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📗 [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?
📗 Answer (comma separated vector): .
📗 [2 points] You performed PCA (Principal Component Analysis) 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 (new features created by PCA) for the point \(x\) = ?
📗 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 \(y\).
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