Convert Linear SDE Parameters to State Space Parameters

This function converts linear stochastic differential equation (SDE) parameters to discrete-time state space model parameters by exact discretization over a specified sampling interval.

Usage

LinSDE2SSM(iota, phi, sigma_l, delta_t = 1)

Arguments

iota: Numeric vector of length $q$. An unobserved term that is constant over time ($\boldsymbol{\iota}$).
phi: Numeric $q \times q$ 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 $q \times q$ matrix. Lower triangular factor of the continuous-time diffusion covariance: sigma_l %*% t(sigma_l). For a positive-definite covariance matrix sigma, use t(chol(sigma)). The factor must be lower triangular with a non-negative diagonal. Zero and positive-semidefinite covariances are allowed.
delta_t: Positive finite numeric scalar. Sampling interval $\Delta t$.

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. Lower triangular factor of the process noise covariance matrix $\boldsymbol{\Psi}$.

Details

The linear stochastic differential equation is $$ \mathrm{d}\boldsymbol{\eta}(t) = (\boldsymbol{\iota} + \boldsymbol{\Phi}\boldsymbol{\eta}(t)) \mathrm{d}t + \mathbf{L}\,\mathrm{d}\mathbf{W}(t), $$ where $\mathbf{L}$ is sigma_l and $\boldsymbol{\Sigma} = \mathbf{L}\mathbf{L}^{\prime}$. Over a sampling interval $\Delta t$, the state equation is $$ \boldsymbol{\eta}(t + \Delta t) = \boldsymbol{\alpha} + \boldsymbol{\beta}\boldsymbol{\eta}(t) + \boldsymbol{\zeta}(t), \qquad \boldsymbol{\zeta}(t) \sim \mathcal{N}(\mathbf{0}, \boldsymbol{\Psi}). $$ The exact discrete-time parameters are $$ \boldsymbol{\beta} = \exp(\Delta t\boldsymbol{\Phi}), \qquad \boldsymbol{\alpha} = \left[\int_0^{\Delta t} \exp(s\boldsymbol{\Phi})\,\mathrm{d}s\right]\boldsymbol{\iota}, $$ and $$ \boldsymbol{\Psi} = \int_0^{\Delta t} \exp(s\boldsymbol{\Phi})\boldsymbol{\Sigma} \exp(s\boldsymbol{\Phi}^{\prime})\,\mathrm{d}s. $$ The implementation evaluates these integrals using block matrix exponentials, so zero and singular drift matrices are supported. 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: Exact sampling 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

LinSDE2SSM(
  iota = iota,
  phi = phi,
  sigma_l = sigma_l,
  delta_t = delta_t
)
#> $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
#> 

# Example 2: Zero drift (Brownian motion with a constant drift term)
LinSDE2SSM(
  iota = 0.2,
  phi = matrix(0),
  sigma_l = matrix(0.5),
  delta_t = 0.5
)
#> $alpha
#>      [,1]
#> [1,]  0.1
#> 
#> $beta
#>      [,1]
#> [1,]    1
#> 
#> $psi_l
#>           [,1]
#> [1,] 0.3535534
#>