Fitting Quadratic Response Surfaces with Ridges using Nonlinear Regression
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1 Fttn Quadratc Response Surfaces wth Rdes usn Nonlnear Reresson Copyrht 3 by Bruce Anenman. All rhts reserved. Statonary Rdes n a Response Surface The estence of statonary rdes can often be eploted to mantan hh qualty whle reducn cost or complety. Copyrht 3 by Bruce Anenman. All rhts reserved.
2 Copyrht 3 by Bruce Anenman. All rhts reserved. 3 The Standard Quadratc Model ( B + + y E B Parameters 3, (,,,,,,,,,,,, + Copyrht 3 by Bruce Anenman. All rhts reserved. 4 The A-Canoncal Equaton The columns of the matr D are the eenvectors of B. The s are the eenvalues of B. By Defnn New Varables: D ( + + y E ( + + Λ Λ Λ Λ E y Λ
3 Quadratc Surfaces can only tae Certan Forms (If far from ero, the quadratc terms domnate the lnear terms. Mamum s< Mnmum s> Saddle Pont s> s< Statonary Rde s s< Copyrht 3 by Bruce Anenman. All rhts reserved. 5 Statonary Pont Rsn Rde s< ª < Epermental Reon Copyrht 3 by Bruce Anenman. All rhts reserved. 6
4 The Eenvalues n the Canoncal Form of the Ftted Surface Indcate Rde Behavor - - OR Copyrht 3 by Bruce Anenman. All rhts reserved. 7 The Eenvalues n the Canoncal Form of the Ftted Surface Indcate Rde Behavor - - AND - - Copyrht 3 by Bruce Anenman. All rhts reserved. 8
5 If there are two eenvalues near ero, then there s a plane, not a lne at the top of the rde. T : Temperature (C 7 A Rsn Rde 6 58% t : tme (hours 5 6% % Path of Steepest Ascent On the top of the rde. 6% 54% 35 6% 68% % 6% 5 c : concentraton (% Copyrht 3 by Bruce Anenman. All rhts reserved. 9 The A-Canoncal Model D E ( y + + Parameters (,,,,, (, (,, ( ξ Where are the other parameters? Copyrht 3 by Bruce Anenman. All rhts reserved.
6 Nonlnear Parameteraton Parametere D as a seres of planar rotaton (Gvens matrces. Each Gvens matr loos le an dentty matr n all rows ecept q and r. The (q,q and (r,r elements are cos(θ and the (q,r and (r,q elements are sn(θ and sn(θ, respectvely. All other elements n the two rows are ero. cos( θ3 sn( θ3 cos( θ sn( θ D ( θ cos( sn( θ 3 θ 3 sn( θ cos( θ sn( θ 3 cos( θ 3 sn( θ3 cos( θ3. Vector of rotaton anles, θ Copyrht 3 by Bruce Anenman. All rhts reserved. The Canoncal Model can now be ft wth a non-lnear model. D ( θ ( + ( θ + ( θ Λ ( θ E y D D D ( E y + + Λ Parameters (,,,, θ, θ3,, θ θ3, θ4,, θ θ,,,, ξ Copyrht 3 by Bruce Anenman. All rhts reserved.
7 Assume: < and A -Dmensonal Statonary Rde has: and E + + ( y + + Copyrht 3 by Bruce Anenman. All rhts reserved. 3 But what about those anles? The rotaton of θ 3 doesn t necessarly tae place n the plane of & 3 because only the last rotaton taes place fully n the plane of a par canoncal aes. Copyrht 3 by Bruce Anenman. All rhts reserved. 4
8 However, f an anle s set to ero, then that Gvens matr becomes the dentty and order doesn t matter. θ 3 cos( θ sn( θ D ( θ cos( θ sn( θ sn( θ cos( θ. 3 3 sn( θ3 cos( θ3 Copyrht 3 by Bruce Anenman. All rhts reserved. 5 To ft a -dmensonal statonary rde, the model s reft after settn: θ < Parameters (, +, +,, θ +, +, θ +, + 3,, θ +, θ +, + 3,, θ +, θ, ( +, ( +,, ( ξ ξ - - Copyrht 3 by Bruce Anenman. All rhts reserved. 6
9 Etra Sum of Squares Test Full Model: Statonary Rde Model: ( + ( θ + ( θ Λ ( θ E y D D D θ < ˆ ˆ ' s & ˆ ' s θˆ' s Derees of Freedom used: Full Model Reduced Model Copyrht 3 by Bruce Anenman. All rhts reserved. 7 (- Assume: < and A -Dmensonal Rsn Rde has: and A snle drecton of steepest ascent E + + ( y Copyrht 3 by Bruce Anenman. All rhts reserved. 8
10 To ft a -dmensonal rsn rde, the model s ft wth the follown reductons:. θ < Parameters (,, +, +,, θ, +, θ, +,, θ, θ+, + 3,, θ +, θ, ( +, ( +,, ( ξ -+ - Copyrht 3 by Bruce Anenman. All rhts reserved. 9 Etra Sum of Squares Test Full Model: ( + ( θ + ( θ Λ ( θ E y D D D Rsn Rde Model: θ < ˆ ˆ ' s & ˆ ' s θˆ' s Derees of Freedom used: Full Model Reduced Model Copyrht 3 by Bruce Anenman. All rhts reserved. (-+
11 Hypothess Test for Rsn Rde H H : : Statonary Rde Rsn Rde Copyrht 3 by Bruce Anenman. All rhts reserved. Partal F Test for a Rsn Rde E( y + D( θ + D( θ ΛD ( θ Rsn Rde Model θ < Statonary Rde Model θ < Derees of Freedom used: Larer Model Reduced Model (Rsn (Statonary ˆ ˆ ' s & ˆ ' s (-+ (- θˆ' s Copyrht 3 by Bruce Anenman. All rhts reserved.
12 Paper n current ssue of IIE (Insttute of Industral Enneers Transactons Anenman, B. E. (3, Identfyn rsn rde behavor n quadratc response surfaces, IIE Transactons, 35, Copyrht 3 by Bruce Anenman. All rhts reserved. 3
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