📗 Enter your ID (the wisc email ID without @wisc.edu) here: and click (or hit enter key) 1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25m6
📗 If the questions are not generated correctly, try refresh the page using the button at the top left corner.
📗 The same ID should generate the same set of questions. Your answers are not saved when you close the browser. You could print the page: , solve the problems, then enter all your answers at the end.
📗 Please do not refresh the page: your answers will not be saved.
📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) is a diagonal matrix, what is the smallest value of \(K\) so that the Manhattan distance between the vector \(\begin{bmatrix} 1 \\ 1 \\ ... \\ 1 \end{bmatrix}\) with ones (\(1\)'s) and its reconstruction using the first \(K\) principal components is less than or equal to ?
📗 Answer: .
📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) = , what is the first principal component?
📗 Answer (comma separated vector):
📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) = . If one original data point is \(x\) = . What is the reconstructed vector using only the first principal components?
📗 Answer (comma separated vector):
📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) = , what is the projected variance of the dataset in the direction of the first principal component?
📗 Answer: .
📗 [3 points] Given the variance matrix \(\hat{\Sigma}\) = , what is the first principal component? Enter a unit vector.
📗 Answer (comma separated vector): .
📗 [3 points] Given the variance matrix of a data set \(V\) = , a principal component \(u\) = , what is the projected variance of the data set in the direction \(u\)?
📗 Answer: .
📗 [3 points] Consider a vector \(x\) = , if the principal component is = , what is the reconstruction of \(x\) using only the first principal components? If more information is needed, please enter a vector of all 0's.
📗 Answer (comma separated vector): .
📗 [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): .
📗 [4 points] What is the projected variance of and onto the principal component ? Use the MLE (Maximum Likelihood Estimate) formula for the variance: \(\sigma^{2} = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} \left(x_{i} - \mu\right)^{2}\) with \(\mu = \dfrac{1}{n} \displaystyle\sum_{i=1}^{n} x_{i}\).
📗 Answer: .
📗 [3 points] 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 \(y\).
📗 You could save the text in the above text box to a file using the button or copy and paste it into a file yourself .
📗 You could load your answers from the text (or txt file) in the text box below using the button . The first two lines should be "##m: 6" and "##id: your id", and the format of the remaining lines should be "##1: your answer to question 1" newline "##2: your answer to question 2", etc. Please make sure that your answers are loaded correctly before submitting them.
📗 You can find videos going through the questions on Link.