geigen_cov: Generalized eigenproblem on a covariance matrix

In genpca: Generalized Principal Component Analysis

Generalized eigenproblem on a covariance matrix

Description

Maximises v'Cv subject to v'Rv = 1 and the additional constraint that v lies in the retained range of R. Successive components are R-orthogonal. If P is the orthogonal projector onto that range, the returned vectors satisfy P C v = \lambda R v and V'RV = I. For a full-rank R this is the usual equation C v = \lambda R v. It also holds for a singular R when C maps its retained range into itself. Otherwise the component of C v outside the retained range need not vanish.

Usage

geigen_cov(
  C,
  R = NULL,
  ncomp = NULL,
  constraints_remedy = c("error", "ridge", "clip", "identity"),
  rank_rtol = 1e-06,
  metric_rtol = .metric_rtol_default(),
  verbose = FALSE
)

Arguments

C	A p x p symmetric matrix. Asymmetry beyond roundoff is an error; an indefinite C is allowed (the problem is still defined) and only produces a warning.
R	Variable-side constraint/metric. Can be: NULL: identity matrix (standard PCA on C) a numeric vector of length p: diagonal weights (must be non-negative) a p x p symmetric PSD matrix: general metric/smoothing/structure penalties
ncomp	Number of components to return. Default is all positive eigenvalues.
constraints_remedy	What to do with an indefinite R: "error" (default), "ridge", "clip" or "identity"; a repair emits a genpca_metric_repaired warning. See genpca.
rank_rtol	Relative cutoff for component acceptance on the singular-value scale (components with d_j <= rank_rtol * d_1 are dropped). Default 1e-6.
metric_rtol	Relative tolerance for validating C and R and for detecting the numerical null space in an eigendecomposition of a general R. Every strictly positive diagonal weight is retained without a rank approximation. Default sqrt(.Machine$double.eps).
verbose	Logical. If TRUE, print progress messages. Default FALSE.

Details

This is a different estimator from the GMD of genpca_cov, which uses R^{1/2} C R^{1/2}. If C and R commute, their common eigenvectors can be ordered differently: the GMD weights variances by metric eigenvalues, whereas this estimator divides by them. With R = c * I, the directions and their ordering agree, but the eigenvalue scales differ unless c = 1.

C is validated for symmetry but may be indefinite (the generalized eigenproblem is still defined); a warning is issued when its minimum eigenvalue is below -metric_rtol * scale. R must be positive semi-definite; an indefinite R is subject to constraints_remedy.

Value

A plain list with the same components as genpca_cov (v, d, lambda, k, propv, cumv, R_rank) and method = "geigen". propv is relative to \mathrm{tr}(R^{-1/2} C R^{-1/2}) on the range of R.

See Also

genpca_cov

Examples

C <- cov(scale(iris[,1:4], center=TRUE, scale=FALSE))
w <- c(1, 1, 0.5, 2)
fit_gmd <- genpca_cov(C, R = w, ncomp = 2)
fit_geigen <- geigen_cov(C, R = w, ncomp = 2)
# different estimators: the singular values generally differ
rbind(gmd = fit_gmd$d, geigen = fit_geigen$d)

# With singular R, the equation is projected onto its retained range
C <- matrix(c(2, 1, 1, 2), 2)
R <- diag(c(1, 0))
fit <- geigen_cov(C, R, ncomp = 1)
P <- diag(c(1, 0))
P %*% C %*% fit$v - (R %*% fit$v) * fit$lambda

genpca documentation built on Sept. 17, 2026, 1:09 a.m.