the Efficient modelling of individual animal movement in continuous time

It is difficult to account for environmental covariates, which may affect an animal’s behaviour (and therefore diffusion rate), in a continuous-time framework. This is because, unlike when we take a discrete-time approach, we are no longer dealing with behavioural changes at a predefined grid of possible behavioural switching times. 

The ‘exact approach’ of Blackwell et al. (2015) allows for the modelling of individual animal movement in continuous time by introducing a concept known as the ‘potential behavioural switch’, which occurs at some uniform rate based on the upper bound on transition rates between animal behaviours. At these ‘potential switches’, environmental covariates may be assessed, and the probability of an ‘actual switch’ may be calculated. However, the time and number of potential switches between observations must be sampled along with likelihood parameters. This means that we do not allow for inference via direct likelihood evaluation. Furthermore, because we have a parameter space of high dimensionality, inference via MCMC takes quite a while, with poor mixing. 

The ‘Integrated Continuous time HMM’ (InCH) approach of Blackwell (2020) frames the ‘exact approach’ as a temporally inhomogeneous HMM, defined at potential switching times. In doing so, the InCH approach uses the ‘forward algorithm’ (the source of the computational efficiency of the discrete-time HMM) to indirectly integrate the model likelihood over the animal’s behavioural states. This means that we no longer have to sample an animal’s behavioural state, improving mixing and the overall time taken for inference in an MCMC framework. However, this approach still requires the time and number of potential behavioural changes to be sampled, so direct optimisation is not possible. 

 Please feel free to contact me at dpdgrainger1[at]sheffield.ac.uk