var1: ngme VAR(1) bivariate model specification (Cayley...

In ngme2: Linear Latent Non-Gaussian Models with Flexible Distributions

ngme VAR(1) bivariate model specification (Cayley re-parameterization)

Description

Builds a Vector Autoregressive order-1 (VAR(1)) operator for bivariate time-series data. The model follows:

Y_t = A\,Y_{t-1} + \varepsilon_t, \quad A = \begin{pmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \end{pmatrix}

Usage

var1(mesh = NULL, p1 = 0, p2 = 1, p3 = 0, p4 = 1)

Arguments

mesh	Integer vector or inla.mesh.1d object for the time mesh (length T). Pass NULL for deferred construction.
p1	Unconstrained skew-symmetric parameter (\in \mathbb{R}); controls the rotation part of the Cayley mapping. Default 0.
p2, p3, p4	Unconstrained lower-triangular Cholesky parameters (\in \mathbb{R}); control the contraction (positive-definite) part. Defaults p2 = 1, p3 = 0, p4 = 1.

Value

An ngme_operator (or ngme_operator_def when mesh = NULL) suitable for use inside f().

Cayley re-parameterization

To guarantee stationarity (\rho(A) < 1) at every SGD step, the 2 \times 2 coefficient matrix A is obtained via the Cayley transform:

A = (I + S)(I - S)^{-1}

where S is constructed from four unconstrained real parameters (p_1, p_2, p_3, p_4) \in \mathbb{R}^4 as S = J - R:

Skew-symmetric part (rotation):

J = \begin{pmatrix} 0 & p_1 \\ -p_1 & 0 \end{pmatrix}

Positive-definite part (contraction):

R = L L^{\top} + \varepsilon I, where L = \begin{pmatrix} p_2 & 0 \\ p_3 & p_4 \end{pmatrix} and \varepsilon = 10^{-5}.

Because S + S^{\top} = -2R is negative definite, all eigenvalues of S have strictly negative real parts, and the Cayley transform then guarantees |\lambda_i(A)| < 1 for every i.

The default values (p_1, p_2, p_3, p_4) = (0, 1, 0, 1) give A \approx 0 (no auto-regression at initialization).

Precision operator

The 2T x 2T precision operator is:

K = M_0 + a_{11}\,M_{11} + a_{22}\,M_{22} + a_{12}\,M_{12} + a_{21}\,M_{21}

where the M_{ij} are fixed sparse block matrices constructed from the T \times T first-order difference matrix C_T.

Examples

set.seed(1)
n_obs <- 10
dat <- data.frame(
  y     = rnorm(2 * n_obs),
  idx   = rep(seq_len(n_obs), 2),
  group = factor(rep(c("y1", "y2"), each = n_obs))
)

fit <- ngme(
  y ~ 0 + f(idx, model = var1(mesh = 1:n_obs), group = group,
            noise = noise_nig()),
  data    = dat,
  family  = noise_normal(sigma = 0.01, fix_sigma = TRUE),
  control_opt = control_opt(estimation = FALSE)
)
print(fit)

ngme2 documentation built on May 20, 2026, 9:10 a.m.