## 6.8 A random walk model of animal movement

A simple random walk model of movement with drift (directional movement) but no correlation is $\begin{gather} x_{1,t} = x_{1,t-1} + u_1 + w_{1,t}, \;\; w_{1,t} \sim \,\text{N}(0,\sigma^2_1)\\ x_{2,t} = x_{2,t-1} + u_2 + w_{2,t}, \;\; w_{2,t} \sim \,\text{N}(0,\sigma^2_2) \tag{6.8} \end{gather}$ where $$x_{1,t}$$ is the location at time $$t$$ along one axis (here, longitude) and $$x_{2,t}$$ is for another, generally orthogonal, axis (in here, latitude). The parameter $$u_1$$ is the rate of longitudinal movement and $$u_2$$ is the rate of latitudinal movement. We add errors to our observations of location: $\begin{gather} y_{1,t} = x_{1,t} + v_{1,t}, \;\; v_{1,t} \sim \,\text{N}(0,\eta^2_1)\\ y_{2,t} = x_{2,t} + v_{2,t}, \;\; v_{2,t} \sim \,\text{N}(0,\eta^2_2), \tag{6.9} \end{gather}$

This model is comprised of two separate univariate state-space models. Note that $$y_1$$ depends only on $$x_1$$ and $$y_2$$ depends only on $$x_2$$. There are no actual interactions between these two univariate models. However, we can write the model down in the form of a multivariate model using diagonal variance-covariance matrices and a diagonal design ($$\mathbf{Z}$$) matrix. Because the variance-covariance matrices and $$\mathbf{Z}$$ are diagonal, the $$x_1$$:$$y_1$$ and $$x_2$$:$$y_2$$ processes will be independent as intended. Here are Equations (6.8) and (6.9) written as a MARSS model (in matrix form): $\begin{gather} \begin{bmatrix}x_{1,t}\\x_{2,t}\end{bmatrix} = \begin{bmatrix}x_{1,t-1}\\x_{2,t-1}\end{bmatrix} + \begin{bmatrix}u_1\\u_2\end{bmatrix} + \begin{bmatrix}w_{1,t}\\w_{2,t}\end{bmatrix}, \textrm{ } \mathbf{w}_t \sim \,\text{MVN}\begin{pmatrix}0,\begin{bmatrix}\sigma^2_1&0\\0&\sigma^2_2\end{bmatrix} \end{pmatrix} \tag{6.10} \\ \nonumber \\ \begin{bmatrix}y_{1,t}\\y_{2,t}\end{bmatrix} = \begin{bmatrix}1&0\\0&1\end{bmatrix} \begin{bmatrix}x_{1,t}\\x_{2,t}\end{bmatrix} + \begin{bmatrix}v_{1,t}\\ v_{2,t}\end{bmatrix}, \textrm{ } \mathbf{v}_t \sim \,\text{MVN}\begin{pmatrix}0,\begin{bmatrix}\eta^2_1&0\\0&\eta^2_2\end{bmatrix} \end{pmatrix} \tag{6.11} \end{gather}$ The variance-covariance matrix for $$\mathbf{w}_t$$ is a diagonal matrix with unequal variances, $$\sigma^2_1$$ and $$\sigma^2_2$$. The variance-covariance matrix for $$\mathbf{v}_t$$ is a diagonal matrix with unequal variances, $$\eta^2_1$$ and $$\eta^2_2$$. We can write this succinctly as $\begin{gather} \mathbf{x}_t = \mathbf{x}_{t-1} + \mathbf{u} + \mathbf{w}_t, \;\; \mathbf{w}_t \sim \,\text{MVN}(0,\mathbf{Q}) \\ \mathbf{y}_t = \mathbf{x}_{t} + \mathbf{v}_t, \;\; \mathbf{v}_t \sim \,\text{MVN}(0,\mathbf{R}). \tag{6.12} \end{gather}$