Suppose you have the following two dimensional dataset as shown in Figure 6 and you want to reduce the dimension of this dataset to one dimension using PCA. Suppose the principal component (eigenvector) of dataset corresponding to the largest eigenvalue of its co-variance matrix is (0.6941, 0.7199) and the principal component (eigenvector) corresponding to the second largest eigenvalue is (0.7199, 0.6941). Answer the following questions.

(a) Suppose we perform dimension reduction from R2 to R using only the rst principal component. How do you represent data #2 after this dimension reduction step?

(b) What is the (x,y) coordinates in the original space, i.e., in 2-D, for data #2 after performing dimension reduction using only rst principal component?

(c) What is the (x,y) coordinates in the original space, i.e., in 2-D, for data #2 after performing dimension reduction using only second principal component?

(d) Suppose we perform dimension reduction for data #2 using only rst principal component. What is the reconstruction error? Remember, in this case reconstruction error is simply the Euclidean distance between data #2 in original space i.e., 2D and (x, y) coordinate of data #2 after performing dimension reduction using only rst principle component.

(e) Suppose we decide to represent the data #2 using rst two principal components. Without doing any calculation what do you think will be the reconstruction error?

24, 03 0.77 -0 4 19. 3019. 38 . 242. 77 9. 68 711.59 2. 06 6 9.74 -6, 08 Figure 6: PCA data