A Preview of Principal Component Analysis 1 punto posible (calificable) This problem illustrates some of the main ideas behind principal component analysis, which will be explored in detail later in this lecture as well as the next lecture. Let X1,,XnRd denote a data set, and let X denote the matrix whose i-th row is XiT. Let S=n1XT(Inn1111T)X denote the empirical covariance matrix for this data set. Consider the optimization problem argmaxu:u22=1uTSu Let u denote the unit vector that maximizes uTSu. Which of the following is a correct interpretation of u ? (Choose all that apply.) u gives a direction such that the data set uTX1,uTX2,,uTXn is clustered closely together. u is the direction that maximizes the empirical variance of the (projected) data points uTX1,uTX2,,uTXn If uTSu is very large, then if we project our data set onto the line spanned by u, we expect the projected data set to be fairly 'spread out' (i.e., the projected data set should have relatively large empirical variance)