Consider a large 5-dimensional data set D = {x1,x,...,x"}, x' ER5, i = 1,...,N. N = 100.000.000. By applying Principal Component Analysis (PCA) to D. one obtains eigenvalues 11 = 10,12 = 4, 13 = 1a = ls = 0.05 of the data covariance matrix C, together with their corresponding eigenvectors up,...,U5 R. (a) Describe the nature of the data distribution in the 5-dimensional space. How many eigenvectors are needed so that when the data is projected onto the low-dimensional subspace spanned by them only 1% of data variability is lost? (b) Assume now that you were told that the data will be transmitted to your friend, but unfortunately the communication channel is unreliable and all elements of the data will be corrupted by an additive independent identically distributed (1.i.d.) Gaussian noise of zero mean and variance o2 > 0. Your friend will apply PCA on the received data by calculating the data covariance matrix C' and finding its eigenvectors u... , us and X...., 45. What relation would you expect between the covariance matrices C and C eigenvectors uz, ..., us and u..., u's, eigenvalues 11, ..., and eigenvalues X4..., ? Justify your answer Consider a large 5-dimensional data set D = {x1,x,...,x"}, x' ER5, i = 1,...,N. N = 100.000.000. By applying Principal Component Analysis (PCA) to D. one obtains eigenvalues 11 = 10,12 = 4, 13 = 1a = ls = 0.05 of the data covariance matrix C, together with their corresponding eigenvectors up,...,U5 R. (a) Describe the nature of the data distribution in the 5-dimensional space. How many eigenvectors are needed so that when the data is projected onto the low-dimensional subspace spanned by them only 1% of data variability is lost? (b) Assume now that you were told that the data will be transmitted to your friend, but unfortunately the communication channel is unreliable and all elements of the data will be corrupted by an additive independent identically distributed (1.i.d.) Gaussian noise of zero mean and variance o2 > 0. Your friend will apply PCA on the received data by calculating the data covariance matrix C' and finding its eigenvectors u... , us and X...., 45. What relation would you expect between the covariance matrices C and C eigenvectors uz, ..., us and u..., u's, eigenvalues 11, ..., and eigenvalues X4..., ? Justify your