[SOLVED] STA280 Homework 2

Context for questions 1-20:
For a given clustering of the data, the total sum of squares (TSS) is the sum of squared

distances between all of the cases and the centroid of all of the cases. In symbols, TSS=m p ni (X X )2,

ikl .k.
where Xikl is the value of case l on variable k in cluster i, there are p variables, m clusters, and ni

i 1 k 1 l 1
is the number of cases in cluster i, and X.k. is the mean of variable k over all cases in the dataset.

The between-cluster sum of squares (BSS) is the sum of weighted squared distances between each cluster’s centroid and the centroid of all cases, where the weight given to cluster i

is ni . In symbols,
where Xik is the mean of the ni values of variable k in cluster i.

p ni (X k1 l1

BSS=m p n(X X )2

i 1 k 1
The within-cluster sum of squares for a given cluster i is the sum of squared distances

i ik. .k.

from each of the cases in cluster i to the centroid of cluster i. In symbols, it is

 X

)2 for cluster i. The within-cluster sum of squares for the dataset (WSS) is the sum of within-cluster sums of squares over all m clusters. In symbols,

ikl

ik

WSS=m p ni (X X )2

ikl ik
Note: For questions 1-20, do not standardize the data in HWcluster.xlsx.

For the 8 numbered cases in the Data1 sheet of HWcluster.xlsx, calculate: 1. The within-cluster sum of squares for Cluster 1
2. The within-cluster sum of squares for Cluster 2
3. The within-cluster sum of squares for Cluster 3

4. WSS
5. BSS
6. TSS
7. The explanatory power (“R-square”) of the 3-cluster solution.

i 1 k 1 l 1

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If each of the 8 numbered cases in the Data1 sheet of HWcluster.xlsx were in its own singleton cluster, what would be the values of:

8. TSS
9. BSS
10. WSS
11. The explanatory power (“R-square”) of the 8-cluster solution.

If all 8 numbered cases in the Data1 sheet of HWcluster.xlsx were in one super-cluster, what would be the values of:

12. TSS
13. BSS
14. WSS
15. The explanatory power (“R-square”) of the 1-cluster solution.

Given the clustering of the 8 numbered cases into the three clusters as shown in the Data1 sheet of HWcluster.xlxs, suppose that a 9th case becomes available. Its coordinates are X = 5, Y = 2. To which of the three clusters would the 9th case be assigned by:
[Use squared Euclidean distance as the measure of (dis)similarity for these questions.]

16. Single linkage 17. Complete linkage 18. Average linkage 19. Centroid method 20. Ward’s method

Context for questions 21-35:
Please refer now to the sheet Data2 in HWcluster.xlsx. Data2 contains data on a random

sample of the customers of a Mid-west based chain of clothing stores. The variables are described in the file. The objective is to try to identify market segments of customers to which the chain appeals.

Run the following SAS program, after modifying the first line to suit your set-up:

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Regarding the cluster history, when five clusters became four clusters:
21. The higher number of the two clusters that joined then is Cluster # ___ 22. The higher-numbered cluster had how many customers?
23. The lower number of the two clusters that joined then is Cluster # ___ 24. The lower-numbered cluster had how many customers?
25. What was the value of “R-Square” after the joining?
26. What was the value of “Semipartial R-square” after the joining?

How many clusters are there? Search for all solutions suggested by each of the pseudo F, pseudo T2, and CCC clustering criteria over the final 15 generations of the cluster history. Include possible one-sided peaks at the boundaries if Gen 1 > Gen 2 and/or Gen 15 > Gen 14. (Also include possible one-sided peaks next to missing values.) After finding all possible suggestions, answer the following:

27. What is the largest number of clusters suggested by the pseudo F criterion? [Exclude 15 as a possible answer.]
28. What is the smallest number of clusters suggested by the pseudo F criterion? [Exclude 1 as a possible answer.]

29. What is the largest number of clusters suggested by the pseudo T2 criterion? [Exclude 15 as a possible answer.]
30. What is the smallest number of clusters suggested by the pseudo T2 criterion? [Exclude 1 as a possible answer.]

31. What is the largest number of clusters suggested by the CCC? [Exclude 15 as a possible answer.]
32. What is the smallest number of clusters suggested by the CCC? [Exclude 1 as a possible answer.]

Which one of the four clusters is a customer most likely to belong to, if that customer is described as:

33. Masters degree, medium salary, medium concern about social status 34. High school diploma, low salary, unconcerned about social status 35. Masters degree, high salary, low savings

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