📗 [2 points] You performed PCA (Principal Component Analysis) in \(\mathbb{R}^{3}\). If the first principal component is \(u_{1}\) = \(\approx\) and the second principal component is \(u_{2}\) = \(\approx\) . What is the new 2D coordinates (new features created by PCA) for the point \(x\) = ?

📗 In the diagram, the black axes are the original axes, the green axes are the PCA axes, the red vector is \(x\), the red point is the reconstruction \(\hat{x}\) using the PCA axes.

📗 Answer (comma separated vector): . Calculate

📗 [4 points] 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 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).

📗 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 (distortion after first assign step minus the distortion after the second assign step)? 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: . Calculate

📗 [4 points] Run search algorithm on the following graph, starting from state 0 with the goal state being . Write down the expansion path (in the order of the states expanded). The heuristic function \(h\) is shown as subscripts. Break tie by expanding the state with a smaller index.

📗 In case the diagram is not clear: the weights are (with heuristic values on the diagonal entries): .

📗 Answer (comma separated vector): .

📗 [3 points] Let \(h_{1}\) be an admissible heuristic from a state to the optimal goal, A* search with which ones of the following \(h\) will be admissible?

📗 Choices:

📗 [3 points] Which nodes are pruned by alpha-beta pruning? The player moves first.

📗 Select the nodes:

📗 Note: please do not select any edges.

📗 [4 points] What is the row player's value in a Nash equilibrium of the following zero-sum normal form game? A (row) is the max player, B (col) is the min player. Use the checkboxes in the table to highlight the best responses: the first checkbox for the row player and the second checkbox for the column player.

📗 Answer: .

📗 [4 points] Given the following game payoff table, suppose the row player uses a mixed strategy playing U with probability \(p\), and column player uses a pure strategy. What is the smallest and largest value of \(p\) in a mixed strategy Nash equilibrium?

📗 Answer (comma separated vector): . Calculate

📗 [4 points] Consider the following Markov Decision Process. It has two states \(s\), A and B. It has two actions \(a\): move and stay. The state transition is deterministic: "move" moves to the other state, while "stay" stays at the current state. The reward \(r\) is for move (from A and B), for stay (in A and B). Suppose the discount rate is \(\beta\) = .

📗 Answer (matrix with multiple lines, each line is a comma separated vector): . Calculate

📗 [1 points] Please enter any comments including possible mistakes and bugs with the questions or your answers. If you have no comments, please enter "None": do not leave it blank.

📗 Answer: .