Chemical Engineering Hand Written_Notes_Part_95

Chemical Engineering Hand Written_Notes_Part_95 - 192 5....

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192 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Here s i,k are ( i, k ) th elements of sample covariance matrix S ϕ of the random variable vector ϕ . Alternatively, this matrix can also be estimated as S ϕ = Cov ( ϕ )= 1 N 1 N X j =1 h ϕ ( j ) b ϕ ih ϕ ( j ) b ϕ i T (6.17) = 1 N 1 Φ 1 ³ b ϕ ´ T ¸ T Φ 1 ³ b ϕ ´ T ¸ (6.18) It may be noted that sample covariance matrix S ϕ is an estimated of population covariance matrix Σ . It may be noted that Finally , sample correlation coe cients (normalized or standardized covariances) are de f ned as r i,k = P N j =1 h φ j,i b ϕ i ih φ j,k b ϕ k i r P N j =1 h φ j,i b ϕ i i 2 r P N j =1 h φ j,k b ϕ k i 2 (6.19) = s i,k s i,i s k,k = s i,k s i s k = £ e ( i ) ¤ T e ( k ) q [ e ( i ) ] T e ( i ) q [ e ( k ) ] T e ( k ) =cos( θ ι,k ) i =1 , 2 , ...m ; i =1 , 2 , ...m where s i and s k represents sample standard deviations of ϕ i and ϕ k , respectively,
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This note was uploaded on 11/26/2011 for the course EGN 3840 taught by Professor Mr.shaw during the Fall '11 term at FSU.

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Chemical Engineering Hand Written_Notes_Part_95 - 192 5....

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