nmf.ffb.diagnostics: Structural Diagnostics of a Fitted NMF-FFB Feedback Matrix

In nmfkc: Non-Negative Matrix Factorization with Kernel Covariates

Structural Diagnostics of a Fitted NMF-FFB Feedback Matrix

Description

Two quantities that describe what the selected feedback is, as opposed to whether it is there (which is the job of the calibrated likelihood-ratio test in nmf.ffb.inference).

Cycle structure. The equilibrium operator is A = X\Theta_1 on the indicators and A_Q = \Theta_1X on the factors, with the same spectral radius \rho. The strongly connected components of the directed graph of A_Q are the cycles. \rho = 0 with a non-empty support means A is nilpotent: the selected feedback is a cascade of directed cross-factor paths without a closed loop. Since A \ge 0, Perron–Frobenius gives a non-negative leading eigenvector, and its normalized entries say how much of the loop each factor carries.

Factor-level alignment. Feedback of the form \Theta_1 = aX^\top\Psi^{-1} enters each factor through the factor scores of the factors and is observationally equivalent, under the Gaussian working model, to a change of \Theta_2 and of the factor covariance \Phi; only the departure from that family is identified, through the uniquenesses \Psi. On the free set F of the exclusion restriction the family is a subspace of dimension at most Q(Q-1), and omega is the share of the squared norm of \Theta_1 that lies in it. omega has no natural zero: for an isotropic direction its expectation is omega0 = dim / n.free, so it should be read against omega0 and against a simulation in which the true feedback is indicator-level (there it averages about 1.3 * omega0).

Usage

nmf.ffb.diagnostics(object, which = c("selected", "full"))

Arguments

object	A fitted nmf.ffb object from nmf.ffb with method = "fiml".
which	Which feedback matrix to describe: "selected" (default, the BIC-selected support) or "full" (the unpenalized fit).

Value

A list with cycles (A.factor, rho, cycle, n.cycles, perron) and omega (omega, omega0, dim, n.free).

See Also

nmf.ffb, nmf.ffb.inference

Examples

set.seed(1)
Y2 <- matrix(runif(2 * 200), 2, 200)
X <- matrix(0.02, 6, 2); X[1:3, 1] <- 1; X[4:6, 2] <- 1
X <- sweep(X, 2, colSums(X), "/")
T1 <- matrix(0, 2, 6); T1[1, 5] <- 0.6; T1[2, 2] <- 0.5
L <- solve(diag(6) - X %*% T1)
Y1 <- pmax(L %*% (X %*% (matrix(c(1, 0.5, 0.3, 1), 2, 2) %*% Y2) +
                  matrix(rnorm(6 * 200, 0, 0.05), 6, 200)), 0)
fit <- nmf.ffb(Y1, Y2, Q = 2)
d <- nmf.ffb.diagnostics(fit)
d$cycles$rho          # spectral radius
d$cycles$cycle        # which factors lie on a cycle
c(d$omega$omega, d$omega$omega0)

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