Chemical Engineering Hand Written_Notes_Part_97

Chemical Engineering Hand Written_Notes_Part_97 - 196 5....

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196 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES that y i = £ ϕ ( i ) ¤ T θ + e i for i =1 , 2 ...N (6.41) cov ( e i ,e j )= 0 when i 6 = j ; i =1 , 2 ...N, j =1 , 2 ...N (6.42) var ( e i )= σ 2 for i =1 , 2 ...N (6.43) The parameter vector θ and the error vector e are not correlated. Thus, statistical model for experimental data given by equation (6.41) can be expressed as y = Φ θ T + e (6.44) Φ = £ ϕ (1) ¤ T ........ £ ϕ ( N ) ¤ T (6.45) where θ T represent the true parameter values and vectors y and e have been de f ned by equations (4.12) and (4.13), respectively. Taking expectation (mean) on both sides of the above equation., we have (6.46) E ( y )= E ( Φ θ T + e )= Φ θ T From least square analysis, we have least square solution , given as b θ =( Φ T Φ ) 1 Φ T y (6.47) E ( b θ )=( Φ T Φ ) 1 Φ T E ( y )=( Φ T Φ ) 1 Φ T E ( Φ θ T + e ) (6.48) =( Φ T Φ ) 1 Φ T Φ θ T = θ T (6.49) The above result guarantees that, if we collect su
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Chemical Engineering Hand Written_Notes_Part_97 - 196 5....

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