Hidden Markov Models (HMM) describe the evolution of a sequence of random variables, \(\{S_t\}\) (i.e. behavioural states), which are not directly observable, but can be inferred from another sequence of random variables, \(\{Y_t\}\), that are observable (i.e. locations). The two main characteristics of HMMs are (1) each observation is assumed to be generated by one of \(N\) distributions, and (2) the hidden state sequence that determines which of the \(N\) distributions is chosen at time \(t\) is modelled as a Markov chain, where the probability of being in each state at time \(t\) depends only on the state value at the previous time step.
The state process \(\{S_t\}\) of a \(N\)-state HMM for \(T\) time steps is characterised by its state transition probability matrix \(\Gamma^{(t)} = (\gamma^{(t)}_{i,j})\), where \(i,j = 1, \dots, N\) and \(\gamma^{(t)}_{i,j} = \text{Pr}(s_{t+1} = j|s_t = i)\). The probability of transitioning to state \(s_t\) from state \(s_{t-1}\) is
\[ s_t \sim \text{Categorical}(\Gamma^{(t-1)}), \quad 1 \leq t \leq T. \]
The process equation for the true locations of the animal at regular time intervals \(t\), \(z_t = \begin{bmatrix} z_{t, \text{lon}} \\ z_{t, \text{lat}} \end{bmatrix}\), assumes that the animal’s location at time \(t\) is not only dependent on the previous location, \(z_{t-1}\), but also on the animal’s previous displacement in each coordinate, \(z_{t-1} - z_{t-2}\):
\[ z_t = z_{t-1} + \lambda_n (z_{t-1} - z_{t-2}) + \epsilon_t, \quad \epsilon_t \sim \text{N}(0, \Omega), \quad 1 \leq n \leq N, \]
where
\[ \Omega = \begin{bmatrix} \tau^2_{\epsilon, \text{lon}} & 0 \\ 0 & \tau^2_{\epsilon, \text{lat}} \end{bmatrix}. \]
The state-depended parameter, \(\lambda_n\), can take values between 0 and
1 (i.e., \(0 \leq \lambda \leq 1\)),
and controls the degree of correlation between steps. By default,
movetrack estimates track-specific \(\lambda_n\) values, but it is also possible
to use the same \(\lambda_n\) for all
tracks by setting i_lambda = FALSE.
The observed locations of an animal, \(y_j = \begin{bmatrix} y_{j, \text{lon}} \\ y_{j, \text{lat}} \end{bmatrix}\), often have irregular time intervals \(j\), with \(J\) representing the total number of observed locations. Therefore, the true location of the animal is linearly interpolated to the time of the observation, with \(w_j\) representing the proportion of the regular time interval between \(t-1\) and \(t\) when the observation \(y_j\) was made:
\[ y_j = w_j z_t + (1 - w_j) z_{t-1} + \theta_j, \quad \theta_j \sim \text{T}(0, \sigma), \quad 1 \leq j \leq J, \]
where \(\text{T}(0, \sigma)\) denotes a bivariate Student’s \(t\)-distribution with measurement error \(\sigma = \begin{bmatrix} \sigma_{\text{lon}} \\ \sigma_{\text{lat}} \end{bmatrix}\).
Auger-Méthé, M., Newman, K., Cole, D., Empacher, F., Gryba, R., King, A. A., … & Thomas, L. (2021). A guide to state–space modeling of ecological time series. Ecological Monographs, 91(4), e01470. doi: 10.1002/ecm.1470