Simulate Data from a State Space Model (Individual-Varying Parameters)

This function simulates data using a state space model. It assumes that the parameters can vary across individuals.

Usage

SimSSMIVary(
  n,
  time,
  delta_t = 1,
  mu0,
  sigma0_l,
  alpha,
  beta,
  psi_l,
  nu,
  lambda,
  theta_l,
  type = 0,
  x = NULL,
  gamma = NULL,
  kappa = NULL
)

Arguments

n: Positive integer. Number of individuals.
time: Positive integer. Number of time points.
delta_t: Numeric. Time interval. The default value is 1.0 with an option to use a numeric value for the discretized state space model parameterization of the linear stochastic differential equation model.
mu0: List of numeric vectors. Each element of the list is the mean of initial latent variable values (\(\boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0}\)).
sigma0_l: List of numeric matrices. Each element of the list is the Cholesky factorization (t(chol(sigma0))) of the covariance matrix of initial latent variable values (\(\boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0}\)).
alpha: List of numeric vectors. Each element of the list is the vector of constant values for the dynamic model (\(\boldsymbol{\alpha}\)).
beta: List of numeric matrices. Each element of the list is the transition matrix relating the values of the latent variables at the previous to the current time point (\(\boldsymbol{\beta}\)).
psi_l: List of numeric matrices. Each element of the list is the Cholesky factorization (t(chol(psi))) of the covariance matrix of the process noise (\(\boldsymbol{\Psi}\)).
nu: List of numeric vectors. Each element of the list is the vector of intercept values for the measurement model (\(\boldsymbol{\nu}\)).
lambda: List of numeric matrices. Each element of the list is the factor loading matrix linking the latent variables to the observed variables (\(\boldsymbol{\Lambda}\)).
theta_l: List of numeric matrices. Each element of the list is the Cholesky factorization (t(chol(theta))) of the covariance matrix of the measurement error (\(\boldsymbol{\Theta}\)).
type: Integer. State space model type. See Details in SimSSMFixed() 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: List of numeric matrices. Each element of the list is the matrix linking the covariates to the latent variables at current time point (\(\boldsymbol{\Gamma}\)).
kappa: List of numeric matrices. Each element of the list is the matrix linking the covariates to the observed variables at current time point (\(\boldsymbol{\kappa}\)).

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.

Details

Parameters can vary across individuals by providing a list of parameter values. If the length of any of the parameters (mu0, sigma0_l, alpha, beta, psi_l, nu, lambda, theta_l, gamma, or kappa) is less the n, the function will cycle through the available values.

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

Author

Ivan Jacob Agaloos Pesigan

Examples

# prepare parameters
# In this example, beta varies across individuals.
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
## dynamic structure
p <- 3
mu0 <- list(
  rep(x = 0, times = p)
)
sigma0 <- 0.001 * diag(p)
sigma0_l <- list(
  t(chol(sigma0))
)
alpha <- list(
  rep(x = 0, times = p)
)
beta <- list(
  0.1 * diag(p),
  0.2 * diag(p),
  0.3 * diag(p),
  0.4 * diag(p),
  0.5 * diag(p)
)
psi <- 0.001 * diag(p)
psi_l <- list(
  t(chol(psi))
)
## measurement model
k <- 3
nu <- list(
  rep(x = 0, times = k)
)
lambda <- list(
  diag(k)
)
theta <- 0.001 * diag(k)
theta_l <- list(
  t(chol(theta))
)
## covariates
j <- 2
x <- lapply(
  X = seq_len(n),
  FUN = function(i) {
    matrix(
      data = stats::rnorm(n = time * j),
      nrow = j,
      ncol = time
    )
  }
)
gamma <- list(
  diag(x = 0.10, nrow = p, ncol = j)
)
kappa <- list(
  diag(x = 0.10, nrow = k, ncol = j)
)

# Type 0
ssm <- SimSSMIVary(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0
)

plot(ssm)




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

plot(ssm)




# Type 2
ssm <- SimSSMIVary(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

plot(ssm)