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10.7 Exercises 413 differ: for instance, cluster 4 in K-means clustering contains a portion of the observations assigned to cluster 1 by hierarchical clustering, as well as all of the observations assigned to cluster 2 by hierarchical clustering. Rather than performing hierarchical clustering on the entire data matrix, we can simply perform hierarchical clustering on the first few principal component score vectors, as follows: > hc.out=hclust(dist(pr.out$x[,1:5])) > plot(hc.out, labels=nci.labs, main="Hier. Clust. on First Five Score Vectors") > table(cutree(hc.out ,4), nci.labs) Not surprisingly, these results are different from the ones that we obtained when we performed hierarchical clustering on the full data set. Sometimes performing clustering on the first few principal component score vectors can give better results than performing clustering on the full data. In this situation, we might view the principal component step as one of denoising the data. We could also perform K-means clustering on the first few principal component score vectors rather than the full data set. 10.7 Exercises Conceptual 1. This problem involves the K-means clustering algorithm. (a) Prove (10.12). (b) On the basis of this identity, argue that the K-means clustering algorithm (Algorithm 10.1) decreases the objective (10.11) at each iteration. 2. Suppose that we have four observations, for which we compute a dissimilarity matrix, given by ⎡ ⎤ 0.3 0.4 0.7 ⎢ 0.3 0.5 0.8 ⎥ ⎣ 0.4 0.5 0.45 ⎦ . 0.7 0.8 0.45 For instance, the dissimilarity between the first and second observations is 0.3, and the dissimilarity between the second and fourth observations is 0.8. (a) On the basis of this dissimilarity matrix, sketch the dendrogram that results from hierarchically clustering these four observations using complete linkage. Be sure to indicate on the plot the height at which each fusion occurs, as well as the observations corresponding to each leaf in the dendrogram.
414 10. Unsupervised Learning (b) Repeat (a), this time using single linkage clustering. (c) Suppose that we cut the dendogram obtained in (a) such that two clusters result. Which observations are in each cluster? (d) Suppose that we cut the dendogram obtained in (b) such that two clusters result. Which observations are in each cluster? (e) It is mentioned in the chapter that at each fusion in the dendrogram, the position of the two clusters being fused can be swapped without changing the meaning of the dendrogram. Draw a dendrogram that is equivalent to the dendrogram in (a), for which two or more of the leaves are repositioned, but for which the meaning of the dendrogram is the same. 3. In this problem, you will perform K-means clustering manually, with K =2,onasmallexamplewithn = 6 observations and p = 2 features. The observations are as follows. Obs. X 1 X 2 1 1 4 2 1 3 3 0 4 4 5 1 5 6 2 6 4 0 (a) Plot the observations. (b) Randomly assign a cluster label to each observation. You can use the sample() command in R to do this. Report the cluster labels for each observation. (c) Compute the centroid for each cluster. (d) Assign each observation to the centroid to which it is closest, in terms of Euclidean distance. Report the cluster labels for each observation. (e) Repeat (c) and (d) until the answers obtained stop changing. (f) In your plot from (a), color the observations according to the cluster labels obtained. 4. Suppose that for a particular data set, we perform hierarchical clustering using single linkage and using complete linkage. We obtain two dendrograms. (a) At a certain point on the single linkage dendrogram, the clusters {1, 2, 3} and {4, 5} fuse. On the complete linkage dendrogram, the clusters {1, 2, 3} and {4, 5} also fuse at a certain point. Which fusion will occur higher on the tree, or will they fuse at the same height, or is there not enough information to tell?
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Springer Texts in Statistics Gareth
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Gareth James • Daniela Witten •
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To our parents: Alison and Michael
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viii Preface disciplines who wish t
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x Contents 3 Linear Regression 59 3
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