Generalized eigenproblem on a covariance matrix

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.

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.

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.

Examples