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#' Gustafson Kessel Improved Covariance Estimation
#'
#' @description This function used to perform Gustafson Kessel Clustering of X dataset.
#'
#' @param X data frame n x p
#' @param K specific number of cluster (must be >1)
#' @param m fuzzifier / degree of fuzziness
#' @param max.iteration maximum iteration to convergence
#' @param threshold threshold of convergence
#' @param RandomNumber specific seed
#' @param gamma tuning parameter of covariance
#' @param rho cluster volume
#'
#' @return func.obj objective function that calculated.
#' @return U matrix n x K consist fuzzy membership matrix
#' @return V matrix K x p consist fuzzy centroid
#' @return D matrix n x K consist distance of data to centroid that calculated
#' @return Clust.desc cluster description (dataset with additional column of cluster label)
#'
#'
#' @details This function perform Fuzzy C-Means algorithm by Gustafson Kessel (1968) that improved by Babuska et al (2002).
#' Gustafson Kessel (GK) is one of fuzzy clustering methods to clustering dataset
#' become K cluster. Number of cluster (K) must be greater than 1. To control the overlaping
#' or fuzziness of clustering, parameter m must be specified.
#' Maximum iteration and threshold is specific number for convergencing the cluster.
#' Random Number is number that will be used for seeding to firstly generate fuzzy membership matrix.
#' @details Clustering will produce fuzzy membership matrix (U) and fuzzy cluster centroid (V).
#' The greatest value of membership on data point will determine cluster label.
#' Centroid or cluster center can be use to interpret the cluster. Both membership and centroid produced by
#' calculating mathematical distance. Fuzzy C-Means calculate distance with Covariance Cluster norm distance. So it can be said that cluster
#' will have both sperichal and elipsodial shape of geometry.
#' @details Babuska improve the covariance estimation via tuning covariance cluster
#' with covariance of data. Tuning parameter determine proportion of covariance data and covariance cluster
#' that will be used to estimate new covariance cluster. Beside improving via tuning, Basbuka improve
#' the algorithm with decomposition of covariance so it will become non singular matrix.
#'
#' @references Babuska, R., Veen, P. v., & Kaymak, U. (2002). Improved Covarians Estimation for Gustafson Kessel Clustering. IEEE, 1081-1084.
#' @references Balasko, B., Abonyi, J., & Feil, B. (2002). Fuzzy Clustering and Data Analysis Toolbox: For Use with Matlab. Veszprem, Hungary.
#' @references Gustafson, D. E., & Kessel, W. C. (1978). Fuzzy Clustering With A Fuzzy Covariance Matrix. 761-766.
#' @export
#' @import MASS
#' @importFrom stats cov runif
#' @examples
#' library(RcmdrPlugin.FuzzyClust)
#' data(iris)
#' fuzzy.GK(X=iris[,1:4],K = 3,m = 2,RandomNumber = 1234,gamma=0, max.iteration=20)->cl
fuzzy.GK<-function(X,K=2,m=1.5,max.iteration=100,
threshold=10^-5,RandomNumber=0,rho=rep(1,K),
gamma=0) {
data.X <- as.matrix(X)
n <- nrow(data.X)
p <- ncol(data.X)
##Initiation Parameter##
if (
(K <= 1) || !(is.numeric(K)) || (K %% ceiling(K) > 0))
K = 2
if ( (m <= 1) || !(is.numeric(m)))
m = 2
if (RandomNumber > 0)
set.seed(RandomNumber)
if(length(rho)!=K)
rho = rep(1,K)
if(gamma<0||gamma>1)
gamma=0
## Membership Matrix U (n x K)
U <- matrix(runif(n * K,0,1),n,K)
## Prerequirement of U:
## Sum of membership on datum is 1
U <- U / rowSums(U)
## Centroid Matrix V (K x p)
V <- matrix(0,K,p)
## Covariance Cluster
F <- array(0,c(p,p,K))
## Distance Matrix
D <- matrix(0,n,K)
U.old <- U + 1
iteration = 0
while ((max(abs(U.old - U)) > threshold) &&
(iteration < max.iteration))
{
U.old <- U
V.old<-V
D.old<-D
## Calculate Centroid
V <- t(U ^ m) %*% data.X / colSums(U ^ m)
for (k in 1:K)
{
F[,,k] = as.matrix(0,p,p)
F.bantu <- F[,,k]
for (i in 1:n)
{
F.bantu = (U[i,k] ^ m) * (data.X[i,] - V[k,]) %*%
t((data.X[i,] - V[k,]))+F.bantu
}
F.bantu = F.bantu / sum(U[,k] ^ m)
F.bantu = (1 - gamma) * F.bantu + (gamma * (det(cov(data.X))) ^ (1 / p)) * diag(p)
if (kappa(F.bantu) > 10 ^ 15)
{
eig <- eigen(F.bantu)
eig.values <- eig$values
eig.vec <- eig$vectors
eig.val.max <- max(eig.values)
eig.values[eig.values*(10^15)<eig.val.max]=eig.val.max/(10^15)
F.bantu = eig.vec %*% diag(eig.values) %*% solve(eig.vec)
}
detMat= det(F.bantu)
#Distance calculation
for (i in 1:n)
{
D[i,k] = t(data.X[i,] - V[k,]) %*% (
(rho[k] * (detMat ^ (1 / p)))*ginv(F.bantu,tol=0)) %*%
(data.X[i,] -V[k,])
}
}
##FUZZY PARTITION MATRIX
for (i in 1:n)
{
U[i,] <- 1 /
(((D[i,]) ^ (1 / (m - 1))) *
sum((1 / (D[i,])) ^ (1 /(m - 1))))
}
if(any(is.na(U))==T||any(is.infinite(U))==T)
{
U<-U.old
V<-V.old
D<-D.old
}
for (i in 1:n)
for (k in 1:K) {
if (U[i,k] < 0)
U[i,k] = 0
else if (U[i,k] > 1)
U[i,k] = 1
}
func.obj = 0
func.obj = sum(U ^ m * D)
iteration = iteration + 1
}
func.obj -> func.Obj.opt
U -> U.opt
V -> V.opt
D -> D.opt
for (k in 1:K)
{
F[,,k] = as.matrix(0,p,p)
F.bantu <- F[,,k]
for (i in 1:n)
{
F.bantu = (U[i,k] ^ m) * (data.X[i,] - V[k,]) %*%
t((data.X[i,] - V[k,]))+F.bantu
}
F.bantu = F.bantu / sum(U[,k] ^ m)}
F->F.opt
###Labelling###
colnames(V.opt)<-colnames(X)
colnames(U.opt) = paste("Clust",1:K,sep = " ")
Clust.desc <- matrix(0,n,p + 1)
rownames(Clust.desc) <- rownames(X)
colnames(Clust.desc) <- c(colnames(X),"cluster")
Clust.desc[,1:p] <- data.X
for (i in 1:n)
Clust.desc[i,p + 1] <- which.max(U.opt[i,])
result <- list()
result$func.obj <- func.Obj.opt
result$U <- U.opt
result$V <- V.opt
result$D <- D.opt
result$m <- m
result$call<-match.call()
result$Clust.desc <- Clust.desc
class(result)<-"fuzzyclust"
print(result)
return(result)
}
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