The Annals of Applied Statistics

A multivariate mixed hidden Markov model for blue whale behaviour and responses to sound exposure

Stacy L. DeRuiter, Roland Langrock, Tomas Skirbutas, Jeremy A. Goldbogen, John Calambokidis, Ari S. Friedlaender, and Brandon L. Southall

Full-text: Open access


Characterization of multivariate time series of behaviour data from animal-borne sensors is challenging. Biologists require methods to objectively quantify baseline behaviour, and then assess behaviour changes in response to environmental stimuli. Here, we apply hidden Markov models (HMMs) to characterize blue whale movement and diving behaviour, identifying latent states corresponding to three main underlying behaviour states: shallow feeding, travelling, and deep feeding. The model formulation accounts for inter-whale differences via a computationally efficient discrete random effect, and measures potential effects of experimental acoustic disturbance on between-state transition probabilities. We identify clear differences in blue whale disturbance response depending on the behavioural context during exposure, with whales less likely to initiate deep foraging behaviour during exposure. Findings are consistent with earlier studies using smaller samples, but the HMM approach provides a more nuanced characterization of behaviour changes.

Article information

Ann. Appl. Stat. Volume 11, Number 1 (2017), 362-392.

Received: February 2016
Revised: December 2016
First available in Project Euclid: 8 April 2017

Permanent link to this document

Digital Object Identifier

Forward algorithm hidden Markov model multivariate time series numerical maximum likelihood random effects blue whales


DeRuiter, Stacy L.; Langrock, Roland; Skirbutas, Tomas; Goldbogen, Jeremy A.; Calambokidis, John; Friedlaender, Ari S.; Southall, Brandon L. A multivariate mixed hidden Markov model for blue whale behaviour and responses to sound exposure. Ann. Appl. Stat. 11 (2017), no. 1, 362--392. doi:10.1214/16-AOAS1008.

Export citation


  • Altman, R. M. (2007). Mixed hidden Markov models: An extension of the hidden Markov model to the longitudinal data setting. J. Amer. Statist. Assoc. 102 201–210.
  • Antunes, R., Kvadsheim, P. H., Lam, F. P. A., Tyack, P. L., Thomas, L., Wensveen, P. J. and Miller, P. J. O. (2014). High thresholds for avoidance of sonar by free-ranging long-finned pilot whales (Globicephala melas). Mar. Pollut. Bull. 83 165–180.
  • Bagniewska, J. M., Hart, T., Harrington, L. A. and Macdonald, D. W. (2013). Hidden Markov analysis describes dive patterns in semiaquatic animals. Behav. Ecol. 24 659–667.
  • Bulla, J., Lagona, F., Maruotti, A. and Picone, M. (2012). A multivariate hidden Markov model for the identification of sea regimes from incomplete skewed and circular time series. J. Agric. Biol. Environ. Stat. 17 544–567.
  • Dean, B., Freeman, R., Kirk, H., Leonard, K., Phillips, R. A., Perrins, C. M. and Guilford, T. (2012). Behavioural mapping of a pelagic seabird: Combining multiple sensors and a hidden Markov model reveals the distribution of at-sea behaviour. J. R. Soc. Interface 20120570. DOI:10.1098/rsif.2012.0570.
  • DeRuiter, S. L. (2010). Marine animal acoustics. In An Introduction to Underwater Acoustics (X. Lurton, ed.) 425–474. Praxis Publishing Limited, Chichester, UK.
  • DeRuiter, S. L., Southall, B. L., Calambokidis, J., Zimmer, W. M. X., Sadykova, D., Falcone, E. A., Friedlaender, A. S., Joseph, J. E., Moretti, D., Schorr, G. S., Thomas, L. and Tyack, P. L. (2013). First direct measurements of behavioural responses by Cuvier’s beaked whales to mid-frequency active sonar. Biol. Lett. 9 20130223. DOI:10.1098/rsbl.2013.0223.
  • DeRuiter, S. L., Langrock, R., Skirbutas, T., Goldbogen, J. A., Calambokidis, J., Friedlaender, A. S. and Southall, B. L. (2017a). Supplement to “A multivariate mixed hidden Markov model for blue whale behaviour and responses to sound exposure.” DOI:10.1214/16-AOAS1008SUPPA.
  • DeRuiter, S. L., Langrock, R., Skirbutas, T., Goldbogen, J. A., Calambokidis, J., Friedlaender, A. S. and Southall, B. L. (2017b). Supplement to “A multivariate mixed hidden Markov model for blue whale behaviour and responses to sound exposure.” DOI:10.1214/16-AOAS1008SUPPB.
  • Donovan, C. R., Harris, C., Harwood, J. and Milazzo, L. (2012). A simulation-based method for quantifying and mitigating the effects of anthropogenic sound on marine mammals. In Proceedings of Meetings on Acoustics 17 070043. Acoustical Society of America, Melville, NY.
  • Eddelbuettel, D. (2013). Seamless R and C++ Integration with Rcpp. Springer, New York.
  • Ellison, W. T., Southall, B. L., Clark, C. W. and Frankel, A. S. (2012). A new context-based approach to assess marine mammal behavioral responses to anthropogenic sounds. Conserv. Biol. 26 21–28.
  • Friedlaender, A. S., Goldbogen, J. A., Hazen, E. L., Calambokidis, J. and Southall, B. L. (2015). Feeding performance by sympatric blue and fin whales exploiting a common prey resource. Mar. Mamm. Sci. 31 345–354.
  • Friedlaender, A. S., Hazen, E. L., Goldbogen, J. A., Stimpert, A. K., Calambokidis, J. and Southall, B. L. (2016). Prey-mediated behavioral responses of feeding blue whales in controlled sound exposure experiments. Ecol. Appl. 26 1075–1085.
  • Goldbogen, J. A., Southall, B. L., DeRuiter, S. L., Calambokidis, J., Friedlaender, A. S., Hazen, E. L., Falcone, E. A., Schorr, G. S., Douglas, A., Moretti, D. J., Kyburg, C., McKenna, M. F. and Tyack, P. L. (2013a). Blue whales respond to simulated mid-frequency military sonar. Proc. Biol. Sci. 280 20130657.
  • Goldbogen, J. A., Friedlaender, A. S., Calambodkis, J., McKenna, M. F., Simon, M. and Nowacek, D. P. (2013b). Integrative approaches to the study of baleen whale diving behavior, feeding performance, and foraging ecology. Bioscience 63 90–100.
  • Goldbogen, J. A., Hazen, E. L., Friedlaender, A. S., Calambokidis, J., DeRuiter, S. L., Stimpert, A. K. and Southall, B. L. (2015). Prey density and distribution drive the three-dimensional foraging strategies of the largest filter feeder. Funct. Ecol. 29 951–961.
  • Hart, T., Mann, R., Coulson, T., Pettorelli, N. and Trathan, P. (2010). Behavioural switching in a central place forager: Patterns of diving behaviour in the macaroni penguin (Eudyptes chrysolophus). Mar. Biol. 157 1543–1553.
  • Hazen, E. L., Friedlaender, A. S. and Goldbogen, J. A. (2015). Blue whales (Balaenoptera musculus) optimize foraging efficiency by balancing oxygen use and energy gain as a function of prey density. Sci. Adv. 1 e1500469.
  • Holyoak, M., Casagrandi, R., Nathan, R., Revilla, E. and Spiegel, O. (2008). Trends and missing parts in the study of movement ecology. Proc. Natl. Acad. Sci. USA 105 19060–19065.
  • Houser, D. S. (2006). A method for modeling marine mammal movement and behavior for environmental impact assessment. IEEE J. Oceanic Eng. 31 76–81.
  • Isojunno, S. and Miller, P. J. O. (2015). Sperm whale response to tag boat presence: Biologically informed hidden state models quantify lost feeding opportunities. Ecosphere 6 1–46.
  • James, F. C. and McCulloch, C. E. (1990). Multivariate analysis in ecology and systematics: Panacea or Pandora’s box?. Ann. Rev. Ecolog. Syst. 21 129–166.
  • Johnson, M. P. and Tyack, P. L. (2003). A digital acoustic recording tag for measuring the response of wild marine mammals to sound. IEEE J. Oceanic Eng. 28 3–12.
  • Johnson, M. P., Hickmott, L. S., Soto, N. A. and Madsen, P. T. (2008). Echolocation behaviour adapted to prey in foraging Blainville’s beaked whale (Mesoplodon densirostris). Proceedings of the Royal Society of London B: Biological Sciences 275 133–139.
  • King, S. L., Schick, R. S., Donovan, C., Booth, C. G., Burgman, M., Thomas, L. and Harwood, J. (2015). An interim framework for assessing the population consequences of disturbance. Methods Ecol. Evol. 6 1150–1158.
  • 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 2336–2342.
  • Langrock, R., Marques, T. A., Baird, R. W. and Thomas, L. (2014). Modeling the diving behavior of whales: A latent-variable approach with feedback and semi-Markovian components. J. Agric. Biol. Environ. Stat. 19 82–100.
  • Langrock, R., Kneib, T., Sohn, A. and DeRuiter, S. L. (2015). Nonparametric inference in hidden Markov models using P-splines. Biometrics 71 520–528.
  • MacDonald, I. L. (2014). Numerical maximisation of likelihood: A neglected alternative to EM? Int. Stat. Rev. 82 296–308.
  • Maruotti, A. and Rydén, T. (2009). A semiparametric approach to hidden Markov models under longitudinal observations. Stat. Comput. 19 381–393.
  • 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.
  • McClintock, B. T., Russell, D. J. F., Matthiopoulos, J. and King, R. (2013). Combining individual animal movement and ancillary biotelemetry data to investigate population-level activity budgets. Ecology 94 838–849.
  • McKellar, A. E., Langrock, R., Walters, J. R. and Kesler, D. C. (2014). Using mixed hidden Markov models to examine behavioral states in a cooperatively breeding bird. Behav. Ecol. 26 148–157.
  • Miller, P. J. O., Kvadsheim, P. H., Lam, F.-P. A., Wensveen, P. J., Antunes, R., Alves, A. C., Visser, F., Kleivane, L., Tyack, P. L. and Sivle, L. D. (2012). The severity of behavioral changes observed during experimental exposures of killer (Orcinus orca), long-finned pilot (Globicephala melas), and sperm (Physeter macrocephalus) whales to naval sonar. Aquat. Mamm. 38 362–401.
  • Miller, P. J. O., Kvadsheim, P. H., Lam, F. P. A., Tyack, P. L., Cure, C., DeRuiter, S. L., Kleivane, L., Sivle, L. D., van IJsselmuide, S. P., Visser, F., Wensveen, P. J., von Benda-Beckmann, A. M., Martin Lopez, L. M., Narazaki, T. and Hooker, S. K. (2015). First indications that northern bottlenose whales are sensitive to behavioural disturbance from anthropogenic noise. R. Soc. Open Sci. 2 140484.
  • 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.
  • National Research Council (2005). Marine Mammal Populations and Ocean Noise: DeterminIng When Noise Causes Biologically Significant Effects. National Academies Press, Washington, DC.
  • New, L. F., Moretti, D. J., Hooker, S. K., Costa, D. P. and Simmons, S. E. (2013). Using energetic models to investigate the survival and reproduction of beaked whales (family Ziphiidae). PLoS ONE 8 e68725.
  • New, L. F., Clark, J. S., Costa, D. P., Fleishman, E., Hindell, M. A., Klanjšček, T., Lusseau, D., Kraus, S., McMahon, C. R., Robinson, P. W., Schick, R. S., Schwarz, L. K., Simmons, S. E., Thomas, L., Tyack, P. and Harwood, J. (2014). Using short-term measures of behaviour to estimate long-term fitness of southern elephant seals. Mar. Ecol. Prog. Ser. 496 99–108.
  • Nowacek, D. P., Johnson, M. P. and Tyack, P. L. (2004). Proceedings of the Royal Society of London B: Biological Sciences. Proceedings of the Royal Society B—Biological Sciences 271 227–231.
  • Patterson, T. A., Thomas, L., Wilcox, C., Ovaskainen, O. and Matthiopoulos, J. (2008). State-space models of individual animal movement. Trends Ecol. Evol. 23 87–94.
  • R Core Team (2015). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. Available at
  • Schick, R. S., New, L. F., Thomas, L., Costa, D. P., Hindell, M. A., McMahon, C. R., Robinson, P. W., Simmons, S. E., Thums, M., Harwood, J. and Clark, J. S. (2013). Estimating resource acquisition and at-sea body condition of a marine predator. J. Anim. Ecol. 82 1300–1315.
  • Shannon, G., McKenna, M. F., Angeloni, L. M., Crooks, K. R., Fristrup, K. M., Brown, E., Warner, K. A., Nelson, M. D., White, C., Briggs, J., McFarland, S. and Wittemyer, G. (2015). A synthesis of two decades of research documenting the effects of noise on wildlife. Biol. Rev. Camb. Philos. Soc. 91 982–1005.
  • Sivle, L. D., Kvadsheim, P. H., Curé, C., Isojunno, S., Wensveen, P. J., Lam, F.-P. A., Visser, F., Kleivane, L., Tyack, P. L., Harris, C. M. and Miller, P. J. O. (2015). Severity of expert-identified behavioural responses of humpback whale, minke whale and northern bottlenose whale to naval sonar. Aquat. Mamm. 41 469–502.
  • Southall, B. L., Bowles, A. E., Ellison, W. T., Finneran, J. J., Gentry, R. L., Greene, C. R., Kastak, D., Ketten, D. R., Miller, J. H., Nachtigall, P. E., Richardson, W. J., Thomas, J. A. and Tyack, P. L. (2007). Marine mammal noise exposure criteria: Initial scientific recommendations. Aquat. Mamm. 33 411–521.
  • Southall, B. L., Moretti, D., Abraham, B., Calambokidis, J., DeRuiter, S. L. and Tyack, P. L. (2012). Marine mammal behavioral response studies in southern California: Advances in technology and experimental methods. Mar. Technol. Soc. J. 46 48–59.
  • Stimpert, A. K., DeRuiter, S. L., Southall, B. L., Moretti, D. J., Falcone, E. A., Goldbogen, J. A., Friedlaender, A., Schorr, G. S. and Calambokidis, J. (2014). Acoustic and foraging behavior of a Baird’s beaked whale, Berardius bairdii, exposed to simulated sonar. Sci. Rep. 4 7031.
  • Towner, A., Leos-Barajas, V., Langrock, R., Schick, R. S., Smale, M. J., Jewell, O., Kaschke, T. and Papastamatiou, Y. P. (2016). Sex-specific and individual preferences for hunting strategies in white sharks. Funct. Ecol. 30 1397–1407.
  • Tyack, P., Gordon, J. and Thompson, D. (2003). Controlled exposure experiments to determine the effects of noise on marine mammals. Mar. Technol. Soc. J. 37 41–53.
  • Tyack, P. L., Zimmer, W. M. X., Moretti, D., Southall, B. L., Claridge, D. E., Durban, J. W., Clark, C. W., D’Amico, A., DiMarzio, N., Jarvis, S., McCarthy, E., Morrissey, R., Ward, J. and Boyd, I. L. (2011). Beaked whales respond to simulated and actual navy sonar. PLoS ONE 6 e17009.
  • Van de Kerk, M., Onorato, D. P., Criffield, M. A., Bolker, B. M., Augustine, B. C., McKinley, S. A. and Oli, M. K. (2015). Hidden semi-Markov models reveal multiphasic movement of the endangered Florida panther. J. Anim. Ecol. 84 576–585.
  • Venzon, D. J. and Moolgavkar, S. H. (1988). A method for computing profile-likelihood-based confidence intervals. Appl. Stat. 37 87.
  • Wood, S. N. (2001). Minimizing model fitting objectives that contain spurious local minima by bootstrap restarting. Biometrics 57 240–244.
  • Zucchini, W., MacDonald, I. L. and Langrock, R. (2016). Hidden Markov Models for Time Series: An Introduction Using R (Second Edition). Chapman & Hall, London.
  • Zucchini, W., Raubenheimer, D. and MacDonald, I. L. (2008). Modeling time series of animal behavior by means of a latent-state model with feedback. Biometrics 64 807–815.

Supplemental materials

  • Supplement A: Detailed experimental methods for “A multivariate mixed hidden Markov model for blue whale behaviour and responses to sound exposure”. This supplement provides additional detail on the field data collection protocols used to collect the dataset.
  • Supplement B: Supplemental figures for “A multivariate mixed hidden Markov model for blue whale behaviour and responses to sound exposure”. This supplement provides figures of the input data (and decoded states from the final model) for all 37 whales in the dataset.