Linear Discriminant Analysis

Linear discriminant analysis.

lda(x, ...)

# S3 method for formula
lda(formula, data, ..., subset, na.action)

# S3 method for default
lda(x, grouping, prior = proportions, tol = 1.0e-4,
    method, CV = FALSE, nu, ...)

# S3 method for data.frame
lda(x, ...)

# S3 method for matrix
lda(x, grouping, ..., subset, na.action)

Arguments

formula	A formula of the form `groups ~ x1 + x2 + ...` That is, the response is the grouping factor and the right hand side specifies the (non-factor) discriminators.
data	Data frame from which variables specified in `formula` are preferentially to be taken.
x	(required if no formula is given as the principal argument.) a matrix or data frame or Matrix containing the explanatory variables.
grouping	(required if no formula principal argument is given.) a factor specifying the class for each observation.
prior	the prior probabilities of class membership. If unspecified, the class proportions for the training set are used. If present, the probabilities should be specified in the order of the factor levels.
tol	A tolerance to decide if a matrix is singular; it will reject variables and linear combinations of unit-variance variables whose variance is less than `tol^2`.
subset	An index vector specifying the cases to be used in the training sample. (NOTE: If given, this argument must be named.)
na.action	A function to specify the action to be taken if `NA`s are found. The default action is for the procedure to fail. An alternative is `na.omit`, which leads to rejection of cases with missing values on any required variable. (NOTE: If given, this argument must be named.)
method	`"moment"` for standard estimators of the mean and variance, `"mle"` for MLEs, `"mve"` to use `cov.mve`, or `"t"` for robust estimates based on a \(t\) distribution.
CV	If true, returns results (classes and posterior probabilities) for leave-one-out cross-validation. Note that if the prior is estimated, the proportions in the whole dataset are used.
nu	degrees of freedom for `method = "t"`.
...	arguments passed to or from other methods.

Value

If CV = TRUE the return value is a list with components class, the MAP classification (a factor), and posterior, posterior probabilities for the classes.

Otherwise it is an object of class "lda" containing the following components:

prior

the prior probabilities used.

means

the group means.

scaling

a matrix which transforms observations to discriminant functions, normalized so that within groups covariance matrix is spherical.

svd

the singular values, which give the ratio of the between- and within-group standard deviations on the linear discriminant variables. Their squares are the canonical F-statistics.

The number of observations used.

call

The (matched) function call.

Details

The function tries hard to detect if the within-class covariance matrix is singular. If any variable has within-group variance less than tol^2 it will stop and report the variable as constant. This could result from poor scaling of the problem, but is more likely to result from constant variables.

Specifying the prior will affect the classification unless over-ridden in predict.lda. Unlike in most statistical packages, it will also affect the rotation of the linear discriminants within their space, as a weighted between-groups covariance matrix is used. Thus the first few linear discriminants emphasize the differences between groups with the weights given by the prior, which may differ from their prevalence in the dataset.

If one or more groups is missing in the supplied data, they are dropped with a warning, but the classifications produced are with respect to the original set of levels.

Note

This function may be called giving either a formula and optional data frame, or a matrix and grouping factor as the first two arguments. All other arguments are optional, but subset= and na.action=, if required, must be fully named.

If a formula is given as the principal argument the object may be modified using update() in the usual way.

References

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

Ripley, B. D. (1996) Pattern Recognition and Neural Networks. Cambridge University Press.

Examples

Iris <- data.frame(rbind(iris3[,,1], iris3[,,2], iris3[,,3]),
                   Sp = rep(c("s","c","v"), rep(50,3)))
train <- sample(1:150, 75)
table(Iris$Sp[train])
#> 
#>  c  s  v 
#> 22 31 22 
## your answer may differ
##  c  s  v
## 22 23 30
z <- lda(Sp ~ ., Iris, prior = c(1,1,1)/3, subset = train)
predict(z, Iris[-train, ])$class
#>  [1] 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 v c c c c v c
#> [39] c c c c c c c c c v v v v v v v v v v c v v v v v v v c v v v v v v v v v
#> Levels: c s v
##  [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 s s c c c
## [31] c c c c c c c v c c c c v c c c c c c c c c c c c v v v v v
## [61] v v v v v v v v v v v v v v v
(z1 <- update(z, . ~ . - Petal.W.))
#> Call:
#> lda(Sp ~ Sepal.L. + Sepal.W. + Petal.L., data = Iris, prior = c(1, 
#>     1, 1)/3, subset = train)
#> 
#> Prior probabilities of groups:
#>         c         s         v 
#> 0.3333333 0.3333333 0.3333333 
#> 
#> Group means:
#>   Sepal.L. Sepal.W. Petal.L.
#> c 5.968182 2.763636 4.254545
#> s 4.974194 3.448387 1.458065
#> v 6.668182 2.950000 5.600000
#> 
#> Coefficients of linear discriminants:
#>                 LD1       LD2
#> Sepal.L.  1.7449211 -1.254634
#> Sepal.W.  0.0379671  3.754167
#> Petal.L. -3.7525160  1.032709
#> 
#> Proportion of trace:
#>    LD1    LD2 
#> 0.9905 0.0095