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📗 [4 points] Suppose K-Means with \(K = 2\) is used to cluster the data set and initial cluster centers are \(c_{1}\) = and \(c_{2}\) = \(x\). What is the smallest value of \(x\) if cluster 1 has \(n\) = points initially (before updating the cluster centers). Break ties by assigning the point to cluster 2.
📗 Note: the red points are the cluster centers and the other points are the training items.
📗 Answer: .
📗 [4 points] Suppose K-Means with \(K = 2\) is used to cluster the data set and initial cluster centers are \(c_{1}\) = and \(c_{2}\) = \(x\). What is the largest value of \(x\) if cluster 1 has \(n\) = points initially (before updating the cluster centers). Break ties by assigning the point to cluster 2.
📗 Answer: .
📗 [4 points] Suppose K-Means with \(K = 2\) is used to cluster the data set and initial cluster centers are \(c_{1}\) = and \(c_{2}\) = \(x\). What is the value of \(x\) if cluster 1 has \(n\) = points initially (before updating the cluster centers). Break ties by assigning the point to cluster 2.
📗 Answer: .
📗 [4 points] Perform hierarchical clustering with linkage in one-dimensional space on the following points: , , , , , . Break ties in distances by first combining the instances with the smallest index (appears earliest in the list). Draw the cluster tree.
📗 Note: the node \(C_{1}\) should be the first cluster formed, \(C_{2}\) should be the second and so on. All edges to point to the instances (or other clusters) that belong to the cluster.
📗 Answer:
graph
📗 Note: to erase an edge, draw the same edge again.
📗 [4 points] Given the dataset , the cluster centers are computed by k-means clustering algorithm with \(k = 2\). The first cluster center is \(x\) and the second cluster center is . What is the imum value of \(x\) such that the second cluster is empty (contains 0 instances). In case of a tie in distance, the point belongs to cluster 1.
📗 Answer: .
📗 [3 points] Consider the 1D data set: \(x_{i} = i\) for \(i\) = to . To select good initial centers for k-means where \(k\) = , let's set \(c_{1}\) = . Then select \(c_{j}\) from the unused points in the data set, so that it is farthest from any already-selected centers \(c_{1}, ..., c_{j-1}\) (i.e. \(c_{j} = \mathop{\mathrm{argmax}}_{x_{i}} \displaystyle\min\left\{d\left(c_{1}, x_{i}\right), d\left(c_{2}, x_{i}\right), ..., d\left(c_{j-1}, x_{i}\right)\right\}\)). Enter the initial centers (including \(c_{1}\)) in increasing order (from the smallest to the largest). In case of ties, select the smaller number.
📗 Answer (comma separated vector): .
📗 [4 points] Consider the four points: \(x_{1}\) = , \(x_{2}\) = , \(x_{3}\) = , \(x_{4}\) = . Let there be two initial cluster centers \(c_{1}\) = , \(c_{2}\) = . Use Euclidean distance. Break ties in distances by putting the point in the cluster with the smaller index (i.e. favor cluster 1). If a cluster contains no points, do not move the cluster center (it stays at the initial position). Write down the cluster centers after one iteration of k-means, the first cluster center (comma separated vector) on the first line and the second cluster center (comma separated vector) on the second line.
📗 Note: the red points are the cluster centers and the other points are the training items.
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
📗 [3 points] Perform k-means clustering on six points: \(x_{1}\) = , \(x_{2}\) = , \(x_{3}\) = , \(x_{4}\) = , \(x_{5}\) = , \(x_{6}\) = . Initially the cluster centers are at \(c_{1}\) = , \(c_{2}\) = . Run k-means for one iteration (assign the points, update center once and reassign the points once). Break ties in distances by putting the point in the cluster with the smaller index (i.e. favor cluster 1). What is the reduction in total distortion? Use Euclidean distance and calculate the total distortion by summing the squares of the individual distances to the center.
📗 Note: the red points are the cluster centers and the other points are the training items.
📗 Answer: .
📗 [3 points] Given data and initial k-means cluster centers \(c_{1}\) = and \(c_{2}\) = , what is the initial total distortion (do not take the square root). Use Euclidean distance. Break ties by assigning points to the first cluster.
📗 Answer: .
📗 [3 points] You have a dataset with unique data points which you want to use k-means clustering on. You setup the experiment as follows: you apply k-means with different k's: \(k\) = . Which \(k\) value will minimize the total distortion? Enter -1 if the answer depends on the data points.
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