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781405779_1_homework2 - Assignment of"Pattern...

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Assignment of “Pattern Recognition” (Ch.2 Bayes Decision Theory) 1 Prove that Mahalanobis metric (distance) indeed possesses the four properties required of all metrics (distances), i.e. r (1) Nonnegativity: ( ) , 0 r a b (2) Reflexivity: ( ) ( ) , , r a b r b a = (3) Symmetry: ( ) , 0 if and only if r a b a b = = (4) Triangle Inequality: ( ) ( ) ( ) , , r a c r a b r b c + , 2. Consider two-dimensional Gaussian distribution, ( ) ( ) 1 2 1,0 , 1,0 , T T = − = μ μ 1 2 1 1 2 1 1 2 , 1 2 1 1 2 1 Σ = Σ = 2 and ( ) ( ) 1 P P ϖ ϖ = . Give the decision rule in form of
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