nmfkc.ar.latent: Latent transition matrices of an NMF-VAR model

In nmfkc: Non-Negative Matrix Factorization with Kernel Covariates

Latent transition matrices of an NMF-VAR model

Description

This function is experimental. The interface may change in future versions: argument names, defaults and the contents of the returned object are not yet stable.

Reduces a fitted NMF-VAR to the Q\times Q operators that drive its coefficient vector. Writing \bm b_t=\Theta\bm a_t for the fitted scores, the residual-free recursion is

\bm b_t=\sum_{d=1}^{D}G_d\,\bm b_{t-d}+\bm\theta,\qquad G_d=\Theta_d X \ \ (Q\times Q),

and the G_d carry the model's whole deterministic structure — transition weights, propagation, roots, the long-run mean. They stay readable where the P\times P matrices \Xi_d = X\Theta_d do not: 4\times4 instead of 47\times47.

Use plot.nmfkc.ar.latent to draw them, nmfkc.ar.latent.inference to attach standard errors, and nmfkc.ar.stationarity if the stationarity verdict is all that is wanted.

Usage

nmfkc.ar.latent(x)

Arguments

x	A fitted "nmfkc" object obtained from an nmfkc.ar design (nmfkc(ar$Y, ar$A, rank = Q)).

Details

The latent process is VARMA, not VAR. The display above drops the observation residual. With \bm y_t = X\bm b_t + \bm e_t the exact identity is

\bm b_t=\sum_{d=1}^{D}G_d\,\bm b_{t-d}+\bm\theta +\sum_{d=1}^{D}\Theta_d\,\bm e_{t-d},

i.e. a VARMA(D,D) in general (verified to machine precision; dropping the moving-average term leaves an error of order the residuals themselves). G_d is therefore the autoregressive operator of the fitted VAR reduced to the latent space, not the coefficient matrix of a latent VAR that the scores obey exactly. In particular an entry of G_d should not be read as Granger causality between latent conditions: the moving-average term is common to all coordinates and is not conditioned on.

Direction convention: (G_d)_{q q'} is the effect of b_{q',\,t-d} on b_{q,\,t} — row = effect (at t), column = cause (at t-d).

Identification. The factorization is unique only up to (X,\Theta)\to(XT,T^{-1}\Theta), under which G_d\to T^{-1}G_dT. The eigenvalues, \rho, \Xi_d and \bm\mu_y are therefore invariant, while individual entries of G_d are not. Column normalization removes only the scale part of T, and a separable fitted X does not certify the rest: with X=I_2, \Theta=\left(\begin{smallmatrix}2&3\\1&1\end{smallmatrix}\right) and T=\left(\begin{smallmatrix}0.9&0.1\\0.1&0.9\end{smallmatrix}\right) the basis is perfectly separable and column-normalized, yet XT\ge0, T^{-1}\Theta\ge0, the product is unchanged and T is no permutation. Read the entries of G_d as descriptive and the invariants as inferential; see nmfkc.ar.latent.inference.

Oscillation. A complex pair means a cycle, but a negative real root also oscillates — it alternates in sign each period, i.e. with period 2. cycle.period covers the first case and alternating the second; non.oscillatory is TRUE only when neither occurs. Positive real roots rule out oscillation but not overshoot: a non-normal G can send a coordinate up before it comes down, which is exactly what the off-diagonal impulse responses of the Canada fit do. Note also that for Q=2, D=1 and G\ge0 the roots are real by construction (the discriminant is (a-d)^2+4bc\ge0), so in that setting the absence of a cycle is a property of the design, not a finding.

Value

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

G	List of D latent transition matrices G_d (Q\times Q), named lag1, ..., with dimnames = basis labels (row = effect, column = cause).
G.sum	\sum_d G_d.
eigenvalues	The DQ eigenvalues of the latent companion matrix. The PD\times PD companion of the \Xi_d has the same non-zero eigenvalues, so these are all of the model's roots.
spectral.radius	\rho, the largest modulus among them.
spectral.radius.sum	\rho(\sum_d G_d); below 1 iff stationary.
stationary	Logical; TRUE when spectral.radius < 1.
cycle.period	Period 2\pi/|\arg\lambda| implied by the dominant complex root, or NA when no root is complex.
alternating	TRUE when some root is real and negative, i.e. the decay alternates in sign (period 2).
non.oscillatory	TRUE only when every root is real and positive. Named after the roots, not the trajectory — overshoot is still possible.
theta0	Latent intercept \bm\theta (or NULL).
mu.b, mu.y	Long-run means (I-\sum_d G_d)^{-1}\bm\theta and X\bm\mu_b; NA when the fit is not stationary or has no intercept.
p.star	Composition of the long-run mean, \bm\mu_b/(\bm 1'\bm\mu_b) — equivalently the fixed point of the residual-free composition dynamics. It is not in general the limit of \bm b_t/(\bm 1'\bm b_t) for the stochastic process, nor its expectation: the ratio of expectations is not the expectation of the ratio.
colsum	Column sums c_j of \sum_d\Xi_d = X\sum_d\Theta_d, computed as colSums(X) %*% Lambda (length P). Being column sums of a non-negative matrix whose radius is \rho(\sum_d G_d), they bracket it. The unweighted colSums(Lambda) would only be correct when \bm 1'X=\bm 1'.
colsum.max	\max_j c_j; below 1 it certifies stationarity.
df	Effective dimension Q(P+m-Q) of the fit, where m is the number of columns of \Theta (PD+1 with an intercept, PD without). These are the degrees of freedom to use when comparing R^2 or information criteria across Q: the naive count Q(P+m) ignores the Q^2 rotational indeterminacy.
separability	Per-basis maximum of the row-normalised X; a value near 1 means that basis owns a variable loading almost only on it. This is a necessary-looking but not sufficient condition for uniqueness — see the counterexample under Identification.
X, Theta	The basis and the full coefficient matrix the quantities above were computed from, carried along so that nmfkc.ar.latent.inference does not need the fitted object a second time.
X.restriction	The constraint nmfkc placed on the columns of X, which decides how much of T is pinned down.
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

plot.nmfkc.ar.latent, nmfkc.ar.latent.inference, nmfkc.ar.stationarity, nmfkc.ar, nmfkc.ar.DOT

Examples

d <- AirPassengers
ar_data <- nmfkc.ar(d, degree = 2)
result <- nmfkc(ar_data$Y, ar_data$A, rank = 1)
lat <- nmfkc.ar.latent(result)
lat$G$lag1          # latent transition matrix of lag 1
lat$eigenvalues     # all roots of the model
lat                 # full report

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