VIEWS: 6 PAGES: 1 POSTED ON: 2/21/2011
Q1. Consider a two-category classification problem with two-dimensional feature vector X = ( x1, x 2 ). The two categories are ω1 and ω 2 , with equal prior and 0-1 loss function. ⎡0⎤ p( X ω1 ) ~ N ( ⎢ ⎥, Σ1 ) ⎣1⎦ ⎡2⎤ p( X ω 2 ) ~ N ( ⎢ ⎥, Σ 2 ) ⎣0⎦ 1 P(ω1 ) = P(ω 2 ) = 2 ⎡1 0⎤ ⎡1 1 ⎤ Σ1 = ⎢ ⎥, Σ 2 = ⎢1 2⎥, ⎣0 1⎦ ⎣ ⎦ Generate 100 bi-variate random training samples from each of the two densities. (a) Plot these samples in the two-dimensional feature space. Sketch the Bayes decision boundary when the true parameters are known. (b) Now we assume that µ1, µ2 , Σ1, Σ 2 are not known. Find the maximum-likelihood estimates of µ1, µ2 , Σ1, Σ 2 using these training samples. (c) Find the decision boundary using the estimated parameters. And draw the decision boundary in the same figure as part (a). Whether the two decision boundary are the same, why? (d) Repeat parts (b) and (c) by drawing another set of 100 random samples from each class. Whether the decision boundary using different training sets are the same, why?
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