AR1 temporal structure (First-order Autoregressive)

Specify a first-order autoregressive temporal random effect. AR1 models temporal correlation where each time point depends on the previous one: phi[t] = rho * phi[t-1] + epsilon[t].

Unlike RW1/RW2, AR1 is a proper (stationary) model with an estimated autocorrelation parameter rho.

Usage

temporal_ar1(time_var, group_var = NULL, shared = NULL, rho_prior = NULL)

Arguments

time_var: A formula (~ time) or single character string naming the time variable in the data.
group_var: Optional formula (~ g) or character string naming a grouping variable. When supplied, a separate random walk is fit per group; NULL (default) fits a single walk shared across all observations.
shared: Whether the temporal effect is shared across processes in a multi-process model. NULL (default) shares the effect; FALSE fits process-specific effects and emits a warning about unshared confounding.
rho_prior: Prior for the AR(1) autocorrelation. Default NULL is a Uniform(-1, 1) prior. Supply a prior_beta(alpha, beta) to place a Beta(alpha, beta) prior on u = (rho + 1)/2 for a more informative prior (e.g. prior_beta(5, 2) favours positive autocorrelation). Honoured on both the exact-sampler and nested-Laplace paths.

Value

A tulpa_temporal object

Details

The AR1 process has marginal variance sigma^2 / (1 - rho^2) and correlation between time points t and s of rho^|t-s|.

The precision matrix is tridiagonal and full rank, so no constraints are needed.

Examples

# Create temporal AR1 specification
temporal_ar1("year")
temporal_ar1("year", group = "site")

# \donttest{
# AR1 temporal correlation
set.seed(127)
df <- data.frame(
  year = rep(1:20, each = 3),
  x = rnorm(60)
)
f <- as.numeric(arima.sim(list(ar = 0.7), 20, sd = 0.4))
df$count <- rpois(60, exp(1 + 0.3 * df$x + f[df$year]))

fit <- tulpa(
  count ~ x,
  data = df,
  family = "poisson",
  temporal = temporal_ar1("year"),
  mode = "auto"
)
summary(fit)
# }