Structural Diagnostics of a Fitted NMF-FFB Feedback Matrix

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

# \donttest{
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
#> [1] 0.17824
d$cycles$cycle        # which factors lie on a cycle
#> Factor1 Factor2 
#>       1       1 
c(d$omega$omega, d$omega$omega0)
#> [1] 0.2960730 0.3333333
# }