# Simulate Data from the Vector Autoregressive Model (Individual-Varying Parameters) This function simulates data from the vector autoregressive model using a state space model parameterization. It assumes that the parameters can vary across individuals. ```r SimSSMVARIVary( 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`: 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 ($\boldsymbol{\Psi}$). - `type`: Integer. State space model type. See Details in `SimSSMVARFixed()` 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 ($\boldsymbol{\Gamma}$). ## Returns 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`, `gamma`, or `kappa`) is less the `n`, the function will cycle through the available values. ## Examples ```r # 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) ``` ## 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. tools:::Rd_expr_doi("10.1080/10705511003661553") ## See Also Other Simulation of State Space Models Data Functions: `LinSDE2SSM()`, `LinSDECovEta()`, `LinSDECovY()`, `LinSDEMeanEta()`, `LinSDEMeanY()`, `ProjectToHurwitz()`, `ProjectToStability()`, `SSMCovEta()`, `SSMCovY()`, `SSMMeanEta()`, `SSMMeanY()`, `SimAlphaN()`, `SimBetaN()`, `SimBetaN2()`, `SimCovDiagN()`, `SimCovN()`, `SimIotaN()`, `SimNuN()`, `SimPhiN()`, `SimPhiN2()`, `SimSSMFixed()`, `SimSSMIVary()`, `SimSSMLinGrowth()`, `SimSSMLinGrowthIVary()`, `SimSSMLinSDEFixed()`, `SimSSMLinSDEIVary()`, `SimSSMOUFixed()`, `SimSSMOUIVary()`, `SimSSMVARFixed()`, `SpectralRadius()`, `TestPhi()`, `TestPhiHurwitz()`, `TestStability()`, `TestStationarity()` ## Author(s) Ivan Jacob Agaloos Pesigan

SimSSMVARIVary function