SimSSMVARFixed {simStateSpace}

Simulate Data from the Vector Autoregressive Model (Fixed Parameters)

Description

This function simulates data from the vector autoregressive model using a state space model parameterization. It assumes that the parameters remain constant across individuals and over time.

Usage

SimSSMVARFixed(
  n,
  time,
  mu0,
  sigma0_l,
  alpha,
  beta,
  psi_l,
  type = 0,
  x = NULL,
  gamma = NULL
)

Arguments

n

Positive integer. Number of individuals.

time

Positive integer. Number of time points.

mu0

Numeric vector. Mean of initial latent variable values (\boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0}).

sigma0_l

Numeric matrix. Cholesky factorization (t(chol(sigma0))) of the covariance matrix of initial latent variable values (\boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0}).

alpha

Numeric vector. Vector of constant values for the dynamic model (\boldsymbol{\alpha}).

beta

Numeric matrix. Transition matrix relating the values of the latent variables at the previous to the current time point (\boldsymbol{\beta}).

psi_l

Numeric matrix. Cholesky factorization (t(chol(psi))) of the covariance matrix of the process noise (\boldsymbol{\Psi}).

type

Integer. State space model type. See Details for more information.

x

List. Each element of the list is a matrix of covariates for each individual i in n. The number of columns in each matrix should be equal to time.

gamma

Numeric matrix. Matrix linking the covariates to the latent variables at current time point (\boldsymbol{\Gamma}).

Details

Type 0

The measurement model is given by

\mathbf{y}_{i, t} = \boldsymbol{\eta}_{i, t}

where \mathbf{y}_{i, t} represents a vector of observed variables and \boldsymbol{\eta}_{i, t} a vector of latent variables for individual i and time t. Since the observed and latent variables are equal, we only generate data from the dynamic structure.

The dynamic structure is given by

\boldsymbol{\eta}_{i, t} = \boldsymbol{\alpha} + \boldsymbol{\beta} \boldsymbol{\eta}_{i, t - 1} + \boldsymbol{\zeta}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\zeta}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Psi} \right)

where \boldsymbol{\eta}_{i, t}, \boldsymbol{\eta}_{i, t - 1}, and \boldsymbol{\zeta}_{i, t} are random variables, and \boldsymbol{\alpha}, \boldsymbol{\beta}, and \boldsymbol{\Psi} are model parameters. Here, \boldsymbol{\eta}_{i, t} is a vector of latent variables at time t and individual i, \boldsymbol{\eta}_{i, t - 1} represents a vector of latent variables at time t - 1 and individual i, and \boldsymbol{\zeta}_{i, t} represents a vector of dynamic noise at time t and individual i. \boldsymbol{\alpha} denotes a vector of intercepts, \boldsymbol{\beta} a matrix of autoregression and cross regression coefficients, and \boldsymbol{\Psi} the covariance matrix of \boldsymbol{\zeta}_{i, t}.

An alternative representation of the dynamic noise is given by

\boldsymbol{\zeta}_{i, t} = \boldsymbol{\Psi}^{\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 \left( \boldsymbol{\Psi}^{\frac{1}{2}} \right) \left( \boldsymbol{\Psi}^{\frac{1}{2}} \right)^{\prime} = \boldsymbol{\Psi} .

Type 1

The measurement model is given by

\mathbf{y}_{i, t} = \boldsymbol{\eta}_{i, t} .

The dynamic structure is given by

\boldsymbol{\eta}_{i, t} = \boldsymbol{\alpha} + \boldsymbol{\beta} \boldsymbol{\eta}_{i, t - 1} + \boldsymbol{\Gamma} \mathbf{x}_{i, t} + \boldsymbol{\zeta}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\zeta}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Psi} \right)

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.

Value

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

• call: Function call.

• args: Function arguments.

• data: Generated data which is a list of length n. Each element of data is a list with the following elements:

  • id: A vector of ID numbers with length l, where l is the value of the function argument time.

  • time: A vector time points of length l.

  • y: A l by k matrix of values for the manifest variables.

  • eta: A l by p matrix of values for the latent variables.

  • x: A l by j matrix of values for the covariates (when covariates are included).

• fun: Function used.

Author(s)

Ivan Jacob Agaloos Pesigan

References

Chow, S.-M., Ho, M. R., Hamaker, E. L., & Dolan, C. V. (2010). Equivalence and differences between structural equation modeling and state-space modeling techniques. Structural Equation Modeling: A Multidisciplinary Journal, 17(2), 303–332. doi:10.1080/10705511003661553

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), SimBetaN(), SimPhiN(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARIVary(), TestPhi(), TestStability(), TestStationarity()

Examples

# prepare parameters
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
## dynamic structure
p <- 3
mu0 <- rep(x = 0, times = p)
sigma0 <- 0.001 * diag(p)
sigma0_l <- t(chol(sigma0))
alpha <- rep(x = 0, times = p)
beta <- 0.50 * diag(p)
psi <- 0.001 * diag(p)
psi_l <- t(chol(psi))
## covariates
j <- 2
x <- lapply(
  X = seq_len(n),
  FUN = function(i) {
    matrix(
      data = stats::rnorm(n = time * j),
      nrow = j,
      ncol = time
    )
  }
)
gamma <- diag(x = 0.10, nrow = p, ncol = j)

# Type 0
ssm <- SimSSMVARFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  type = 0
)

plot(ssm)

# Type 1
ssm <- SimSSMVARFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  type = 1,
  x = x,
  gamma = gamma
)

plot(ssm)