Type 0
The measurement model is given by
$$
\mathbf{y}_{i, t}
=
\boldsymbol{\nu}
+
\boldsymbol{\Lambda}
\boldsymbol{\eta}_{i, t}
+
\boldsymbol{\varepsilon}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\varepsilon}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\boldsymbol{\Theta}
\right)
$$
where
\(\mathbf{y}_{i, t}\),
\(\boldsymbol{\eta}_{i, t}\),
and
\(\boldsymbol{\varepsilon}_{i, t}\)
are random variables
and
\(\boldsymbol{\nu}\),
\(\boldsymbol{\Lambda}\),
and
\(\boldsymbol{\Theta}\)
are model parameters.
\(\mathbf{y}_{i, t}\)
represents a vector of observed random variables,
\(\boldsymbol{\eta}_{i, t}\)
a vector of latent random variables,
and
\(\boldsymbol{\varepsilon}_{i, t}\)
a vector of random measurement errors,
at time \(t\) and individual \(i\).
\(\boldsymbol{\nu}\)
denotes a vector of intercepts,
\(\boldsymbol{\Lambda}\)
a matrix of factor loadings,
and
\(\boldsymbol{\Theta}\)
the covariance matrix of
\(\boldsymbol{\varepsilon}\).
An alternative representation of the measurement error
is given by
$$
\boldsymbol{\varepsilon}_{i, t}
=
\boldsymbol{\Theta}^{\frac{1}{2}}
\mathbf{z}_{i, t},
\quad
\mathrm{with}
\quad
\mathbf{z}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\mathbf{I}
\right)
$$
where
\(\mathbf{z}_{i, t}\) is a vector of
independent standard normal random variables and
\(
\left( \boldsymbol{\Theta}^{\frac{1}{2}} \right)
\left( \boldsymbol{\Theta}^{\frac{1}{2}} \right)^{\prime}
=
\boldsymbol{\Theta} .
\)
The dynamic structure is given by
$$
\mathrm{d} \boldsymbol{\eta}_{i, t}
=
\left(
\boldsymbol{\iota}
+
\boldsymbol{\Phi}
\boldsymbol{\eta}_{i, t}
\right)
\mathrm{d}t
+
\boldsymbol{\Sigma}^{\frac{1}{2}}
\mathrm{d}
\mathbf{W}_{i, t}
$$
where
\(\boldsymbol{\iota}\)
is a term which is unobserved and constant over time,
\(\boldsymbol{\Phi}\)
is the drift matrix
which represents the rate of change of the solution
in the absence of any random fluctuations,
\(\boldsymbol{\Sigma}\)
is the matrix of volatility
or randomness in the process, and
\(\mathrm{d}\boldsymbol{W}\)
is a Wiener process or Brownian motion,
which represents random fluctuations.
Type 1
The measurement model is given by
$$
\mathbf{y}_{i, t}
=
\boldsymbol{\nu}
+
\boldsymbol{\Lambda}
\boldsymbol{\eta}_{i, t}
+
\boldsymbol{\varepsilon}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\varepsilon}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\boldsymbol{\Theta}
\right) .
$$
The dynamic structure is given by
$$
\mathrm{d} \boldsymbol{\eta}_{i, t}
=
\left(
\boldsymbol{\iota}
+
\boldsymbol{\Phi}
\boldsymbol{\eta}_{i, t}
\right)
\mathrm{d}t
+
\boldsymbol{\Gamma}
\mathbf{x}_{i, t}
+
\boldsymbol{\Sigma}^{\frac{1}{2}}
\mathrm{d}
\mathbf{W}_{i, t}
$$
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
$$
\mathbf{y}_{i, t}
=
\boldsymbol{\nu}
+
\boldsymbol{\Lambda}
\boldsymbol{\eta}_{i, t}
+
\boldsymbol{\kappa}
\mathbf{x}_{i, t}
+
\boldsymbol{\varepsilon}_{i, t},
\quad
\mathrm{with}
\quad
\boldsymbol{\varepsilon}_{i, t}
\sim
\mathcal{N}
\left(
\mathbf{0},
\boldsymbol{\Theta}
\right)
$$
where
\(\boldsymbol{\kappa}\) represents the coefficient matrix
linking the covariates to the observed variables.
The dynamic structure is given by
$$
\mathrm{d} \boldsymbol{\eta}_{i, t}
=
\left(
\boldsymbol{\iota}
+
\boldsymbol{\Phi}
\boldsymbol{\eta}_{i, t}
\right)
\mathrm{d}t
+
\boldsymbol{\Gamma}
\mathbf{x}_{i, t}
+
\boldsymbol{\Sigma}^{\frac{1}{2}}
\mathrm{d}
\mathbf{W}_{i, t} .
$$
State Space Parameterization
The state space parameters
as a function of the linear stochastic differential equation model
parameters
are given by
$$
\boldsymbol{\beta}_{\Delta t_{{l_{i}}}}
=
\exp{
\left(
\Delta t
\boldsymbol{\Phi}
\right)
}
$$
$$
\boldsymbol{\alpha}_{\Delta t_{{l_{i}}}}
=
\boldsymbol{\Phi}^{-1}
\left(
\boldsymbol{\beta} - \mathbf{I}_{p}
\right)
\boldsymbol{\iota}
$$
$$
\mathrm{vec}
\left(
\boldsymbol{\Psi}_{\Delta t_{{l_{i}}}}
\right)
=
\left[
\left(
\boldsymbol{\Phi} \otimes \mathbf{I}_{p}
\right)
+
\left(
\mathbf{I}_{p} \otimes \boldsymbol{\Phi}
\right)
\right]
\left[
\exp
\left(
\left[
\left(
\boldsymbol{\Phi} \otimes \mathbf{I}_{p}
\right)
+
\left(
\mathbf{I}_{p} \otimes \boldsymbol{\Phi}
\right)
\right]
\Delta t
\right)
-
\mathbf{I}_{p \times p}
\right]
\mathrm{vec}
\left(
\boldsymbol{\Sigma}
\right)
$$
where \(p\) is the number of latent variables and
\(\Delta t\) is the time interval.