Convert CTVAR Parameters to State Space Parameters

This function converts continuous-time vector autoregressive (CTVAR) parameters to discrete-time state space model parameters by exact sampling of the derivative companion state.

Usage

CTVAR2SSM(iota, phi, sigma_l, delta_t = 1)

Arguments

iota: Numeric vector of length \(q\). Constant term entering the highest derivative of the CTVAR model.
phi: List of length \(p\). Numeric \(q \times q\) matrices containing the CTVAR derivative coefficient matrices \(\mathbf{S}_{0}, \ldots, \mathbf{S}_{p - 1}\), in increasing derivative order.
sigma_l: Numeric \(q \times q\) matrix. Lower triangular factor of the continuous-time diffusion covariance: sigma_l %*% t(sigma_l). Noise enters the highest derivative. For a positive-definite covariance matrix sigma, use t(chol(sigma)). Zero and positive-semidefinite covariances are allowed.
delta_t: Positive finite numeric scalar. Sampling interval \(\Delta t\).

Value

Returns a list of state space model parameters:

alpha: Numeric vector of length \(pq\). Intercept of the sampled derivative state.
beta: Numeric \(pq \times pq\) matrix. Transition matrix of the sampled derivative state.
psi_l: Numeric \(pq \times pq\) matrix. Lower triangular factor of the interval process noise covariance, reconstructed as psi_l %*% t(psi_l).
lambda: Numeric \(q \times pq\) matrix. Observation matrix selecting the first \(q\) state components.

Details

The transformation uses Equations (4)–(7) and Table 1, block A, of Chow et al. (2023). For \(q\) observed variables and derivative order \(p\), the state has length \(pq\) and is ordered as \((\mathbf{y}, \mathbf{y}^{(1)}, \ldots, \mathbf{y}^{(p-1)})\). The drift coefficients are \(\mathbf{S}_0, \ldots, \mathbf{S}_{p-1}\); the intercept and diffusion enter the last block of the state.

For orders above one, the sampled observed process generally requires a state space representation. For order one, this transformation corresponds to LinSDE2SSM() with phi = phi[[1]] and an identity observation matrix.

References

Chow, S.-M., Losardo, D., Park, J. J., & Molenaar, P. C. M. (2023). Continuous-time dynamic models: Connections to structural equation models and other discrete-time models. In R. H. Hoyle (Ed.), Handbook of structural equation modeling (2nd ed.). The Guilford Press.

Author

Ivan Jacob Agaloos Pesigan

Examples

# Example 1: CTVAR(1) to a state space model
iota <- c(0.32, 0.23)
phi <- list(
  matrix(
    data = c(
      -0.10, 0.05,
      0.05, -0.10
    ),
    nrow = 2
  )
)
sigma <- matrix(
  data = c(
    2.79, 0.06,
    0.06, 3.27
  ),
  nrow = 2
)
sigma_l <- t(chol(sigma))
CTVAR2SSM(
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  delta_t = 1.0
)
#> $alpha
#>           [,1]
#> [1,] 0.3100258
#> [2,] 0.2264506
#> 
#> $beta
#>            [,1]       [,2]
#> [1,] 0.90596870 0.04526072
#> [2,] 0.04526072 0.90596870
#> 
#> $psi_l
#>           [,1]     [,2]
#> [1,] 1.5923846 0.000000
#> [2,] 0.1176299 1.719563
#> 
#> $lambda
#>      [,1] [,2]
#> [1,]    1    0
#> [2,]    0    1
#> 

# Example 2: CTVAR(2) to a state space model
iota <- c(0.32, 0.23)
phi <- list(
  matrix(
    data = c(
      -0.10, 0.05,
      0.05, -0.10
    ),
    nrow = 2
  ),
  matrix(
    data = c(
      -0.30, 0.02,
      0.02, -0.30
    ),
    nrow = 2
  )
)
sigma <- matrix(
  data = c(
    2.79, 0.06,
    0.06, 3.27
  ),
  nrow = 2
)
sigma_l <- t(chol(sigma))
CTVAR2SSM(
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  delta_t = 1.0
)
#> $alpha
#>           [,1]
#> [1,] 0.1450384
#> [2,] 0.1049712
#> [3,] 0.2754204
#> [4,] 0.2003077
#> 
#> $beta
#>             [,1]        [,2]       [,3]       [,4]
#> [1,]  0.95524659  0.02202526 0.84977288 0.01518730
#> [2,]  0.02202526  0.95524659 0.01518730 0.84977288
#> [3,] -0.08421792  0.04096991 0.70061848 0.03446453
#> [4,]  0.04096991 -0.08421792 0.03446453 0.70061848
#> 
#> $psi_l
#>            [,1]       [,2]       [,3]     [,4]
#> [1,] 0.85637465 0.00000000 0.00000000 0.000000
#> [2,] 0.04062557 0.92617013 0.00000000 0.000000
#> [3,] 1.17763926 0.01543529 0.80688762 0.000000
#> [4,] 0.06937570 1.27291298 0.01905466 0.873332
#> 
#> $lambda
#>      [,1] [,2] [,3] [,4]
#> [1,]    1    0    0    0
#> [2,]    0    1    0    0
#>