Nathalie Peyrard - Statistical Approaches for Hidden Variables in Ecology

Figure 1.12shows the evolution of the probability of state transitions as the distance from the nest changes. We see that, overall, there is a high level of persistence within states at any distance from the nest. Nevertheless, it appears that the closer the bird is to the nest (to the right on the x axis), the less likely it is to transition from activity 2 to activity 3. Note that the interest of the chosen covariate in this specific case is debatable; based on the AIC, a model without this covariate would be preferred.

Figure 1.12. Evolution of estimated transition probabilities as a function of distance from the nest. The figure should be read as a transition matrix. The graph in the second line, third column represents the evolution of the probability of a transition from state 2 to state 3 as a function of distance from the nest. As the distance variable has been centered and reduced, the origin represents the mean distance from the nest across all data points. For a color version of this figure, see www.iste.co.uk/peyrard/ecology.zip

The calculation of model selection criteria is valuable in helping to chose the number of states to use, as is the AIC. Table 1.1shows AIC and ICL scores for different numbers of activities across our three trajectories.

Table 1.1. Evolution of model selection criteria (AIC and ICL) as a function of the number of hidden states J. In both cases, the best scores are attained for a model with six hidden states

From a purely statistical perspective, a 6-state model appears preferable here.

Figure 1.13shows states along a trajectory (using the bivariate velocity model) alongside the speed characteristics of these states. We see that a classification into six activities broadly corresponds to the creation of subdivisions in the intermediate state. States previously characterized as belonging to activity 2 or 3 ( Figure 1.9, top left) are divided into four different groups in the new model. In our view, the choice of an optimum number of states in this case should be guided by our capacity to interpret the model, rather than by purely statistical considerations.

Figure 1.13. Study zone (red dot on the map) and three trajectories of three different red-footed boobies. Measured over a time step of 10 s. For a color version of this figure, see www.iste.co.uk/peyrard/ecology.zip

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