## 10.4 Different error structures

The example observation equation we used above had what we refer to as an “unconstrained” variance-covariance matrix \(\mathbf{R}\) wherein all of the parameters are unique. In certain applications, however, we may want to change our assumptions about the forms for \(\mathbf{R}\). For example, we might have good reason to believe that all of the observations have different error variances and they were independent of one another (e.g., different methods were used for sampling), in which case

\[ \mathbf{R} = \begin{bmatrix} r_1&0&0&0&0 \\ 0&r_2&0&0&0 \\ 0&0&r_3&0&0 \\ 0&0&0&r_4&0 \\ 0&0&0&0&r_5 \end{bmatrix}. \]

Alternatively, we might have a situation where all of the observation errors had the same variance \(r\), but they were not independent from one another. In that case we would have to include a covariance parameter \(k\), such that

\[ \mathbf{R} = \begin{bmatrix} r&k&k&k&k \\ k&r&k&k&k \\ k&k&r&k&k \\ k&k&k&r&k \\ k&k&k&k&r \end{bmatrix}. \]

Any of these options for \(\mathbf{R}\) (and other custom options as well) are available to us in a DFA model, just as they were in the MARSS models used in previous chapters.