nmfkc.ar.stationarity: Stationarity of an NMF-VAR model

In nmfkc: Non-Negative Matrix Factorization with Kernel Covariates

Stationarity of an NMF-VAR model

Description

Reports whether a fitted NMF-VAR is stationary, and how cheaply that could be decided. Three numbers are returned, in increasing order of sharpness: colsum.max =\max_j c_j, the largest column sum of \sum_d\Xi_d = X\sum_d\Theta_d, certifies stationarity when below 1 with no eigenvalue computation at all (a sufficient condition only, so a value above 1 is merely inconclusive); spectral.radius.sum =\rho(\sum_d G_d) is below 1 exactly when the model is stationary (and is bracketed by \min_j c_j and \max_j c_j); and spectral.radius is the spectral radius of the companion matrix itself.

Note that the c_j are weighted by \bm 1'X, i.e. colSums(X) %*% Lambda. Using the raw column sums of \sum_d\Theta_d is correct only when \bm 1'X=\bm 1' and can certify stationarity for a non-stationary fit otherwise.

This function answers the yes/no question only. For the dynamics behind the number — the transition matrices, the roots, the impulse responses, the long-run mean — use nmfkc.ar.latent, and for the standard error of \rho use nmfkc.ar.latent.inference.

Usage

nmfkc.ar.stationarity(x, method = c("latent", "companion"))

Arguments

x

A fitted "nmfkc" object obtained from an nmfkc.ar design, or an object from nmfkc.ar.latent.

method

Route used for spectral.radius. "latent" (default) uses the DQ\times DQ companion matrix of the G_d; "companion" uses the PD\times PD companion matrix of the \Xi_d. The two companion matrices share the same non-zero eigenvalues, so the results agree up to floating-point rounding, but "latent" costs O((DQ)^3) instead of O((PD)^3) — for a 47-variable, 7-lag, rank-4 model that is a 28\times28 eigenproblem instead of a 329\times329 one. Use "companion" to reproduce the value bit-for-bit as computed by earlier versions of the package.

Value

An object of class "nmfkc.ar.stationarity": a list with

spectral.radius	Spectral radius of the companion matrix (route selected by method). A value below 1 indicates stationarity.
stationary	Logical; TRUE when spectral.radius < 1.
method	The route actually used.
spectral.radius.sum	\rho(\sum_d G_d); below 1 iff stationary.
colsum	Column sums c_j of \sum_d\Xi_d, i.e. colSums(X) %*% Lambda (length P).
colsum.max	\max_j c_j; below 1 it certifies stationarity.
dims	Named vector of P, Q, D.

References

Satoh, K. (2025). Applying non-negative matrix factorization with covariates to multivariate time series data as a vector autoregression model. Japanese Journal of Statistics and Data Science. arXiv:2501.17446. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s42081-025-00314-0")}

See Also

nmfkc.ar.latent, nmfkc.ar.latent.inference, nmfkc.ar

Examples

# Check stationarity of fitted AR model
d <- AirPassengers
ar_data <- nmfkc.ar(d, degree = 2)
result <- nmfkc(ar_data$Y, ar_data$A, rank = 1)
st <- nmfkc.ar.stationarity(result)
st$spectral.radius
st$stationary
st                 # full report

nmfkc documentation built on Oct. 30, 2026, 9:06 a.m.