MCPhiSigma: Generate Random Drift Matrices and Process Noise Covariance Matrices Using the Monte Carlo Method

Description

This function generates random drift matrices $\boldsymbol{\Phi}$ and process noise covariance matrices $\boldsymbol{\Sigma}$ using the Monte Carlo method.

Usage

MCPhiSigma(
  phi,
  sigma,
  vcov_theta,
  R,
  test_phi = TRUE,
  sigma_diag = FALSE,
  ncores = NULL,
  seed = NULL
)

Value

Returns an object of class ctmedmcphi which is a list with the following elements:

call: Function call.
args: Function arguments.
fun: Function used ("MCPhiSigma").
output: A list of simulated draws. Each element contains a drift matrix and a process noise covariance matrix.

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 ($\boldsymbol{\Sigma}$).
vcov_theta: Numeric matrix. The sampling variance-covariance matrix of $\mathrm{vec} \left( \boldsymbol{\Phi} \right)$ and $\mathrm{vech} \left( \boldsymbol{\Sigma} \right)$
R: Positive integer. Number of replications.
test_phi: Logical. If test_phi = TRUE, the function tests the stability of the generated drift matrix $\boldsymbol{\Phi}$. If the test returns FALSE, the function generates a new drift matrix $\boldsymbol{\Phi}$ and runs the test recursively until the test returns TRUE.
sigma_diag: Logical. If sigma_diag = TRUE, treat $\boldsymbol{\Sigma}$ as a diagonal matrix.
ncores: Positive integer. Number of cores to use. If ncores = NULL, use a single core. Consider using multiple cores when number of replications R is a large value.
seed: Random seed.

Author

Ivan Jacob Agaloos Pesigan

Details

Monte Carlo Method

Let $\boldsymbol{\theta}$ be a vector that combines $\mathrm{vec} \left( \boldsymbol{\Phi} \right)$, that is, the elements of the $\boldsymbol{\Phi}$ matrix in vector form sorted column-wise and $\mathrm{vech} \left( \boldsymbol{\Sigma} \right)$, that is, the unique elements of the $\boldsymbol{\Sigma}$ matrix in vector form sorted column-wise. Let $\hat{\boldsymbol{\theta}}$ be a vector that combines $\mathrm{vec} \left( \hat{\boldsymbol{\Phi}} \right)$ and $\mathrm{vech} \left( \hat{\boldsymbol{\Sigma}} \right)$. Based on the asymptotic properties of maximum likelihood estimators, we can assume that estimators are normally distributed around the population parameters. $$ \hat{\boldsymbol{\theta}} \sim \mathcal{N} \left( \boldsymbol{\theta}, \mathbb{V} \left( \hat{\boldsymbol{\theta}} \right) \right) $$ Using this distributional assumption, a sampling distribution of $\hat{\boldsymbol{\theta}}$ which we refer to as $\hat{\boldsymbol{\theta}}^{\ast}$ can be generated by replacing the population parameters with sample estimates, that is, $$ \hat{\boldsymbol{\theta}}^{\ast} \sim \mathcal{N} \left( \hat{\boldsymbol{\theta}}, \hat{\mathbb{V}} \left( \hat{\boldsymbol{\theta}} \right) \right) . $$

Other Continuous-Time Mediation Functions: BootBeta(), BootBetaStd(), BootDirectCentral(), BootDirectCentralStd(), BootIndirectCentral(), BootIndirectCentralStd(), BootMed(), BootMedStd(), BootTotalCentral(), BootTotalCentralStd(), DeltaBeta(), DeltaBetaStd(), DeltaDirectCentral(), DeltaDirectCentralStd(), DeltaIndirectCentral(), DeltaMed(), DeltaMedStd(), DeltaTotalCentral(), DeltaTotalCentralStd(), Direct(), DirectCentral(), DirectCentralStd(), DirectStd(), Indirect(), IndirectCentral(), IndirectCentralStd(), IndirectStd(), MCBeta(), MCBetaStd(), MCDirectCentral(), MCDirectCentralStd(), MCIndirectCentral(), MCIndirectCentralStd(), MCMed(), MCMedStd(), MCPhi(), MCTotalCentral(), MCTotalCentralStd(), Med(), MedStd(), PosteriorBeta(), PosteriorBetaStd(), PosteriorDirectCentral(), PosteriorDirectCentralStd(), PosteriorIndirectCentral(), PosteriorIndirectCentralStd(), PosteriorMed(), PosteriorMedStd(), PosteriorTotalCentral(), PosteriorTotalCentralStd(), Total(), TotalCentral(), TotalCentralStd(), TotalStd(), Trajectory()

Examples

Run this code

set.seed(42)
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(
    0.24455556, 0.02201587, -0.05004762,
    0.02201587, 0.07067800, 0.01539456,
    -0.05004762, 0.01539456, 0.07553061
  ),
  nrow = 3
)
MCPhiSigma(
  phi = phi,
  sigma = sigma,
  vcov_theta = 0.1 * diag(15),
  R = 100L # use a large value for R in actual research
)

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