Package 'simStateSpace'

Coerce an Object of Class simstatespace to a Data Frame

Description

Coerce an Object of Class simstatespace to a Data Frame

Usage

## S3 method for class 'simstatespace'
as.data.frame(
  x,
  row.names = NULL,
  optional = FALSE,
  eta = FALSE,
  long = TRUE,
  burnin = 0,
  reset_time = TRUE,
  ...
)

Arguments

x	Object of class simstatespace.
row.names	NULL or character vector giving the row names for the data frame. Missing values are not allowed.
optional	Logical. If TRUE, setting row names and converting column names is optional.
eta	Logical. If eta = TRUE, include eta. If eta = FALSE, exclude eta.
long	Logical. If long = TRUE, use long format. If long = FALSE, use wide format.
burnin	Positive integer. Initial data points to discard. Default is zero.
reset_time	Logical. Reset the time index after burnin.
...	Additional arguments.

Author(s)

Ivan Jacob Agaloos Pesigan

Examples

# prepare parameters
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
## dynamic structure
p <- 3
mu0 <- rep(x = 0, times = p)
sigma0 <- diag(p)
sigma0_l <- t(chol(sigma0))
alpha <- rep(x = 0, times = p)
beta <- 0.50 * diag(p)
psi <- diag(p)
psi_l <- t(chol(psi))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.50 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0
)

head(as.data.frame(ssm))
head(as.data.frame(ssm, long = FALSE))

# Type 1
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

head(as.data.frame(ssm))
head(as.data.frame(ssm, long = FALSE))

# Type 2
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

head(as.data.frame(ssm))
head(as.data.frame(ssm, long = FALSE))

---

Coerce an Object of Class simstatespace to a Matrix

Description

Coerce an Object of Class simstatespace to a Matrix

Usage

## S3 method for class 'simstatespace'
as.matrix(x, eta = FALSE, long = TRUE, burnin = 0, reset_time = TRUE, ...)

Arguments

x	Object of class simstatespace.
eta	Logical. If eta = TRUE, include eta. If eta = FALSE, exclude eta.
long	Logical. If long = TRUE, use long format. If long = FALSE, use wide format.
burnin	Positive integer. Initial data points to discard. Default is zero.
reset_time	Logical. Reset the time index after burnin.
...	Additional arguments.

Author(s)

Ivan Jacob Agaloos Pesigan

Examples

# prepare parameters
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
## dynamic structure
p <- 3
mu0 <- rep(x = 0, times = p)
sigma0 <- diag(p)
sigma0_l <- t(chol(sigma0))
alpha <- rep(x = 0, times = p)
beta <- 0.50 * diag(p)
psi <- diag(p)
psi_l <- t(chol(psi))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.50 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0
)

head(as.matrix(ssm))
head(as.matrix(ssm, long = FALSE))

# Type 1
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

head(as.matrix(ssm))
head(as.matrix(ssm, long = FALSE))

# Type 2
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

head(as.matrix(ssm))
head(as.matrix(ssm, long = FALSE))

---

Convert Parameters from the Linear Stochastic Differential Equation Model to State Space Model Parameterization

Description

This function converts parameters from the linear stochastic differential equation model to state space model parameterization.

Usage

LinSDE2SSM(iota, phi, sigma_l, delta_t)

Arguments

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 ($\boldsymbol{\Sigma}$).
delta_t	Numeric. Time interval ($\Delta_t$).

Details

Let the linear stochastic equation model be given by

$\mathrm{d} \boldsymbol{\eta}_{i, t} = \left( \boldsymbol{\iota} + \boldsymbol{\Phi} \boldsymbol{\eta}_{i, t} \right) \mathrm{d} t + \boldsymbol{\Sigma}^{\frac{1}{2}} \mathrm{d} \mathbf{W}_{i, t}$

for individual $i$ and time $t$. The discrete-time state space model given below represents the discrete-time solution for the linear stochastic differential equation.

$\boldsymbol{\eta}_{i, t_{{l_{i}}}} = \boldsymbol{\alpha}_{\Delta t_{{l_{i}}}} + \boldsymbol{\beta}_{\Delta t_{{l_{i}}}} \boldsymbol{\eta}_{i, t_{l_{i} - 1}} + \boldsymbol{\zeta}_{i, t_{{l_{i}}}}, \quad \mathrm{with} \quad \boldsymbol{\zeta}_{i, t_{{l_{i}}}} \sim \mathcal{N} \left( \mathbf{0}, \boldsymbol{\Psi}_{\Delta t_{{l_{i}}}} \right)$

with

$\boldsymbol{\beta}_{\Delta t_{{l_{i}}}} = \exp{ \left( \Delta t \boldsymbol{\Phi} \right) },$

$\boldsymbol{\alpha}_{\Delta t_{{l_{i}}}} = \boldsymbol{\Phi}^{-1} \left( \boldsymbol{\beta} - \mathbf{I}_{p} \right) \boldsymbol{\iota}, \quad \mathrm{and}$

$\mathrm{vec} \left( \boldsymbol{\Psi}_{\Delta t_{{l_{i}}}} \right) = \left[ \left( \boldsymbol{\Phi} \otimes \mathbf{I}_{p} \right) + \left( \mathbf{I}_{p} \otimes \boldsymbol{\Phi} \right) \right] \left[ \exp \left( \left[ \left( \boldsymbol{\Phi} \otimes \mathbf{I}_{p} \right) + \left( \mathbf{I}_{p} \otimes \boldsymbol{\Phi} \right) \right] \Delta t \right) - \mathbf{I}_{p \times p} \right] \mathrm{vec} \left( \boldsymbol{\Sigma} \right)$

where $t$ denotes continuous-time processes that can be defined by any arbitrary time point, $t_{l_{i}}$ the $l^\mathrm{th}$ observed measurement occassion for individual $i$, $p$ the number of latent variables and $\Delta t$ the time interval.

Value

Returns a list of state space parameters:

• alpha: Numeric vector. Vector of constant values for the dynamic model ($\boldsymbol{\alpha}$).

• beta: Numeric matrix. Transition matrix relating the values of the latent variables from the previous time point to the current time point ($\boldsymbol{\beta}$).

• psi_l: Numeric matrix. Cholesky factorization (t(chol(psi))) of the process noise covariance matrix $\boldsymbol{\Psi}$.

Author(s)

Ivan Jacob Agaloos Pesigan

References

Harvey, A. C. (1990). Forecasting, structural time series models and the Kalman filter. Cambridge University Press. doi:10.1017/cbo9781107049994

See Also

Other Simulation of State Space Models Data Functions: LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

p <- 2
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))
delta_t <- 0.10

LinSDE2SSM(
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  delta_t = delta_t
)

---

Steady-State Covariance Matrix for the Latent Variables in the Linear Stochastic Differential Equation Model

Description

The steady-state covariance matrix for the latent variables in the linear stochastic differential equation model $\mathrm{Cov} \left( \boldsymbol{\eta} \right)$ is the solution to the Sylvester equation, i.e.

$\mathbf{A} \mathbf{X} + \mathbf{X} \mathbf{B} + \mathbf{C} = \mathbf{0} ,$

where $\mathbf{X}$ is unknown, $\mathbf{A} = \boldsymbol{\Phi}$, $\mathbf{B} = \boldsymbol{\Phi}^{\prime}$, and $\mathbf{C} = \boldsymbol{\Sigma}$ where $\boldsymbol{\Phi}$ is the drift matrix and $\boldsymbol{\Sigma}$ is the covariance matrix of volatility or randomness in the process.

Usage

LinSDECovEta(phi, sigma)

Arguments

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	Numeric matrix. The covariance matrix of volatility or randomness in the process ($\boldsymbol{\Sigma}$).

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0.0, 0.0, -0.693
  ),
  nrow = 3
)
sigma <- matrix(
  data = c(
    0.24455556, 0.02201587, -0.05004762,
    0.02201587, 0.07067800, 0.01539456,
    -0.05004762, 0.01539456, 0.07553061
  ),
  nrow = 3
)
LinSDECovEta(
  phi = phi,
  sigma = sigma
)

---

Steady-State Covariance Matrix for the Observed Variables in the Linear Stochastic Differential Equation Model

Description

The steady-state covariance matrix for the observed variables in the linear stochastic differential equation model $\mathrm{Cov} \left( \mathbf{y} \right)$ is given by

$\mathrm{Cov} \left( \mathbf{y} \right) = \boldsymbol{\Lambda} \mathrm{Cov} \left( \boldsymbol{\eta} \right) \boldsymbol{\Lambda}^{\prime} + \boldsymbol{\Theta}$

where $\boldsymbol{\Lambda}$ is the matrix of factor loadings, $\boldsymbol{\Theta}$ is the covariance matrix of the measurement error, and $\mathrm{Cov} \left( \boldsymbol{\eta} \right)$ is the steady-state covariance matrix for the latent variables.

Usage

LinSDECovY(lambda, theta, cov_eta)

Arguments

lambda	Numeric matrix. Factor loading matrix linking the latent variables to the observed variables ($\boldsymbol{\Lambda}$).
theta	Numeric matrix. The covariance matrix of the measurement error ($\boldsymbol{\Theta}$).
cov_eta	Numeric matrix. The steady-state covariance matrix for the latent variables in the linear stochastic differential equation model

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0.0, 0.0, -0.693
  ),
  nrow = 3
)
sigma <- matrix(
  data = c(
    0.24455556, 0.02201587, -0.05004762,
    0.02201587, 0.07067800, 0.01539456,
    -0.05004762, 0.01539456, 0.07553061
  ),
  nrow = 3
)
lambda <- diag(3)
theta <- diag(3)
cov_eta <- LinSDECovEta(
  phi = phi,
  sigma = sigma
)
LinSDECovY(
  lambda = lambda,
  theta = theta,
  cov_eta = cov_eta
)

---

Intercept from Steady-State Mean Vector for the Latent Variables in the Linear Stochastic Differential Equation Model

Description

The intercept vector for the latent variables in the linear stochastic differential equation model $\boldsymbol{\iota}$ is given by

$\boldsymbol{\iota} = - \boldsymbol{\Phi} \mathrm{Mean} \left( \boldsymbol{\eta} \right)$

where $\boldsymbol{\Phi}$ is the drift matrix and $\mathrm{Mean} \left( \boldsymbol{\eta} \right)$ is the steady-state mean vector for the latent variables.

Usage

LinSDEInterceptEta(phi, mean_eta)

Arguments

phi	Numeric matrix. The drift matrix which represents the rate of change of the solution in the absence of any random fluctuations ($\boldsymbol{\Phi}$).
mean_eta	Numeric vector. Steady-state mean vector of the latent variables $\mathrm{Mean} \left( \boldsymbol{\eta} \right)$.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0.0, 0.0, -0.693
  ),
  nrow = 3
)
iota <- rep(x = 1, times = 3)
lambda <- diag(3)
nu <- rep(x = 1, times = 3)
LinSDEMeanEta(
  phi = phi,
  iota = iota
)

---

Intercept from Steady-State Mean Vector for the Observed Variables in the Linear Stochastic Differential Equation Model

Description

The intercept vector for the observed variables in the linear stochastic differential equation model $\boldsymbol{\nu}$ is given by

$\boldsymbol{\nu} = \mathrm{Mean} \left( \mathbf{y} \right) - \boldsymbol{\Lambda} \mathrm{Mean} \left( \boldsymbol{\eta} \right)$

where $\boldsymbol{\Lambda}$ is the matrix of factor loadings, $\mathrm{Mean} \left( \mathbf{y} \right)$ is the steady-state mean vector for the observed variables, and $\mathrm{Mean} \left( \boldsymbol{\eta} \right)$ is the steady-state mean vector for the latent variables.

Usage

LinSDEInterceptY(mean_y, mean_eta, lambda)

Arguments

mean_y	Numeric vector. Steady-state mean vector of the observed variables $\mathrm{Mean} \left( \mathbf{y} \right)$.
mean_eta	Numeric vector. Steady-state mean vector of the latent variables $\mathrm{Mean} \left( \boldsymbol{\eta} \right)$.
lambda	Numeric matrix. Factor loading matrix linking the latent variables to the observed variables ($\boldsymbol{\Lambda}$).

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0.0, 0.0, -0.693
  ),
  nrow = 3
)
iota <- rep(x = 1, times = 3)
lambda <- diag(3)
nu <- rep(x = 1, times = 3)
mean_eta <- LinSDEMeanEta(
  phi = phi,
  iota = iota
)
mean_y <- LinSDEMeanY(
  nu = nu,
  lambda = lambda,
  mean_eta = mean_eta
)
LinSDEInterceptY(
  mean_y = mean_y,
  mean_eta = mean_eta,
  lambda = lambda
)

---

Steady-State Mean Vector for the Latent Variables in the Linear Stochastic Differential Equation Model

Description

The steady-state mean vector for the latent variables in the linear stochastic differential equation model $\mathrm{Mean} \left( \boldsymbol{\eta} \right)$ is given by

$\mathrm{Mean} \left( \boldsymbol{\eta} \right) = -\boldsymbol{\Phi}^{-1} \boldsymbol{\iota}$

where $\boldsymbol{\Phi}$ is the drift matrix, and $\boldsymbol{\iota}$ is an unobserved term that is constant over time.

Usage

LinSDEMeanEta(phi, iota)

Arguments

phi	Numeric matrix. The drift matrix which represents the rate of change of the solution in the absence of any random fluctuations ($\boldsymbol{\Phi}$).
iota	Numeric vector. An unobserved term that is constant over time ($\boldsymbol{\iota}$).

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0.0, 0.0, -0.693
  ),
  nrow = 3
)
iota <- rep(x = 1, times = 3)
LinSDEMeanEta(
  phi = phi,
  iota = iota
)

---

Steady-State Mean Vector for the Observed Variables in the Linear Stochastic Differential Equation Model

Description

The steady-state mean vector for the observed variables in the linear stochastic differential equation model $\mathrm{Mean} \left( \mathbf{y} \right)$ is given by

$\mathrm{Mean} \left( \mathbf{y} \right) = \boldsymbol{\nu} + \boldsymbol{\Lambda} \mathrm{Mean} \left( \boldsymbol{\eta} \right)$

where $\boldsymbol{\nu}$ is the vector of intercept values for the measurement model, $\boldsymbol{\Lambda}$ is the matrix of factor loadings, and $\mathrm{Mean} \left( \boldsymbol{\eta} \right)$ is the steady-state mean vector for the latent variables.

Usage

LinSDEMeanY(nu, lambda, mean_eta)

Arguments

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 ($\boldsymbol{\Lambda}$).
mean_eta	Numeric vector. Steady-state mean vector of the latent variables $\mathrm{Mean} \left( \boldsymbol{\eta} \right)$.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0.0, 0.0, -0.693
  ),
  nrow = 3
)
iota <- rep(x = 1, times = 3)
lambda <- diag(3)
nu <- rep(x = 1, times = 3)
mean_eta <- LinSDEMeanEta(
  phi = phi,
  iota = iota
)
LinSDEMeanY(
  nu = nu,
  lambda = lambda,
  mean_eta = mean_eta
)

---

Plot Method for an Object of Class simstatespace

Description

Plot Method for an Object of Class simstatespace

Usage

## S3 method for class 'simstatespace'
plot(
  x,
  id = NULL,
  time = NULL,
  eta = FALSE,
  type = "b",
  burnin = 0,
  reset_time = TRUE,
  ...
)

Arguments

x	Object of class simstatespace.
id	Numeric vector. Optional id numbers to plot. If id = NULL, plot all available data.
time	Numeric vector. Optional time points to plot. If time = NULL, plot all available data.
eta	Logical. If eta = TRUE, plot the latent variables. If eta = FALSE, plot the observed variables.
type	Character indicating the type of plotting; actually any of the types as in plot.default().
burnin	Positive integer. Initial data points to discard. Default is zero.
reset_time	Logical. Reset the time index after burnin.
...	Additional arguments.

Author(s)

Ivan Jacob Agaloos Pesigan

Examples

# prepare parameters
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
## dynamic structure
p <- 3
mu0 <- rep(x = 0, times = p)
sigma0 <- diag(p)
sigma0_l <- t(chol(sigma0))
alpha <- rep(x = 0, times = p)
beta <- 0.50 * diag(p)
psi <- diag(p)
psi_l <- t(chol(psi))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.50 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0
)

plot(ssm)
plot(ssm, id = 1:3, time = 0:9)

# Type 1
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

plot(ssm)
plot(ssm, id = 1:3, time = 0:9)

# Type 2
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

plot(ssm)
plot(ssm, id = 1:3, time = 0:9)

---

Print Method for an Object of Class simstatespace

Description

Print Method for an Object of Class simstatespace

Usage

## S3 method for class 'simstatespace'
print(x, ...)

Arguments

x	Object of Class simstatespace.
...	Additional arguments.

Value

Prints simulated data in long format.

Author(s)

Ivan Jacob Agaloos Pesigan

Examples

# prepare parameters
set.seed(42)
## number of individuals
n <- 5
## time points
time <- 50
## dynamic structure
p <- 3
mu0 <- rep(x = 0, times = p)
sigma0 <- diag(p)
sigma0_l <- t(chol(sigma0))
alpha <- rep(x = 0, times = p)
beta <- 0.50 * diag(p)
psi <- diag(p)
psi_l <- t(chol(psi))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.50 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0
)

print(ssm)

# Type 1
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

print(ssm)

# Type 2
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

print(ssm)

---

Project Matrix to Hurwitz Stability

Description

Shifts a square matrix left on the real axis so that its spectral abscissa (maximum real part of the eigenvalues) is strictly less than -margin. This is useful for ensuring that continuous-time drift matrices (e.g. in linear SDEs/state-space models) are Hurwitz-stable. If the matrix already satisfies the margin, it is returned unchanged.

Usage

ProjectToHurwitz(x, margin = 0.001)

Arguments

x	Numeric square matrix.
margin	Positive numeric. Target buffer inside the Hurwitz region; the result satisfies $\max \Re\{\lambda_i(x^\star)\} \le -\text{margin}$ (default 1e-3).

Details

The projection is performed by subtracting a multiple of the identity:

$x^\star = x - (\alpha + \text{margin}) I,$

where $\alpha = \max \Re\{\lambda_i(x)\}$ is the spectral abscissa.

Value

A numeric matrix of the same dimensions as x, shifted if necessary to satisfy the Hurwitz stability constraint.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

# Unstable (spectral abscissa >= 0):
x <- matrix(
  data = c(
    0.10, -0.40,
    0.50, 0.20
  ),
  nrow = 2
)
SpectralAbscissa(x = x) # >= 0
SpectralAbscissa(x = ProjectToHurwitz(x = x)) # <= -1e-3 (default margin)

# Already Hurwitz-stable is returned unchanged up to numerics:
x <- matrix(
  data = c(
    -0.50, -0.20,
     1.00, -0.30
  ),
  nrow = 2
)
SpectralAbscissa(x = x) # < 0
identical(ProjectToHurwitz(x = x), x)

---

Project Matrix to Stability

Description

Scales a square matrix so that its spectral radius is strictly less than 1 by a specified stability margin. This is useful for ensuring that transition matrices in state space or vector autoregressive (VAR) models are stationary. If the matrix is already within the margin, it is returned unchanged.

Usage

ProjectToStability(x, margin = 0.98, tol = 1e-12)

Arguments

x	Numeric square matrix.
margin	Double in $(0, 1)$. Target upper bound for the spectral radius (default = 0.98).
tol	Small positive double added to the denominator in the scaling factor to avoid division by zero (default 1e-12).

Details

The projection is performed by multiplying the matrix by a constant factor $c = \frac{\text{margin}}{\rho + \text{tol}}$, where $\rho$ is the spectral radius and tol is a small positive number to prevent division by zero.

Value

A numeric matrix of the same dimensions as x, scaled if necessary to satisfy the stability constraint.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

# Matrix with eigenvalues greater than 1
x <- matrix(
  data = c(
    1.2, 0.3,
    0.4, 0.9
  ),
  nrow = 2
)
SpectralRadius(x = x) # > 1
SpectralRadius(x = ProjectToStability(x = x))  # < 1

# Matrix already stable is returned unchanged
x <- matrix(
  data = c(
    0.5, 0.3,
    0.2, 0.4
  ),
  nrow = 2
)
identical(ProjectToStability(x = x), x)

---

Simulate Intercept Vectors in a Discrete-Time Vector Autoregressive Model from the Multivariate Normal Distribution

Description

This function simulates random intercept vectors in a discrete-time vector autoregressive model from the multivariate normal distribution.

Usage

SimAlphaN(n, alpha, vcov_alpha_l)

Arguments

n	Positive integer. Number of replications.
alpha	Numeric vector. Intercept ($\boldsymbol{\alpha}$).
vcov_alpha_l	Numeric matrix. Cholesky factorization (t(chol(vcov_alpha))) of the sampling variance-covariance matrix of $\boldsymbol{\alpha}$.

Value

Returns a list of random intercept vectors.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 10
alpha <- c(0, 0, 0)
vcov_alpha_l <- t(chol(0.001 * diag(3)))
SimAlphaN(n = n, alpha = alpha, vcov_alpha_l = vcov_alpha_l)

---

Simulate Transition Matrices from the Multivariate Normal Distribution

Description

This function simulates random transition matrices from the multivariate normal distribution. The function ensures that the generated transition matrices are stationary using TestStationarity() with a rejection sampling approach.

Usage

SimBetaN(
  n,
  beta,
  vcov_beta_vec_l,
  margin = 1,
  beta_lbound = NULL,
  beta_ubound = NULL,
  bound = FALSE,
  max_iter = 100000L
)

Arguments

n	Positive integer. Number of replications.
beta	Numeric matrix. The transition matrix ($\boldsymbol{\beta}$).
vcov_beta_vec_l	Numeric matrix. Cholesky factorization (t(chol(vcov_beta_vec))) of the sampling variance-covariance matrix of $\mathrm{vec} \left( \boldsymbol{\beta} \right)$.
margin	Numeric scalar specifying the stationarity threshold. Values less than 1 indicate stricter stationarity criteria.
beta_lbound	Optional numeric matrix of same dim as beta. Use NA for no lower bound.
beta_ubound	Optional numeric matrix of same dim as beta. Use NA for no upper bound.
bound	Logical; if TRUE, resample until all elements respect bounds (NA bounds ignored).
max_iter	Safety cap on resampling attempts per draw.

Value

Returns a list of random transition matrices.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 10
beta <- matrix(
  data = c(
    0.7, 0.5, -0.1,
    0.0, 0.6, 0.4,
    0, 0, 0.5
  ),
  nrow = 3
)
vcov_beta_vec_l <- t(chol(0.001 * diag(9)))
SimBetaN(n = n, beta = beta, vcov_beta_vec_l = vcov_beta_vec_l)

---

Simulate Transition Matrices from the Multivariate Normal Distribution and Project to Stability

Description

This function simulates random transition matrices from the multivariate normal distribution then projects each draw to the stability region using ProjectToStability().

Usage

SimBetaN2(n, beta, vcov_beta_vec_l, margin = 0.98, tol = 1e-12)

Arguments

n	Positive integer. Number of replications.
beta	Numeric matrix. The transition matrix ($\boldsymbol{\beta}$).
vcov_beta_vec_l	Numeric matrix. Cholesky factorization (t(chol(vcov_beta_vec))) of the sampling variance-covariance matrix of $\mathrm{vec} \left( \boldsymbol{\beta} \right)$.
margin	Double in $(0, 1)$. Target upper bound for the spectral radius (default = 0.98).
tol	Small positive double added to the denominator in the scaling factor to avoid division by zero (default = 1e-12).

Value

Returns a list of random transition matrices.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 10
beta <- matrix(
  data = c(
    0.7, 0.5, -0.1,
    0.0, 0.6, 0.4,
    0, 0, 0.5
  ),
  nrow = 3
)
vcov_beta_vec_l <- t(chol(0.001 * diag(9)))
SimBetaN2(n = n, beta = beta, vcov_beta_vec_l = vcov_beta_vec_l)

---

Simulate Transition Matrices with a Covariate from the Multivariate Normal Distribution

Description

This function simulates random transition matrices from a multivariate normal distribution, allowing the mean transition matrix to vary as a linear function of a covariate. The function ensures that the generated transition matrices are stationary using TestStationarity() with a rejection sampling approach.

Usage

SimBetaNCovariate(
  n,
  beta0,
  vcov_beta_vec_l,
  beta1,
  x,
  margin = 1,
  beta_lbound = NULL,
  beta_ubound = NULL,
  bound = FALSE,
  max_iter = 100000L
)

Arguments

n	Positive integer. Number of replications.
beta0	Numeric matrix. Baseline transition matrix $\boldsymbol{\beta}_0$ corresponding to $\mathbf{x} = \mathbf{0}$.
vcov_beta_vec_l	Numeric matrix. Cholesky factorization (t(chol(vcov_beta_vec))) of the sampling variance-covariance matrix of $\mathrm{vec} \left( \boldsymbol{\beta} \right)$.
beta1	Numeric matrix. Matrix of covariate effects mapping $\mathbf{x}$ to $\mathrm{vec}(\boldsymbol{\beta})$.
x	List of numeric vectors. Covariate values.
margin	Numeric scalar specifying the stationarity threshold. Values less than 1 indicate stricter stationarity criteria.
beta_lbound	Optional numeric matrix of same dim as beta. Use NA for no lower bound.
beta_ubound	Optional numeric matrix of same dim as beta. Use NA for no upper bound.
bound	Logical; if TRUE, resample until all elements respect bounds (NA bounds ignored).
max_iter	Safety cap on resampling attempts per draw.

Value

Returns a list of random transition matrices.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 5
beta0 <- matrix(
  data = c(
    0.7, 0.5, -0.1,
    0.0, 0.6, 0.4,
    0, 0, 0.5
  ),
  nrow = 3
)
vcov_beta_vec_l <- t(chol(0.001 * diag(9)))
# One scalar covariate per replication
beta1 <- matrix(data = 0, nrow = 9, ncol = 1)
beta1[1, 1] <- 0.10  # x shifts beta[1,1]
x <- list(c(0), c(1), c(-1), c(0.5), c(2))

SimBetaNCovariate(
  n = n,
  beta0 = beta0,
  vcov_beta_vec_l = vcov_beta_vec_l,
  beta1 = beta1,
  x = x
)

---

Simulate Diagonal Covariance Matrices from the Multivariate Normal Distribution

Description

This function simulates random diagonal covariance matrices from the multivariate normal distribution. The function ensures that the generated covariance matrices are positive semi-definite.

Usage

SimCovDiagN(n, sigma_diag, vcov_sigma_diag_l)

Arguments

n	Positive integer. Number of replications.
sigma_diag	Numeric matrix. The covariance matrix ($\boldsymbol{\Sigma}$).
vcov_sigma_diag_l	Numeric matrix. Cholesky factorization (t(chol(vcov_sigma_vech))) of the sampling variance-covariance matrix of $\mathrm{vech} \left( \boldsymbol{\Sigma} \right)$.

Value

Returns a list of random diagonal covariance matrices.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 10
sigma_diag <- c(1, 1, 1)
vcov_sigma_diag_l <- t(chol(0.001 * diag(3)))
SimCovDiagN(
  n = n,
  sigma_diag = sigma_diag,
  vcov_sigma_diag_l = vcov_sigma_diag_l
)

---

Simulate Covariance Matrices from the Multivariate Normal Distribution

Description

This function simulates random covariance matrices from the multivariate normal distribution. The function ensures that the generated covariance matrices are positive semi-definite.

Usage

SimCovN(n, sigma, vcov_sigma_vech_l)

Arguments

n	Positive integer. Number of replications.
sigma	Numeric matrix. The covariance matrix ($\boldsymbol{\Sigma}$).
vcov_sigma_vech_l	Numeric matrix. Cholesky factorization (t(chol(vcov_sigma_vech))) of the sampling variance-covariance matrix of $\mathrm{vech} \left( \boldsymbol{\Sigma} \right)$.

Value

Returns a list of random covariance matrices.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 10
sigma <- matrix(
  data = c(
    1.0, 0.5, 0.3,
    0.5, 1.0, 0.4,
    0.3, 0.4, 1.0
  ),
  nrow = 3
)
vcov_sigma_vech_l <- t(
  chol(
    0.001 * diag(3 * (3 + 1) / 2)
  )
)
SimCovN(
  n = n,
  sigma = sigma,
  vcov_sigma_vech_l = vcov_sigma_vech_l
)

---

Simulate Intercept Vectors in a Continuous-Time Vector Autoregressive Model from the Multivariate Normal Distribution

Description

This function simulates random intercept vectors in a continuous-time vector autoregressive model from the multivariate normal distribution.

Usage

SimIotaN(n, iota, vcov_iota_l)

Arguments

n	Positive integer. Number of replications.
iota	Numeric vector. Intercept ($\boldsymbol{\iota}$).
vcov_iota_l	Numeric matrix. Cholesky factorization (t(chol(vcov_iota))) of the sampling variance-covariance matrix of $\boldsymbol{\iota}$.

Value

Returns a list of random intercept vectors.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 10
iota <- c(0, 0, 0)
vcov_iota_l <- t(chol(0.001 * diag(3)))
SimIotaN(n = n, iota = iota, vcov_iota_l = vcov_iota_l)

---

Simulate Set Point Vectors from the Multivariate Normal Distribution

Description

This function simulates random set point vectors from the multivariate normal distribution.

Usage

SimMuN(n, mu, vcov_mu_l)

Arguments

n	Positive integer. Number of replications.
mu	Numeric vector. Set point ($\boldsymbol{\mu}$).
vcov_mu_l	Numeric matrix. Cholesky factorization (t(chol(vcov_mu))) of the sampling variance-covariance matrix of $\boldsymbol{\mu}$.

Value

Returns a list of random set point vectors.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 10
mu <- c(0, 0, 0)
vcov_mu_l <- t(chol(0.001 * diag(3)))
SimMuN(n = n, mu = mu, vcov_mu_l = vcov_mu_l)

---

Simulate Vectors from the Multivariate Normal Distribution

Description

This function simulates random vectors from the multivariate normal distribution.

Usage

SimMVN(n, mu, sigma_l)

Arguments

n	Positive integer. Number of replications.
mu	Numeric vector. Mean vector ($\boldsymbol{\nu}$).
sigma_l	Numeric matrix. Cholesky factorization (t(chol(sigma))) of the variance-covariance matrix $\boldsymbol{\Sigma}$.

Value

Returns a list of random vectors.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 10
mu <- c(0, 0, 0)
sigma_l <- t(chol(0.001 * diag(3)))
SimMVN(n = n, mu = mu, sigma_l = sigma_l)

---

Simulate Intercept Vectors in a Discrete-Time Vector Autoregressive Model from the Multivariate Normal Distribution

Description

This function simulates random intercept vectors in a discrete-time vector autoregressive model from the multivariate normal distribution.

Usage

SimNuN(n, nu, vcov_nu_l)

Arguments

n	Positive integer. Number of replications.
nu	Numeric vector. Intercept ($\boldsymbol{\nu}$).
vcov_nu_l	Numeric matrix. Cholesky factorization (t(chol(vcov_nu))) of the sampling variance-covariance matrix of $\boldsymbol{\nu}$.

Value

Returns a list of random intercept vectors.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 10
nu <- c(0, 0, 0)
vcov_nu_l <- t(chol(0.001 * diag(3)))
SimNuN(n = n, nu = nu, vcov_nu_l = vcov_nu_l)

---

Simulate Random Drift Matrices from the Multivariate Normal Distribution

Description

This function simulates random drift matrices from the multivariate normal distribution. The function ensures that the generated drift matrices are stable using TestPhi().

Usage

SimPhiN(
  n,
  phi,
  vcov_phi_vec_l,
  margin = 0,
  auto_ubound = 0,
  phi_lbound = NULL,
  phi_ubound = NULL,
  bound = FALSE,
  max_iter = 100000L
)

Arguments

n	Positive integer. Number of replications.
phi	Numeric matrix. The drift matrix ($\boldsymbol{\Phi}$).
vcov_phi_vec_l	Numeric matrix. Cholesky factorization (t(chol(vcov_phi_vec))) of the sampling variance-covariance matrix of $\mathrm{vec} \left( \boldsymbol{\Phi} \right)$.
margin	Numeric scalar specifying the stability threshold for the real part of the eigenvalues. The default 0.0 corresponds to the imaginary axis; values less than 0.0 enforce a stricter stability margin.
auto_ubound	Numeric scalar specifying the upper bound for the diagonal elements of $\boldsymbol{\Phi}$. Default is 0.0, requiring all diagonal values to be $\leq 0$.
phi_lbound	Optional numeric matrix of same dim as phi. Use NA for no lower bound.
phi_ubound	Optional numeric matrix of same dim as phi. Use NA for no upper bound.
bound	Logical; if TRUE, resample until all elements respect bounds (NA bounds ignored).
max_iter	Safety cap on resampling attempts per draw.

Value

Returns a list of random drift matrices.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN2(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 10
phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0, 0, -0.693
  ),
  nrow = 3
)
vcov_phi_vec_l <- t(chol(0.001 * diag(9)))
SimPhiN(n = n, phi = phi, vcov_phi_vec_l = vcov_phi_vec_l)

---

Simulate Random Drift Matrices from the Multivariate Normal Distribution and Project to Hurwitz

Description

This function simulates random dirft matrices from the multivariate normal distribution then projects each draw to the Hurwitz-stable region using ProjectToHurwitz().

Usage

SimPhiN2(n, phi, vcov_phi_vec_l, margin = 0.001)

Arguments

n	Positive integer. Number of replications.
phi	Numeric matrix. The drift matrix ($\boldsymbol{\Phi}$).
vcov_phi_vec_l	Numeric matrix. Cholesky factorization (t(chol(vcov_phi_vec))) of the sampling variance-covariance matrix of $\mathrm{vec} \left( \boldsymbol{\Phi} \right)$.
margin	Positive numeric. Target buffer so that the spectral abscissa is $\le -\text{margin}$ (default 1e-3).

Value

Returns a list of random drift matrices.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiNCovariate(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 10
phi <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0, 0, -0.693
  ),
  nrow = 3
)
vcov_phi_vec_l <- t(chol(0.001 * diag(9)))
SimPhiN2(n = n, phi = phi, vcov_phi_vec_l = vcov_phi_vec_l)

---

Simulate Random Drift Matrices with a Covariate from the Multivariate Normal Distribution

Description

This function simulates random drift matrices from the multivariate normal distribution, allowing the mean drift matrix to vary as a linear function of a covariate The function ensures that the generated drift matrices are stable using TestPhi().

Usage

SimPhiNCovariate(
  n,
  phi0,
  vcov_phi_vec_l,
  phi1,
  x,
  margin = 0,
  auto_ubound = 0,
  phi_lbound = NULL,
  phi_ubound = NULL,
  bound = FALSE,
  max_iter = 100000L
)

Arguments

n	Positive integer. Number of replications.
phi0	Numeric matrix. Baseline drift matrix ($\boldsymbol{\Phi}_0$).
vcov_phi_vec_l	Numeric matrix. Cholesky factorization (t(chol(vcov_phi_vec))) of the sampling variance-covariance matrix of $\mathrm{vec} \left( \boldsymbol{\Phi} \right)$.
phi1	Numeric matrix. Matrix of covariate effects mapping $\mathbf{x}$ to $\mathrm{vec}(\boldsymbol{\Phi})$.
x	List of numeric vectors. Covariate values.
margin	Numeric scalar specifying the stability threshold for the real part of the eigenvalues. The default 0.0 corresponds to the imaginary axis; values less than 0.0 enforce a stricter stability margin.
auto_ubound	Numeric scalar specifying the upper bound for the diagonal elements of $\boldsymbol{\Phi}$. Default is 0.0, requiring all diagonal values to be $\leq 0$.
phi_lbound	Optional numeric matrix of same dim as phi. Use NA for no lower bound.
phi_ubound	Optional numeric matrix of same dim as phi. Use NA for no upper bound.
bound	Logical; if TRUE, resample until all elements respect bounds (NA bounds ignored).
max_iter	Safety cap on resampling attempts per draw.

Value

Returns a list of random drift matrices.

Author(s)

Ivan Jacob Agaloos Pesigan

See Also

Other Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimSSMFixed(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

n <- 5
phi0 <- matrix(
  data = c(
    -0.357, 0.771, -0.450,
    0.0, -0.511, 0.729,
    0, 0, -0.693
  ),
  nrow = 3
)
vcov_phi_vec_l <- t(chol(0.001 * diag(9)))
# One scalar covariate per replication
phi1 <- matrix(data = 0, nrow = 9, ncol = 1)
phi1[1, 1] <- 0.10  # x shifts phi[1,1]
x <- list(c(0), c(1), c(-1), c(0.5), c(2))
SimPhiNCovariate(
  n = n,
  phi0 = phi0,
  vcov_phi_vec_l = vcov_phi_vec_l,
  phi1 = phi1,
  x = x
)

---

Simulate Data from a State Space Model (Fixed Parameters)

Description

This function simulates data using a state space model. It assumes that the parameters remain constant across individuals and over time.

Usage

SimSSMFixed(
  n,
  time,
  delta_t = 1,
  mu0,
  sigma0_l,
  alpha,
  beta,
  psi_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	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 ($\boldsymbol{\Sigma}_{\boldsymbol{\eta} \mid 0}$).
alpha	Numeric vector. Vector of constant values for the dynamic model ($\boldsymbol{\alpha}$).
beta	Numeric matrix. Transition matrix relating the values of the latent variables at the previous to the current time point ($\boldsymbol{\beta}$).
psi_l	Numeric matrix. Cholesky factorization (t(chol(psi))) of the covariance matrix of the process noise ($\boldsymbol{\Psi}$).
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 ($\boldsymbol{\kappa}$).

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 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.

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 Simulation of State Space Models Data Functions: LinSDE2SSM(), LinSDECovEta(), LinSDECovY(), LinSDEInterceptEta(), LinSDEInterceptY(), LinSDEMeanEta(), LinSDEMeanY(), ProjectToHurwitz(), ProjectToStability(), SSMCovEta(), SSMCovY(), SSMInterceptEta(), SSMInterceptY(), SSMMeanEta(), SSMMeanY(), SimAlphaN(), SimBetaN(), SimBetaN2(), SimBetaNCovariate(), SimCovDiagN(), SimCovN(), SimIotaN(), SimMVN(), SimMuN(), SimNuN(), SimPhiN(), SimPhiN2(), SimPhiNCovariate(), SimSSMIVary(), SimSSMLinGrowth(), SimSSMLinGrowthIVary(), SimSSMLinSDEFixed(), SimSSMLinSDEIVary(), SimSSMOUFixed(), SimSSMOUIVary(), SimSSMVARFixed(), SimSSMVARIVary(), SpectralRadius(), TestPhi(), TestPhiHurwitz(), TestStability(), TestStationarity()

Examples

# prepare parameters
set.seed(42)
## 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))
## measurement model
k <- 3
nu <- rep(x = 0, times = k)
lambda <- diag(k)
theta <- 0.001 * diag(k)
theta_l <- 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 <- diag(x = 0.10, nrow = p, ncol = j)
kappa <- diag(x = 0.10, nrow = k, ncol = j)

# Type 0
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 0
)

plot(ssm)

# Type 1
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 1,
  x = x,
  gamma = gamma
)

plot(ssm)

# Type 2
ssm <- SimSSMFixed(
  n = n,
  time = time,
  mu0 = mu0,
  sigma0_l = sigma0_l,
  alpha = alpha,
  beta = beta,
  psi_l = psi_l,
  nu = nu,
  lambda = lambda,
  theta_l = theta_l,
  type = 2,
  x = x,
  gamma = gamma,
  kappa = kappa
)

plot(ssm)

---