## 7.4 Four subpopulations with temporally uncorrelated errors

The model for one well-mixed population was not very good. Another reasonable assumption is that the different census regions are measuring four different temporally independent subpopulations. We write a model of the log subpopulation abundances for this case as: \[\begin{equation} \begin{gathered} \begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}_t = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}_{t-1} + \begin{bmatrix}u\\u\\u\\u\end{bmatrix} + \begin{bmatrix}w_1\\w_2\\w_3\\w_4\end{bmatrix}_t \\ \textrm{ where } \mathbf{w}_t \sim \,\text{MVN}\begin{pmatrix}0, \begin{bmatrix} q & 0 & 0 & 0 \\ 0 & q & 0 & 0\\ 0 & 0 & q & 0 \\ 0 & 0 & 0 & q \end{bmatrix}\end{pmatrix}\\ \begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}_0 = \begin{bmatrix}\mu_1\\\mu_2\\\mu_3\\\mu_4\end{bmatrix}_t \end{gathered} \tag{7.7} \end{equation}\] The \(\mathbf{Q}\) matrix is diagonal with one variance value. This means that the process variance (variance in year-to-year population growth rates) is independent (good and bad years are not correlated) but the level of variability is the same across regions. We made the \(\mathbf{u}\) matrix with one \(u\) value. This means that we assume the population growth rates are the same across regions.

Notice that we set the \(\mathbf{B}\) matrix equal to a diagonal matrix with 1 on the diagonal. This is the “identity” matrix and it is like a 1 but for matrices. We do not need \(\mathbf{B}\) for our model, but `MARSS()`

requires a value.

### 7.4.1 The observation process

In this model, each survey is an observation of a different \(x\): \[\begin{equation} \left[ \begin{array}{c} y_{1} \\ y_{2} \\ y_{3} \\ y_{4} \end{array} \right]_t = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix}x_1\\x_2\\x_3\\x_4\end{bmatrix}_t + \left[ \begin{array}{c} 0 \\ 0 \\ 0 \\ 0 \end{array} \right] + \left[ \begin{array}{c} v_{1} \\ v_{2} \\ v_{3} \\ v_{4} \end{array} \right]_t \tag{7.8} \end{equation}\] No \(a\)’s can be estimated since we do not have multiple observations of a given \(x\) time series. Our \(\mathbf{R}\) matrix doesn’t change; the observation errors are still assumed to the independent with different variances.

Notice that our \(\mathbf{Z}\) matrix changed. \(\mathbf{Z}\) is specifying which \(y_i\) goes to which \(x_j\). The one we have specified means that \(y_1\) is observing \(x_1\), \(y_2\) observes \(x_2\), etc. We could have set up \(\mathbf{Z}\) like so \[\begin{equation} \begin{bmatrix} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{bmatrix} \end{equation}\]

This would mean that \(y_1\) observes \(x_2\), \(y_2\) observes \(x_1\), \(y_3\) observes \(x_4\), and \(y_4\) observes \(x_3\). Which \(x\) goes to which \(y\) is arbitrary; we need to make sure it is one-to-one. We will stay with \(\mathbf{Z}\) as an identity matrix since \(y_i\) observing \(x_i\) makes it easier to remember which \(x\) goes with which \(y\).

### 7.4.2 Fitting the model

We set up the model list for `MARSS()`

as:

```
.1 <- list(B = "identity", U = "equal", Q = "diagonal and equal",
mod.listZ = "identity", A = "scaling", R = "diagonal and unequal",
x0 = "unequal", tinitx = 0)
```

We introduced a few more short-cuts. `"equal"`

means all the values in the matrix are the same. `"diagonal and equal"`

means that the matrix is diagonal with one value on the diagonal. `"unequal"`

means that all values in the matrix are different.

We can then fit our model for 4 subpopulations as:

`.1 <- MARSS::MARSS(dat, model = mod.list.1) fit`