Choose the Reduced Dimension of a Quantile Tensor Regression

Runs pqtr() over a grid of candidate reduced dimensions, scores each one by nfolds-fold cross-validated check loss, and returns the model refitted on all subjects at the best value. Use this when the eigenvalue-ratio rule built into pqtr() is not obviously right, for instance when the coefficient array is not close to rank one.

Usage

pqtr_cv(
  X,
  y,
  tau = 0.5,
  Z = NULL,
  u_grid = NULL,
  nfolds = 5L,
  ridge = 1e-08,
  maxit = 200L,
  tol = 1e-08
)

Arguments

X, y, tau, Z, ridge, maxit, tol

As in pqtr().

u_grid

Optional list of candidate dimension vectors, each of length m (a plain integer vector is read as one candidate per element, used in every mode). Leave NULL (default) for the automatic equal-dimension grid.

nfolds

Number of cross-validation folds (default 5).

Value

The pqtr fit at the selected dimension, with three extra elements: u_grid (the candidates tried), cv (a data frame of u and mean out-of-fold check loss), and u_min (the chosen dimension vector).

Details

The default grid keeps the same number of directions in every mode, u = rep(d, m) for d from 1 up to the largest value for which the reduced quantile regression stays estimable, capped at 12 as in the reference implementation. That is a deliberate simplification: enumerating every combination of per-mode dimensions costs d^m fits. Pass u_grid explicitly, for instance u_grid = apply(expand.grid(1:2, 1:3), 1, as.integer, simplify = FALSE), when the modes should be allowed to differ.

See also

pqtr()

Examples

set.seed(2)
B <- outer(c(2, rep(0, 5)), c(2, rep(0, 4)))
X <- lapply(1:120, function(i) matrix(rnorm(30), 6, 5))
y <- vapply(X, function(xi) sum(B * xi), numeric(1)) + rnorm(120)

fit <- pqtr_cv(X, y, tau = 0.5, nfolds = 3)
fit$u_min
#> [1] 5 5
fit$cv
#>       u     loss
#> 1 1 x 1 27.13937
#> 2 2 x 2 27.28447
#> 3 3 x 3 25.10003
#> 4 4 x 4 21.85729
#> 5 5 x 5 20.12311