Nothing
R/mixtureData.R
In locClassData: Collection of Artificial Classification Problems
Defines functions
mixtureData
mixtureDataHelper
mixtureLabels
mixturePosterior
mixturePosteriorHelper
mixtureBayesClass
Documented in
mixtureBayesClass
mixtureData
mixtureLabels
mixturePosterior
# Copyright (C) 2011 Julia Schiffner
#
# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2 or 3 of the License
# (at your option).
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# A copy of the GNU General Public License is available at
# http://www.r-project.org/Licenses/
#
#' Generation of Gaussian mixture data for classification.
#'
# details
#'
#' @title Generation of Gaussian Mixture Data for Classification
#'
#' @param n Number of observations.
#' @param prior Vector of class prior probabilities.
#' @param lambda The conditional probabilities for the mixture components given the class. Either a vector (if the same number \eqn{m}
#' of mixture components is desired for each class and the conditional probabilities for each class should be equal) or a list as long
#' as the number of classes containing one vector of probabilities for every class. The length of the \eqn{k}-th element is the desired
#' number of mixture components for the \eqn{k}-th class.
#' @param mu The centers of the mixture components. A list containing one \eqn{m_k \times d}{m_k x d} matrix of centers for
#' each class where \eqn{d} is the desired dimensionality of the data set.
#' @param sigma The covariance matrices of the mixture components. Either one single matrix that is used for each mixture
#' component or a list as long as the number of classes. List elements can be matrices (in case that for all mixture components forming
#' one class the same covariance matrix shall be used) or lists of matrices as long as the number of mixture components in the corresponding
#' class.
#' @param data A \code{data.frame}.
#'
#' @return
#' \code{mixtureData} returns an object of class \code{"locClass"}, a list with components:
#' \item{x}{(A matrix.) The explanatory variables.}
#' \item{y}{(A factor.) The class labels.}
#'
#' @examples
#' ## Simplest case:
#' # lambda vector, sigma matrix
#' # Generate a training and a test set
#' mu <- list(matrix(c(-1,-1),1), matrix(rep(c(1,4.5,12),2),3))
#' train <- mixtureData(n = 1000, prior = rep(0.5,2), lambda = list(1, rep(1/3,3)), mu = mu, sigma = 3*diag(2))
#' test <- mixtureData(n = 1000, prior = rep(0.5,2), lambda = list(1, rep(1/3,3)), mu = mu, sigma = 3*diag(2))
#'
#' # Generate a grid of points
#' x.1 <- x.2 <- seq(-7,15,0.1)
#' grid <- expand.grid(x.1 = x.1, x.2 = x.2)
#'
#' # Calculate the posterior probablities for all grid points
#' gridPosterior <- mixturePosterior(grid, prior = rep(0.5,2), lambda = list(1, rep(1/3,3)), mu = mu, sigma = 3*diag(2))
#'
#' # Draw contour lines of posterior probabilities and plot training observations
#' contour(x.1, x.2, matrix(gridPosterior[,1], length(x.1)), col = "gray")
#' points(train$x, col = train$y)
#'
#' # Calculate Bayes error
#' ybayes <- mixtureBayesClass(test$x, prior = rep(0.5,2), lambda = list(1, rep(1/3,3)), mu = mu, sigma = 3*diag(2))
#' mean(ybayes != test$y)
#'
#' if (require(MASS)) {
#'
#' # Fit an LDA model and calculate misclassification rate on the test data set
#' tr <- lda(y ~ ., data = as.data.frame(train))
#' pred <- predict(tr, as.data.frame(test))
#' mean(pred$class != test$y)
#'
#' # Draw decision boundary
#' gridPred <- predict(tr, grid)
#' contour(x.1, x.2, matrix(gridPred$posterior[,1], length(x.1)), col = "red", levels = 0.5, add = TRUE)
#'
#' }
#'
#' ## lambda list, sigma list of lists of matrices
#' mu <- list()
#' mu[[1]] <- matrix(c(1,5,1,5),2)
#' mu[[2]] <- matrix(c(8,11,8,11),2)
#' lambda <- list(c(0.5, 0.5), c(0.1, 0.9))
#' sigma <- list()
#' sigma[[1]] <- diag(2)
#' sigma[[2]] <- list(diag(2), 3*diag(2))
#' data <- mixtureData(n = 100, prior = c(0.3, 0.7), lambda, mu, sigma)
#' plot(data$x, col = data$y)
#'
#' @aliases mixtureData mixtureLabels mixturePosterior mixtureBayesClass
#'
#' @rdname mixtureData
#'
#' @import mvtnorm
#'
#' @export
mixtureData <- function(n, prior, lambda, mu, sigma) {
nclass <- length(prior)
y <- factor(sample(1:nclass, size = n, replace = TRUE, prob = prior), levels = 1:nclass) # class labels
ncl <- as.vector(table(y)) # # observations per class
if (length(mu) != nclass)
stop("'length(mu)' is not 'nclass'")
d <- sapply(mu, ncol)
if (any(d[1] != d))
stop("numbers of columns for elements in 'mu' differs")
d <- d[1] # dimensionality
ncomp <- sapply(mu, nrow) # # of mixture components per class
x <- matrix(0, n, d)
if (is.list(sigma)) { # sigma list or list of lists
if (length(sigma) != nclass)
stop("'length(sigma)' and 'nclass' differ")
if (is.list(lambda)) { # lambda list
if (length(lambda) != nclass)
stop("'length(lambda)' is not 'nclass'")
if (any(sapply(lambda, length) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
for (k in 1:nclass)
x[y == k,] <- mixtureDataHelper(ncl[k], lambda[[k]], mu[[k]], sigma[[k]])
} else if (is.vector(lambda)) {
if (any(length(lambda) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
for (k in 1:nclass)
x[y == k,] <- mixtureDataHelper(ncl[k], lambda, mu[[k]], sigma[[k]])
} else {
stop("'lambda' is neither list nor vector")
}
} else if (is.matrix(sigma)) { # sigma matrix
if (is.list(lambda)) {
if (length(lambda) != nclass)
stop("'length(lambda)' is not 'nclass'")
if (any(sapply(lambda, length) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
for (k in 1:nclass)
x[y == k,] <- mixtureDataHelper(ncl[k], lambda[[k]], mu[[k]], sigma)
} else if (is.vector(lambda)) {
if (any(length(lambda) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
for (k in 1:nclass)
x[y == k,] <- mixtureDataHelper(ncl[k], lambda, mu[[k]], sigma)
} else {
stop("'lambda' is neither list nor vector")
}
}
data <- list(x = x, y = y)
class(data) <- c("locClass.mixtureData", "locClass")
attr(data, "prior") <- prior
attr(data, "lambda") <- lambda
attr(data, "mu") <- mu
attr(data, "sigma") <- sigma
return(data)
}
#' @noRd
mixtureDataHelper <- function(n_k, lambda_k, mu_k, sigma_k) {
ncomp_k <- length(lambda_k)
comp_k <- sample(1:ncomp_k, size = n_k, replace = TRUE, prob = lambda_k)
tab <- as.vector(table(comp_k))
d <- ncol(mu_k)
x_k <- matrix(0, n_k, d)
if (is.list(sigma_k)) {
if (length(sigma_k) != ncomp_k)
stop("lengths of 'lambda' and 'sigma' do not fit")
if (any(as.vector(sapply(sigma_k, dim)) != d))
stop("dimensionality of elements in 'sigma' incorrect")
for (i in 1:ncomp_k)
x_k[comp_k == i,] <- rmvnorm(tab[i], mu_k[i,], sigma_k[[i]])
} else if (is.matrix(sigma_k)) {
if (any(dim(sigma_k) != d))
stop("dimensionality of 'sigma_k' incorrect")
for (i in 1:ncomp_k)
x_k[comp_k == i,] <- rmvnorm(tab[i], mu_k[i,], sigma_k)
} else
stop("'sigma_k' is neither list nor matrix")
return(x_k)
}
#' @return \code{mixtureLabels} returns a factor of class labels.
#'
#' @rdname mixtureData
#'
#' @export
mixtureLabels <- function(data, prior, lambda, mu, sigma) {
nclass <- length(prior)
posterior <- mixturePosterior(data, prior, lambda, mu, sigma)
y <- factor(apply(posterior, 1, function(x) sample(1:nclass, size = 1, prob = x)), levels = 1:nclass)
return(y)
}
#' @return \code{mixturePosterior} returns a matrix of posterior probabilities.
#'
#' @rdname mixtureData
#'
#' @export
mixturePosterior <- function(data, prior, lambda, mu, sigma) {
nclass <- length(prior)
if (length(mu) != nclass)
stop("'length(mu)' is not 'nclass'")
d <- sapply(mu, ncol)
if (any(d[1] != d))
stop("numbers of columns for elements in 'mu' differs")
d <- d[1] # dimensionality
ncomp <- sapply(mu, nrow) # # of mixture components per class
if (is.list(sigma)) { # sigma list or list of lists
if (length(sigma) != nclass)
stop("'length(sigma)' and 'nclass' differ")
if (is.list(lambda)) { # lambda list
if (length(lambda) != nclass)
stop("'length(lambda)' is not 'nclass'")
if (any(sapply(lambda, length) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
dens <- sapply(1:nclass, function(k) prior[k] * mixturePosteriorHelper(data, lambda[[k]], mu[[k]], sigma[[k]]))
} else if (is.vector(lambda)) {
if (any(length(lambda) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
dens <- sapply(1:nclass, function(k) prior[k] * mixturePosteriorHelper(data, lambda, mu[[k]], sigma[[k]]))
} else {
stop("'lambda' is neither list nor vector")
}
} else if (is.matrix(sigma)) { # sigma matrix
if (is.list(lambda)) {
if (length(lambda) != nclass)
stop("'length(lambda)' is not 'nclass'")
if (any(sapply(lambda, length) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
dens <- sapply(1:nclass, function(k) prior[k] * mixturePosteriorHelper(data, lambda[[k]], mu[[k]], sigma))
} else if (is.vector(lambda)) {
if (any(length(lambda) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
dens <- sapply(1:nclass, function(k) prior[k] * mixturePosteriorHelper(data, lambda, mu[[k]], sigma))
} else {
stop("'lambda' is neither list nor vector")
}
}
colnames(dens) <- paste("posterior", 1:nclass, sep = ".")
rownames(dens) <- rownames(data)
posterior <- dens/rowSums(dens)
posterior
}
#' @noRd
mixturePosteriorHelper <- function(data, lambda_k, mu_k, sigma_k) {
ncomp_k <- length(lambda_k)
d <- ncol(mu_k)
if (is.list(sigma_k)) {
if (length(sigma_k) != ncomp_k)
stop("lengths of 'lambda' and 'sigma' do not fit")
if (any(as.vector(sapply(sigma_k, dim)) != d))
stop("dimensionality of elements in 'sigma' incorrect")
dens_k <- rowSums(sapply(1:ncomp_k, function(i) lambda_k[i] * dmvnorm(data, mu_k[i,], sigma_k[[i]])))
} else if (is.matrix(sigma_k)) {
if (any(dim(sigma_k) != d))
stop("dimensionality of 'sigma_k' incorrect")
dens_k <- rowSums(sapply(1:ncomp_k, function(i) lambda_k[i] * dmvnorm(data, mu_k[i,], sigma_k)))
} else
stop("'sigma_k' is neither list nor matrix")
return(dens_k)
}
#' @return \code{mixtureBayesClass} returns a factor of Bayes predictions.
#'
#' @rdname mixtureData
#'
#' @export
mixtureBayesClass <- function(data, prior, lambda, mu, sigma) {
nclass <- length(prior)
posterior <- mixturePosterior(data, prior, lambda, mu, sigma)
bayesclass <- factor(max.col(posterior), levels = 1:nclass)
return(bayesclass)
}
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locClassData documentation built on May 2, 2019, 5:26 p.m.
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Defines functions mixtureData mixtureDataHelper mixtureLabels mixturePosterior mixturePosteriorHelper mixtureBayesClass
Documented in mixtureBayesClass mixtureData mixtureLabels mixturePosterior
# Copyright (C) 2011 Julia Schiffner
#
# This program is free software; you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation; either version 2 or 3 of the License
# (at your option).
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# A copy of the GNU General Public License is available at
# http://www.r-project.org/Licenses/
#
#' Generation of Gaussian mixture data for classification.
#'
# details
#'
#' @title Generation of Gaussian Mixture Data for Classification
#'
#' @param n Number of observations.
#' @param prior Vector of class prior probabilities.
#' @param lambda The conditional probabilities for the mixture components given the class. Either a vector (if the same number \eqn{m}
#' of mixture components is desired for each class and the conditional probabilities for each class should be equal) or a list as long
#' as the number of classes containing one vector of probabilities for every class. The length of the \eqn{k}-th element is the desired
#' number of mixture components for the \eqn{k}-th class.
#' @param mu The centers of the mixture components. A list containing one \eqn{m_k \times d}{m_k x d} matrix of centers for
#' each class where \eqn{d} is the desired dimensionality of the data set.
#' @param sigma The covariance matrices of the mixture components. Either one single matrix that is used for each mixture
#' component or a list as long as the number of classes. List elements can be matrices (in case that for all mixture components forming
#' one class the same covariance matrix shall be used) or lists of matrices as long as the number of mixture components in the corresponding
#' class.
#' @param data A \code{data.frame}.
#'
#' @return
#' \code{mixtureData} returns an object of class \code{"locClass"}, a list with components:
#' \item{x}{(A matrix.) The explanatory variables.}
#' \item{y}{(A factor.) The class labels.}
#'
#' @examples
#' ## Simplest case:
#' # lambda vector, sigma matrix
#' # Generate a training and a test set
#' mu <- list(matrix(c(-1,-1),1), matrix(rep(c(1,4.5,12),2),3))
#' train <- mixtureData(n = 1000, prior = rep(0.5,2), lambda = list(1, rep(1/3,3)), mu = mu, sigma = 3*diag(2))
#' test <- mixtureData(n = 1000, prior = rep(0.5,2), lambda = list(1, rep(1/3,3)), mu = mu, sigma = 3*diag(2))
#'
#' # Generate a grid of points
#' x.1 <- x.2 <- seq(-7,15,0.1)
#' grid <- expand.grid(x.1 = x.1, x.2 = x.2)
#'
#' # Calculate the posterior probablities for all grid points
#' gridPosterior <- mixturePosterior(grid, prior = rep(0.5,2), lambda = list(1, rep(1/3,3)), mu = mu, sigma = 3*diag(2))
#'
#' # Draw contour lines of posterior probabilities and plot training observations
#' contour(x.1, x.2, matrix(gridPosterior[,1], length(x.1)), col = "gray")
#' points(train$x, col = train$y)
#'
#' # Calculate Bayes error
#' ybayes <- mixtureBayesClass(test$x, prior = rep(0.5,2), lambda = list(1, rep(1/3,3)), mu = mu, sigma = 3*diag(2))
#' mean(ybayes != test$y)
#'
#' if (require(MASS)) {
#'
#' # Fit an LDA model and calculate misclassification rate on the test data set
#' tr <- lda(y ~ ., data = as.data.frame(train))
#' pred <- predict(tr, as.data.frame(test))
#' mean(pred$class != test$y)
#'
#' # Draw decision boundary
#' gridPred <- predict(tr, grid)
#' contour(x.1, x.2, matrix(gridPred$posterior[,1], length(x.1)), col = "red", levels = 0.5, add = TRUE)
#'
#' }
#'
#' ## lambda list, sigma list of lists of matrices
#' mu <- list()
#' mu[[1]] <- matrix(c(1,5,1,5),2)
#' mu[[2]] <- matrix(c(8,11,8,11),2)
#' lambda <- list(c(0.5, 0.5), c(0.1, 0.9))
#' sigma <- list()
#' sigma[[1]] <- diag(2)
#' sigma[[2]] <- list(diag(2), 3*diag(2))
#' data <- mixtureData(n = 100, prior = c(0.3, 0.7), lambda, mu, sigma)
#' plot(data$x, col = data$y)
#'
#' @aliases mixtureData mixtureLabels mixturePosterior mixtureBayesClass
#'
#' @rdname mixtureData
#'
#' @import mvtnorm
#'
#' @export
mixtureData <- function(n, prior, lambda, mu, sigma) {
nclass <- length(prior)
y <- factor(sample(1:nclass, size = n, replace = TRUE, prob = prior), levels = 1:nclass) # class labels
ncl <- as.vector(table(y)) # # observations per class
if (length(mu) != nclass)
stop("'length(mu)' is not 'nclass'")
d <- sapply(mu, ncol)
if (any(d[1] != d))
stop("numbers of columns for elements in 'mu' differs")
d <- d[1] # dimensionality
ncomp <- sapply(mu, nrow) # # of mixture components per class
x <- matrix(0, n, d)
if (is.list(sigma)) { # sigma list or list of lists
if (length(sigma) != nclass)
stop("'length(sigma)' and 'nclass' differ")
if (is.list(lambda)) { # lambda list
if (length(lambda) != nclass)
stop("'length(lambda)' is not 'nclass'")
if (any(sapply(lambda, length) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
for (k in 1:nclass)
x[y == k,] <- mixtureDataHelper(ncl[k], lambda[[k]], mu[[k]], sigma[[k]])
} else if (is.vector(lambda)) {
if (any(length(lambda) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
for (k in 1:nclass)
x[y == k,] <- mixtureDataHelper(ncl[k], lambda, mu[[k]], sigma[[k]])
} else {
stop("'lambda' is neither list nor vector")
}
} else if (is.matrix(sigma)) { # sigma matrix
if (is.list(lambda)) {
if (length(lambda) != nclass)
stop("'length(lambda)' is not 'nclass'")
if (any(sapply(lambda, length) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
for (k in 1:nclass)
x[y == k,] <- mixtureDataHelper(ncl[k], lambda[[k]], mu[[k]], sigma)
} else if (is.vector(lambda)) {
if (any(length(lambda) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
for (k in 1:nclass)
x[y == k,] <- mixtureDataHelper(ncl[k], lambda, mu[[k]], sigma)
} else {
stop("'lambda' is neither list nor vector")
}
}
data <- list(x = x, y = y)
class(data) <- c("locClass.mixtureData", "locClass")
attr(data, "prior") <- prior
attr(data, "lambda") <- lambda
attr(data, "mu") <- mu
attr(data, "sigma") <- sigma
return(data)
}
#' @noRd
mixtureDataHelper <- function(n_k, lambda_k, mu_k, sigma_k) {
ncomp_k <- length(lambda_k)
comp_k <- sample(1:ncomp_k, size = n_k, replace = TRUE, prob = lambda_k)
tab <- as.vector(table(comp_k))
d <- ncol(mu_k)
x_k <- matrix(0, n_k, d)
if (is.list(sigma_k)) {
if (length(sigma_k) != ncomp_k)
stop("lengths of 'lambda' and 'sigma' do not fit")
if (any(as.vector(sapply(sigma_k, dim)) != d))
stop("dimensionality of elements in 'sigma' incorrect")
for (i in 1:ncomp_k)
x_k[comp_k == i,] <- rmvnorm(tab[i], mu_k[i,], sigma_k[[i]])
} else if (is.matrix(sigma_k)) {
if (any(dim(sigma_k) != d))
stop("dimensionality of 'sigma_k' incorrect")
for (i in 1:ncomp_k)
x_k[comp_k == i,] <- rmvnorm(tab[i], mu_k[i,], sigma_k)
} else
stop("'sigma_k' is neither list nor matrix")
return(x_k)
}
#' @return \code{mixtureLabels} returns a factor of class labels.
#'
#' @rdname mixtureData
#'
#' @export
mixtureLabels <- function(data, prior, lambda, mu, sigma) {
nclass <- length(prior)
posterior <- mixturePosterior(data, prior, lambda, mu, sigma)
y <- factor(apply(posterior, 1, function(x) sample(1:nclass, size = 1, prob = x)), levels = 1:nclass)
return(y)
}
#' @return \code{mixturePosterior} returns a matrix of posterior probabilities.
#'
#' @rdname mixtureData
#'
#' @export
mixturePosterior <- function(data, prior, lambda, mu, sigma) {
nclass <- length(prior)
if (length(mu) != nclass)
stop("'length(mu)' is not 'nclass'")
d <- sapply(mu, ncol)
if (any(d[1] != d))
stop("numbers of columns for elements in 'mu' differs")
d <- d[1] # dimensionality
ncomp <- sapply(mu, nrow) # # of mixture components per class
if (is.list(sigma)) { # sigma list or list of lists
if (length(sigma) != nclass)
stop("'length(sigma)' and 'nclass' differ")
if (is.list(lambda)) { # lambda list
if (length(lambda) != nclass)
stop("'length(lambda)' is not 'nclass'")
if (any(sapply(lambda, length) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
dens <- sapply(1:nclass, function(k) prior[k] * mixturePosteriorHelper(data, lambda[[k]], mu[[k]], sigma[[k]]))
} else if (is.vector(lambda)) {
if (any(length(lambda) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
dens <- sapply(1:nclass, function(k) prior[k] * mixturePosteriorHelper(data, lambda, mu[[k]], sigma[[k]]))
} else {
stop("'lambda' is neither list nor vector")
}
} else if (is.matrix(sigma)) { # sigma matrix
if (is.list(lambda)) {
if (length(lambda) != nclass)
stop("'length(lambda)' is not 'nclass'")
if (any(sapply(lambda, length) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
dens <- sapply(1:nclass, function(k) prior[k] * mixturePosteriorHelper(data, lambda[[k]], mu[[k]], sigma))
} else if (is.vector(lambda)) {
if (any(length(lambda) != ncomp))
stop("length of 'lambda' and 'mu' does not match")
dens <- sapply(1:nclass, function(k) prior[k] * mixturePosteriorHelper(data, lambda, mu[[k]], sigma))
} else {
stop("'lambda' is neither list nor vector")
}
}
colnames(dens) <- paste("posterior", 1:nclass, sep = ".")
rownames(dens) <- rownames(data)
posterior <- dens/rowSums(dens)
posterior
}
#' @noRd
mixturePosteriorHelper <- function(data, lambda_k, mu_k, sigma_k) {
ncomp_k <- length(lambda_k)
d <- ncol(mu_k)
if (is.list(sigma_k)) {
if (length(sigma_k) != ncomp_k)
stop("lengths of 'lambda' and 'sigma' do not fit")
if (any(as.vector(sapply(sigma_k, dim)) != d))
stop("dimensionality of elements in 'sigma' incorrect")
dens_k <- rowSums(sapply(1:ncomp_k, function(i) lambda_k[i] * dmvnorm(data, mu_k[i,], sigma_k[[i]])))
} else if (is.matrix(sigma_k)) {
if (any(dim(sigma_k) != d))
stop("dimensionality of 'sigma_k' incorrect")
dens_k <- rowSums(sapply(1:ncomp_k, function(i) lambda_k[i] * dmvnorm(data, mu_k[i,], sigma_k)))
} else
stop("'sigma_k' is neither list nor matrix")
return(dens_k)
}
#' @return \code{mixtureBayesClass} returns a factor of Bayes predictions.
#'
#' @rdname mixtureData
#'
#' @export
mixtureBayesClass <- function(data, prior, lambda, mu, sigma) {
nclass <- length(prior)
posterior <- mixturePosterior(data, prior, lambda, mu, sigma)
bayesclass <- factor(max.col(posterior), levels = 1:nclass)
return(bayesclass)
}
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