#' Linear Discriminant Analysis using the Thomaz-Kitani-Gillies Covariance
#' Matrix Estimator
#'
#' 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. This function replaces the pooled sample covariance matrix
#' with a regularized estimator from Thomaz et al. (2006), where the smallest
#' eigenvalues are replaced with the average eigenvalue. Specifically, small
#' eigenvalues here means that the eigenvalues are less than the average
#' eigenvalue.
#'
#' The matrix of training observations are given in \code{x}. The rows of \code{x}
#' contain the sample observations, and the columns contain the features for each
#' training observation.
#'
#' The vector of class labels given in \code{y} are coerced to a \code{factor}.
#' The length of \code{y} should match the number of rows in \code{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, \code{prior}, contains the \emph{a priori} class membership for
#' each class. If \code{prior} is NULL (default), the class membership
#' probabilities are estimated as the sample proportion of observations belonging
#' to each class. Otherwise, \code{prior} should be a vector with the same length
#' as the number of classes in \code{y}. The \code{prior} probabilities should be
#' nonnegative and sum to one.
#'
#' @export
#'
#' @param x matrix containing the training data. The rows are the sample
#' observations, and the columns are the features.
#' @param y vector of class labels for each training observation
#' @param prior vector with prior probabilities for each class. If NULL
#' (default), then equal probabilities are used. See details.
#' @return \code{lda_thomaz} object that contains the trained classifier
#' @examples
#' n <- nrow(iris)
#' train <- sample(seq_len(n), n / 2)
#' lda_thomaz_out <- lda_thomaz(Species ~ ., data = iris[train, ])
#' predicted <- predict(lda_thomaz_out, iris[-train, -5])$class
#'
#' lda_thomaz_out2 <- lda_thomaz(x = iris[train, -5], y = iris[train, 5])
#' predicted2 <- predict(lda_thomaz_out2, iris[-train, -5])$class
#' all.equal(predicted, predicted2)
#' @references Thomaz, C. E., Kitani, E. C., and Gillies, D. F. (2006). "A
#' maximum uncertainty LDA-based approach for limited sample size problems with
#' application to face recognition," J. Braz. Comp. Soc., 12, 2, 7-18.
lda_thomaz <- function(x, ...) {
UseMethod("lda_thomaz")
}
#' @rdname lda_thomaz
#' @export
lda_thomaz.default <- function(x, y, prior = NULL, ...) {
x <- as.matrix(x)
y <- as.factor(y)
obj <- regdiscrim_estimates(x = x, y = y, prior = prior, cov = TRUE)
# Computes eigenvalue decomposition of pooled sample covariance matrix
# Then regularizes the estimator based on Thomaz et al.'s (2006) method
cov_eigen <- eigen(obj$cov_pool, symmetric = TRUE)
evals <- cov_eigen$values
mean_eval <- mean(evals)
evals[evals < mean_eval] <- mean_eval
if (obj$p > 1) {
obj$cov_pool <- with(cov_eigen,
tcrossprod(vectors %*% diag(evals), vectors))
# Computes the inverse of the resulting covariance matrix estimator
obj$cov_inv <- with(cov_eigen,
tcrossprod(vectors %*% diag(1 / evals), vectors))
} else {
obj$cov_pool <- with(cov_eigen,
tcrossprod(vectors %*% as.matrix(evals), vectors))
obj$cov_inv <- with(cov_eigen,
tcrossprod(vectors %*% as.matrix(1 / evals), vectors))
}
# Creates an object of type 'lda_thomaz' and adds the 'match.call' to the object
obj$call <- match.call()
class(obj) <- "lda_thomaz"
obj
}
#' @param formula A formula of the form \code{groups ~ x1 + x2 + ...} That is,
#' the response is the grouping factor and the right hand side specifies the
#' (non-factor) discriminators.
#' @param data data frame from which variables specified in \code{formula} are
#' preferentially to be taken.
#' @rdname lda_thomaz
#' @importFrom stats model.frame model.matrix model.response
#' @export
lda_thomaz.formula <- function(formula, data, prior = NULL, ...) {
# The formula interface includes an intercept. If the user includes the
# intercept in the model, it should be removed. Otherwise, errors and doom
# happen.
# To remove the intercept, we update the formula, like so:
# (NOTE: The terms must be collected in case the dot (.) notation is used)
formula <- no_intercept(formula, data)
mf <- model.frame(formula = formula, data = data)
x <- model.matrix(attr(mf, "terms"), data = mf)
y <- model.response(mf)
est <- lda_thomaz.default(x = x, y = y, prior = prior)
est$call <- match.call()
est$formula <- formula
est
}
#' Outputs the summary for a lda_thomaz classifier object.
#'
#' Summarizes the trained lda_thomaz classifier in a nice manner.
#'
#' @param x object to print
#' @param ... unused
#' @export
print.lda_thomaz <- function(x, ...) {
cat("Call:\n")
print(x$call)
cat("Sample Size:\n")
print(x$N)
cat("Number of Features:\n")
print(x$p)
cat("Classes:\n")
print(x$groups)
cat("Prior Probabilities:\n")
print(sapply(x$est, function(z) z$prior))
}
#' Predicts of class membership of a matrix of new observations using Linear
#' Discriminant Analysis (LDA) using the Schafer-Strimmer Covariance Matrix
#' Estimator
#'
#' 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. This function replaces the pooled sample covariance matrix
#' with a regularized estimator from Thomaz et al. (2006), where the smallest
#' eigenvalues are replaced with the average eigenvalue. Specifically, small
#' eigenvalues here means that the eigenvalues are less than the average
#' eigenvalue.
#'
#' @rdname lda_thomaz
#' @export
#'
#' @references Thomaz, C. E., Kitani, E. C., and Gillies, D. F. (2006). "A
#' maximum uncertainty LDA-based approach for limited sample size problems with
#' application to face recognition," J. Braz. Comp. Soc., 12, 2, 7-18.
#' @param object trained lda_thomaz object
#' @param newdata matrix of observations to predict. Each row corresponds to a
#' new observation.
#' @param ... additional arguments
#' @return list predicted class memberships of each row in newdata
predict.lda_thomaz <- function(object, newdata, ...) {
if (!inherits(object, "lda_thomaz")) {
stop("object not of class 'lda_thomaz'")
}
newdata <- as.matrix(newdata)
# Calculates the discriminant scores for each test observation
scores <- apply(newdata, 1, function(obs) {
sapply(object$est, function(class_est) {
with(class_est, quadform(object$cov_inv, obs - xbar) + log(prior))
})
})
if (is.vector(scores)) {
min_scores <- which.min(scores)
} else {
min_scores <- apply(scores, 2, which.min)
}
# Posterior probabilities via Bayes Theorem
means <- lapply(object$est, "[[", "xbar")
covs <- replicate(n=object$num_groups, object$cov_pool, simplify=FALSE)
priors <- lapply(object$est, "[[", "prior")
posterior <- posterior_probs(x=newdata,
means=means,
covs=covs,
priors=priors)
class <- factor(object$groups[min_scores], levels = object$groups)
list(class = class, scores = scores, posterior = posterior)
}
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