Given a dataset ,..., n, where the dimension of each observation is 3. The sample mean...
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Given a dataset ,..., n, where the dimension of each observation is 3. The sample mean and sample covariance matrix are X H = 1 (3) 2 S = 10.5 -5 -2.1 -5 6.3 1.4 -2.1 1.4 9.9 The spectral decomposition on S is S = HAHT, where 0.553 -0.366 -0.748 0.833 0.243 0.497 -0.898 0.440 0 A = 2.98 0 0 0 0 0 0 15.06 8.67 (a) What is the percentage of the total variance by the first, second and the third principal component? (b) Consider an observation x = (-1,2,2)T, what are the scores for the first and second principal component for x? (c) What is the reconstructed x for x, in (b) using both the first and the second principal components? (d) Find a 3 1 vector u to minimize TS-lu. What is the corresponding minimum value? Is this vector u unique? Explain your answer. 10 (e) denote y = (X1, X2, 1X3), where X, X2, and X3 are the three variables in the random vector a. Compute the sample covariance and the sample correlation matrix for y. Given a dataset ,..., n, where the dimension of each observation is 3. The sample mean and sample covariance matrix are X H = 1 (3) 2 S = 10.5 -5 -2.1 -5 6.3 1.4 -2.1 1.4 9.9 The spectral decomposition on S is S = HAHT, where 0.553 -0.366 -0.748 0.833 0.243 0.497 -0.898 0.440 0 A = 2.98 0 0 0 0 0 0 15.06 8.67 (a) What is the percentage of the total variance by the first, second and the third principal component? (b) Consider an observation x = (-1,2,2)T, what are the scores for the first and second principal component for x? (c) What is the reconstructed x for x, in (b) using both the first and the second principal components? (d) Find a 3 1 vector u to minimize TS-lu. What is the corresponding minimum value? Is this vector u unique? Explain your answer. 10 (e) denote y = (X1, X2, 1X3), where X, X2, and X3 are the three variables in the random vector a. Compute the sample covariance and the sample correlation matrix for y.
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a To find the percentage of total variance explained by each principal component we can use the eigenvalues ... View the full answer
Related Book For
Income Tax Fundamentals 2013
ISBN: 9781285586618
31st Edition
Authors: Gerald E. Whittenburg, Martha Altus Buller, Steven L Gill
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