Cluster Dendrogram. SolincoTour8. WilsonJuice108 VolklOrganixV1Midplus. YonexEZone98 DonnayProOne. HeadPrestigeMid. HeadPrestigeMidplus.

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1 STAT 530 HW 5 Example solutions Problem 1: Answers may vary according to which hierarchical method and which partitioning method you choose. Below is the dendrogram for a complete linkage solution (note that I have scaled the data before doing the clustering that does make a difference): The choice of the number of clusters is subjective. I might choose k=3 clusters. If so, here are the partitions of the racquets into clusters: [[1]] [1] "Asics109" "DonnayProOne" "GammaRZR98" [4] "GammaRZR98T" "HeadPrestigeMid" "HeadPrestigeMidplus" [7] "HeadPrestigePro" "HeadPrestigeS" "HeadRadicalMidplus" [10] "HeadRadicalOversize" "HeadRadicalPro" "PrinceRebel95" [13] "PrinceRebel98" "PrinceWarrior100" "WilsonJuice100" [16] "WilsonJuice108" "WilsonSteam100" "YonexEZone98" [19] "YonexEZone100" [[2]] [1] "Asics116" "Asics125" "BabolatPureDrive" [4] "BabolatPureDrive107" "DunlopBiomimetic700" "VolklOrganixV1Midplus" [7] "VolklOrganixV1Oversize" "YonexEZone107" [[3]] [1] "BabolatPureDriveLite" "HeadRadicalS" [3] "SolincoTour8" "WilsonProStaffSix.One100" Cluster Dendrogram Height Asics116 DunlopBiomimetic700 Asics125 VolklOrganixV1Oversize BabolatPureDriveLite HeadRadicalS WilsonProStaffSix.One100 WilsonSteam100 BabolatPureDrive107 Asics109 WilsonJuice108 VolklOrganixV1Midplus YonexEZone107 PrinceWarrior100 WilsonJuice100 BabolatPureDrive YonexEZone100 SolincoTour8 PrinceRebel95 GammaRZR98T HeadPrestigePro HeadPrestigeMid HeadPrestigeMidplus YonexEZone98 DonnayProOne HeadRadicalPro HeadRadicalOversize HeadRadicalMidplus PrinceRebel98 GammaRZR98 HeadPrestigeS racq.dist hclust (*, "complete")

2 To get an idea of the nature of the clusters, I do a PCA and plot the data in the space of the first 2 PC scores, separated by cluster. The first PC seems to measure the headsize, lightness, and head-heavy balance of the racquets. The second PC measures how light the racquet FEELS to swing. Loadings: Comp.1 Comp.2 Comp.3 Comp.4 Comp.5 Comp.6 length static.weight balance swingweight headsize beamwidth PC 2 SolincoTour8 HeadRadicalS BabolatPureDriv elite Asics116 WilsonProStaf f Six.One100 HeadPrestigeMidplus GammaRZR98 BabolatPureDriv e107 VolklOrganixV1Midplus adprestigemid BabolatPureDriv e HeadPrestigeS Donnay ProOneHeadRadicalOv WilsonSteam100 ersize YonexEZone100 YonexEZone107 HeadRadicalMidplus DunlopBiomimetic700 PrinceRebel98 HeadPrestigePro WilsonJuice100 YonexEZone98 PrinceWarrior100 PrinceRebel95 WilsonJuice108 GammaRZR98T Asics125 HeadRadicalPro Asics109 VolklOrganixV1Ov ersize PC 1 Looking at the plot, the red cluster seems to consist of racquets with large headsize, light weight, and head-heavy balance. The green cluster seems to consist of racquets with smaller headsize, heavier weight, and head-light balance. The blue clusters is racquets that are medium in these respects. The swingweight does not play a major role in the clustering structure, at least in this solution. For the partitioning method, I do a K-medoids clustering. The highest average silhouette width is actually given by k=2: my.k.choices avg.sil.width [1,] [2,] [3,] [4,] [5,] [6,] [7,]

3 PC 2 SolincoTour8 HeadRadicalS BabolatPureDriv elite Asics116 WilsonProStaf f Six.One100 HeadPrestigeMidplus GammaRZR98 BabolatPureDriv e107 VolklOrganixV1Midplus adprestigemid BabolatPureDriv e HeadPrestigeS Donnay ProOneHeadRadicalOv WilsonSteam100 ersize YonexEZone100 YonexEZone107 HeadRadicalMidplus DunlopBiomimetic700 PrinceRebel98 HeadPrestigePro WilsonJuice100 YonexEZone98 PrinceWarrior100 PrinceRebel95 WilsonJuice108 GammaRZR98T Asics125 HeadRadicalPro Asics109 VolklOrganixV1Ov ersize PC 1 The clusters (plotted in the space of the first 2 PCs again) appear similar as before, but without the middle group: The blue cluster seems to consist of racquets with large headsize, light weight, and headheavy balance. The red cluster seems to consist of racquets with smaller headsize, heavier weight, and head-light balance. PROBLEM 2: Use linear discriminant analysis (LDA) to build a classification rule to classifying the Bumpus bird data into two groups ("survived" and "died") based on the 5 numerical measurements. Assume equal prior probabilities of surviving and dying. (a) Use the LDA rule to predict the survival status for a hypothetical bird with: tot.length=156, alar.length=242, beak.head.length=31.4, humerus.length=18.1, keel.stern.length=19.4 Give the probability of surviving for such a bird. For the bird with total length=156, alar length=242, beak-head length=31.4, humerus length=18.1, and keelstern length=19.4, this bird is predicted to die. The posterior probability that the bird survives is , given equal prior probabilities of surviving and dying. (b) Find the plug-in misclassification rate and the cross-validation misclassification rate for the LDA classification rule.

4 The plug-in misclassification rate for LDA here is 17/49 = The CV rate is PROBLEM 3: Do Problem 7.4 in the Everitt textbook to perform supervised classification on the Skulls data we looked at in class, but use the CLASSIFICATION TREE approach of the Skulls data to obtain the classification tree (show the plot of the tree) and classify into an Epoch the new skull with the measurements given in problem 7.4 (MB=133.0, BH=130.0, BL=95.0, NH=50.0). You may assume equal prior probabilities of being in each category. The classification tree for these data is given below. (The plug-in misclassification rate is ) We predict the epoch of a new skull with MB = 133, BH = 130, BL = 95, NH = 50 (assuming equal prior probabilities of being in each category): c1850bc c200bc c3300bc c4000bc cad The predicted category for this skull is c3300bc. We can see this graphically by proceeding down the branches of the tree, based on the logical conditions, until we reach the c3300bc node near the bottom right of the tree.

5 BL>=97.5 BL< 98.5 BH>=129.5 MB>= BCBL< NH< 51.5 BH< c1850bc c3300bc c3300bc c3300bc c4000bc MB< MB>=135.5 MB>=129.5 c1850bc BH>=134.5 c4000bc c200bc c3300bc c200bc cad150 PROBLEM 3: Graduate Student Problem: [NOTE: In some newer implementations of the mclust package, the 3-cluster solution is actually given as best according to BIC. It is fine if your best solution is actually the 3-cluster solution.] In the model-based clustering, the best solution according to BIC is the 6-cluster solution with the VEV structure. The clustering partition for this is: [[1]] [1]

6 [[2]] [1] [[3]] [1] [[4]] [1] [[5]] [1] [[6]] [1] BIC EII VII EEI VEI EVI VVI EEE EEV VEV VVV Number of components

7 PC PC 1 The data do not seems to be very well separated in the space of the first 2 PCs, which explain about 74% of the data s variance. If we force a 3-cluster solution, we see much better separation: [[1]]

8 [1] [[2]] [1] [[3]] [1] PC PC 1 The green cluster has high scores on PC1 (i.e., low,, and ). The red cluster has low scores on PC1 (i.e., high,, and ). The blue cluster is somewhere in the middle.