PS#9

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California Lutheran University *

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Apr 3, 2024

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Q1 K-means 55 Points Q1.1 5 Points Which of following tasks might be suitable to apply K-means algorithm? Q1.2 7 Points Suppose our current three cluster centroids are :(1,2), :(-3,0) and :(4,2). And we have a training example : (-1,2). After a cluster assignment step, what will be? Q1.3 7 Points Given historical weather records, predict if tomorrow's weather will be sunny or rainy.  Given many emails, you want to determine if they are Spam or Non- Spam emails.  From the user usage patterns on a website, figure out what different groups of users exists.  Given sales data from a large number of products in a supermarket, estimate future sales for each of these products.  μ 1 μ 2 μ 3 x ( i ) c ( i ) 1  2  3  not assigned 

Select all correct about K-means: Q1.4 12 Points Given the following 6 data points, simulate K-means clustering manually with K = 2. Each example consists of two features and initially assigned to the cluster k. Once an example has been assigned to a particular centroid, it will never be reassigned to another centroid K-Means will always give the same results regardless of the initialization of the centroids. On every iteration of K-means, the cost funtion J(c(1), ..., c(m), μ1, ..., μk (the distortion function) should either stay the same or decrease; in particular, it should not increase  For some datasets, the "right" or "correct" value of K (the number of clusters) can be ambiguous, and hard even for a human expert looking carefully at the data to decide. 

Compute the centroid of each cluster. mu1={3, 11/4}, mu2={5/2, 3/2} Q1.5 12 Points Continue with above question. Assign each data point to its closest centroid, reporting the new cluster label k. (in format of a list in order of i, eg. [1,2,1,1,1,2] [1,1,1,2,1,2] Q1.6 12 Points Repeat above two steps until convergence. Once the centroids and the cluster labels stop changing, report the cluster label k for each data point and final centroids cluster labels: [1,1,1,2,2,2] Final Centroids: k=1 centroid (2/3, 11/3) k=2 centroid (5, 1) Q2 PCA 45 Points Q2.1 7 Points Which of following tasks might be suitable to apply PCA? (select all correct)

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Q2.2 7 Points What happens when you get features in lower dimensions using PCA? (select all correct) Q2.3 7 Points Consider the following dataset, which vector could be the first principal compoment ? (select all possible ones) Data compression: Reduce the dimension of your data  As a replacement for (or alternative to) linear regression Data visualization: To take 2D data, and find a different way of plotting it in 2D (using k=2) The features will still have interpretability The features will lose interpretability  The features must carry all information present in data The features may not carry all information present in data  μ μ 1

Q2.4 5 Points Suppose we have eigenvalue and vector pairs of the scatter matrix S as , and . What should be the first principal component direction?  μ A  μ B μ C μ D (0.1, μ ) a (5, μ ) b (1, μ ) c  μ a  μ b  μ c

Q2.5 7 Points If and are the first two principal components vectors, what statements are correct about them? (select all) Q2.6 7 Points Consider we are using PCA to generalize k-dim representations of our data. We tried different ks and have variability of each principal component as shown in the figure where the x-axis is k, and the y- axis is . Which k might be a good cutoff? μ 1 μ 2 is orthogonal to  μ 1 μ 2 is parallel to μ 1 μ 2 variance along is bigger than variance along  μ 1 μ 2 variance along is bigger than variance along μ 2 μ 1 λ j λ j

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Q2.7 5 Points When do we need kernel PCA? 1  2  5  10  When we need to capture non-linearity in the data  When there are kernels in the data  When any two features are highly correlated  When we need to reduce the number of dimensions to half of the original dimension.  GRADED Problem Set (PS) #09 STUDENT Urvashiben Patel TOTAL POINTS 100 / 100 pts QUESTION 1 K-means 55 / 55 pts 1.1 (no title) 5 / 5 pts 1.2 (no title) 7 / 7 pts 

1.3 (no title) 7 / 7 pts 1.4 (no title) 12 / 12 pts 1.5 (no title) 12 / 12 pts 1.6 (no title) 12 / 12 pts QUESTION 2 PCA 45 / 45 pts 2.1 (no title) 7 / 7 pts 2.2 (no title) 7 / 7 pts 2.3 (no title) 7 / 7 pts 2.4 (no title) 5 / 5 pts 2.5 (no title) 7 / 7 pts 2.6 (no title) 7 / 7 pts 2.7 (no title) 5 / 5 pts

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