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4241First Principal Direction from 2x2 Covariance 1A centered two-feature dataset has covariance matrix [[4.2, 1.6], [1.6, 1.8]]. What is the first principal-component direction and its variance?机器学习中等数值题未尝试面试订阅4246Explained Variance Ratio 1PCA on a covariance matrix yields eigenvalues 12, 3, and 1. What fraction of total variance is explained by the first principal component?机器学习简单数值题未尝试面试订阅4247Explained Variance Ratio 2PCA produces eigenvalues 12, 3, and 1. What is the smallest number of principal components needed to explain at least 90% of the variance?机器学习简单数值题未尝试面试订阅4248Explained Variance Ratio 3A centered point is x=(3,1), and the first principal-component loading is v=(2,1)/sqrt(5). What is the PC1 score of x?机器学习简单数值题未尝试面试订阅4249Explained Variance Ratio 4A rank-1 PCA approximation keeps only score 4 on loading vector v=(1,2)/sqrt(5). What reconstructed centered point does it produce?机器学习简单数值题未尝试面试订阅4250Explained Variance Ratio 5A PCA model has component variances 4 and 1. After whitening, what variance should the second whitened component have?机器学习简单数值题未尝试面试订阅4251Rank-1 PCA Reconstruction 1A centered two-feature dataset has covariance matrix [[1.8, 2.4], [2.4, 8.2]]. What is the second principal-component direction and its variance?机器学习中等数值题未尝试面试订阅4256PCA Whitening Coordinates 1A centered point has PC scores (2, -1) on orthonormal loadings v1=(1,0) and v2=(0,1). If the original mean is (5,7), what reconstructed point do the first two PCs produce?机器学习中等数值题未尝试面试订阅4257PCA Whitening Coordinates 2A PCA model has eigenvalues 5, 2, and 1. If you keep the first two PCs, what fraction of variance is discarded?机器学习中等数值题未尝试面试订阅4258PCA Whitening Coordinates 3A whitened PCA transform scales raw PC scores by dividing by the square roots of eigenvalues 9 and 4. If the raw scores are (6,4), what are the whitened scores?机器学习中等数值题未尝试面试订阅4259PCA Whitening Coordinates 4A centered point is x=(5,0), and the first PC loading is v=(0.6,0.8). What point do you get by projecting x onto PC1 alone?机器学习中等数值题未尝试面试订阅4261Why Standardization Can Change PCA DramaticallyWhy is standardizing features often important before PCA when the raw variables use very different units?机器学习中等essay未尝试面试订阅4262Why Whitening Can Amplify NoiseWhy can aggressive whitening make downstream models numerically noisier?机器学习中等essay未尝试面试订阅4263Why PCA Helps Before Some ModelsWhy does flipping the sign of a principal-component loading vector not change the PCA solution in substance?机器学习中等essay未尝试面试订阅4264When PCA Can HurtWhen can PCA actually hurt a predictive pipeline?机器学习中等essay未尝试面试订阅4265A Fast Sanity Check for PCA QuestionsWhy might the first principal component be poor for predicting a label even though it explains the most variance?机器学习中等essay未尝试面试订阅