MA321 Problem Sheet - Assignment PDF

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MA321 Problem Sheet 1. Let (X1 , . . . , Xq ) be a q-dimensional random variable with V ar [(X1 , . . . , Xq )] = Σ . The variance of the first principal component a1 of Σ is   V ar a1 t (X1 , . . . , Xq ) = a1 t Σa1 . Show mathematically that V ar [a1 t (X1 , . . . , Xq )] = λ1 with λ1 to be the largest eigenvalue of Σ.

2. (same as the example in the short video of K-means on moodle) The table below displays four items of two-dimensional data together. The cluster initial assignment is given that Cluster 1 = {Observation 1, Observation 2} = {(0, 0), (2, 2)}, Cluster 2 = {Observation 3, Observation 4} = {(1, 0), (3, 2)}. Carry out 2-means clustering on this data set with the Euclidean measure as the distance metric. Observation number 1 Observation (0,0)

2 (2,2)

3 (1,0)

4 (3,2)

3. (a) There are four observations of two-dimensional data in the table below. Calculate the pairwise dissimilarity matrix using Euclidean distance. Observation number 1 Observation (0,1)

2 (2,1)

3 (2,4)

4 (3,2)

(b) Use complete-linkage cluster analysis on the disimilarity matrix above, and draw the associated dendrogram....