We will be using the MARSS package to fit univariate and multivariate state-space models.

The main function is MARSS():

fit <- MARSS(data, model=list())

data are a vector or a matrix with time going along the columns.

model list is a list with the structure of all the parameters.

\[x_t = \mathbf{B}x_{t-1} + \mathbf{U} + w_t, \,\,\, w_t \sim N(0,\mathbf{Q})\] \[y_t = \mathbf{Z}x_{t} + \mathbf{A} + v_t, \,\,\, v_t \sim N(0,\mathbf{R})\]

The MARSS model list follows this notation one-to-one.

\[\begin{bmatrix}x_1 \\ x_2\end{bmatrix}_t = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\begin{bmatrix}x_1 \\ x_2\end{bmatrix}_{t-1} + \begin{bmatrix}u \\ u\end{bmatrix}+\begin{bmatrix}w_1 \\ w_2\end{bmatrix}_t\]

\[\begin{bmatrix}w_1 \\ w_2\end{bmatrix}_t \sim N\left( 0, \begin{bmatrix}q & 0 \\ 0 & q\end{bmatrix} \right)\]

\[\mathbf{B} = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\]

B <- diag(1,2)
B

##      [,1] [,2]
## [1,]    1    0
## [2,]    0    1

\[\mathbf{U} = \begin{bmatrix}u \\ u\end{bmatrix}\]

U <- matrix("u",2,1)
U

##      [,1]
## [1,] "u" 
## [2,] "u"

A value in quotes is a character vector and is the name of a parameter to be estimated.

\[\mathbf{Q} = \begin{bmatrix}q & c \\ c & q\end{bmatrix}\]

Q <- matrix(c("q","c","c","q"),2,2)
Q

##      [,1] [,2]
## [1,] "q"  "c" 
## [2,] "c"  "q"

or use

Q <- "equalvarcov"

\[\begin{bmatrix}y_1 \\ y_2\end{bmatrix}_t = \begin{bmatrix}1 & 0 \\ 0 & 1\end{bmatrix}\begin{bmatrix}x_1 \\ x_2\end{bmatrix}_t + \begin{bmatrix}0 \\ 0\end{bmatrix}+\begin{bmatrix}v_1 \\ v_2\end{bmatrix}_t\]

\[\begin{bmatrix}v_1 \\ v_2\end{bmatrix}_t \sim N\left( 0, \begin{bmatrix}r & 0 \\ 0 & r\end{bmatrix} \right)\]

The \(\mathbf{Z}\) and \(\mathbf{A}\) are just like \(\mathbf{B}\) and \(\mathbf{U}\), but \(\mathbf{R}\) is different.

\[\mathbf{R} = \begin{bmatrix}r & 0 \\ 0 & r\end{bmatrix}\]

R <- matrix(list(0),2,2)
diag(Q) <- "r"
R

##      [,1] [,2]
## [1,] 0    0   
## [2,] 0    0

or use

R <- "diagonal and equal"

\(\mathbf{R}\) has both fixed values (the 0s) and estimated values (the \(r\)), so we need both numbers and charaters in our matrix.