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Unformatted text preview: Problem Set 3 MAS 622J/1.126J: Pattern Recognition and Analysis Due Monday, 16 October 2006 [Note: All instructions to plot data or write a program should be carried out using either Python accompanied by the matplotlib package or Matlab. Feel free to use either or both, but in order to maintain a reasonable level of consistency and simplicity we ask that you do not use other software tools.] Problem 1: (DHS 2.6) Optimal Decision Boundaries Your friend has built a system to recognize into which of two categories, 1 or 2 , her advisors email can be classified. She has brilliantly identified two features such that her training data is well approximated by two Gaussians: p ( x | 1 ) N ( 1 , 1 ) p ( x | 2 ) N ( 2 , 2 ) where 1 = [8 9] T , 2 = [0 9] T , 1 = I, and 2 = 16 , where I is the identity matrix. 16 a. Plot the one-sigma ellipses for these two classes in the place x = [ x 1 x 2 ] T . b. Your friend finds that choosing a threshold at x 1 = 4 perfectly separates the training examples she has; thus, she proposes that this should be the best classifier. Show her an expression, in terms of x , which can improve her classifier with respect to minimizing the Bayes probability of error. Assume that email from class 2 is twice as likely as email from class 1 . c. The shape of this optimal decision boundary is: a line a parabola a hyperbola a circle an ellipse none of the above (explain) 1 Be sure to justify your answer.answer....
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