Linear Discriminant Analysis (LDA) with the Moore-Penrose Pseudo-Inverse

Description

Given a set of training data, this function builds the Linear Discriminant Analysis (LDA) classifier, where the distributions of each class are assumed to be multivariate normal and share a common covariance matrix. When the pooled sample covariance matrix is singular, the linear discriminant function is incalculable. A common method to overcome this issue is to replace the inverse of the pooled sample covariance matrix with the Moore-Penrose pseudo-inverse, which is unique and always exists. Note that when the pooled sample covariance matrix is nonsingular, it is equal to the pseudo-inverse.

The Linear Discriminant Analysis (LDA) classifier involves the assumption that the distributions of each class are assumed to be multivariate normal and share a common covariance matrix. When the pooled sample covariance matrix is singular, the linear discriminant function is incalculable. A common method to overcome this issue is to replace the inverse of the pooled sample covariance matrix with the Moore-Penrose pseudo-inverse, which is unique and always exists. Note that when the pooled sample covariance matrix is nonsingular, it is equal to the pseudo-inverse.

Usage

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lda_pseudo(x, ...)

## Default S3 method:
lda_pseudo(x, y, prior = NULL, tol = 1e-08, ...)

## S3 method for class 'formula'
lda_pseudo(formula, data, prior = NULL, tol = 1e-08, ...)

## S3 method for class 'lda_pseudo'
predict(object, newdata, ...)

Arguments

x

matrix containing the training data. The rows are the sample observations, and the columns are the features.

...

additional arguments

y

vector of class labels for each training observation

prior

vector with prior probabilities for each class. If NULL (default), then equal probabilities are used. See details.

tol

tolerance value below which eigenvalues are considered numerically equal to 0

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.

object

trained lda_pseudo object

newdata

matrix of observations to predict. Each row corresponds to a new observation.

Details

The matrix of training observations are given in x. The rows of x contain the sample observations, and the columns contain the features for each training observation.

The vector of class labels given in y are coerced to a factor. The length of y should match the number of rows in x.

An error is thrown if a given class has less than 2 observations because the variance for each feature within a class cannot be estimated with less than 2 observations.

The vector, prior, contains the a priori class membership for each class. If prior is NULL (default), the class membership probabilities are estimated as the sample proportion of observations belonging to each class. Otherwise, prior should be a vector with the same length as the number of classes in y. The prior probabilties should be nonnegative and sum to one.

Value

lda_pseudo object that contains the trained lda_pseudo classifier

list predicted class memberships of each row in newdata

Examples

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n <- nrow(iris)
train <- sample(seq_len(n), n / 2)
lda_pseudo_out <- lda_pseudo(Species ~ ., data = iris[train, ])
predicted <- predict(lda_pseudo_out, iris[-train, -5])$class

lda_pseudo_out2 <- lda_pseudo(x = iris[train, -5], y = iris[train, 5])
predicted2 <- predict(lda_pseudo_out2, iris[-train, -5])$class
all.equal(predicted, predicted2)

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