Question 1

In the K-means algorithm, cluster centroids $(\mu_k)$ are updated at each iteration to minimize the sum of squared distances between data points and their assigned centroids. The formula used for this update is: $$\mu_k = \frac{\sum x_i \mathbb{1}(z_i = k)}{\sum \mathbb{1}(z_i = k)}$$ Here: - $x_i$ represents $i^{\text{th}}$ data point in the dataset. - $z_i$ represents the cluster assignment for data point $x_i$, where $z_i$ is an integer indicating the cluster index $(k)$. Suppose you have a dataset of 9 two-dimensional data points and are performing K-means clustering with $K = 3$ clusters. After the first iteration, clusters $C_1, C_2$, and $C_3$ have the following data points: $C_1$: $\{(2, 2), (3, 3), (4,4) \}$\ $C_2$: $\{ (6, 6), (7, 7),(8,8 )\}$\ $C_3$: $\{(10, 10), (11, 11), (12, 12)\}$ After the second iteration, which of the following options represents the updated cluster centroid $\mu_k$ for $C_1$ using the provided formula? Consider 9 data points $x_1, x_2, \ldots, x_9$: $(2,3),(1,2),(3,1),(5,6),(7,8),(9,9),(6,7),(10,10),(11,11)$. Suppose we aim to form 3 clusters from these points using the K-Means algorithm. After the first iteration, clusters $C_1, C_2, C_3$ have the following data points: $C_1$: $(1, 2), (3, 1)$\ $C_2$: $(2, 3), (5, 6), (6,7)$\ $C_3$: $(9, 9), (7, 8), (10, 10), (11, 11)$ After the second iteration, to which cluster will the data point $(2, 3)$ be reassigned? Which of the following statements are true regarding applying standard PCA on a centered dataset?