Choose the number of factors for NMF-FFB by element-wise cross-validation

Step 1 of the NMF-FFB workflow: select \(Q\) by element-wise cross-validation of the feed-forward fit. Held-out entries of \(Y_1\) are predicted from the remaining ones, so the criterion measures how well a \(Q\)-factor non-negative basis describes the outcomes, which is what \(Q\) is for.

Usage

nmf.ffb.ecv(
  Y1,
  Y2,
  rank = NULL,
  X.init = "nndsvd",
  X.L2.ortho = 100,
  epsilon = 1e-06,
  maxit = 5000,
  ...
)

Arguments

Y1, Y2: Endogenous and exogenous matrices (variables in rows, units in columns).
rank: Candidate values of \(Q\). Defaults to seq_len(min(3, nrow(Y1))).
X.init: Stage-1 initialization, passed to nmfkc.
X.L2.ortho: Orthogonality penalty on the basis in stage 1. Keep the value that will be used in nmf.ffb: \(Q\) and the penalty are not separable.
epsilon, maxit: Convergence tolerance and iteration cap for stage 1.
...: Passed to nmfkc.ecv: nfolds, seed, and cores (or ncores) to evaluate the rank x fold grid in parallel, defaulting to getOption("mc.cores", 1L) as elsewhere in the package.

Value

The object returned by nmfkc.ecv: the held-out error by rank, with the selected rank.

Details

Feedback cannot be cross-validated, and that is not a limitation of the implementation. By the reduced-form equivalence, the equilibrium mapping \(M = (I - X\Theta_1)^{-1}X\Theta_2\) is reproduced exactly by a feed-forward model with the exogenous matrix \(L_Q\Theta_2\): the two models predict \(Y_1\) from \(Y_2\) identically, so no prediction criterion – column-wise CV, element-wise CV, or a test-sample error – can prefer one over the other. Feedback is identified by the conditional covariance, not by prediction, which is why it is addressed by nmf.ffb.test and not here.

Workflow


ecv <- nmf.ffb.ecv(Y1, Y2, rank = 1:5)   # 1. choose Q            <- this function
fit <- nmf.ffb(Y1, Y2, rank = ecv$rank)  # 2. estimate; BIC selects the support
tst <- nmf.ffb.test(fit, Y1, Y2)         # 3. test the feed-forward null
dgn <- nmf.ffb.diagnostics(fit)          # 4. cycles, spectral radius, identifiability
inf <- nmf.ffb.inference(fit, Y1, Y2)    # 5. intervals for the retained entries

Examples

set.seed(1)
Y <- t(as.matrix(iris[, 1:4]))
Y1 <- Y[1:2, ]; Y2 <- Y[3:4, ]
ecv <- nmf.ffb.ecv(Y1, Y2, rank = 1:2, nfolds = 3)
#> Performing Element-wise CV for Q = 1,2 (3-fold)...
ecv$rank
#> [1] 1 2