Regularised / Generalised Partial Least Squares (RPLS / GPLS)

Implements the algorithm of Allen et al. (2013) for supervised dimension-reduction with optional sparsity ($\ell_1$) or ridge ($\ell_2$) penalties and the generalised extension that operates in a user-supplied quadratic form $Q$.

Usage

rpls(
  X,
  Y,
  K = 2,
  lambda = 0.1,
  penalty = c("l1", "ridge"),
  Q = NULL,
  nonneg = FALSE,
  preproc_x = multivarious::pass(),
  preproc_y = multivarious::pass(),
  tol = 1e-06,
  maxiter = 200,
  verbose = FALSE,
  ...
)

Arguments

X: Numeric matrix $(n \times p)$ — predictors.
Y: Numeric matrix $(n \times q)$ — responses.
K: Integer, number of latent factors to extract. Default 2.
lambda: Scalar or length-K numeric vector of penalties.
penalty: Either "l1" (lasso) or "ridge".
Q: Optional positive-(semi)definite $p \times p$ matrix inducing generalised PLS. NULL means identity.
nonneg: Logical, force non-negative loadings when penalty = "l1". Note: This option is currently ignored when penalty = "ridge".
preproc_x, preproc_y: Optional multivarious preprocessing objects (see fit_transform). By default they pass the data through unchanged using pass().
tol: Relative tolerance for the inner iterations convergence check. Default 1e-6.
maxiter: Maximum number of inner iterations per component. Default 200.
verbose: Logical; print progress messages during component extraction. Default FALSE.
...: Further arguments (e.g., custom stopping criteria if implemented) are stored in the returned object (they are not used by fit_rpls).

Value

An object of class c("rpls","cross_projector","projector") with at least the elements

vx: $p \times K$ matrix of X-loadings.
vy: $q \times K$ matrix of Y-loadings.
ncomp: Number of components actually extracted (may be < K).
penalty: Penalty type used ("l1" or "ridge").
lambda: The lambda value(s) used.
tol: The convergence tolerance used.
maxiter: The maximum number of inner iterations used per component.
nonneg: Logical flag for the non-negativity constraint on l1.
Q_used: Logical; TRUE if a custom Q metric was supplied to induce generalised RPLS, FALSE for standard (identity-metric) RPLS. Note the field is named Q_used, not Q: the metric matrix itself is not retained on the returned object.
verbose: The verbose flag used.
preproc_x, preproc_y: Pre-processing transforms used.

rpls objects store no row (X-)scores: the factors $z_k$ are computed internally during fitting to drive deflation and are then discarded, so use multivarious::project() on X to obtain scores for any rows of interest.

The object supports predict(), project(), transfer(), coef() and other multivarious generics.

Details

Unlike genpls(), which handles separate row and column metrics (Mx, Ax, My, Ay) with a Gram–Schmidt orthogonalisation step, rpls() uses a single metric Q and the simpler penalised updates of Allen et al.

Method

The routine follows Algorithm 1 of Allen et al. (2013, Stat. Anal. Data Min., 6 : 302–314) — see the paper for details. Briefly, with $C = X^\top Y$ the cross-product of the (preprocessed) blocks, each component maximises $$\max_{u,v}\; v^\top Q C u - \lambda \, P(v)$$ with $Q = I_p$ for standard RPLS. The alternating updates are: $u \leftarrow C^\top Q v / \|C^\top Q v\|_2$, then a penalised (possibly non-negative) regression for $v$, normalised in the $Q$-norm.

References

Allen, G. I., Peterson, C., Vannucci, M., & Maletić-Savatić, M. (2013). Regularized Partial Least Squares with an Application to NMR Spectroscopy. Statistical Analysis and Data Mining, 6(4), 302-314. DOI:10.1002/sam.11169.

Examples

# Generate sample data
set.seed(123)
n <- 50
p <- 20
q <- 10
X <- matrix(rnorm(n * p), n, p)
Y <- X[, 1:5] %*% matrix(rnorm(5 * q), 5, q) + matrix(rnorm(n * q), n, q)

# Fit regularized PLS with L1 penalty
fit_l1 <- rpls(X, Y, K = 3, lambda = 0.1, penalty = "l1")
print(fit_l1)
#> cross projector:  rpls cross_projector projector 
#> input dim (X):  20 
#> output dim (X):  3 
#> input dim (Y):  10 
#> output dim (Y):  3 

# Fit regularized PLS with ridge penalty
fit_ridge <- rpls(X, Y, K = 3, lambda = 0.1, penalty = "ridge")
print(fit_ridge)
#> cross projector:  rpls cross_projector projector 
#> input dim (X):  20 
#> output dim (X):  3 
#> input dim (Y):  10 
#> output dim (Y):  3