Quadratic Discriminant Analysis with multiple gips-projected covariances

Quadratic Discriminant Analysis (QDA) in which each class covariance matrix is projected using the gipsmult framework, allowing for structured permutation symmetry across multiple covariance matrices.

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: A factor specifying the class for each observation.
prior: Prior probabilities of class membership. Must sum to one.
nu: Degrees of freedom parameter used internally during covariance projection.
MAP: Logical; if TRUE, a maximum a posteriori covariance projection is used.
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 projection optimizer
call: the matched call

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 the gipsmult framework. 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 the covariance projection are documented in the gipsmult package.

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 v v v v v v v v v v v v v v
#> Levels: c s v