# Parametric Bootstrap for the Linear Stochastic Differential Equation Model using a State Space Model Parameterization (Fixed Parameters) This function simulates data from a linear stochastic differential equation model using a state-space model parameterization and fits the model using the `dynr` package. The process is repeated `R` times. It assumes that the parameters remain constant across individuals and over time. At the moment, the function only supports `type = 0`. ```r PBSSMLinSDEFixed( R, path, prefix, n, time, delta_t = 0.1, mu0, sigma0_l, iota, phi, sigma_l, nu, lambda, theta_l, type = 0, x = NULL, gamma = NULL, kappa = NULL, mu0_fixed = FALSE, sigma0_fixed = FALSE, alpha_level = 0.05, optimization_flag = TRUE, hessian_flag = FALSE, verbose = FALSE, weight_flag = FALSE, debug_flag = FALSE, perturb_flag = FALSE, xtol_rel = 1e-07, stopval = -9999, ftol_rel = -1, ftol_abs = -1, maxeval = as.integer(-1), maxtime = -1, ncores = NULL, seed = NULL, clean = TRUE ) ``` ## Arguments - `R`: Positive integer. Number of bootstrap samples. - `path`: Path to a directory to store bootstrap samples and estimates. - `prefix`: Character string. Prefix used for the file names for the bootstrap samples and estimates. - `n`: Positive integer. Number of individuals. - `time`: Positive integer. Number of time points. - `delta_t`: Numeric. Time interval ($\Delta_t$). - `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 ($\boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0}$). - `iota`: Numeric vector. An unobserved term that is constant over time ($\boldsymbol{\iota}$). - `phi`: Numeric matrix. The drift matrix which represents the rate of change of the solution in the absence of any random fluctuations ($\boldsymbol{\Phi}$). - `sigma_l`: Numeric matrix. Cholesky factorization (`t(chol(sigma))`) of the covariance matrix of volatility or randomness in the process ($\boldsymbol{\Sigma}$). - `nu`: Numeric vector. Vector of intercept values for the measurement model ($\boldsymbol{\nu}$). - `lambda`: Numeric matrix. Factor loading matrix linking the latent variables to the observed variables ($\boldsymbol{\Lambda}$). - `theta_l`: Numeric matrix. Cholesky factorization (`t(chol(theta))`) of the covariance matrix of the measurement error ($\boldsymbol{\Theta}$). - `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 ($\boldsymbol{\Gamma}$). - `kappa`: Numeric matrix. Matrix linking the covariates to the observed variables at current time point ($\boldsymbol{\kappa}$). - `mu0_fixed`: Logical. If `mu0_fixed = TRUE`, fix the initial mean vector to `mu0`. If `mu0_fixed = FALSE`, `mu0` is estimated. - `sigma0_fixed`: Logical. If `sigma0_fixed = TRUE`, fix the initial covariance matrix to `tcrossprod(sigma0_l)`. If `sigma0_fixed = FALSE`, `sigma0` is estimated. - `alpha_level`: Numeric vector. Significance level $\alpha$. - `optimization_flag`: a flag (TRUE/FALSE) indicating whether optimization is to be done. - `hessian_flag`: a flag (TRUE/FALSE) indicating whether the Hessian matrix is to be calculated. - `verbose`: a flag (TRUE/FALSE) indicating whether more detailed intermediate output during the estimation process should be printed - `weight_flag`: a flag (TRUE/FALSE) indicating whether the negative log likelihood function should be weighted by the length of the time series for each individual - `debug_flag`: a flag (TRUE/FALSE) indicating whether users want additional dynr output that can be used for diagnostic purposes - `perturb_flag`: a flag (TRUE/FLASE) indicating whether to perturb the latent states during estimation. Only useful for ensemble forecasting. - `xtol_rel`: Stopping criteria option for parameter optimization. See `dynr::dynr.model()` for more details. - `stopval`: Stopping criteria option for parameter optimization. See `dynr::dynr.model()` for more details. - `ftol_rel`: Stopping criteria option for parameter optimization. See `dynr::dynr.model()` for more details. - `ftol_abs`: Stopping criteria option for parameter optimization. See `dynr::dynr.model()` for more details. - `maxeval`: Stopping criteria option for parameter optimization. See `dynr::dynr.model()` for more details. - `maxtime`: Stopping criteria option for parameter optimization. See `dynr::dynr.model()` for more details. - `ncores`: Positive integer. Number of cores to use. If `ncores = NULL`, use a single core. Consider using multiple cores when number of bootstrap samples `R` is a large value. - `seed`: Random seed. - `clean`: Logical. If `clean = TRUE`, delete intermediate files generated by the function. ## Returns Returns an object of class `bootstatespace` which is a list with the following elements: - **call**: Function call. - **args**: Function arguments. - **thetahatstar**: Sampling distribution of $\boldsymbol{\hat{\theta}}$. - **vcov**: Sampling variance-covariance matrix of $\boldsymbol{\hat{\theta}}$. - **est**: Vector of estimated $\boldsymbol{\hat{\theta}}$. - **fun**: Function used ("PBSSMLinSDEFixed"). - **method**: Bootstrap method used ("parametric"). ## 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 c("$\n$", "$ \\left( \\boldsymbol{\\Theta}^{\\frac{1}{2}} \\right)\n$", "$ \\left( \\boldsymbol{\\Theta}^{\\frac{1}{2}} \\right)^{\\prime}\n$", "$ =\n$", "$ \\boldsymbol{\\Theta} .\n$", "$ $") 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. ## Examples ```r # prepare parameters ## 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)) path <- tempdir() pb <- PBSSMLinSDEFixed( R = 10L, # use at least 1000 in actual research path = path, prefix = "lse", 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, ncores = 1, # consider using multiple cores seed = 42 ) print(pb) summary(pb) confint(pb) vcov(pb) coef(pb) print(pb, type = "bc") # bias-corrected summary(pb, type = "bc") confint(pb, type = "bc") ``` ## 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 Bootstrap for State Space Models Functions: `PBSSMFixed()`, `PBSSMOUFixed()`, `PBSSMVARFixed()` ## Author(s) Ivan Jacob Agaloos Pesigan

PBSSMLinSDEFixed function