ExpCov {cTMed} | R Documentation |
Model-Implied State Covariance Matrix
Description
The function returns the model-implied state covariance matrix
for a particular time interval \Delta t
given by
\mathrm{vec} \left( \mathrm{Cov} \left( \boldsymbol{\eta} \right) \right)
=
\left(
\mathbf{J} -
\boldsymbol{\beta}_{\Delta t} \otimes \boldsymbol{\beta}_{\Delta t}
\right)^{-1}
\mathrm{vec} \left( \boldsymbol{\Psi}_{\Delta t} \right)
where
\boldsymbol{\beta}_{\Delta t}
=
\exp \left( \Delta t \boldsymbol{\Phi} \right) ,
\boldsymbol{\Psi}_{\Delta t}
=
\boldsymbol{\Phi}^{\#}
\left(
\exp \left( \Delta t \boldsymbol{\Phi} \right) - \mathbf{J}
\right)
\mathrm{vec} \left( \boldsymbol{\Sigma} \right) , \quad \mathrm{and}
\boldsymbol{\Phi}^{\#}
=
\left(
\boldsymbol{\Phi} \otimes \mathbf{I}
\right) +
\left(
\mathbf{I} \otimes \boldsymbol{\Phi}
\right).
Note that \mathbf{I}
and \mathbf{J}
are identity matrices.
Usage
ExpCov(phi, sigma, delta_t)
Arguments
phi |
Numeric matrix.
The drift matrix ( |
sigma |
Numeric matrix.
The process noise covariance matrix ( |
delta_t |
Numeric.
Time interval
( |
Details
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.
Value
Returns a numeric matrix.
Author(s)
Ivan Jacob Agaloos Pesigan
See Also
Other Continuous Time Mediation Functions:
DeltaBeta()
,
DeltaBetaStd()
,
DeltaIndirectCentral()
,
DeltaMed()
,
DeltaMedStd()
,
DeltaTotalCentral()
,
Direct()
,
DirectStd()
,
ExpMean()
,
Indirect()
,
IndirectCentral()
,
IndirectStd()
,
MCBeta()
,
MCBetaStd()
,
MCIndirectCentral()
,
MCMed()
,
MCMedStd()
,
MCPhi()
,
MCTotalCentral()
,
Med()
,
MedStd()
,
PosteriorBeta()
,
PosteriorIndirectCentral()
,
PosteriorMed()
,
PosteriorTotalCentral()
,
Total()
,
TotalCentral()
,
TotalStd()
,
Trajectory()
Examples
phi <- matrix(
data = c(
-0.357, 0.771, -0.450,
0.0, -0.511, 0.729,
0, 0, -0.693
),
nrow = 3
)
colnames(phi) <- rownames(phi) <- c("x", "m", "y")
sigma <- matrix(
data = c(
0.24, 0.02, -0.05,
0.02, 0.07, 0.02,
-0.05, 0.02, 0.08
),
nrow = 3
)
delta_t <- 1
ExpCov(
phi = phi,
sigma = sigma,
delta_t = delta_t
)