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 (\boldsymbol{\Phi}). phi should have row and column names pertaining to the variables in the system.

sigma

Numeric matrix. The process noise covariance matrix (\boldsymbol{\Sigma}).

delta_t

Numeric. Time interval (\Delta t).

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: BootBeta(), BootBetaStd(), BootMed(), BootMedStd(), 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
)


[Package cTMed version 1.0.4 Index]