Fig 1.
The hidden states Ct represent behavioural states that influence the distribution of the observed variables Xt.
Fig 2.
Hidden Markov Model with feedback processes.
The transition probabilities between hidden states Ct depends on the observed covariate processes , Dt and Ht.
Fig 3.
Response variables and covariate processes.
Time series plot of maximum depth (MD), duration of dive (DT), and post-dive duration (PD) from dive number 3890 to 3950 and the covariate processes counting the time since last deep dive (τt), number of deep dives in a row (dt), and the hour at initiation of dive (ht). The symbols indicate the decoded hidden states from a model fitted to a dependent log-normal distribution (Model 1).
Fig 4.
Representative part of the narwhal diving data, covering 24 hours of dives on August 15th 2013. The red parts are where a lower temperature in the stomach has been registered, indicating that the narwhal has swallowed a prey. The blue line indicates a depth of 350m, the threshold for a deep dive used in the definition of the covariates.
Fig 5.
Histograms of response variables MD, DT and PD. The fit of Model 1 is indicated with black curves, for dependent lognormal (DL), independent lognormal (IL), dependent gamma (DG) and independent gamma (IG). The distribution of the fitted states are indicated with colours as given in the legend. State 1 corresponds to near surface, state 2 medium depths, and state 3 large depths.
Table 1.
Runtimes and number of variables for different state distributions and for 2, 3 and 4 states for covariate model 1. Runtimes are on Intel Xeon E5-2697v2 @ 2.7 GHz.
Table 2.
Different models for covariate effects on the transition probabilities between behavioural states.
The predictors ηij relate to the transition probabilities as given in Eq (7). The spline effects of hour are denoted by , of τt by
, and of dt by
for k = 1, 2, 3 and i, j = 1, 2, 3; i ≠ j. A list of all explored models can be found in S1 Table in the Supporting Information.
Table 3.
Differences in AIC values, ΔAIC = AIC—AICmin, between the different models with 3 hidden states, where AICmin is the value of the model with the lowest AIC. The best fit is given by the minimum AIC. For all the tested state distributions, covariate model 1 was preferred, and for all covariate models, the dependent log-normal state distribution was preferred. Because the runtimes for the correlated gamma model are high, only Model 1 was fitted. The best model is highlighted in bold. np: number of parameters.
Fig 6.
Quantile-quantile residual plots.
QQ-plots of forecast pseudo-residuals from covariate model 1 with correlated log-normal state distribution.
Fig 7.
A: Transition probabilities between behavioural states depending on covariates related to deep dives of correlated log-normal model 1, at approximately 12 pm. B: Transition probabilities depending on diurnal effects in model 1 with correlated log-normal state distributions, calculated for τt = 0.58 and dt = 0 (the medians).
Fig 8.
The estimated hidden state per dive for 12 hours of the data, starting on 22 September 2013 at 14:18:39.
Table 4.
Summary measures of Model 1 with 3 states.
Means and standard deviations based on correlated Log-normal, correlated Gamma, independent Log-normal and independent Gamma distribution. MD: Maximum Depth; DT: Diving Time; PD: Post-Dive duration. E: mean; SD: standard deviation; Corr1: Correlation between MD and and DT. Corr2: Correlation between MD and and PD. Corr3: Correlation between DT and and PD. The empirical distribution is the empirical measures in three subgroups of the data classified according to MD, state 1: MD between 20 and 50 m, state 2: MD between 50 and 350 m, state 3: MD above 350 m.
Fig 9.
The estimated hidden state per dive for each of the three observed variables under covariate model 1 and state distribution the correlated log-normal. The longest pause of no deep dives starts from the 1345th dive until the 1894th dive, and it lasts approximately 2 days and 17.5 hours.