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📗 (Fall 2018 Midterm Q13) You performed PCA in \(\mathbb{R}^{3}\). If the first principal component is \(v_{1}\) = and the second principal component is \(v_{2}\) = . What is the new 2D coordinates for the point \(x\) = ?
📗 Answer (comma separated vector): .
📗 (Fall 2018 Midterm Q14) Let \(x\) = and \(v\) = . The projection of \(x\) onto \(v\) is the point \(y\) on the direction of \(v\) such that the line connecting \(x, y\) is perpendicular to \(v\). Compute \(y\):
📗 Answer (comma separated vector): .
📗 (Fall 2016 Final Q9, Fall 2014 Midterm Q5, Fall 2012 Final Q3) 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?
📗 Note: in 1D, use the Manhattan distances, so do not square the distances when computing the distortion.
📗 Answer: .
📗 (Fall 2017 Final Q22, Fall 2014 Final Q20, Fall 2013 Final Q14) 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). 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.
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
📗 (Fall 2017 Final Q17, Fall 2016 Midterm Q10, Fall 2016 Final Q8, Fall 2014 Midterm Q1, Fall 2012 Final Q2, Fall 2010 Final Q12) Perform hierarchical clustering with single 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:
📗 Note: to use the eraser, drag it from one node to another to remove the (directed) edge in between. This answer cannot be saved and loaded using the "download" and "load" buttons at the bottom of the page.
📗 (Fall 2017 Final Q17, Fall 2016 Midterm Q10, Fall 2016 Final Q8, Fall 2014 Midterm Q1, Fall 2012 Final Q2, Fall 2010 Final Q12) Perform hierarchical clustering with complete 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:
📗 Note: to use the eraser, drag it from one node to another to remove the (directed) edge in between. This answer cannot be saved and loaded using the "download" and "load" buttons at the bottom of the page.
📗 (Fall 2011 Midterm Q2) 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. \(\displaystyle\max_{c_{j}} \displaystyle\min\left\{d\left(c_{1}, c_{j}\right), d\left(c_{2}, c_{j}\right), ..., d\left(c_{j-1}, c_{j}\right)\right\}\)). 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): .
📗 (Spring 2017 Midterm Q4) You are given the distance table. Consider the next iteration of hierarchical agglomerative clustering (another name for the hierarchical clustering method we covered in the lectures) using linkage. What will the new values be in the resulting distance table corresponding to the four new clusters? If you merge two columns (rows), put the new distances in the column (row) with the smaller index. For example, if you merge columns 2 and 4, the new column 2 should contain the new distances and column 4 should be removed, i.e. the columns and rows should be in the order (1), (2 and 4), (3), (5).
\(d\) =
📗 Hint: the resulting matrix should have 4 columns and 4 rows.
📗 Answer (matrix with multiple lines, each line is a comma separated vector): .
📗 Go to the K-means clustering demo Link and find an example in which the algorithm converges to a local minimum that is not a global minimum. (You can use the same one I gave during the lecture, but make sure you can replicate it.) Post a screen shot on Piazza.
📗 The data set you used is (e.g. mine is Gaussian Mixture): and I have participated in the discussion on Piazza.
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📗 Answer: .
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