Stationarity of an NMF-VAR model

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. doi:10.1007/s42081-025-00314-0

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)
#> Y(1,142)~X(1,1)C(1,3)A(3,142)=XB(1,142)...
#> 0sec
st <- nmfkc.ar.stationarity(result)
st$spectral.radius
#> [1] 0.968468
st$stationary
#> [1] TRUE
st                 # full report
#> rho(companion, 2x2, method="latent") = 0.9685  ->  stationary
#> rho(sum_d G_d)        = 0.9675   (< 1 iff stationary)
#> bracket: min_j c_j = 0.9675  <=  rho(sum_d G_d)  <=  max_j c_j = 0.9675
#>    max_j c_j < 1: stationarity certified without an eigenvalue
#> 
#> For the dynamics behind this number see nmfkc.ar.latent(), for a test of H0: rho >= 1 see nmfkc.ar.latent.inference().

Usage

Arguments

Value

References

See also

Examples