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.

Usage

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.

Value

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 $ \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 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.

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.

References

Chow, S.-M., Ho, M. R., Hamaker, E. L., & Dolan, C. V. (2010). Equivalence and differences between structural equation modeling and state-space modeling techniques. Structural Equation Modeling: A Multidisciplinary Journal, 17(2), 303–332. doi:10.1080/10705511003661553

Author

Ivan Jacob Agaloos Pesigan

Examples

if (FALSE) { # \dontrun{
# 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")
} # }