Model formulation

State process

The state process {S_t} of a N-state HMM for T time steps is characterised by its state transition probability matrix Γ^(t) = (γ_i, j^(t)), where i, j = 1, …, N and γ_i, j^(t) = Pr(s_t + 1 = j|s_t = i). The probability of transitioning to state s_t from state s_t − 1 is

s_t ∼ Categorical(Γ^{(t − 1)}),  1 ≤ t ≤ T.

Process model

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 + λ_n(z_t − 1 − z_t − 2) + ϵ_t,  ϵ_t ∼ N(0, Ω),  1 ≤ n ≤ N,

where

$$ \Omega = \begin{bmatrix} \tau^2_{\epsilon, \text{lon}} & 0 \\ 0 & \tau^2_{\epsilon, \text{lat}} \end{bmatrix}. $$

The state-depended parameter, λ_n, can take values between 0 and 1 (i.e., 0 ≤ λ ≤ 1), and controls the degree of correlation between steps. By default, movetrack estimates track-specific λ_n values, but it is also possible to use the same λ_n for all tracks by setting i_lambda = FALSE.

Observation model

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_jz_t + (1 − w_j)z_t − 1 + θ_j,  θ_j ∼ T(0, σ_j),  1 ≤ j ≤ J,

where T(0, σ_j) denotes a bivariate Student’s t-distribution with measurement error $\sigma_j = \begin{bmatrix} \sigma_{j, \text{lon}} \\ \sigma_{j, \text{lat}} \end{bmatrix}$.