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10.5 Lab 2: Clustering 405 Cluster means: [,1] [,2] 1 2.3001545 -2.69622023 2 -0.3820397 -0.08740753 3 3.7789567 -4.56200798 Clustering vector: [1] 3 1 3 1 3 3 3 1 3 1 3 1 3 1 3 1 3 3 3 3 3 1 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 2 1 2 2 2 2 Within cluster sum of squares by cluster: [1] 19.56137 52.67700 25.74089 (between_SS / total_SS = 79.3 %) Available components : [1] "cluster" "centers" "totss" "withinss" "tot.withinss" "betweenss " "size" > plot(x, col=(km.out$cluster +1), main="K-Means Clustering Results with K=3", xlab="", ylab="", pch=20, cex=2) When K =3,K-means clustering splits up the two clusters. To run the kmeans() function in R with multiple initial cluster assignments, we use the nstart argument. If a value of nstart greater than one is used, then K-means clustering will be performed using multiple random assignments in Step 1 of Algorithm 10.1, and the kmeans() function will report only the best results. Here we compare using nstart=1 to nstart=20. > set.seed(3) > km.out=kmeans(x,3,nstart=1) > km.out$tot.withinss [1] 104.3319 > km.out=kmeans(x,3,nstart=20) > km.out$tot.withinss [1] 97.9793 Note that km.out$tot.withinss is the total within-cluster sum of squares, which we seek to minimize by performing K-means clustering (Equation 10.11). The individual within-cluster sum-of-squares are contained in the vector km.out$withinss. We strongly recommend always running K-means clustering with a large value of nstart, such as 20 or 50, since otherwise an undesirable local optimum may be obtained. When performing K-means clustering, in addition to using multiple initial cluster assignments, it is also important to set a random seed using the set.seed() function. This way, the initial cluster assignments in Step 1 can be replicated, and the K-means output will be fully reproducible.
406 10. Unsupervised Learning 10.5.2 Hierarchical Clustering The hclust() function implements hierarchical clustering in R. Inthefol- hclust() lowing example we use the data from Section 10.5.1 to plot the hierarchical clustering dendrogram using complete, single, and average linkage clustering, with Euclidean distance as the dissimilarity measure. We begin by clustering observations using complete linkage. The dist() function is used dist() to compute the 50 × 50 inter-observation Euclidean distance matrix. > hc.complete=hclust(dist(x), method="complete ") We could just as easily perform hierarchical clustering with average or single linkage instead: > hc.average=hclust(dist(x), method="average") > hc.single=hclust(dist(x), method="single") We can now plot the dendrograms obtained using the usual plot() function. The numbers at the bottom of the plot identify each observation. > par(mfrow=c(1,3)) > plot(hc.complete ,main="Complete Linkage", xlab="", sub="", cex=.9) > plot(hc.average , main="Average Linkage", xlab="", sub="", cex=.9) > plot(hc.single , main="Single Linkage", xlab="", sub="", cex=.9) To determine the cluster labels for each observation associated with a given cut of the dendrogram, we can use the cutree() function: > cutree(hc.complete , 2) [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 [30] 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 > cutree(hc.average , 2) [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 2 2 2 [30] 2 2 2 1 2 2 2 2 2 2 2 2 2 2 1 2 1 2 2 2 2 > cutree(hc.single , 2) [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 1 1 1 1 [30] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 For this data, complete and average linkage generally separate the observations into their correct groups. However, single linkage identifies one point as belonging to its own cluster. A more sensible answer is obtained when four clusters are selected, although there are still two singletons. > cutree(hc.single , 4) [1] 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1 1 3 3 3 3 [30] 3 3 3 3 3 3 3 3 3 3 3 3 4 3 3 3 3 3 3 3 3 To scale the variables before performing hierarchical clustering of the observations, we use the scale() function: > xsc=scale(x) > plot(hclust(dist(xsc), method="complete "), main="Hierarchical Clustering with Scaled Features ") cutree() scale()
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Springer Texts in Statistics Gareth
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Gareth James • Daniela Witten •
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viii Preface disciplines who wish t
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