Title:	Simulate Data from State Space Models
Version:	1.2.10
Description:	Provides a streamlined and user-friendly framework for simulating data in state space models, particularly when the number of subjects/units (n) exceeds one, a scenario commonly encountered in social and behavioral sciences. For an introduction to state space models in social and behavioral sciences, refer to Chow, Ho, Hamaker, and Dolan (2010) <doi:10.1080/10705511003661553>.
URL:	https://github.com/jeksterslab/simStateSpace, https://jeksterslab.github.io/simStateSpace/
BugReports:	https://github.com/jeksterslab/simStateSpace/issues
License:	GPL (≥ 3)
Encoding:	UTF-8
Depends:	R (≥ 3.5.0)
LinkingTo:	Rcpp, RcppArmadillo
Imports:	Rcpp, stats
Suggests:	knitr, rmarkdown, testthat, expm
RoxygenNote:	7.3.2
NeedsCompilation:	yes
Packaged:	2025-03-28 16:49:44 UTC; root
Author:	Ivan Jacob Agaloos Pesigan [aut, cre, cph]
Maintainer:	Ivan Jacob Agaloos Pesigan <r.jeksterslab@gmail.com>
Repository:	CRAN
Date/Publication:	2025-03-30 02:40:02 UTC

simStateSpace: Simulate Data from State Space Models

Description

Provides a streamlined and user-friendly framework for simulating data in state space models, particularly when the number of subjects/units (n) exceeds one, a scenario commonly encountered in social and behavioral sciences. For an introduction to state space models in social and behavioral sciences, refer to Chow, Ho, Hamaker, and Dolan (2010) doi:10.1080/10705511003661553.

Author(s)

Maintainer: Ivan Jacob Agaloos Pesigan r.jeksterslab@gmail.com (ORCID) [copyright holder]

See Also

Useful links:

• https://github.com/jeksterslab/simStateSpace

• https://jeksterslab.github.io/simStateSpace/

• Report bugs at https://github.com/jeksterslab/simStateSpace/issues

Convert Parameters from the Linear Stochastic Differential Equation Model to State Space Model Parameterization

Description

This function converts parameters from the linear stochastic differential equation model to state space model parameterization.

Usage

LinSDE2SSM(iota, phi, sigma_l, delta_t)

Arguments

Details

Let the linear stochastic equation model be 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}

for individual i and time t. The discrete-time state space model given below represents the discrete-time solution for the linear stochastic differential equation.

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

with

\boldsymbol{\beta}_{\Delta t_{{l_{i}}}} = \exp{ \left( \Delta t \boldsymbol{\Phi} \right) },

\boldsymbol{\alpha}_{\Delta t_{{l_{i}}}} = \boldsymbol{\Phi}^{-1} \left( \boldsymbol{\beta} - \mathbf{I}_{p} \right) \boldsymbol{\iota}, \quad \mathrm{and}

\mathrm{vec} \left( \boldsymbol{\Psi}_{\Delta t_{{l_{i}}}} \right) = \left[ \left( \boldsymbol{\Phi} \otimes \mathbf{I}_{p} \right) + \left( \mathbf{I}_{p} \otimes \boldsymbol{\Phi} \right) \right] \left[ \exp \left( \left[ \left( \boldsymbol{\Phi} \otimes \mathbf{I}_{p} \right) + \left( \mathbf{I}_{p} \otimes \boldsymbol{\Phi} \right) \right] \Delta t \right) - \mathbf{I}_{p \times p} \right] \mathrm{vec} \left( \boldsymbol{\Sigma} \right)

where t denotes continuous-time processes that can be defined by any arbitrary time point, t_{l_{i}} the l^\mathrm{th} observed measurement occassion for individual i, p the number of latent variables and \Delta t the time interval.

Value

Returns a list of state space parameters:

• 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 from the previous time point to the current time point. (\boldsymbol{\beta}).

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

Author(s)

Ivan Jacob Agaloos Pesigan

References

Harvey, A. C. (1990). Forecasting, structural time series models and the Kalman filter. Cambridge University Press. doi:10.1017/cbo9781107049994

See Also

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

Examples

p <- 2
iota <- c(0.317, 0.230)
phi <- matrix(
  data = c(
   -0.10,
   0.05,
   0.05,
   -0.10
 ),
 nrow = p
)
sigma <- matrix(
  data = c(
    2.79,
    0.06,
    0.06,
    3.27
  ),
  nrow = p
)
sigma_l <- t(chol(sigma))
delta_t <- 0.10

LinSDE2SSM(
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  delta_t = delta_t
)

Steady-State Covariance Matrix for the Linear Stochastic Differential Equation Model

Description

The steady-state covariance matrix is the solution to the Sylvester equation, i.e.

\mathbf{A} \mathbf{X} + \mathbf{X} \mathbf{B} + \mathbf{C} = \mathbf{0} ,

where \mathbf{X} is unknown, \mathbf{A} = \boldsymbol{\Phi}, \mathbf{B} = \boldsymbol{\Phi}^{\prime}, and \mathbf{C} = \boldsymbol{\Sigma}.

Usage

LinSDECov(phi, sigma)

Arguments

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

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

Examples

phi <- matrix(
  data = c(
    -0.10,
    0.05,
    0.05,
    -0.10
  ),
  nrow = 2
)
sigma <- matrix(
  data = c(
    2.79,
    0.06,
    0.06,
    3.27
  ),
  nrow = 2
)
LinSDECov(phi = phi, sigma = sigma)

Steady-State Mean Vector for the Linear Stochastic Differential Equation Model

Description

The steady-state mean vector is given by

-\boldsymbol{\Phi}^{-1} \boldsymbol{\iota}

.

Usage

LinSDEMean(phi, iota)

Arguments

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

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

Examples

iota <- c(0.317, 0.230)
phi <- matrix(
  data = c(
    -0.10,
    0.05,
    0.05,
    -0.10
  ),
  nrow = 2
)
LinSDEMean(phi = phi, iota = iota)

Simulate Transition Matrices from the Multivariate Normal Distribution

Description

This function simulates random transition matrices from the multivariate normal distribution. The function ensures that the generated transition matrices are stationary using TestStationarity().

Usage

SimBetaN(n, beta, vcov_beta_vec_l)

Arguments

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

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

Examples

n <- 10
beta <- matrix(
  data = c(
    0.7, 0.5, -0.1,
    0.0, 0.6, 0.4,
    0, 0, 0.5
  ),
  nrow = 3
)
vcov_beta_vec_l <- t(chol(0.001 * diag(9)))
SimBetaN(n = n, beta = beta, vcov_beta_vec_l = vcov_beta_vec_l)

Simulate Random Drift Matrices from the Multivariate Normal Distribution

Description

This function simulates random drift matrices from the multivariate normal distribution. The function ensures that the generated drift matrices are stable using TestPhi().

Usage

SimPhiN(n, phi, vcov_phi_vec_l)

Arguments

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

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

Examples

n <- 10
phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0, 0, -0.693
  ),
  nrow = 3
)
vcov_phi_vec_l <- t(chol(0.001 * diag(9)))
SimPhiN(n = n, phi = phi, vcov_phi_vec_l = vcov_phi_vec_l)

Simulate Data from a State Space Model (Fixed Parameters)

Description

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

Usage

SimSSMFixed(
  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

Details

Type 0

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

\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{\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) .

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.

Type 2

The measurement model is given by

\mathbf{y}_{i, t} = \boldsymbol{\nu} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\kappa} \mathbf{x}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right)

where \boldsymbol{\kappa} represents the coefficient matrix linking the covariates to the observed variables.

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) .

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(), LinSDECov(), LinSDEMean(), SimBetaN(), SimPhiN(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), 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))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.001 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMFixed(
  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 <- SimSSMFixed(
  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 <- SimSSMFixed(
  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)

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

Description

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

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.

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(), LinSDECov(), LinSDEMean(), SimBetaN(), SimPhiN(), SimSSMFixed(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), TestPhi(), TestStability(), TestStationarity()

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)

Simulate Data from the Linear Growth Curve Model

Description

This function simulates data from the linear growth curve model.

Usage

SimSSMLinGrowth(
  n,
  time,
  mu0,
  sigma0_l,
  theta_l,
  type = 0,
  x = NULL,
  gamma = NULL,
  kappa = NULL
)

Arguments

Details

Type 0

The measurement model is given by

Y_{i, t} = \left( \begin{array}{cc} 1 & 0 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( 0, \theta \right)

where Y_{i, t}, \eta_{0_{i, t}}, \eta_{1_{i, t}}, and \boldsymbol{\varepsilon}_{i, t} are random variables and \theta is a model parameter. Y_{i, t} is the observed random variable at time t and individual i, \eta_{0_{i, t}} (intercept) and \eta_{1_{i, t}} (slope) form a vector of latent random variables at time t and individual i, and \boldsymbol{\varepsilon}_{i, t} a vector of random measurement errors at time t and individual i. \theta is the variance of \boldsymbol{\varepsilon}.

The dynamic structure is given by

\left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t - 1}} \\ \eta_{1_{i, t - 1}} \\ \end{array} \right) .

The mean vector and covariance matrix of the intercept and slope are captured in the mean vector and covariance matrix of the initial condition given by

\boldsymbol{\mu}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{c} \mu_{\eta_{0}} \\ \mu_{\eta_{1}} \\ \end{array} \right) \quad \mathrm{and,}

\boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0} = \left( \begin{array}{cc} \sigma^{2}_{\eta_{0}} & \sigma_{\eta_{0}, \eta_{1}} \\ \sigma_{\eta_{1}, \eta_{0}} & \sigma^{2}_{\eta_{1}} \\ \end{array} \right) .

Type 1

The measurement model is given by

Y_{i, t} = \left( \begin{array}{cc} 1 & 0 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( 0, \theta \right) .

The dynamic structure is given by

\left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t - 1}} \\ \eta_{1_{i, t - 1}} \\ \end{array} \right) + \boldsymbol{\Gamma} \mathbf{x}_{i, t}

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.

Type 2

The measurement model is given by

Y_{i, t} = \left( \begin{array}{cc} 1 & 0 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) + \boldsymbol{\kappa} \mathbf{x}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( 0, \theta \right)

where \boldsymbol{\kappa} represents the coefficient matrix linking the covariates to the observed variables.

The dynamic structure is given by

\left( \begin{array}{c} \eta_{0_{i, t}} \\ \eta_{1_{i, t}} \\ \end{array} \right) = \left( \begin{array}{cc} 1 & 1 \\ 0 & 1 \\ \end{array} \right) \left( \begin{array}{c} \eta_{0_{i, t - 1}} \\ \eta_{1_{i, t - 1}} \\ \end{array} \right) + \boldsymbol{\Gamma} \mathbf{x}_{i, t} .

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(), LinSDECov(), LinSDEMean(), SimBetaN(), SimPhiN(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), TestPhi(), TestStability(), TestStationarity()

Examples

# prepare parameters
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 5
## dynamic structure
p <- 2
mu0 <- c(0.615, 1.006)
sigma0 <- matrix(
  data = c(
    1.932,
    0.618,
    0.618,
    0.587
  ),
  nrow = p
)
sigma0_l <- t(chol(sigma0))
## measurement model
k <- 1
theta <- 0.50
theta_l <- sqrt(theta)
## covariates
j <- 2
x <- lapply(
  X = seq_len(n),
  FUN = function(i) {
    matrix(
      data = rnorm(n = j * time),
      nrow = j
    )
  }
)
gamma <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMLinGrowth(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  theta_l = theta_l,
  type = 0
)

plot(ssm)

# Type 1
ssm <- SimSSMLinGrowth(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

plot(ssm)

# Type 2
ssm <- SimSSMLinGrowth(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

plot(ssm)

Simulate Data from the Linear Growth Curve Model (Individual-Varying Parameters)

Description

This function simulates data from the linear growth curve model. It assumes that the parameters can vary across individuals.

Usage

SimSSMLinGrowthIVary(
  n,
  time,
  mu0,
  sigma0_l,
  theta_l,
  type = 0,
  x = NULL,
  gamma = NULL,
  kappa = NULL
)

Arguments

Details

Parameters can vary across individuals by providing a list of parameter values. If the length of any of the parameters (mu0, sigma0, mu, theta_l, gamma, or kappa) is less the n, the function will cycle through the available values.

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(), LinSDECov(), LinSDEMean(), SimBetaN(), SimPhiN(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), TestPhi(), TestStability(), TestStationarity()

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

Examples

# prepare parameters
# In this example, the mean vector of the intercept and slope vary.
# Specifically,
# there are two sets of values representing two latent classes.
set.seed(42)
## number of individuals
n <- 10
## time points
time <- 5
## dynamic structure
p <- 2
mu0_1 <- c(0.615, 1.006) # lower starting point, higher growth
mu0_2 <- c(1.000, 0.500) # higher starting point, lower growth
mu0 <- list(mu0_1, mu0_2)
sigma0 <- matrix(
  data = c(
    1.932,
    0.618,
    0.618,
    0.587
  ),
  nrow = p
)
sigma0_l <- list(t(chol(sigma0)))
## measurement model
k <- 1
theta <- 0.50
theta_l <- list(sqrt(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 <- SimSSMLinGrowthIVary(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  theta_l = theta_l,
  type = 0
)

plot(ssm)

# Type 1
ssm <- SimSSMLinGrowthIVary(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

plot(ssm)

# Type 2
ssm <- SimSSMLinGrowthIVary(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

plot(ssm)

Simulate Data from the Linear Stochastic Differential Equation Model using a State Space Model Parameterization (Fixed Parameters)

Description

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

Usage

SimSSMLinSDEFixed(
  n,
  time,
  delta_t = 1,
  mu0,
  sigma0_l,
  iota,
  phi,
  sigma_l,
  nu,
  lambda,
  theta_l,
  type = 0,
  x = NULL,
  gamma = NULL,
  kappa = NULL
)

Arguments

Details

Type 0

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.

Type 1

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) .

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{\Gamma} \mathbf{x}_{i, t} + \boldsymbol{\Sigma}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t}

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.

Type 2

The measurement model is given by

\mathbf{y}_{i, t} = \boldsymbol{\nu} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\kappa} \mathbf{x}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right)

where \boldsymbol{\kappa} represents the coefficient matrix linking the covariates to the observed variables.

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{\Gamma} \mathbf{x}_{i, t} + \boldsymbol{\Sigma}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t} .

State Space Parameterization

The state space parameters as a function of the linear stochastic differential equation model parameters are given by

\boldsymbol{\beta}_{\Delta t_{{l_{i}}}} = \exp{ \left( \Delta t \boldsymbol{\Phi} \right) }

\boldsymbol{\alpha}_{\Delta t_{{l_{i}}}} = \boldsymbol{\Phi}^{-1} \left( \boldsymbol{\beta} - \mathbf{I}_{p} \right) \boldsymbol{\iota}

\mathrm{vec} \left( \boldsymbol{\Psi}_{\Delta t_{{l_{i}}}} \right) = \left[ \left( \boldsymbol{\Phi} \otimes \mathbf{I}_{p} \right) + \left( \mathbf{I}_{p} \otimes \boldsymbol{\Phi} \right) \right] \left[ \exp \left( \left[ \left( \boldsymbol{\Phi} \otimes \mathbf{I}_{p} \right) + \left( \mathbf{I}_{p} \otimes \boldsymbol{\Phi} \right) \right] \Delta t \right) - \mathbf{I}_{p \times p} \right] \mathrm{vec} \left( \boldsymbol{\Sigma} \right)

where p is the number of latent variables and \Delta t is the time interval.

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

Chow, S.-M., Losardo, D., Park, J., & Molenaar, P. C. M. (2023). Continuous-time dynamic models: Connections to structural equation models and other discrete-time models. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (2nd ed.). The Guilford Press.

Harvey, A. C. (1990). Forecasting, structural time series models and the Kalman filter. Cambridge University Press. doi:10.1017/cbo9781107049994

See Also

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

Examples

# prepare parameters
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
delta_t <- 0.10
## dynamic structure
p <- 2
mu0 <- c(-3.0, 1.5)
sigma0 <- 0.001 * diag(p)
sigma0_l <- t(chol(sigma0))
iota <- c(0.317, 0.230)
phi <- matrix(
  data = c(
    -0.10,
    0.05,
    0.05,
    -0.10
  ),
  nrow = p
)
sigma <- matrix(
  data = c(
    2.79,
    0.06,
    0.06,
    3.27
  ),
  nrow = p
)
sigma_l <- t(chol(sigma))
## measurement model
k <- 2
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.001 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMLinSDEFixed(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0
)

plot(ssm)

# Type 1
ssm <- SimSSMLinSDEFixed(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

plot(ssm)

# Type 2
ssm <- SimSSMLinSDEFixed(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

plot(ssm)

Simulate Data from the Linear Stochastic Differential Equation Model using a State Space Model Parameterization (Individual-Varying Parameters)

Description

This function simulates data from the linear stochastic differential equation model using a state space model parameterization. It assumes that the parameters can vary across individuals.

Usage

SimSSMLinSDEIVary(
  n,
  time,
  delta_t = 1,
  mu0,
  sigma0_l,
  iota,
  phi,
  sigma_l,
  nu,
  lambda,
  theta_l,
  type = 0,
  x = NULL,
  gamma = NULL,
  kappa = NULL
)

Arguments

Details

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

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

Chow, S.-M., Losardo, D., Park, J., & Molenaar, P. C. M. (2023). Continuous-time dynamic models: Connections to structural equation models and other discrete-time models. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (2nd ed.). The Guilford Press.

Harvey, A. C. (1990). Forecasting, structural time series models and the Kalman filter. Cambridge University Press. doi:10.1017/cbo9781107049994

See Also

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

Examples

# prepare parameters
# In this example, phi varies across individuals.
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
delta_t <- 0.10
## dynamic structure
p <- 2
mu0 <- list(
  c(-3.0, 1.5)
)
sigma0 <- 0.001 * diag(p)
sigma0_l <- list(
  t(chol(sigma0))
)
iota <- list(
  c(0.317, 0.230)
)
phi <- list(
  -0.1 * diag(p),
  -0.2 * diag(p),
  -0.3 * diag(p),
  -0.4 * diag(p),
  -0.5 * diag(p)
)
sigma <- matrix(
  data = c(
    2.79,
    0.06,
    0.06,
    3.27
  ),
  nrow = p
)
sigma_l <- list(
  t(chol(sigma))
)
## measurement model
k <- 2
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 <- SimSSMLinSDEIVary(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0
)

plot(ssm)

# Type 1
ssm <- SimSSMLinSDEIVary(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

plot(ssm)

# Type 2
ssm <- SimSSMLinSDEIVary(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

plot(ssm)

Simulate Data from the Ornstein–Uhlenbeck Model using a State Space Model Parameterization (Fixed Parameters)

Description

This function simulates data from the Ornstein–Uhlenbeck (OU) model using a state space model parameterization. It assumes that the parameters remain constant across individuals and over time.

Usage

SimSSMOUFixed(
  n,
  time,
  delta_t = 1,
  mu0,
  sigma0_l,
  mu,
  phi,
  sigma_l,
  nu,
  lambda,
  theta_l,
  type = 0,
  x = NULL,
  gamma = NULL,
  kappa = NULL
)

Arguments

Details

Type 0

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} = \boldsymbol{\Phi} \left( \boldsymbol{\eta}_{i, t} - \boldsymbol{\mu} \right) \mathrm{d}t + \boldsymbol{\Sigma}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t}

where \boldsymbol{\mu} is the long-term mean or equilibrium level, \boldsymbol{\Phi} is the rate of mean reversion, determining how quickly the variable returns to its mean, \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.

Type 1

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) .

The dynamic structure is given by

\mathrm{d} \boldsymbol{\eta}_{i, t} = \boldsymbol{\Phi} \left( \boldsymbol{\eta}_{i, t} - \boldsymbol{\mu} \right) \mathrm{d}t + \boldsymbol{\Gamma} \mathbf{x}_{i, t} + \boldsymbol{\Sigma}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t}

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.

Type 2

The measurement model is given by

\mathbf{y}_{i, t} = \boldsymbol{\nu} + \boldsymbol{\Lambda} \boldsymbol{\eta}_{i, t} + \boldsymbol{\kappa} \mathbf{x}_{i, t} + \boldsymbol{\varepsilon}_{i, t}, \quad \mathrm{with} \quad \boldsymbol{\varepsilon}_{i, t} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Theta} \right)

where \boldsymbol{\kappa} represents the coefficient matrix linking the covariates to the observed variables.

The dynamic structure is given by

\mathrm{d} \boldsymbol{\eta}_{i, t} = \boldsymbol{\Phi} \left( \boldsymbol{\eta}_{i, t} - \boldsymbol{\mu} \right) \mathrm{d}t + \boldsymbol{\Gamma} \mathbf{x}_{i, t} + \boldsymbol{\Sigma}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t} .

The OU model as a linear stochastic differential equation model

The OU model is a first-order linear stochastic differential equation model in the form of

\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{\mu} = - \boldsymbol{\Phi}^{-1} \boldsymbol{\iota} and, equivalently \boldsymbol{\iota} = - \boldsymbol{\Phi} \boldsymbol{\mu}.

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

Chow, S.-M., Losardo, D., Park, J., & Molenaar, P. C. M. (2023). Continuous-time dynamic models: Connections to structural equation models and other discrete-time models. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (2nd ed.). The Guilford Press.

Harvey, A. C. (1990). Forecasting, structural time series models and the Kalman filter. Cambridge University Press. doi:10.1017/cbo9781107049994

Oravecz, Z., Tuerlinckx, F., & Vandekerckhove, J. (2011). A hierarchical latent stochastic differential equation model for affective dynamics. Psychological Methods, 16 (4), 468–490. doi:10.1037/a0024375

Uhlenbeck, G. E., & Ornstein, L. S. (1930). On the theory of the brownian motion. Physical Review, 36 (5), 823–841. doi:10.1103/physrev.36.823

See Also

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

Examples

# prepare parameters
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
delta_t <- 0.10
## dynamic structure
p <- 2
mu0 <- c(-3.0, 1.5)
sigma0 <- 0.001 * diag(p)
sigma0_l <- t(chol(sigma0))
mu <- c(5.76, 5.18)
phi <- matrix(
  data = c(
    -0.10,
    0.05,
    0.05,
    -0.10
  ),
  nrow = p
)
sigma <- matrix(
  data = c(
    2.79,
    0.06,
    0.06,
    3.27
  ),
  nrow = p
)
sigma_l <- t(chol(sigma))
## measurement model
k <- 2
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.001 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMOUFixed(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  mu = mu,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0
)

plot(ssm)

# Type 1
ssm <- SimSSMOUFixed(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  mu = mu,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

plot(ssm)

# Type 2
ssm <- SimSSMOUFixed(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  mu = mu,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

plot(ssm)

Simulate Data from the Ornstein–Uhlenbeck Model using a State Space Model Parameterization (Individual-Varying Parameters)

Description

This function simulates data from the Ornstein–Uhlenbeck model using a state space model parameterization. It assumes that the parameters can vary across individuals.

Usage

SimSSMOUIVary(
  n,
  time,
  delta_t = 1,
  mu0,
  sigma0_l,
  mu,
  phi,
  sigma_l,
  nu,
  lambda,
  theta_l,
  type = 0,
  x = NULL,
  gamma = NULL,
  kappa = NULL
)

Arguments

Details

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

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

Chow, S.-M., Losardo, D., Park, J., & Molenaar, P. C. M. (2023). Continuous-time dynamic models: Connections to structural equation models and other discrete-time models. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (2nd ed.). The Guilford Press.

Harvey, A. C. (1990). Forecasting, structural time series models and the Kalman filter. Cambridge University Press. doi:10.1017/cbo9781107049994

Oravecz, Z., Tuerlinckx, F., & Vandekerckhove, J. (2011). A hierarchical latent stochastic differential equation model for affective dynamics. Psychological Methods, 16 (4), 468–490. doi:10.1037/a0024375

Uhlenbeck, G. E., & Ornstein, L. S. (1930). On the theory of the brownian motion. Physical Review, 36 (5), 823–841. doi:10.1103/physrev.36.823

See Also

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

Examples

# prepare parameters
# In this example, phi varies across individuals.
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
delta_t <- 0.10
## dynamic structure
p <- 2
mu0 <- list(
  c(-3.0, 1.5)
)
sigma0 <- 0.001 * diag(p)
sigma0_l <- list(
  t(chol(sigma0))
)
mu <- list(
  c(5.76, 5.18)
)
phi <- list(
  -0.1 * diag(p),
  -0.2 * diag(p),
  -0.3 * diag(p),
  -0.4 * diag(p),
  -0.5 * diag(p)
)
sigma <- matrix(
  data = c(
    2.79,
    0.06,
    0.06,
    3.27
  ),
  nrow = p
)
sigma_l <- list(
  t(chol(sigma))
)
## measurement model
k <- 2
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 <- SimSSMOUIVary(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  mu = mu,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0
)

plot(ssm)

# Type 1
ssm <- SimSSMOUIVary(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  mu = mu,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

plot(ssm)

# Type 2
ssm <- SimSSMOUIVary(
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  mu = mu,
  phi = phi,
  sigma_l = sigma_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

plot(ssm)

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

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(), LinSDECov(), LinSDEMean(), 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)

Simulate Data from the Vector Autoregressive Model (Individual-Varying Parameters)

Description

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

Usage

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

Arguments

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, gamma, or kappa) is less the n, the function will cycle through the available values.

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(), LinSDECov(), LinSDEMean(), SimBetaN(), SimPhiN(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), TestPhi(), TestStability(), TestStationarity()

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))
)
## 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)
)

# Type 0
ssm <- SimSSMVARIVary(
  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 <- SimSSMVARIVary(
  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)

Test the Drift Matrix

Description

Both have to be true for the function to return TRUE.

• Test that the real part of all eigenvalues of \boldsymbol{\Phi} are less than zero.

• Test that the diagonal values of \boldsymbol{\Phi} are between 0 to negative inifinity.

Usage

TestPhi(phi)

Arguments

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

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

Examples

phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0, 0, -0.693
  ),
  nrow = 3
)
TestPhi(phi = phi)

Test Stability

Description

The function computes the eigenvalues of the input matrix x. It checks if the real part of all eigenvalues is negative. If all eigenvalues have negative real parts, the system is considered stable.

Usage

TestStability(x)

Arguments

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

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

Examples

x <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0, 0, -0.693
  ),
  nrow = 3
)
TestStability(x)

Test Stationarity

Description

The function computes the eigenvalues of the input matrix x. It checks if all eigenvalues have moduli less than 1. If all eigenvalues have moduli less than 1, the system is considered stationary.

Usage

TestStationarity(x)

Arguments

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

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

Examples

x <- matrix(
  data = c(0.5, 0.3, 0.2, 0.4),
  nrow = 2
)
TestStationarity(x)

x <- matrix(
  data = c(0.9, -0.5, 0.8, 0.7),
  nrow = 2
)
TestStationarity(x)

Coerce an Object of Class simstatespace to a Data Frame

Description

Coerce an Object of Class simstatespace to a Data Frame

Usage

## S3 method for class 'simstatespace'
as.data.frame(
  x,
  row.names = NULL,
  optional = FALSE,
  eta = FALSE,
  long = TRUE,
  ...
)

Arguments

Author(s)

Ivan Jacob Agaloos Pesigan

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 <- diag(p)
sigma0_l <- t(chol(sigma0))
alpha <- rep(x = 0, times = p)
beta <- 0.50 * diag(p)
psi <- diag(p)
psi_l <- t(chol(psi))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.50 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMFixed(
  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
)

head(as.data.frame(ssm))
head(as.data.frame(ssm, long = FALSE))

# Type 1
ssm <- SimSSMFixed(
  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
)

head(as.data.frame(ssm))
head(as.data.frame(ssm, long = FALSE))

# Type 2
ssm <- SimSSMFixed(
  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
)

head(as.data.frame(ssm))
head(as.data.frame(ssm, long = FALSE))

Coerce an Object of Class simstatespace to a Matrix

Description

Coerce an Object of Class simstatespace to a Matrix

Usage

## S3 method for class 'simstatespace'
as.matrix(x, eta = FALSE, long = TRUE, ...)

Arguments

Author(s)

Ivan Jacob Agaloos Pesigan

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 <- diag(p)
sigma0_l <- t(chol(sigma0))
alpha <- rep(x = 0, times = p)
beta <- 0.50 * diag(p)
psi <- diag(p)
psi_l <- t(chol(psi))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.50 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMFixed(
  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
)

head(as.matrix(ssm))
head(as.matrix(ssm, long = FALSE))

# Type 1
ssm <- SimSSMFixed(
  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
)

head(as.matrix(ssm))
head(as.matrix(ssm, long = FALSE))

# Type 2
ssm <- SimSSMFixed(
  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
)

head(as.matrix(ssm))
head(as.matrix(ssm, long = FALSE))

Plot Method for an Object of Class simstatespace

Description

Plot Method for an Object of Class simstatespace

Usage

## S3 method for class 'simstatespace'
plot(x, id = NULL, time = NULL, eta = FALSE, type = "b", ...)

Arguments

Author(s)

Ivan Jacob Agaloos Pesigan

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 <- diag(p)
sigma0_l <- t(chol(sigma0))
alpha <- rep(x = 0, times = p)
beta <- 0.50 * diag(p)
psi <- diag(p)
psi_l <- t(chol(psi))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.50 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMFixed(
  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)
plot(ssm, id = 1:3, time = 0:9)

# Type 1
ssm <- SimSSMFixed(
  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)
plot(ssm, id = 1:3, time = 0:9)

# Type 2
ssm <- SimSSMFixed(
  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)
plot(ssm, id = 1:3, time = 0:9)

Print Method for an Object of Class simstatespace

Description

Print Method for an Object of Class simstatespace

Usage

## S3 method for class 'simstatespace'
print(x, ...)

Arguments

Value

Prints simulated data in long format.

Author(s)

Ivan Jacob Agaloos Pesigan

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 <- diag(p)
sigma0_l <- t(chol(sigma0))
alpha <- rep(x = 0, times = p)
beta <- 0.50 * diag(p)
psi <- diag(p)
psi_l <- t(chol(psi))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.50 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMFixed(
  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
)

print(ssm)

# Type 1
ssm <- SimSSMFixed(
  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
)

print(ssm)

# Type 2
ssm <- SimSSMFixed(
  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
)

print(ssm)