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midterm_solutions_2009_11_12_01 - EE263 S Lall...

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EE263 S. Lall 2009.11.12.01. Midterm exam 09-10 Solutions This is a 24-hour take-home exam. You may use any books, lecture-notes or computer programs that you wish, but you may not discuss this exam with anyone until Saturday Nov 7, after everyone has taken it. The only exception is that you can ask the TA’s or Sanjay Lall for clarification. Web page for typos and datafiles. The web page http://junction.stanford.edu/~lall/ee263/midterm contains any needed files, and it should always contain all known typos; check this before contacting one of us for clarification. Please note that we have tried pretty hard to make the exam unambiguous and clear, so we are unlikely to say much. If you choose to send out an email for clarification, please use the staff email address [email protected] so that you can get reply as soon as possible. Since you have 24 hours, we expect your solutions to be legible, neat, and clear. Do not hand in your rough notes, and please try to simplify your solutions as much as possible. Good Luck! 1. Minimizing the perpendicular distance 0025 Suppose c R n and v R n , and v is a unit vector. The set L = { vz + c | z R } is a line in R n . For any point x R n the distance between x and L is defined by d ( x,L ) = min y L bardbl x y bardbl We have m points x 1 ,...,x m R n , and we would like to find the line L that minimizes the total squared distance d total = m summationdisplay i =1 d ( x i ,L ) 2 This is illustrated below. x 4 x 3 x 2 x 1 That is, we’d like to fit the points to a straight line, as close as possible in the sense that d total is minimized. (a) Show that d ( x,L ) = bardbl ( I vv T )( x c ) bardbl (b) Define Q = I vv T . Show that Q is a projection matrix. What are its eigenvalues? What is the dimension of null( Q )? 1
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EE263 S. Lall 2009.11.12.01. (c) We now proceed to solve the above optimization problem. Show that the optimal c must satisfy Q ( c μ ) = 0 where we define μ to the be centroid of the points, given by μ = 1 m m summationdisplay i =1 x i Hence show that one optimal c is given by c opt = μ Are there any other choices of c which are optimal? (d) With the above choice of c , the total distance squared is given by d total = m summationdisplay i =1 bardbl ( I vv T )( x i μ ) bardbl 2 What is the optimal v , in terms of the x i and μ ? (e) The file min perp distance.m contains a matrix X R 2 × m . Let x i be the i ’th column of X . Find the optimal c and v , and plot the resulting line and the data points. Solution. (a) We have d = min z bardbl vz + c x bardbl and so the least-squares solution is z = ( v T v ) 1 v T ( x c ) = v T ( x c ) which gives the result. (b) Pick a matrix U 2 so that W = bracketleftbig v U 2 bracketrightbig is orthogonal. Then it is easy to verify that Q = W bracketleftbigg 0 0 0 I bracketrightbigg W T and so the eigenvalues of Q are 0 and 1.
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