Title:	Bootstrap for State Space Models
Version:	1.0.2
Description:	Provides a streamlined and user-friendly framework for bootstrapping 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/bootStateSpace, https://jeksterslab.github.io/bootStateSpace/
BugReports:	https://github.com/jeksterslab/bootStateSpace/issues
License:	GPL (≥ 3)
Encoding:	UTF-8
Depends:	R (≥ 3.5.0)
Imports:	stats, simStateSpace, dynr
Suggests:	knitr, rmarkdown, testthat
RoxygenNote:	7.3.2
NeedsCompilation:	no
Packaged:	2025-02-14 20:32:15 UTC; root
Author:	Ivan Jacob Agaloos Pesigan [aut, cre, cph]
Maintainer:	Ivan Jacob Agaloos Pesigan <r.jeksterslab@gmail.com>
Repository:	CRAN
Date/Publication:	2025-02-14 22:20:02 UTC

bootStateSpace: Bootstrap for State Space Models

Description

Provides a streamlined and user-friendly framework for bootstrapping 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/bootStateSpace

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

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

Parametric Bootstrap for the State Space Model (Fixed Parameters)

Description

This function simulates data from a state-space model 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

PBSSMFixed(
  R,
  path,
  prefix,
  n,
  time,
  delta_t = 1,
  mu0,
  sigma0_l,
  alpha,
  beta,
  psi_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

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 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 ("PBSSMFixed").

method

Bootstrap method used ("parametric").

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 Bootstrap for State Space Models Functions: PBSSMLinSDEFixed(), PBSSMOUFixed(), PBSSMVARFixed()

Examples

# prepare parameters
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
delta_t <- 1
## 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))

path <- tempdir()

pb <- PBSSMFixed(
  R = 10L, # use at least 1000 in actual research
  path = path,
  prefix = "ssm",
  n = n,
  time = time,
  delta_t = delta_t,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_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")

Parametric Bootstrap for the Linear Stochastic Differential Equation Model using a State Space Model Parameterization (Fixed Parameters)

Description

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

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 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").

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 Bootstrap for State Space Models Functions: PBSSMFixed(), PBSSMOUFixed(), PBSSMVARFixed()

Examples

# 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")

Parametric Bootstrap for the Ornstein–Uhlenbeck Model using a State Space Model Parameterization (Fixed Parameters)

Description

This function simulates data from a Ornstein–Uhlenbeck (OU) 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

PBSSMOUFixed(
  R,
  path,
  prefix,
  n,
  time,
  delta_t = 0.1,
  mu0,
  sigma0_l,
  mu,
  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

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 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 ("PBSSMOUFixed").

method

Bootstrap method used ("parametric").

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 Bootstrap for State Space Models Functions: PBSSMFixed(), PBSSMLinSDEFixed(), PBSSMVARFixed()

Examples

# 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))
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))

path <- tempdir()

pb <- PBSSMOUFixed(
  R = 10L, # use at least 1000 in actual research
  path = path,
  prefix = "ou",
  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,
  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")

Parametric Bootstrap for the Vector Autoregressive Model (Fixed Parameters)

Description

This function simulates data from a vector autoregressive 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

PBSSMVARFixed(
  R,
  path,
  prefix,
  n,
  time,
  mu0,
  sigma0_l,
  alpha,
  beta,
  psi_l,
  type = 0,
  x = NULL,
  gamma = 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

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 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 ("PBSSMVARFixed").

method

Bootstrap method used ("parametric").

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 Bootstrap for State Space Models Functions: PBSSMFixed(), PBSSMLinSDEFixed(), PBSSMOUFixed()

Examples

# prepare parameters
## 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))

path <- tempdir()

pb <- PBSSMVARFixed(
  R = 10L, # use at least 1000 in actual research
  path = path,
  prefix = "var",
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_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")

Estimated Parameter Method for an Object of Class bootstatespace

Description

Estimated Parameter Method for an Object of Class bootstatespace

Usage

## S3 method for class 'bootstatespace'
coef(object, ...)

Arguments

Value

Returns a vector of estimated parameters.

Author(s)

Ivan Jacob Agaloos Pesigan

Confidence Intervals Method for an Object of Class bootstatespace

Description

Confidence Intervals Method for an Object of Class bootstatespace

Usage

## S3 method for class 'bootstatespace'
confint(object, parm = NULL, level = 0.95, type = "pc", ...)

Arguments

Value

Returns a matrix of confidence intervals.

Author(s)

Ivan Jacob Agaloos Pesigan

Extract Generic Function

Description

A generic function for extracting elements from objects.

Usage

extract(object, what)

Arguments

Value

A value determined by the specific method for the object's class.

Extract Method for an Object of Class bootstatespace

Description

Extract Method for an Object of Class bootstatespace

Usage

## S3 method for class 'bootstatespace'
extract(object, what = NULL)

Arguments

Value

Returns a list. Each element of the list is a list of bootstrap estimates in matrix format.

Author(s)

Ivan Jacob Agaloos Pesigan

Print Method for an Object of Class bootstatespace

Description

Print Method for an Object of Class bootstatespace

Usage

## S3 method for class 'bootstatespace'
print(x, alpha = NULL, type = "pc", digits = 4, ...)

Arguments

Value

Prints a matrix of estimates, standard errors, number of bootstrap replications, and confidence intervals.

Author(s)

Ivan Jacob Agaloos Pesigan

Summary Method for an Object of Class bootstatespace

Description

Summary Method for an Object of Class bootstatespace

Usage

## S3 method for class 'bootstatespace'
summary(object, alpha = NULL, type = "pc", digits = 4, ...)

Arguments

Value

Returns a matrix of estimates, standard errors, number of bootstrap replications, and confidence intervals.

Author(s)

Ivan Jacob Agaloos Pesigan

Sampling Variance-Covariance Matrix Method for an Object of Class bootstatespace

Description

Sampling Variance-Covariance Matrix Method for an Object of Class bootstatespace

Usage

## S3 method for class 'bootstatespace'
vcov(object, ...)

Arguments

Value

Returns the variance-covariance matrix of estimates.

Author(s)

Ivan Jacob Agaloos Pesigan