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Unformatted text preview: 1 Stat Stat 430/830: 430/830: Analysis Analysis of of Variance Variance Lecture November 3, 2010 Response Surface Methods (RSM) Response Surface Methods (RSM) and Designs and Designs Techniques used to optimize the response influenced by several controllable variables. Example: suppose we need to optimize the yield in a process influenced by temperature ( x 1 ) and pressure ( x 2 ). The process yield can be represented as y = η (x 1 , x 2 ) + ε where ε represents a noise or error in the observed response. The function η (x 1 , x 2 ) = E(y) is called the response surface RSM RSM Contour Plot Response Surface RSM RSM Usually we do not know the form of η . Then we approximate it using loworder polynomial responses. Firstorder (linear) model y = β o + β 1 ⋅ x 1 + … + β k ⋅ x k + ε Secondorder (quadratic) model ε β β β β + + + + = ∑ ∑ ∑ ∑ < = = j i j i ij k i i ii k i i i o x x x x y 1 2 1 2 RSM RSM – Bias and Variance Bias and Variance bias. x x x x x x x x x or error c systemati the is ) f( ) ( ) ( where ) ( ) ( f ) ( y Then, ). R( interest of region the over ) ( f ~ E(y) e approximat we Assume . x (x let and variables coded the are ( 0.5( x Assume k 1 i i i i i i η δ ε δ ε η ξ ξ ξ ξ ξ = + + = + = = + = + + ) , ... , ) ) RSM RSM – Bias and Variance Bias and Variance ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 1 2 2 2 2 2 2 2 2 2 2 2 1 ˆ . / ) ( 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ x x N i i x x 1 x V )/ y NV( variance normalized The N x x with x N 1 x)' (1 X) (X' x) (1 bx) V(a ) y V( E(b)x E(a) ) y E( Then bx. a y equation square least the provides x with points data N of set particular a Suppose . ) ( ) y E( y V ) ( ) y E( ) y ( E y E ) ( ) y E( ) y E( y E ) ( y E is ) f( y ion approximat the of error square mean normalized The σ σ σ σ σ η η η η + = = = + = ⋅ ⋅ = + = + = + = = + = + = + = = ∑ = Example x x x x x RSM RSM – Bias and Variance Bias and Variance 32 27 1 3 / 4 8 27 1 3 / 2 5 . 1 1 2 2 2 2 x V x : (b) Design x V x : (a) Design x x V Then ). x ,0, (x at points three just with designs two Consider . x o x o o x o o + = = + = = + = Example RMS Sequential Procedure RMS Sequential Procedure First Order Model First Order Model Objective Find a path of improvement towards the vicinity of the optimum 3 Steepest Ascent / Descent Steepest Ascent / Descent First Order Model First Order Model Steepest Ascent : move in the direction of maximum increase. Maximization is desired Steepest Descent : move in the direction of maximum decrease. Minimization is desired For a linear model the contours are parallel lines. Steps proportional to the regression coefficients β i ....
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 Fall '03
 StefanSteiner
 Variance, Response surface methodology, order model, Second Order Models

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