15.5 Multivariate responses

Let’s imagine that we have two sites \(y_1\) and \(y_2\). We can model relationship between the seasonality in these two sites in many different ways. How we model it depends on our assumptions about our site or might reflect different relationships that we want to test.

15.5.1 Same seasonality and same level

In this case \(x_t\) is shared and the \(\beta\)’s. We can allow the data to be scaled (translated up or down) relative to each other however.

\[\begin{bmatrix}x \\ \beta_1 \\ \beta_2 \end{bmatrix}_t = \begin{bmatrix}x \\ \beta_1 \\ \beta_2 \end{bmatrix}_{t-1} + \begin{bmatrix}w_1 \\ w_2 \\ w_3 \end{bmatrix}_t\]

\[\begin{bmatrix}y_1 \\ y_2\end{bmatrix}_t = \begin{bmatrix}1& \sin(2\pi t/p)&\cos(2\pi t/p)\\1& \sin(2\pi t/p)&\cos(2\pi t/p)\end{bmatrix} \begin{bmatrix}x \\ \beta_1 \\ \beta_2 \end{bmatrix}_t + \begin{bmatrix}0 \\ a_2\end{bmatrix} + \begin{bmatrix}v_1 \\ v_2\end{bmatrix}_t\]

15.5.2 Same seasonality and same level but scaled

Same as the model above but we allow that the \(y\) level are stretched versions of \(x\) (so \(z x_t\)) ). We only change one of the \(y\) to have a scaled \(x\) or else the model would have infinite number of solutions.

\[\begin{bmatrix}y_1 \\ y_2\end{bmatrix}_t = \begin{bmatrix}1& \sin(2\pi t/p)&\cos(2\pi t/p)\\z& \sin(2\pi t/p)&\cos(2\pi t/p)\end{bmatrix} \begin{bmatrix}x \\ \beta_1 \\ \beta_2 \end{bmatrix}_t + \begin{bmatrix}0 \\ a_2\end{bmatrix} + \begin{bmatrix}v_1 \\ v_2\end{bmatrix}_t\]

15.5.3 Scaled seasonality and same level

We could also say that the are affect by the same seasonality but the amplitude is different. So covariate will be \(z(\sin(2\pi t/12)+\cos(2\pi t/12))\) for the second \(y\).

\[\begin{bmatrix}y_1 \\ y_2\end{bmatrix}_t = \begin{bmatrix}1& \sin(2\pi t/p)&\cos(2\pi t/p)\\1& z\sin(2\pi t/p)&z\cos(2\pi t/p)\end{bmatrix} \begin{bmatrix}x \\ \beta_1 \\ \beta_2 \end{bmatrix}_t + \begin{bmatrix}0 \\ a_2\end{bmatrix} + \begin{bmatrix}v_1 \\ v_2\end{bmatrix}_t\]

15.5.4 Different seasonality but correlated

We might imagine that the seasonality is different between the sites but that the changes in seasonality are allowed to covary.

\[\begin{bmatrix}x \\ \beta_{1a} \\ \beta_{1b} \\ \beta_{2a} \\ \beta_{2b} \end{bmatrix}_t = \begin{bmatrix}x \\ \beta_{1a} \\ \beta_{1b} \\ \beta_{2a} \\ \beta_{2b} \end{bmatrix}_{t-1} + \begin{bmatrix}w \\ w_1 \\ w_2 \\w_3\\w_4 \end{bmatrix}_t\]

\[\begin{bmatrix}w \\ w_1 \\ w_2 \\w_3\\w_4 \end{bmatrix}_t \sim \text{MVN}\left(0, \begin{bmatrix}q & 0 & 0 & 0 & 0 \\ 0 & q_1 & c_1 & 0 & 0 \\ 0 & c_1 & q_1 & 0 & 0 \\ 0 & 0 & 0 & q_2 & c_2 \\ 0 & 0 & 0 & c_2 & q_2 \end{bmatrix} \right)\]

\[\begin{bmatrix}y_1 \\ y_2\end{bmatrix}_t = \begin{bmatrix} 1& \sin(2\pi t/p)&\cos(2\pi t/p) & 0 & 0 \\1& 0 & 0 & \sin(2\pi t/p)&\cos(2\pi t/p) \end{bmatrix} \begin{bmatrix}x \\ \beta_{1a} \\ \beta_{1b} \\ \beta_{2a} \\ \beta_{2b} \end{bmatrix}_t + \begin{bmatrix}0 \\ a_2\end{bmatrix} + \begin{bmatrix}v_1 \\ v_2\end{bmatrix}_t\]

15.5.5 Same seasonality and different level

In this case there is a different \(x_t\) for each \(y_t\).

\[\begin{bmatrix}x_1 \\ x_2 \\ \beta_1 \\ \beta_2 \end{bmatrix}_t = \begin{bmatrix}x_1\\x_2 \\ \beta_1 \\ \beta_2 \end{bmatrix}_{t-1} + \begin{bmatrix}w_1 \\ w_2 \\ w_3\\w_4 \end{bmatrix}_t\]

\[\begin{bmatrix}y_1 \\ y_2\end{bmatrix}_t = \begin{bmatrix}1& 0&\sin(2\pi t/p)&\cos(2\pi t/p)\\0&1& \sin(2\pi t/p)&\cos(2\pi t/p)\end{bmatrix} \begin{bmatrix}x_1\\x_2 \\ \beta_1 \\ \beta_2 \end{bmatrix}_t + \begin{bmatrix}0 \\ a_2\end{bmatrix} + \begin{bmatrix}v_1 \\ v_2\end{bmatrix}_t\]