ou2: Two-dimensional discrete-time Ornstein-Uhlenbeck process

Description

ou2() constructs a ‘pomp’ object encoding a bivariate discrete-time Ornstein-Uhlenbeck process with noisy observations.

Usage

ou2(
  alpha_1 = 0.8,
  alpha_2 = -0.5,
  alpha_3 = 0.3,
  alpha_4 = 0.9,
  sigma_1 = 3,
  sigma_2 = -0.5,
  sigma_3 = 2,
  tau = 1,
  x1_0 = -3,
  x2_0 = 4,
  times = 1:100,
  t0 = 0
)

Value

A ‘pomp’ object with simulated data.

Arguments

alpha_1, alpha_2, alpha_3, alpha_4: entries of the $\alpha$ matrix, in column-major order. That is, alpha_2 is in the lower-left position.
sigma_1, sigma_2, sigma_3: entries of the lower-triangular $\sigma$ matrix. sigma_2 is the entry in the lower-left position.
tau: measurement error s.d.
x1_0, x2_0: latent variable values at time t0
times: vector of observation times
t0: the zero time

Details

If the state process is $X(t) = (X_{1}(t),X_{2}(t))$, then $$X(t+1) = \alpha X(t) + \sigma \epsilon(t),$$ where $\alpha$ and $\sigma$ are 2x2 matrices, $\sigma$ is lower-triangular, and $\epsilon(t)$ is standard bivariate normal. The observation process is $Y(t) = (Y_1(t),Y_2(t))$, where $Y_i(t) \sim \mathrm{normal}(X_i(t),\tau)$.

Examples

Run this code

po <- ou2()
plot(po)
coef(po)
x <- simulate(po)
plot(x)
pf <- pfilter(po,Np=1000)
logLik(pf)