## 9.3 Stochastic level models

The most simple DLM is a stochastic level model, where the level is a random walk without drift, and this level is observed with error. We will write it first in using regression notation where the intercept is $$\alpha$$ and then in MARSS notation. In the latter, $$\alpha_t=x_t$$.

$$$\begin{gathered} \tag{9.5} y_t = \alpha_t + e_t \\ \alpha_t = \alpha_{t-1} + w_t \\ \Downarrow \\ y_t = x_t + v_t \\ x_t = x_{t-1} + w_t \end{gathered}$$$

Using this model, we can model the Nile River level and fit the model using MARSS().

## load Nile flow data
data(Nile, package = "datasets")

## define model list
mod_list <- list(B = "identity", U = "zero", Q = matrix("q"),
Z = "identity", A = matrix("a"), R = matrix("r"))

## fit the model with MARSS
fit <- MARSS(matrix(Nile, nrow = 1), mod_list)
Success! abstol and log-log tests passed at 82 iterations.
Alert: conv.test.slope.tol is 0.5.
Test with smaller values (<0.1) to ensure convergence.

Estimation method: kem
Convergence test: conv.test.slope.tol = 0.5, abstol = 0.001
Estimation converged in 82 iterations.
Log-likelihood: -637.7569
AIC: 1283.514   AICc: 1283.935

Estimate
A.a      -0.338
R.r   15135.796
Q.q    1381.153
x0.x0  1111.791
Initial states (x0) defined at t=0

Standard errors have not been calculated.
Use MARSSparamCIs to compute CIs and bias estimates.

### 9.3.1 Stochastic level with drift

We can add a drift term to the level model to allow the level to tend upward or downward with a deterministic rate $$\eta$$. This is a random walk with bias.

$$$\begin{gathered} \tag{9.6} y_t = \alpha_t + e_t \\ \alpha_t = \alpha_{t-1} + \eta + w_t \\ \Downarrow \\ y_t = x_t + v_t \\ x_t = x_{t-1} + u + w_t \end{gathered}$$$

We can allow that the drift term $$\eta$$ evolves over time along with the level. In this case, $$\eta$$ is modeled as a random walk along with $$\alpha$$. This model is

$$$\begin{gathered} \tag{9.7} y_t = \alpha_t + e_t \\ \alpha_t = \alpha_{t-1} + \eta_{t-1} + w_{\alpha,t} \\ \eta_t = \eta_{t-1} + w_{\eta,t} \end{gathered}$$$ Equation (9.7) can be written in matrix form as: $$$\begin{gathered} \tag{9.8} y_t = \begin{bmatrix}1&0\end{bmatrix}\begin{bmatrix} \alpha \\ \eta \end{bmatrix}_t + v_t \\ \begin{bmatrix} \alpha \\ \eta \end{bmatrix}_t = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix}\begin{bmatrix} \alpha \\ \eta \end{bmatrix}_{t-1} + \begin{bmatrix} w_{\alpha} \\ w_{\eta} \end{bmatrix}_t \end{gathered}$$$ Equation (9.8) is a MARSS model.

$$$y_t = \mathbf{Z}\mathbf{x}_t + v_t \\ \mathbf{x}_t = \mathbf{B}\mathbf{x}_{t-1} + \mathbf{w}_t$$$

where $$\mathbf{B}=\begin{bmatrix} 1 & 1 \\ 0 & 1\end{bmatrix}$$, $$\mathbf{x}=\begin{bmatrix}\alpha \\ \eta\end{bmatrix}$$ and $$\mathbf{Z}=\begin{bmatrix}1&0\end{bmatrix}$$.

See Section 6.2 for more discussion of stochastic level models and Section @ref() to see how to fit this model with the StructTS(sec-uss-the-structts-function) function in the stats package.