29 Jan 2019
We want to understand how covariates drive the hidden process. For example, population growth rate from year to year or movement rates from step to step.
We want to test hypotheses for what caused a perturbation or change in the dynamics.
We want to forecast using covariates.
We want to model the autocorrelation in the process errors using the known driver.
Auto-correlated observation errors
Model your \(v_t\) as a AR-1 process. hard numerically
If know what is causing the auto-correlation, include that as a covariate. Easier.
Correlated observation errors across sites or species (y rows)
Use a R matrix with off-diagonal terms. really hard numerically
If you know or suspect what is causing the correlation, include that as a covariate. Easier.
"hard numerically" = you need a lot of data
eg, nutrients affects growth, high temps kill…
\[\mathbf{x}_t = \mathbf{B}_t\mathbf{x}_{t-1}+\mathbf{u}+\boxed{\mathbf{C}\mathbf{c}_t}+\mathbf{w}_t\]
\(\boxed{\mathbf{C}\mathbf{c}_t}\) The covariate is in \(\mathbf{c}_t\) and the effect is in matrix \(\mathbf{C}\). This is similar to how the covariate appears in a multivariate regression.
\[\begin{bmatrix}x_1 \\ x_2\end{bmatrix}_t = \begin{bmatrix}x_1 \\ x_2\end{bmatrix}_{t-1} + \begin{bmatrix}C_a & C_b \\ C_a & C_b\end{bmatrix}\begin{bmatrix}temp \\ wind\end{bmatrix}_t + \begin{bmatrix}w_1 \\ w_2\end{bmatrix}_t\] The model for \(x_t\) in site 1 is:
\[x_{1,t}=x_{1,t-1}+C_a temp_t + C_b wind_t + w_{1,t}\] There is an effect of the prior \(x_t\) and an effect of temp and wind.
The structure of \(\mathbf{C}\) can model different effect structures
Effect of temp and wind is the same
\[\begin{bmatrix}C & C \\ C & C\end{bmatrix}\begin{bmatrix}temp \\ wind\end{bmatrix}_t\]
Effect of temp and wind is different but the same across sites, species, whatever the row in \(\mathbf{x}\) is
\[\begin{bmatrix}C_a & C_b \\ C_a & C_b\end{bmatrix}\begin{bmatrix}temp \\ wind\end{bmatrix}_t\]
Effect of temp and wind is different across sites or whatever the row in \(\mathbf{x}\) is
\[\begin{bmatrix}C_{a1} & C_{b1} \\ C_{a2} & C_{b2}\end{bmatrix}\begin{bmatrix}temp \\ wind\end{bmatrix}_t\]
Effect of temp is the same across sites but wind is not
\[\begin{bmatrix}C_{a} & C_{b1} \\ C_{a} & C_{b2}\end{bmatrix}\begin{bmatrix}temp \\ wind\end{bmatrix}_t\]
eg, vegetation obscures individuals, temperature affects behavior making animals more or less visible
\[\mathbf{y}_t = \mathbf{Z}\mathbf{x}_{t}+\mathbf{a}+\boxed{\mathbf{D}\mathbf{d}_t}+\mathbf{w}_t\]
\[\begin{bmatrix}y_1 \\ y_2 \\y_3\end{bmatrix}_t = \begin{bmatrix}1 & 0 \\ 1 & 0 \\ 0 & 1\end{bmatrix}\begin{bmatrix}x_1 \\ x_2\end{bmatrix}_{t} + \begin{bmatrix}D_a & D_b \\ D_a & D_b \\D_a & D_b\end{bmatrix}\begin{bmatrix}temp \\ wind\end{bmatrix}_t + \begin{bmatrix}v_1 \\ v_2 \\v_3\end{bmatrix}_t\] In this case the covariate does not affect the state \(x\). It only affects the observation of the state.
The model for \(y_t\) in site 1 is:
\[y_{1,t}=x_{1,t}+D_a temp_t + D_b wind_t + v_{1,t}\]
The structure of \(\mathbf{D}\) can model many different structures of the effects.
Effect of temp and wind is the same across sites 1 & 2 but different for site 3. In site 3, temp has an effect but wind does not
\[\begin{bmatrix}D_a & D_b \\ D_a & D_b \\ D_c & 0\end{bmatrix}\begin{bmatrix}temp \\ wind\end{bmatrix}_t\]