RW2 temporal structure (Second-order Random Walk)

Specify a second-order random walk temporal random effect. RW2 penalizes deviations from linearity, resulting in smoother trends than RW1: phi[t] - 2*phi[t-1] + phi[t-2] ~ N(0, sigma^2).

Usage

temporal_rw2(time_var, group_var = NULL, cyclic = FALSE, shared = 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.
cyclic: Logical. If TRUE, the random walk wraps around so the last time point is a neighbour of the first (cyclic boundary, e.g. month of year). Default FALSE.
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.

Value

A tulpa_temporal object

Details

RW2 produces smoother curves than RW1 because it penalizes the second derivative (curvature) rather than the first derivative (slope). It requires at least 3 time points.

The precision matrix is rank T-2 (two constraints needed).

Examples

# Create temporal RW2 specification
temporal_rw2("year")

# \donttest{
# Smooth temporal trend
set.seed(126)
df <- data.frame(
  year = rep(1:20, each = 4),
  x = rnorm(80)
)
trend <- sin(seq(0, 2, length.out = 20))
df$count <- rpois(80, exp(1 + 0.3 * df$x + trend[df$year]))

fit <- tulpa(
  count ~ x,
  data = df,
  family = "poisson",
  temporal = temporal_rw2("year"),
  mode = "auto"
)
summary(fit)
# }