Chapter7-1.pdf - Chapter 7 Solutions to Review Questions 7.1 Given to us is a two dimensional Gaussian distribution with mean 1 X = 0 and variance 1 0.5

# Chapter7-1.pdf - Chapter 7 Solutions to Review Questions...

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1 Chapter 7 Solutions to Review Questions 7.1 Given to us is a two dimensional Gaussian distribution with mean - = 0 1 X and variance (a) Mahalanobis distance for a point X is given by the expression - ( ) ( ) X X K X X 1 T 2 = Here = 5 . 0 5 . 0 X − = 5 . 0 5 . 0 X X and ( ) 1 5 . 0 5 . 0 33 . 1 67 . 0 67 . 0 33 . 1 5 . 0 5 . 0 2 = − = (b) Probability density at this point is given by - ( ) ( ) ( ) ( ) 0241 . 0 exp 75 . 0 2 1 exp K 2 1 x p 2 1 2 2 1 2 = π = π = 7.2 Class 1 has points: = = = = 5 . 1 2 X , 1 2 X 5 . 1 5 . 1 X , 1 5 . 1 X 4 3 2 1 Total number of points are 4. Therefore estimated mean matrix is given by – = 1 5 . 0 5 . 0 1 K = 33 . 1 67 . 0 67 . 0 33 . 1 K 1
2 = = + + + = = = 25 . 1 75 . 1 5 7 4 1 5 . 1 2 1 2 5 . 1 5 . 1 1 5 . 1 4 1 X 4 1 X 4 1 R R Covariance matrix is given by – T R R R X X X X X K ) )( ( 4 1 4 1 = = ( ) ( ) ( ) ( ) + + − + = 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 25 . 0 4 1 + + + = 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 0625 . 0 4 1 = 0625 . 0 0 0 0625 . 0 Class 2 has four points: = − = = = 1 0 y , 0 1 y 1 0 y , 0 1 y 4 3 2 1 Therefore estimated mean matrix is given by – = + − + + = = = 0 0 1 0 0 1 1 0 0 1 4 1 Y 4 1 Y 4 1 R R Covariance matrix is given by – T R R R Y Y Y Y Y K ) ( ) ( 4 1 4 1 = = ( ) ( ) ( ) ( ) + − + + = 1 0 1 0 0 1 0 1 1 0 1 0 0 1 0 1 4 1 + + + = 1 0 0 0 0 0 0 1 1 0 0 0 0 0 0 1 4 1
3 = = 5 . 0 0 0

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• Summer '18
• Shiv
• Trigraph, Covariance matrix

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