KFAS (version 1.5.1)

simulateSSM: Simulation of a Gaussian State Space Model

Description

Function simulateSMM simulates states, signals, disturbances or missing observations of the Gaussian state space model either conditional on the data (simulation smoother) or unconditionally.

Usage

simulateSSM(
  object,
  type = c("states", "signals", "disturbances", "observations", "epsilon", "eta"),
  filtered = FALSE,
  nsim = 1,
  antithetics = FALSE,
  conditional = TRUE
)

Value

An n x k x nsim array containing the simulated series, where k is number of observations, signals, states or disturbances.

Arguments

object

Gaussian state space object of class SSModel.

type

What to simulate.

filtered

Simulate from \(p(\alpha_t|y_{t-1},...,y_1)\) instead of \(p(\alpha|y)\).

nsim

Number of independent samples. Default is 1.

antithetics

Use antithetic variables in simulation. Default is FALSE.

conditional

Simulations are conditional to data. If FALSE, the states having exact diffuse initial distribution (as defined by P1inf are fixed to corresponding values of a1. See details.

Details

Simulation smoother algorithm is based on article by J. Durbin and S.J. Koopman (2002). The simulation filter (filtered = TRUE) is a straightforward modification of the simulations smoother, where only filtering steps are performed.

Function can use two antithetic variables, one for location and other for scale, so output contains four blocks of simulated values which correlate which each other (ith block correlates negatively with (i+1)th block, and positively with (i+2)th block etc.).

Note that KFAS versions 1.2.0 and older, for unconditional simulation the initial distribution of states was fixed so that a1 was set to the smoothed estimates of the first state and the initial variance was set to zero. Now original a1 and P1 are used, and P1inf is ignored (i.e. diffuse states are fixed to corresponding elements of a1).

References

Durbin J. and Koopman, S.J. (2002). A simple and efficient simulation smoother for state space time series analysis, Biometrika, Volume 89, Issue 3

Examples

set.seed(123)
# simulate new observations from the "fitted" model
model <- SSModel(Nile ~ SSMtrend(1, Q = 1469), H = 15099)
# signal conditional on the data i.e. samples from p(theta | y)
# unconditional simulation is not reasonable as the model is nonstationary
signal_sim <- simulateSSM(model, type = "signals", nsim = 10)
# and add unconditional noise term i.e samples from p(epsilon)
epsilon_sim <- simulateSSM(model, type = "epsilon", nsim = 10,
  conditional = FALSE)
observation_sim <- signal_sim + epsilon_sim

ts.plot(observation_sim[,1,], Nile, col = c(rep(2, 10), 1),
  lty = c(rep(2, 10), 1), lwd = c(rep(1, 10), 2))

# fully unconditional simulation:
observation_sim2 <- simulateSSM(model, type = "observations", nsim = 10,
  conditional = FALSE)
ts.plot(observation_sim[,1,], observation_sim2[,1,], Nile,
col = c(rep(2:3, each = 10), 1), lty = c(rep(2, 20), 1),
lwd = c(rep(1, 20), 2))

# illustrating use of antithetics
model <- SSModel(matrix(NA, 100, 1) ~ SSMtrend(1, 1, P1inf = 0), H = 1)

set.seed(123)
sim <- simulateSSM(model, "obs", nsim = 2, antithetics = TRUE)
# first time points
sim[1,,]
# correlation structure between simulations with two antithetics
cor(sim[,1,])

out_NA <- KFS(model, filtering = "none", smoothing = "state")
model["y"] <- sim[, 1, 1]
out_obs <- KFS(model, filtering = "none", smoothing = "state")

set.seed(40216)
# simulate states from the p(alpha | y)
sim_conditional <- simulateSSM(model, nsim = 10, antithetics = TRUE)

# mean of the simulated states is exactly correct due to antithetic variables
mean(sim_conditional[2, 1, ])
out_obs$alpha[2]
# for variances more simulations are needed
var(sim_conditional[2, 1, ])
out_obs$V[2]

set.seed(40216)
# no data, simulations from p(alpha)
sim_unconditional <- simulateSSM(model, nsim = 10, antithetics = TRUE,
  conditional = FALSE)
mean(sim_unconditional[2, 1, ])
out_NA$alpha[2]
var(sim_unconditional[2, 1, ])
out_NA$V[2]

ts.plot(cbind(sim_conditional[,1,1:5], sim_unconditional[,1,1:5]),
  col = rep(c(2,4), each = 5))
lines(out_obs$alpha, lwd=2)