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University of Illinois, Urbana Champaign *

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Statistics

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Feb 20, 2024

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Homework Assignment 2 – [30 points] STAT437 Unsupervised Learning – Spring 2024 Due: Friday, February 2 on Canvas Questions #1-#3: Refer to the attached Jupyter notebook to complete questions #1-#3 of this assignment. Question #4: [3 pt] Plotted and shown below is a two-dimensional dataset with 10 objects. Your two initial medoids have been randomly initialized to be object 6 and object 7 . What will be the NEXT two medoids in the k-medoids algorithm? Show your work. Note 1: In this k-medoids algorithm, we will be using the Euclidean distance metric. A pairwise Euclidean distance matrix for each pair of objects in the dataset is also given below. Note 2: Many algorithms that we will learn in this class may encounter a “tie” scenario for certain datasets. For instance, when it comes to assigning a particular object to a particular cluster based on the rules stipulated by the general algorithm , the object could technically be assigned to more than one cluster. In scenarios such these, it’s often useful to designate a “tie-breaker” rule. For instance, let’s designate the following “tie-breaker rule” for the cluster assignment step of the k-Medoids algorithm. Tie-Breaker Rule: If an observation is equally close to two or more centroids (medoids), assign it to the centroid with the “lowest index” (ie. “object 3” < “object 4”). Data x y Object 1 1 1 Object 2 2 2 Object 3 1 2 Object 4 2 1 Object 5 1.5 1.5 Object 6 2.5 4.5 Object 7 3 6 Object 8 4 5 Object 9 4 6 Object 10 3.5 5.5 Euclidean Distance Between Each Pair of Objects Object 1 Object 2 Object 3 Object 4 Object 5 Object 6 Object 7 Object 8 Object 9 Object 10 Object 1 0.00 1.41 1.00 1.00 0.71 3.81 5.39 5.00 5.83 5.15 Object 2 1.41 0.00 1.00 1.00 0.71 2.55 4.12 3.61 4.47 3.81 Object 3 1.00 1.00 0.00 1.41 0.71 2.92 4.47 4.24 5.00 4.30 Object 4 1.00 1.00 1.41 0.00 0.71 3.54 5.10 4.47 5.39 4.74 Object 5 0.71 0.71 0.71 0.71 0.00 3.16 4.74 4.30 5.15 4.47 Object 6 3.81 2.55 2.92 3.54 3.16 0.00 1.58 1.58 2.12 1.41 Object 7 5.39 4.12 4.47 5.10 4.74 1.58 0.00 1.41 1.00 0.71 Object 8 5.00 3.61 4.24 4.47 4.30 1.58 1.41 0.00 1.00 0.71 Object 9 5.83 4.47 5.00 5.39 5.15 2.12 1.00 1.00 0.00 0.71 Object 10 5.15 3.81 4.30 4.74 4.47 1.41 0.71 0.71 0.71 0.00

Cost with Object 7: 1.41 The second new medoid is Object 10 with a cost of 0.71. The next medoid after Object 6 is Object 9 with a cost of 1.00. Cost with Object 7: 0.71 Cost with Object 6: 1.41 For Object 10: Cost with Object 7: 1.00 Cost with Object 6: 2.12 For Object 9: For Object 1: Cost with Object 6: 3.81 For Object 8: Cost with Object 7: 4.74 Cost with Object 6: 3.16 For Object 5: Cost with Object 7: 5.10 Cost with Object 6: 3.54 For Object 4: Cost with Object 7: 4.47 Cost with Object 6: 2.92 For Object 3: Cost with Object 7: 4.12 Cost with Object 6: 2.55 For Object 2: Cost with Object 7: 5.39 Cost with Object 6: 1.58

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