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CH2MultiObs.pdf

# CH2MultiObs.pdf - Chapter 2 Multivariate Observations 1...

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Chapter 2 Multivariate Observations 1. Random Vectors Expectation : For X ( X 1 , · · · , X k ) t : vector of random variables, E ( X ) ( EX 1 , · · · , EX k ) t For W ( W ij ) : matrix of random variables, E ( W ) ( EW ij ) Variance and Covariance matrices : For X ( X 1 , · · · , X m ) t and Y ( Y 1 , · · · , Y n ) t , Cov( X, Y ) (Cov( X i , Y j )) ( m × n ) Var( X ) Cov( X, X ) ( m × m ) Alternative expressions and formulae : (i) Cov( Y, X ) = (Cov( X, Y )) t (ii) Cov( X, Y ) = E { ( X µ )( Y ξ ) t } , µ E ( X ) , ξ E ( Y ) = EXY t E ( X )( E ( Y )) t (iii) Var( X ) = E { ( X µ )( Y ξ ) } = EXX t E ( X )( E ( X )) t Linearity and bilinearity : (i) E ( X + Y ) = E ( X ) + E ( Y ) E ( AX ) = AE ( X ) , E ( XB ) = E ( X ) B ( A, B : constants) 1

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(ii) Cov( X + Y, Z ) = Cov( X, Z ) + Cov( Y, Z ) Cov( X, Y + Z ) = Cov( X, Y ) + Cov( X, Z ) Cov( AX, Y ) = A Cov( X, Y ) Cov( X, BY ) = Cov( X, Y ) B t ( A, B : constants) (iii) Cov( AX 1 + BX 2 ) = A Var( X 1 ) A T + A Cov( X 1 , X 2 ) B T + B Cov( X 2 , X 1 ) A T + B Var( X 2 ) B T . Variance matrix : (i) Var( AX ) = A Var( X ) A t Var( X + b ) = Var( X ) ( A, b : constants) (ii) Var( X + Y ) = Var( X ) + Cov( X, Y ) + Cov( Y, X ) + Var( Y ) (iii) Var( X ) : non-negative definite a Var( X ) a = Var( a X ) 0 a All eigen values of Var( X ) are non-negative. Var( X ) = Σ 1 / 2 Σ 1 / 2 for a symmetric Σ 1 / 2 ; Σ 1 / 2 P diag( p λ i ) P where Var( X ) = P diag( λ i ) P , PP = P P = I . (iv) Var( X ) : singular ⇔ ∃ a ̸ = 0 : P ( a ( X µ ) = 0) = 1 (v) Var( X ) : non-singular Var( X ) : positive definite a Var( X ) a > 0 a ̸ = 0 All eigen values of Var( X ) are positive. Var( X ) = Σ 1 / 2 Σ 1 / 2 for invertible symmetric Σ 1 / 2 ; Σ 1 / 2 1 / 2 ) 1 2
Correlation matrix : Corr( X i , X j ) = Cov( X i , X j ) p Var( X i ) p Var( X j )

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