# Simulate Data from the Ornstein-Uhlenbeck Model using a State Space Model Parameterization (Individual-Varying Parameters) This function simulates data from the Ornstein-Uhlenbeck model using a state space model parameterization. It assumes that the parameters can vary across individuals. ```r 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 - `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 ($\boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0}$). - `mu`: List of numeric vectors. Each element of the list is the long-term mean or equilibrium level ($\boldsymbol{\mu}$). - `phi`: List of numeric matrix. Each element of the list is the drift matrix which represents the rate of change of the solution in the absence of any random fluctuations ($\boldsymbol{\Phi}$). It also represents the rate of mean reversion, determining how quickly the variable returns to its mean. - `sigma_l`: List of numeric matrix. Each element of the list is the Cholesky factorization (`t(chol(sigma))`) of the covariance matrix of volatility or randomness in the process $\boldsymbol{\Sigma}$. - `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 `SimSSMOUFixed()` 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 ($\boldsymbol{\kappa}$). ## 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`, `mu`, `phi`, `sigma_l`, `nu`, `lambda`, `theta_l`, `gamma`, or `kappa`) is less the `n`, the function will cycle through the available values. ## Examples ```r # 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) ``` ## 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") 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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.1037/a0024375") Uhlenbeck, G. E., & Ornstein, L. S. (1930). On the theory of the brownian motion. Physical Review, 36 (5), 823-841. tools:::Rd_expr_doi("10.1103/physrev.36.823") ## See Also Other Simulation of State Space Models Data Functions: `LinSDE2SSM()`, `LinSDECovEta()`, `LinSDECovY()`, `LinSDEMeanEta()`, `LinSDEMeanY()`, `ProjectToHurwitz()`, `ProjectToStability()`, `SSMCovEta()`, `SSMCovY()`, `SSMInterceptEta()`, `SSMInterceptY()`, `SSMMeanEta()`, `SSMMeanY()`, `SimAlphaN()`, `SimBetaN()`, `SimBetaN2()`, `SimBetaNCovariate()`, `SimCovDiagN()`, `SimCovN()`, `SimIotaN()`, `SimNuN()`, `SimPhiN()`, `SimPhiN2()`, `SimPhiNCovariate()`, `SimSSMFixed()`, `SimSSMIVary()`, `SimSSMLinGrowth()`, `SimSSMLinGrowthIVary()`, `SimSSMLinSDEFixed()`, `SimSSMLinSDEIVary()`, `SimSSMOUFixed()`, `SimSSMVARFixed()`, `SimSSMVARIVary()`, `SpectralRadius()`, `TestPhi()`, `TestPhiHurwitz()`, `TestStability()`, `TestStationarity()` ## Author(s) Ivan Jacob Agaloos Pesigan

SimSSMOUIVary function