4. Cluster validation
A. Internal criteria
To validate using an internal criteria, a silhoutte and shadow value (CSV) indices can be applied. The silhoutte produces silhoutte index and a silhoutte plot, while the shadow function produces and plots shadow value. The validated clustering result consists of three arguments, i.e. the distance/ matrix, medoids id, and cluster membership. Then, the output is a list of silhoutte/ shadow index and plot.
Silhoutte plot:
rownames(mrwdist) <- colnames(mrwdist) <- 1:nrow(mrwdist)
siliris <- silhoutte(mrwdist, result$medoid, result$cluster)
siliris$result[c(1:3,70:75,101:103),]## silhoutte cluster
## 1 0.8244574 1
## 2 0.7491049 1
## 3 0.7855731 1
## 70 0.6479288 2
## 71 0.1344483 3
## 72 0.5455987 2
## 73 0.2186731 2
## 74 0.4462472 2
## 75 0.3542140 2
## 101 0.4871723 3
## 102 0.0241100 3
## 103 0.5690102 3
Shadow value (CSV) plot:
## shadval cluster
## 1 0.1158460 1
## 2 0.2888389 1
## 3 0.2493415 1
## 70 0.3422222 2
## 71 0.8000000 3
## 72 0.4744792 2
## 73 0.9568075 2
## 74 0.6676907 2
## 75 0.7713964 2
## 101 0.5912951 3
## 102 0.9042793 3
## 103 0.4388626 3
B. Relative criteria
1. Bootstrap replicate matrix
To evaluate a clustering algorithm, a bootstrap evaluation function (clustboot) can be applied. Before applying clustboot, a clustering algorihtm that will be evaluated must be created first. The evaluated clustering algorithm must consist of two arguments, i.e. the distance/ matrix and the number of cluster. Then, the output must be a vector of membership. We create two functions of clustering algorihtm that will be evaluated by the bootstrap. The first is a simple and fast k-medoids and the second is k-means algorithm, which is get from kmeans function in stats package.
k <- 3
# a simple and fast k-medoids function for bootstrap evaluation
parkboot <- function(x, nclust) {
res <- fastkmed(x, nclust, iterate = 50)
return(res$cluster)
}
# k-means function for bootstrap evaluation
kmboot <- function(x, nclust) {
res <- kmeans(x, nclust)
return(res$cluster)
}The result is a n x b matrix, where n is the number of objects and b is the number of bootstrap replicates.
fastkmedboot <- clustboot(mrwdist, nclust=k, parkboot, nboot=50)
kmeansboot <- clustboot(num, nclust=k, kmboot, nboot=50, diss = FALSE)
fastkmedboot[1:5,c(1:5,46:50)]## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0 0 1 0 2 3 0 3 1 1
## [2,] 3 0 1 1 2 3 0 0 1 1
## [3,] 0 0 0 0 2 3 3 0 0 1
## [4,] 3 0 1 0 0 3 3 0 0 1
## [5,] 0 3 0 1 2 0 3 3 1 1## [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
## [1,] 0 2 0 3 3 0 1 2 2 0
## [2,] 1 0 3 3 3 2 1 2 2 2
## [3,] 1 2 3 3 3 2 1 2 2 2
## [4,] 0 2 0 3 0 2 1 2 0 2
## [5,] 1 0 3 0 0 1 1 2 0 02. Consensus matrix
A consensus matrix (n x n) can be produced from a bootstrap replicate matrix. To create the consensus matrix, a function to order the objects is required. The consensusmatrix function provides reorder in which a reorder method can be supplied. The reorder method must have two input arguments, namely a distance matrix and a number of clusters. Meanwhile, the output is only a membership. For example, hierarchical cluster algorithm with ward linkage is applied to order the objects in the consensus matrix.
wardorder <- function(x, nclust) {
res <- hclust(x, method = "ward.D2")
member <- cutree(res, nclust)
return(member)
}
consensusfastkmed <- consensusmatrix(fastkmedboot, nclust = k, wardorder)
consensusfastkmed[c(1:5,51:55,101:105),c(1:5,51:55,101:105)]## 1 1 1 1 1 2 2 2 2 2 2
## 1 1 1 1 1 1 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 1 1 1 1 1 1 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 1 1 1 1 1 1 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 1 1 1 1 1 1 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 1 1 1 1 1 1 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000 0.0000000
## 2 0 0 0 0 0 1.0000000 0.8750000 0.9310345 0.5555556 0.9047619 0.7666667
## 2 0 0 0 0 0 0.8750000 1.0000000 0.8095238 0.9000000 1.0000000 0.6000000
## 2 0 0 0 0 0 0.9310345 0.8095238 1.0000000 0.4545455 0.8500000 0.7931034
## 2 0 0 0 0 0 0.5555556 0.9000000 0.4545455 1.0000000 0.7777778 0.3200000
## 2 0 0 0 0 0 0.9047619 1.0000000 0.8500000 0.7777778 1.0000000 0.7037037
## 2 0 0 0 0 0 0.7666667 0.6000000 0.7931034 0.3200000 0.7037037 1.0000000
## 2 0 0 0 0 0 0.7727273 0.6000000 0.7500000 0.3684211 0.8125000 1.0000000
## 2 0 0 0 0 0 0.9354839 0.8333333 0.9200000 0.4761905 0.8800000 0.8064516
## 2 0 0 0 0 0 0.7407407 0.6190476 0.7000000 0.3500000 0.7368421 1.0000000
## 2 0 0 0 0 0 0.7083333 0.6000000 0.7272727 0.3000000 0.7000000 1.0000000
## 2 2 2 2
## 1 0.0000000 0.0000000 0.0000000 0.0000000
## 1 0.0000000 0.0000000 0.0000000 0.0000000
## 1 0.0000000 0.0000000 0.0000000 0.0000000
## 1 0.0000000 0.0000000 0.0000000 0.0000000
## 1 0.0000000 0.0000000 0.0000000 0.0000000
## 2 0.7727273 0.9354839 0.7407407 0.7083333
## 2 0.6000000 0.8333333 0.6190476 0.6000000
## 2 0.7500000 0.9200000 0.7000000 0.7272727
## 2 0.3684211 0.4761905 0.3500000 0.3000000
## 2 0.8125000 0.8800000 0.7368421 0.7000000
## 2 1.0000000 0.8064516 1.0000000 1.0000000
## 2 1.0000000 0.8421053 1.0000000 1.0000000
## 2 0.8421053 1.0000000 0.8214286 0.8000000
## 2 1.0000000 0.8214286 1.0000000 1.0000000
## 2 1.0000000 0.8000000 1.0000000 1.0000000This consensus matrix can then be visualized using heatmap directly.
3. Visualization of cluster consensus matrix
To produce a heatmap of consensus matrix clustheatmap can be applied in the consensus matrix. The consensus matrix heatmap of Iris data by simple and fast k-medoids is produced.
We can also create a heatmap of the consensus matrix fromm the k-means algorithm bootstraping.
consensuskmeans <- consensusmatrix(kmeansboot, nclust = k, wardorder)
clustheatmap(consensuskmeans, "Iris Data via K-means")
