Convert State Space Parameters to Linear SDE Parameters

This function converts parameters from the discrete-time state space model parameterization to the linear stochastic differential equation model using the exact matrix-logarithm mapping.

Usage

SSM2LinSDE(
  alpha,
  beta,
  psi_l,
  delta_t,
  tol = sqrt(.Machine$double.eps),
  tol_psi_l = sqrt(.Machine$double.eps)
)

Arguments

alpha: Numeric vector of length $q$. Intercept of the discrete-time state equation.
beta: Numeric $q \times q$ matrix. Transition matrix of the discrete-time state equation.
psi_l: Numeric $q \times q$ matrix. Lower triangular factor of the interval process noise covariance: psi_l %*% t(psi_l). For a positive-definite covariance matrix psi, use t(chol(psi)). Zero and positive-semidefinite covariances are allowed.
delta_t: Positive finite numeric scalar. Sampling interval $\Delta t$.
tol: Non-negative finite numeric scalar. Tolerance for the largest absolute imaginary component of the matrix logarithm.
tol_psi_l: Non-negative finite numeric scalar. Tolerance for checking that psi_l is lower triangular with a non-negative diagonal.

Value

Returns a list of linear stochastic differential equation model parameters:

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. Lower triangular factor of the covariance matrix of volatility or randomness in the process $\boldsymbol{\Sigma}$.

Details

The state equation and its continuous-time counterpart are described in LinSDE2SSM(). The drift matrix is computed using the principal matrix logarithm: $$ \boldsymbol{\Phi} = \frac{1}{\Delta t}\log(\boldsymbol{\beta}). $$ The intercept and diffusion covariance are recovered by solving $$ \boldsymbol{\alpha} = \left[\int_0^{\Delta t} \exp(s\boldsymbol{\Phi})\,\mathrm{d}s\right]\boldsymbol{\iota} $$ and $$ \boldsymbol{\Psi} = \int_0^{\Delta t} \exp(s\boldsymbol{\Phi})\boldsymbol{\Sigma} \exp(s\boldsymbol{\Phi}^{\prime})\,\mathrm{d}s, $$ where psi_l %*% t(psi_l) gives $\boldsymbol{\Psi}$. The integrated exponential operators support zero and singular drifts.

The principal matrix-logarithm branch does not search all possible continuous-time embeddings or resolve temporal aliasing. Inputs are rejected if the logarithm is not finite (singular beta), has an imaginary component exceeding tol, the integral inversions are not unique, or the recovered diffusion covariance is not positive semidefinite. All inputs must contain finite values and have compatible dimensions.

References

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

Author

Ivan Jacob Agaloos Pesigan

Examples

# Example 1: Recover the parameters of a bivariate linear SDE
iota <- c(0.32, 0.23)

phi <- 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))
delta_t <- 1.0

ssm <- LinSDE2SSM(
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  delta_t = delta_t
)

SSM2LinSDE(
  alpha = ssm$alpha,
  beta = ssm$beta,
  psi_l = ssm$psi_l,
  delta_t = delta_t
)
#> $iota
#>      [,1]
#> [1,] 0.32
#> [2,] 0.23
#> 
#> $phi
#>       [,1]  [,2]
#> [1,] -0.10  0.05
#> [2,]  0.05 -0.10
#> 
#> $sigma_l
#>            [,1]     [,2]
#> [1,] 1.67032931 0.000000
#> [2,] 0.03592106 1.807957
#>