., and Morales, J. (2017a). Animal Movement: Statistical Models for Telemetry Data. CRC Press, Boca Raton.

    Book  Google Scholar 

  • Hooten, M. B., King, R., and Langrock, R. (2017b). Guest editor’s introduction to the special issue on “animal movement modeling”. Journal of Agricultural, Biological and Environmental Statistics, 22(3):224–231.

    Article  MathSciNet  Google Scholar 

  • Hussey, N. E., Kessel, S. T., Aarestrup, K., Cooke, S. J., Cowley, P. D., Fisk, A. T., Harcourt, R. G., Holland, K. N., Iverson, S. J., Kocik, J. F., Mills Flemming, J. E., and Whoriskey, F. G. (2015). Aquatic animal telemetry: A panoramic window into the underwater world. Science, 348(6240):1255642.

    Article  Google Scholar 

  • Jonsen, I. D., Flemming, J. M., and Myers, R. A. (2005). Robust state-space modeling of animal movement data. Ecology, 86(11):2874–2880.

    Article  Google Scholar 

  • Kristensen, K., Nielsen, A., Berg, C. W., Skaug, H. J., and Bell, B. M. (2016). TMB: Automatic differentiation and Laplace approximation. Journal of Statistical Software, 70(5):1–21.

    Article  Google Scholar 

  • Langrock, R., King, R., Matthiopoulos, J., Thomas, L., Fortin, D., and Morales, J. M. (2012). Flexible and practical modeling of animal telemetry data: hidden Markov models and extensions. Ecology, 93(11):2336–2342.

    Article  Google Scholar 

  • Leos-Barajas, V., Gangloff, E. J., Adam, T., Langrock, R., van Beest, F. M., Nabe-Nielsen, J., and Morales, J. M. (2017). Multi-scale modeling of animal movement and general behavior data using hidden Markov models with hierarchical structures. Journal of Agricultural, Biological and Environmental Statistics, 22(3):232–248.

    Article  MathSciNet  Google Scholar 

  • McClintock, B. T. (2017). Incorporating telemetry error into hidden Markov models of animal movement using multiple imputation. Journal of Agricultural, Biological and Environmental Statistics, 22(3):249–269.

    Article  MathSciNet  Google Scholar 

  • 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. Ecological Monographs, 82(3):335–349.

    Article  Google Scholar 

  • McClintock, B. T. and Michelot, T. (2018). momentuhmm: R package for generalized hidden Markov models of animal movement. Methods in Ecology and Evolution, 9(6):1518–1530.

    Article  Google Scholar 

  • Michelot, T., Langrock, R., and Patterson, T. A. (2016). movehmm: An R package for the statistical modelling of animal movement data using hidden Markov models. Methods in Ecology and Evolution, 7(11):1308–1315.

    Article  Google Scholar 

  • 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(9):2436–2445.

    Article  Google Scholar 

  • Patterson, T. A., Parton, A., Langrock, R., Blackwell, P. G., Thomas, L., and King, R. (2017). Statistical modelling of individual animal movement: an overview of key methods and a discussion of practical challenges. AStA Advances in Statistical Analysis, 101(4):399–438.

    Article  MathSciNet  Google Scholar 

  • Pohle, J., Langrock, R., van Beest, F. M., and Schmidt, N. M. (2017). Selecting the number of states in hidden Markov models: Pragmatic solutions illustrated using animal movement. Journal of Agricultural, Biological and Environmental Statistics, 22(3):270–293.

    Article  MathSciNet  Google Scholar 

  • Potts, J. R., Marie, A., Karl, M., and Lewis, M. A. (2014). A generalized residual technique for analysing complex movement models using earth mover’s distance. Methods in Ecology and Evolution, 5(10):1012–1022.

    Article  Google Scholar 

  • Scharf, H., Hooten, M. B., and Johnson, D. S. (2017). Imputation approaches for animal movement modeling. Journal of Agricultural, Biological and Environmental Statistics, 22(3):335–352.

    Article  MathSciNet  Google Scholar 

  • Shepherd, B. E., Li, C., and Liu, Q. (2016). Probability-scale residuals for continuous, discrete, and censored data. Canadian Journal of Statistics, 44(4):463–479.

    Article  MathSciNet  Google Scholar 

  • Whoriskey, K., Auger-Méthé, M., Albertsen, C. M., Whoriskey, F. G., Binder, T. R., Krueger, C. C., and Mills Flemming, J. (2017). A hidden Markov movement model for rapidly identifying behavioral states from animal tracks. Ecology and Evolution, 7(7):2112–2121.

    Article  Google Scholar 

  • Zucchini, W., MacDonald, I. L., and Langrock, R. (2016). Hidden Markov models for time series: an introduction using R (Vol. 150). Boca Raton: CRC Press.

    MATH  Google Scholar 

Download references


The authors would like to thank the associate editor and two reviewers who provided immensely helpful comments and, in particular, gave additional focus to the simulation section. We also thank the Ocean Tracking Network, Damian Lidgard at Dalhousie University, and Dan Bowen at the Department of Fisheries and Oceans for allowing the use of the grey seal data used in Sects. 5 and 6. This research was funded by a Canadian Statistical Sciences Institute Collaborative Research Team, and a Vanier Canada Graduate Scholarship and Killam Predoctoral Scholarship to the first author.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Ethan Lawler.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (pdf 616 KB)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lawler, E., Whoriskey, K., Aeberhard, W.H. et al. The Conditionally Autoregressive Hidden Markov Model (CarHMM): Inferring Behavioural States from Animal Tracking Data Exhibiting Conditional Autocorrelation. JABES 24, 651–668 (2019).

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: