Linear Stochastic Differential Equation Model
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.