Quadratic Discriminant Analysis with gips covariance projection

Quadratic discriminant analysis (QDA) using covariance matrices projected via the gips framework to enforce permutation symmetry and improve numerical stability.

Usage

gipsqda(x, ...)

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

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

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

# S3 method for class 'matrix'
gipsqda(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: The prior probabilities of class membership. If omitted, training-set class proportions are used. Supplied values must sum to one and match the number of groups.
nu: Reserved for compatibility with related QDA interfaces; it is not currently used by the covariance projection.
MAP: Logical; if TRUE, 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 depending on the problem dimension is used.
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 "gipsqda" containing the following components:

prior: prior probabilities of the groups
counts: number of observations in each group
means: group means
scaling: 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 projection optimizer for the final class projection
call: the matched call
Formula fits additionally contain terms, contrasts, xlevels, and any recorded na.action.

Details

This function is a minor modification of qda, replacing the classical sample covariance estimators by projected covariance matrices obtained using the gips framework.

Quadratic discriminant analysis models each class with its own covariance matrix. In gipsqda, these covariance matrices are projected using the gips framework, which enforces permutation symmetry and mitigates singularity and overfitting in high-dimensional or small-sample settings.

Classification can be performed using plug-in, predictive, debiased, or leave-one-out cross-validation rules via predict.gipsqda.

Note

The function may be called with either a formula interface or with a matrix and grouping factor. Arguments subset and na.action, if used, must be named.

References

Chojecki, A., et al. (2025). Learning Permutation Symmetry of a Gaussian Vector with gips in R. Journal of Statistical Software, 112(7), 1–38. doi:10.18637/jss.v112.i07

Venables, W. N. and Ripley, B. D. (2002). Modern Applied Statistics with S. Fourth edition. Springer.

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 <- gipsqda(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 v c c c c c c c c c v v v v v v v v v v v v v v v c v v v v v v v v v
#> Levels: c s v