temporal_ar2: AR(2) temporal latent field (second-order autoregressive)

In tulpa: Templated Unified Library for Posterior Approximation in Bayesian Hierarchical Models

AR(2) temporal latent field (second-order autoregressive)

Description

A stationary second-order autoregressive process w_t = \phi_1 w_{t-1} + \phi_2 w_{t-2} + \varepsilon_t as a user-defined GMRF latent block, for use in a model formula via latent(temporal_ar2(...)). It reuses tulpa's nested-Laplace / NUTS machinery (no dedicated C++ kernel). The precision is the exact stationary AR(2) GMRF; stationarity is enforced by the PACF parameterization, so the hyperparameter grid never leaves the stationarity region.

The block's hyperparameters are log_tau (log innovation precision) and atanh_psi1, atanh_psi2 (unconstrained partial autocorrelations); the AR coefficients are recovered as phi2 = tanh(atanh_psi2), phi1 = tanh(atanh_psi1) (1 - phi2).

Usage

temporal_ar2(
  time_idx,
  n_times = NULL,
  prior_tau_sd = 2,
  prior_psi_sd = 1.5,
  name = "ar2"
)

Arguments

time_idx	Integer vector (1-based) of the time point for each observation.
n_times	Number of distinct time points; defaults to max(time_idx).
prior_tau_sd, prior_psi_sd	Prior SDs for log_tau and the two atanh_psi hyperparameters (weakly-informative Gaussian). Defaults 2 / 1.5.
name	Optional block label.

Value

A tgmrf / tulpa_latent_block object.

See Also

temporal_ar1() for the first-order (formula-integrated) process; tgmrf() for the general user-defined GMRF interface.

Examples

set.seed(1)
Tt <- 120L
w <- numeric(Tt); w[1:2] <- rnorm(2)
for (t in 3:Tt) w[t] <- 0.5 * w[t-1] + 0.3 * w[t-2] + rnorm(1, 0, 0.4)
d <- data.frame(t = seq_len(Tt), y = w + rnorm(Tt, 0, 0.3))
# `phi` is the gaussian residual VARIANCE, here the 0.3 the series was
# simulated with. Left unsupplied the fit conditions on 1 and says so.
fit <- tulpa(y ~ latent(temporal_ar2(d$t)), data = d, family = "gaussian",
             phi = 0.3^2, mode = "nested_laplace")

tulpa documentation built on Oct. 19, 2026, 5:09 p.m.