Quadratic Discriminant Analysis with multiple gips-projected covariances

Quadratic Discriminant Analysis (QDA) in which each class covariance matrix participates in one joint gips projection, allowing the covariance matrices to share an estimated permutation symmetry.

Usage

gipsmultqda(x, ...)

# S3 method for class 'formula'
gipsmultqda(formula, data, ..., subset, na.action)

# Default S3 method
gipsmultqda(x, grouping, prior = proportions,
  nu = 5, MAP = TRUE, optimizer = NULL, max_iter = NULL, ...)

# S3 method for class 'data.frame'
gipsmultqda(x, ...)

# S3 method for class 'matrix'
gipsmultqda(x, grouping, ..., subset, na.action)

Arguments

x: (required if no formula is given as the principal argument) a matrix or data frame containing the explanatory variables.
...: Arguments passed to or from other methods.
formula: A formula of the form groups ~ x1 + x2 + .... The response is the grouping factor and the right-hand side specifies the (non-factor) discriminators.
data: An optional data frame, list or environment from which variables specified in formula are preferentially taken.
grouping: (required if no formula is given) a factor specifying the class for each observation.
prior: Prior probabilities of class membership. If omitted, training-set class proportions are used. Supplied values must sum to one.
nu: Reserved for compatibility with related QDA interfaces; it is not currently used by the covariance projection.
MAP: Logical; if TRUE, a maximum a posteriori covariance projection is used. If FALSE, projections are averaged using retained posterior permutation probabilities.
optimizer: Character string specifying the optimization method used for covariance projection. If NULL, a default choice is made based on the problem dimension.
max_iter: Maximum number of iterations for stochastic optimizers.
subset: An index vector specifying the cases to be used in the training sample. (NOTE: must be named.)
na.action: A function specifying the action to be taken if NAs are found.

Value

An object of class "gipsmultqda" containing:

prior: prior probabilities of the groups
counts: number of observations per group
means: group means
scaling: array of group-specific scaling matrices derived from the projected covariance matrices
ldet: log-determinants of the projected covariance matrices
lev: class labels
N: total number of observations
optimization_info: information returned by the covariance optimizer for the joint projection
call: the matched call
Formula fits additionally contain terms, contrasts, xlevels, and any recorded na.action.

Details

This function is a modification of qda in which the class-specific covariance matrices are jointly projected to improve numerical stability and exploit shared symmetry assumptions.

In contrast to classical QDA, which estimates each class covariance matrix independently, gipsmultqda performs a joint projection of all class covariance matrices using gips. This allows the incorporation of shared permutation symmetries and can improve classification performance in high-dimensional or small-sample regimes.

Several classification rules are available via predict.gipsmultqda, including plug-in, predictive, debiased, and leave-one-out cross-validation.

Note

This function is not a drop-in replacement for qda. The covariance estimation, returned object, and classification rules differ substantially.

The theoretical background and details of covariance projection are documented by gips and Chojecki et al. (2025).

Examples

tr <- sample(1:50, 25)
train <- rbind(iris3[tr, , 1], iris3[tr, , 2], iris3[tr, , 3])
test <- rbind(iris3[-tr, , 1], iris3[-tr, , 2], iris3[-tr, , 3])
cl <- factor(c(rep("s", 25), rep("c", 25), rep("v", 25)))
z <- gipsmultqda(train, cl)
predict(z, test)$class
#>  [1] s s s s s s s s s s s s s s s s s s s s s s s s s c c c c c c c c c c c c c
#> [39] c c c c c c c c c c c c v v v v v v v v v v v c v v v v v v v c v v v v v
#> Levels: c s v