Stabilised Linear Discriminant Analysis

Linear discriminant analysis based on left-spherically distributed linear scores.

Usage

slda(y, ...)
# S3 method for class 'formula'
slda(formula, data, subset, na.action=na.rpart, ...)
# S3 method for class 'factor'
slda(y, X, q=NULL, ...)

Arguments

y: the response variable: a factor vector of class labels.
X: a data frame of predictor variables.
q: the number of positive eigenvalues the scores are derived from, see below.
formula: a formula of the form lhs ~ rhs where lhs is the response variable and rhs a set of predictors.
data: optional data frame containing the variables in the model formula.
subset: optional vector specifying a subset of observations to be used.
na.action: function which indicates what should happen when the data contain NAs. Defaults to na.rpart.
...: additional parameters passed to lda.

Details

This function implements the LDA for $q$-dimensional linear scores of the original $p$ predictors derived from the $PC_q$ rule by Laeuter et al. (1998). Based on the product sum matrix $$W = (X - \bar{X})^\top(X - \bar{X})$$ the eigenvalue problem $WD = diag(W)DL$ is solved. The first $q$ columns $D_q$ of $D$ are used as a weight matrix for the original $p$ predictors: $XD_q$. By default, $q$ is the number of eigenvalues greater one. The $q$-dimensional linear scores are left-spherically distributed and are used as predictors for a classical LDA.

This form of reduction of the dimensionality was developed for discriminant analysis problems by Laeuter (1992) and was used for multivariate tests by Laeuter et al. (1998), Kropf (2000) gives an overview. For details on left-spherically distributions see Fang and Zhang (1990).

Value

An object of class slda, a list with components

scores: the weight matrix.
mylda: an object of class lda.

References

Fang Kai-Tai and Zhang Yao-Ting (1990), Generalized Multivariate Analysis, Springer, Berlin.

Siegfried Kropf (2000), Hochdimensionale multivariate Verfahren in der medizinischen Statistik, Shaker Verlag, Aachen (in german).

Juergen Laeuter (1992), Stabile multivariate Verfahren, Akademie Verlag, Berlin (in german).

Juergen Laeuter, Ekkehard Glimm and Siegfried Kropf (1998), Multivariate Tests Based on Left-Spherically Distributed Linear Scores. The Annals of Statistics, 26(5) 1972–1988.

Examples


library("mlbench")
library("MASS")
learn <- as.data.frame(mlbench.twonorm(100))
test <- as.data.frame(mlbench.twonorm(1000))

mlda <- lda(classes ~ ., data=learn)
mslda <- slda(classes ~ ., data=learn)

print(mean(predict(mlda, newdata=test)$class != test$classes))
#> [1] 0.035
print(mean(predict(mslda, newdata=test)$class != test$classes))
#> [1] 0.025