temporal_ar: AR(p) temporal latent field (general-order autoregressive)

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

AR(p) temporal latent field (general-order autoregressive)

Description

A stationary autoregressive process of order p, w_t = \sum_{j=1}^{p} \phi_j w_{t-j} + \varepsilon_t, as a user-defined GMRF latent block for a model formula via latent(temporal_ar(...)) – the general-order extension of temporal_ar2(), sharing the same construction: the precision is the exact stationary AR(p) GMRF (the banded inverse of the Yule-Walker Toeplitz autocovariance), and stationarity is enforced by the partial-autocorrelation parameterization (Levinson-Durbin map of p unconstrained atanh_psi hyperparameters), so the hyperparameter integration never leaves the stationary region.

The block's hyperparameters are log_tau (log innovation precision) and atanh_psi1..⁠atanh_psi<p>⁠; the AR coefficients are recovered through psi_j = tanh(atanh_psi_j) and the Levinson-Durbin recursion. Note the outer integration cost grows with the hyperparameter count p + 1; orders beyond 3-4 are better served by a sampler tier.

Usage

temporal_ar(
  time_idx,
  p = 2L,
  n_times = NULL,
  prior_tau_sd = 2,
  prior_psi_sd = 1.5,
  name = NULL
)

## S3 method for class 'tulpa_temporal_latent_block'
print(x, ...)

Arguments

time_idx	Integer vector (1-based) of the time point for each observation.
p	Autoregressive order (>= 1).
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 atanh_psi hyperparameters (weakly-informative Gaussian). Defaults 2 / 1.5.
name	Optional block label (default ⁠ar<p>⁠).
x	A temporal AR(p) block.
...	Passed to the tgmrf print method.

Value

A tgmrf / tulpa_latent_block object.

See Also

temporal_ar2() (the p = 2 shorthand), temporal_ar1() for the first-order formula-integrated process, tgmrf() for the general user-defined GMRF interface.

Examples

set.seed(1)
Tt <- 150L
w <- numeric(Tt); w[1:3] <- rnorm(3)
for (t in 4:Tt) w[t] <- 0.4 * w[t-1] + 0.2 * w[t-2] - 0.2 * w[t-3] + rnorm(1, 0, 0.4)
d <- data.frame(t = seq_len(Tt), y = w + rnorm(Tt, 0, 0.3))
fit <- tulpa(y ~ latent(temporal_ar(d$t, p = 3)), data = d,
             family = "gaussian", mode = "nested_laplace")

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