pca with guassian noise data

Question 3 Principal Component Analysis Consider a large 5-dimensional data set D {x},x2,...,xM}, X' ERS, 1 = 1, ...,N, N = 100,000,000. By applying Principal Component Analysis (PCA) to D, one obtains eigenvalues , 10, 12 = 4, 13 = /4 = 15 = 0.05 of the data covariance matrix C. together with their corresponding eigenvectors uz, ..., u R5. (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? [7 marks] (b) Assume now that you were told that the data D will be transmitted to your friend, but unfortunately the communication channel is unreliable and all elements of the data will be corrupted an additive independent identially distributed (i.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. , u's and X1, ..., \'s. What relation would you expect between the covariance matrices C and C', eigenvectors ui...., u and u. eigenvalues 11. ..., 15 and eigenvalues 11. .. XG? [13 marks) Justify your answer. Question 3 Principal Component Analysis Consider a large 5-dimensional data set D {x},x2,...,xM}, X' ERS, 1 = 1, ...,N, N = 100,000,000. By applying Principal Component Analysis (PCA) to D, one obtains eigenvalues , 10, 12 = 4, 13 = /4 = 15 = 0.05 of the data covariance matrix C. together with their corresponding eigenvectors uz, ..., u R5. (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? [7 marks] (b) Assume now that you were told that the data D will be transmitted to your friend, but unfortunately the communication channel is unreliable and all elements of the data will be corrupted an additive independent identially distributed (i.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. , u's and X1, ..., \'s. What relation would you expect between the covariance matrices C and C', eigenvectors ui...., u and u. eigenvalues 11. ..., 15 and eigenvalues 11. .. XG? [13 marks) Justify your