Cross-Validation for NMF-FFB

Performs K-fold cross-validation to evaluate the equilibrium mapping of the NMF-FFB model.

For each fold, nmf.ffb is fitted on the training samples, yielding an equilibrium mapping $\hat Y_1 = M_{\mathrm{model}} Y_2$. The held-out endogenous variables $Y_1$ are then predicted from $Y_2$ using this mapping, and the mean absolute error (MAE) over all entries in the test block is computed. The returned value is the average MAE across folds.

This implements the hyperparameter selection strategy described in the paper: hyperparameters are chosen by predictive cross-validation rather than direct inspection of the internal structural matrices.

method = "fiml". When method = "fiml" is passed (through ...), the column-wise scheme above is not run: the equilibrium prediction $M_{\mathrm{model}} Y_2$ is a reduced-form quantity, and the reduced form is the same with and without feedback, so predicting held-out columns cannot select $\Theta_1$ (that is what the BIC path and the bootstrap-calibrated LR test in nmf.ffb / nmf.ffb.inference are for). The only tunable quantity is the stage-1 rank, and the call is delegated to the element-wise CV of the feed-forward fit, nmfkc.ecv(Y = Y1, A = Y2, rank = rank, ...), whose object is returned ($objfunc holds the CV error per rank; rank may be a vector, default 1:min(3, P1)). C1.L1 and C2.L1 are ignored in that case.

Usage

nmf.ffb.cv(
  Y1,
  Y2,
  rank = NULL,
  X.init = "nndsvd",
  X.L2.ortho = 100,
  C1.L1 = 1,
  C2.L1 = 0.1,
  epsilon = 1e-06,
  maxit = 5000,
  ...
)

Arguments

Y1: A non-negative numeric matrix of endogenous variables with rows = variables (P1), columns = samples (N).
Y2: A non-negative numeric matrix of exogenous variables with rows = variables (P2), columns = samples (N). Must satisfy ncol(Y1) == ncol(Y2).
rank: Integer; rank (number of latent factors) passed to nmf.ffb. If NULL, nmf.ffb decides the effective rank (via ... or nrow(Y2)).
X.init: Initialization strategy for X, forwarded to nmf.ffb. One of "nndsvd" (default), "kmeans", "kmeansar", "runif", a numeric $P_1 \times Q$ matrix, or NULL (alias for "nndsvd"). See nmf.ffb for details.
X.L2.ortho: L2 orthogonality penalty for X.
C1.L1: L1 sparsity penalty for C1 ($\Theta_1$).
C2.L1: L1 sparsity penalty for C2 ($\Theta_2$).
epsilon: Convergence threshold for nmf.ffb.
maxit: Maximum number of iterations for nmf.ffb.
...: Additional arguments passed to nmf.ffb (except for rank, seed, div, shuffle, which are handled here). Also accepts: nfolds (number of folds, default 5; div also accepted), seed (master random seed, default NULL), shuffle (logical, default TRUE), cores (integer; number of parallel workers for the fold loop, default getOption("mc.cores", 1L)). Fold partitions and per-fold seeds are drawn before the loop and each nmf.ffb fit self-seeds, so results are identical to the sequential run for any cores.

Value

A numeric scalar: mean MAE across CV folds (the fits use method = "mu" unless method is given). With method = "fiml", the nmfkc.ecv object of the stage-1 rank sweep.

Examples

Y <- t(iris[, -5])
Y1 <- Y[1:2, ]
Y2 <- Y[3:4, ]
mae <- nmf.ffb.cv(Y1, Y2, rank = 2, maxit = 500, nfolds = 3)
#> Warning: maximum iterations (500) reached...
#> Warning: maximum iterations (500) reached...
#> Warning: maximum iterations (500) reached...
mae
#> [1] 1.656582
# \donttest{
# rank selection for the likelihood-based estimator
ecv <- nmf.ffb.cv(Y1, Y2, rank = 1:2, method = "fiml", nfolds = 3)
#> Performing Element-wise CV for Q = 1,2 (3-fold)...
ecv$objfunc
#>      Q=1      Q=2 
#> 2.915778 2.915701 
# }

Usage

Arguments

Value

See also

Examples