Autoregressive node

The AR node (also exported as Autoregressive) encodes a Bayesian autoregressive process of order p:

\[y_t \sim \mathcal{N}(\theta^\top x_t, \, \gamma^{-1})\]

where yₜ is the current observation, xₜ = (yₜ₋₁, …, yₜ₋ₚ) is the vector of p lagged values, θ is the vector of AR coefficients, and γ is the observation precision.

This node is the natural building block for time series models such as AR(p), latent AR processes, and state-space models with autoregressive dynamics.

ReactiveMP.AR — Type

AR

Autoregressive factor node encoding a Bayesian autoregressive process of order p:

\[y_t \sim \mathcal{N}(\theta^\top x_t, \, \gamma^{-1})\]

where yₜ is the current observation, xₜ = (yₜ₋₁, …, yₜ₋ₚ) is the vector of p lagged values, θ is the vector of AR coefficients, and γ is the observation precision. Also available under the alias Autoregressive.

Interfaces

y (alias out) — current observation yₜ.
x — lagged state vector of length p.
θ — AR coefficient vector of length p.
γ — observation precision (scalar).

The node has no default meta and requires an ARMeta to be passed explicitly. See ConjugateAR for a variant that keeps (θ, γ) on a single conjugate MvNormalGamma edge.

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ReactiveMP.ARMeta — Type

ARMeta(form, order, stype)

Metadata for the AR node. Required — the node has no default meta.

Arguments

form — Univariate or Multivariate, the variate form that determines how x and y are interpreted. Univariate forces order to 1 (an AR(1) model); Multivariate uses the companion-matrix representation to handle AR(p) for p > 1.
order — the AR order p.
stype — ARsafe() or ARunsafe(), the numerical-stability mode. ARsafe adds a small regularization to avoid singular matrices; ARunsafe is faster but may be numerically fragile.

meta = ARMeta(Multivariate, order, ARsafe())

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Interfaces

Interface	Alias	Role
`y`	`out`	Current observation `yₜ`
`x`	—	Lagged state vector `(yₜ₋₁, …, yₜ₋ₚ)`
`θ`	—	AR coefficient vector (length `p`)
`γ`	—	Observation precision (scalar)

Metadata

ARMeta is required and must be passed explicitly — the node has no default meta:

y[t] ~ AR(x[t], θ, γ) where { meta = ARMeta(Multivariate, order, ARsafe()) }

The constructor takes:

Univariate or Multivariate — variate form (determines how x and y are interpreted).
order — the AR order p (must equal 1 for Univariate).
ARsafe() or ARunsafe() — numerical stability mode (ARsafe adds a small regularization to avoid singular matrices; ARunsafe is faster but may be numerically fragile).

Univariate vs multivariate

ARMeta{Univariate} treats y and the first element of x as scalars, with order forced to 1. This is an AR(1) model.

ARMeta{Multivariate} uses the full companion-matrix representation to handle AR(p) for p > 1. The state vector x has length p, and the AR process is embedded as a linear state-space model. See CompanionMatrix for the underlying algebraic structure.

State vector slicing

The ReactiveMP.ar_unit and ReactiveMP.ar_slice utilities extract specific parts of the joint state vector in the multivariate setting:

ReactiveMP.ar_unit — returns an appropriately shaped zero vector or matrix for initializing accumulators.
ReactiveMP.ar_slice — extracts a subvector or submatrix from a joint mean/covariance. This is used inside rules to separate the y part from the x part of the joint Gaussian q(y, x).

These are internal helpers that surface when writing custom rules for AR-based models.

ReactiveMP.ar_unit — Function

ar_unit(::T, order)

Returns [ 1.0, 0.0 ... 0.0 ] with length equal to order in case if T is Multivariate, and 1.0 in case if T is Univariate

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ReactiveMP.ar_slice — Function

ar_slice(::T, array, ranges...)

Returns array[ranges...] in case if T is Multivariate, and first(array[ranges...]) in case if T is Univariate

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