Type 0
The measurement model is given by
$$
Y_{i, t}
=
\left(
\begin{array}{cc}
1 & 0 \\
\end{array}
\right)
\left(
\begin{array}{c}
\eta_{0_{i, t}} \\
\eta_{1_{i, t}} \\
\end{array}
\right)
+
\boldsymbol{\varepsilon}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\varepsilon}_{i, t}
\sim
\mathcal{N}
\left(
0,
\theta
\right)
$$
where \(Y_{i, t}\), \(\eta_{0_{i, t}}\),
\(\eta_{1_{i, t}}\),
and \(\boldsymbol{\varepsilon}_{i, t}\)
are random variables and
\(\theta\) is a model parameter.
\(Y_{i, t}\) is the observed random variable
at time \(t\) and individual \(i\),
\(\eta_{0_{i, t}}\) (intercept)
and
\(\eta_{1_{i, t}}\) (slope)
form a vector of latent random variables
at time \(t\) and individual \(i\),
and \(\boldsymbol{\varepsilon}_{i, t}\)
a vector of random measurement errors
at time \(t\) and individual \(i\).
\(\theta\) is the variance of
\(\boldsymbol{\varepsilon}\).
The dynamic structure is given by
$$
\left(
\begin{array}{c}
\eta_{0_{i, t}} \\
\eta_{1_{i, t}} \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
1 & 1 \\
0 & 1 \\
\end{array}
\right)
\left(
\begin{array}{c}
\eta_{0_{i, t - 1}} \\
\eta_{1_{i, t - 1}} \\
\end{array}
\right) .
$$
The mean vector and covariance matrix of the intercept and slope
are captured in the mean vector and covariance matrix
of the initial condition given by
$$
\boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0}
=
\left(
\begin{array}{c}
\mu_{\eta_{0}} \\
\mu_{\eta_{1}} \\
\end{array}
\right) \quad \mathrm{and,}
$$
$$
\boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0}
=
\left(
\begin{array}{cc}
\sigma^{2}_{\eta_{0}} &
\sigma_{\eta_{0}, \eta_{1}} \\
\sigma_{\eta_{1}, \eta_{0}} &
\sigma^{2}_{\eta_{1}} \\
\end{array}
\right) .
$$
Type 1
The measurement model is given by
$$
Y_{i, t}
=
\left(
\begin{array}{cc}
1 & 0 \\
\end{array}
\right)
\left(
\begin{array}{c}
\eta_{0_{i, t}} \\
\eta_{1_{i, t}} \\
\end{array}
\right)
+
\boldsymbol{\varepsilon}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\varepsilon}_{i, t}
\sim
\mathcal{N}
\left(
0,
\theta
\right) .
$$
The dynamic structure is given by
$$
\left(
\begin{array}{c}
\eta_{0_{i, t}} \\
\eta_{1_{i, t}} \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
1 & 1 \\
0 & 1 \\
\end{array}
\right)
\left(
\begin{array}{c}
\eta_{0_{i, t - 1}} \\
\eta_{1_{i, t - 1}} \\
\end{array}
\right)
+
\boldsymbol{\Gamma}
\mathbf{x}_{i, t}
$$
where
\(\mathbf{x}_{i, t}\) represents a vector of covariates
at time \(t\) and individual \(i\),
and \(\boldsymbol{\Gamma}\) the coefficient matrix
linking the covariates to the latent variables.
Type 2
The measurement model is given by
$$
Y_{i, t}
=
\left(
\begin{array}{cc}
1 & 0 \\
\end{array}
\right)
\left(
\begin{array}{c}
\eta_{0_{i, t}} \\
\eta_{1_{i, t}} \\
\end{array}
\right)
+
\boldsymbol{\kappa}
\mathbf{x}_{i, t}
+
\boldsymbol{\varepsilon}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\varepsilon}_{i, t}
\sim
\mathcal{N}
\left(
0,
\theta
\right)
$$
where
\(\boldsymbol{\kappa}\) represents the coefficient matrix
linking the covariates to the observed variables.
The dynamic structure is given by
$$
\left(
\begin{array}{c}
\eta_{0_{i, t}} \\
\eta_{1_{i, t}} \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
1 & 1 \\
0 & 1 \\
\end{array}
\right)
\left(
\begin{array}{c}
\eta_{0_{i, t - 1}} \\
\eta_{1_{i, t - 1}} \\
\end{array}
\right)
+
\boldsymbol{\Gamma}
\mathbf{x}_{i, t} .
$$