Homework 3

Due: October 14

    1. Consider a 2-dimensional, 2-class problem where the patterns are normally distributed with the same covariance matrix, and P(C1) = P(C2) = 0.5.

        C = [2 0;0 2]

      and the mean vectors are:

        μ = [1;1]      μ = [2;1]

      1. Find the decision boundary between the two classes.
      2. Find the probability of an error.

    2. For the previous problem, assume that the classes have different covariance matrices:

        C1 = [2 1;1 4]     C2 = [4 0.5;0.5 1]

      and find the decision boundary between the two classes.

    3. For the previous problems, assume that both classes have the same covariance matrix:

        C = [2 1;1 4]

      and the a priori probabilities P(C1) = 0.25, P(C2) = 0.75 and find the decision boundary.

    4. For the previous problem, assume the following loss values:

        l11 = l (choose 1 | 1) = l22 = 0
        l12 = 0.5     l21 = 2.5
      and determine the decision boundary.