8kl - CSE-5120-Fall-2009 To reduce the number of dimensions...

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CSE-5120-Fall-2009 To reduce the number of dimensions Karhunen-Loeve Expansion Eigenvectors Covariance Matrix Transformation Dimension Elimination For k-NN querying multidimensional data: 1. with all data points, go through dimen- sion reduction process. Keep resulting data points in some search structure such as the R-tree. 2. Given any query point, transform the query in the same way, then look for the nearest neighbors in the search structure. Eigenvalues and Eigenvectors A number λ is called an eigenvalue (or char- acteristic value) of a n × n matrix A if there exists a vector x 6 = 0 such that Ax = λ · x The vector x is then called an eigenvector of A belonging to λ . For example, A = " - 5 2 2 - 2 # det ( A - λI ) = f f f f f - 5 - λ 2 2 - 2 - λ f f f f f = λ 2 + 7 λ + 6 = 0 Its solutions are λ 1 = - 1 and λ 2 = - 6. For λ = λ 1 = - 1, we have ( A - λI ) x = 0 ˆ - 5 - λ 2 2 - 2 - λ x 1 x 2 ! = ˆ 0 0 ! - 4 x 1 + 2 x 2 = 0 2 x 1 - x 2 = 0 A solution of the equations is x 1 = 1 , x 2 = 2. Hence an eigenvector corresponding to λ 1 = - 1 is x = ˆ 1 2 ! Similarly an eigenvector corresponding to λ 2 = - 6 is x = ˆ 2 - 1 ! 55
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Expected vector Expected vector or mean of a set of vectors X . M = E { X } student Assignment Mid-term Final S1 90 100 90 S2 60 60 60 S3 50 50 30 S4 50 60 10 S5 40 80 30 S6 20 30 50 S7 40 40 10 Mean 50 60 40 The given vectors are 90 100 90 , 60 60 60 , 50 50 30 , ... The mean values are: M = 50 60 40 Form new set of vectors: 90 - 50 100 - 60 90 - 40 , 60 - 50 60 - 60 60 - 40 , ... Y = 40 10 0 0 - 10 - 30 - 10 40 0 - 10 0 20 - 30 - 20 50 20 - 10 - 30 - 10 10 - 30 Covariance Matrix : C = 1 7 Y Y T where Y Y T is µ 40 10 0 0 - 10 - 30 - 10 40 0 - 10 0 20 - 30 - 20 50 20 - 10 - 30 - 10 10 - 30 40 40 50 10 0 20 0 - 10 - 10 0 0 - 30 - 10 20 - 10 - 30 - 30 10 - 10 - 20 - 30 C = 1 7 2800 2500 2300 2500 3400 2200 2300 2200 5000 assign midterm final assign midterm final Mid-term Final Positive Covariance Time Bugs Negative Covariance 56
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$ Happiness Covariance 0 Covariance matrix Indicates the dispersion of a distribution. Covariance matrix is deFned by Σ = E { ( X - M )( X - M ) T } = E x 1 - m 1 : : x n - m n [ x 1 - m 1 ...x n - m n ] = E ( x 1 - m 1 )( x 1 - m 1 ) ... ( x 1 - m 1 )( x n - m n ) : : : : ( x n - m n )( x 1 - m 1 ) ... ( x n - m n )( x n - m n ) = E { ( x 1 - m 1 )( x 1 - m 1 ) } ... E { ( x 1 - m 1 )( x n - m n ) } : : : : E { ( x n - m n )( x 1 - m 1 ) } ... E { ( x n - m n )( x n - m n ) } = c 11 ... c 1 n : : c n 1 ... c nn where c ij = E { ( x i - m i )( x
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8kl - CSE-5120-Fall-2009 To reduce the number of dimensions...

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