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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

Question #5 [3 pt] Designing our Own Clustering Algorithms Suppose that we’d like to design ourselves three different types of clustering algorithms.  Clustering Algorithm A: a clustering algorithm that is designed to detect density-based clusters  Clustering Algorithm B: a clustering algorithm that is designed to detect contiguity-based clusters  Clustering Algorithm C: a clustering algorithm that is designed to detect prototype-based clusters Three Datasets and their Desired Clusters Three datasets are shown in the scatterplots below. The clusters of observations that we’d ideally like for the chosen clustering algorithm to find are color-coded. (One exception to this is dataset 2. We can assume that the observations represented with green diamonds (ie. label 6) actually represent “noise” that we would NOT like for the algorithm to put in an “official cluster” and simply just label as “noise”.) Match the Algorithm to the Dataset 1. Select the algorithm (from the A,B,C above) that would be the best at detecting the cluster that we’re looking for in dataset 1. 2. Select the algorithm (from the A,B,C above) that would be the best at detecting the cluster that we’re looking for in dataset 2. 3. Select the algorithm (from the A,B,C above) that would be the best at detecting the cluster that we’re looking for in dataset 3. Match the Algorithm to the Dataset For questions (1, 2, and 3) above also answer the following.  If you selected a clustering algorithm designed to detect density-based clusters for a given dataset , how might you specifically determine if a point should belong to a density-based cluster as opposed to being considered as noise in this given dataset?  If you selected a clustering algorithm designed to detect contiguity-based clusters for a given dataset , how might you specifically determine if two points are connected in this given dataset? (We can also assume that if A and B connect, and B and C connected, then A and C connect as well).  If you selected a clustering algorithm designed to detect prototype-based clusters for a given dataset , what prototype might you use for this given dataset? Dataset 3: C, I think i would picked medians as the prototype for this dataset. connected as if not. Dataset 2: B, if datapoints are within 0.5 euclidean distance of each other you can say they are dimensions so that it avoids detecting label 6 as a official cluster. avoid picking label 6 as a official cluster because it looks like the smallest so maybe increase the points you can disreguard them as noise and if otherwise see it as a density based cluster. You can Dataset 1: A, you can measure the density in a 1 by 1 sqaure and if there are less than, lets say 5 data

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