Chemical Engineering Hand Written_Notes_Part_90

Chemical Engineering Hand Written_Notes_Part_90 - 182 5....

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182 5. OPTIMIZATION AND RELATED NUMERICAL SCHEMES Using su cient condition for optimality , Hessian matrix should be positive de f nite or positive semi-de f nite for the stationary point to be a minimum. Now, (4.24) " 2 £ e T We ¤ θ 2 # =2 ( Φ T W Φ ) It can be easily shown that (4.25) v T ¡ Φ T W Φ ¢ v 0 for any v R m and the su ciency condition is satis f ed and the stationary point is a minimum. As Φ is a convex function, it can be shown that the solution ˆ θ is the global minimum of Φ . Thus, linear least square estimation problem is f nally reduced to solving equation of the form Ax = b where (4.26) A = Φ T W Φ and b = Φ T Wy Note that Φ T W Φ is symmetric and positive semi-de f nite and Cholensky decomposition method can be used to solve the resulting linear system. 5. Projections: Geometric Interpretation of Linear Regression 5.1. Distance of a Point from a Line. Suppose we are given a point y R 3 in space and we want to f
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Chemical Engineering Hand Written_Notes_Part_90 - 182 5....

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