Chapter 11. Supplemental Text Material. The method of steepest ascent can be derived as follows. Suppose that we have fit a firstorder

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1 S-. The Method of Steepet cent Chapter. Supplemental Text Materal The method of teepet acent can be derved a follow. Suppoe that we have ft a frtorder model y = β + β x and we wh to ue th model to determne a path leadng from the center of the degn regon x = that ncreae the predcted repone mot qucly. Snce the frt order model an unbounded functon, we cannot jut fnd the value of the x that maxmze the predcted repone. Suppoe that ntead we fnd the x that maxmze the predcted repone at a pont on a hyperphere of radu r. That The can be formulated a Max ubject to = x = y = β + = r = β x Max G = β + β x λ x r L NM = = where λ a LaGrange multpler. Tang the dervatve of G yeld G = β λx,,, x L NM G = x r λ = Equatng thee dervatve to zero reult n = x x O QP β = =,,, λ = r Now the frt of thee equaton how that the coordnate of the pont on the hyperphere are proportonal to the gn and magntude of the regreon coeffcent (the quantty λ a contant that jut fxe the radu of the hyperphere). The econd equaton jut tate that the pont atfe the contrant. Therefore, the heurtc decrpton of the method of teepet acent can be jutfed from a more formal perpectve. O QP

2 S-. The Canoncal Form of the Second-Order Repone Surface Model Equaton (-9) preent a very ueful reult, the canoncal form of the econd-order repone urface model. We tate that th form of the model produced a a reult of a tranlaton of the orgnal coded varable axe followed by rotaton of thee axe. It eay to demontrate that th true. Wrte the econd-order model a y = β + x β + x x Now tranlate the coded degn varable axe x to a new center, the tatonary pont, by mang the ubttuton z = x x. Th tranlaton produce y = β + ( z+ x ) β + ( z+ x ) ( z+ x ) = β + x β + x x + z β + z z+ x z + zz becaue from Equaton (-7) we have xz = z β. Now rotate thee new axe (z) o that they are parallel to the prncpal axe of the contour ytem. The new varable are w = M z, where MM = The dagonal matrx Λ ha the egenvalue of, λ, λ,, λ on the man dagonal and M a matrx of normalzed egenvector. Therefore, whch Equaton (-9). Λ y + zz + wmmz + w Λw + λ w = S-3. Center Pont n the Central Compote Degn In ecton -4, we dcu degn for fttng the econd-order model. The CCD a very mportant econd-order degn. We have gven ome recommendaton regardng the number of center run for the CCD; namely, 3 5generally gve good reult. The center run erve to tablze the predcton varance, mang t nearly contant over a broad regon near the center of the degn pace. To llutrate, uppoe that we are conderng a CCD n = varable but we only plan to run n c = center run. The followng graph of the tandardzed tandard devaton of the predcted repone wa obtaned from Degn-Expert: n c

3 DESIGN-EXPERT Plot ctual Factor: X = Y = StdErr of Ev aluaton Notce that the plot of the predcton tandard devaton ha a large bump n the center. Th ndcate that the degn wll lead to a model that doe not predct accurately near the center of the regon of exploraton, a regon lely to be of nteret to the expermenter. Th the reult of ung an nuffcent number of center run. Suppoe that the number of center run ncreaed to n c = 4. The predcton tandard devaton plot now loo le th: DESIGN-EXPERT Plot ctual Factor: X = Y = StdErr of Ev aluaton

4 Notce that the addton of two more center run ha reulted n a much flatter (and hence more table) tandard devaton of predcted repone over the regon of nteret. The CCD a phercal degn. Generally, every degn on a phere mut have at leat one center pont or the XX matrx wll be ngular. However, the number of center pont can often nfluence other properte of the degn, uch a predcton varance. S-4. Center Run n the Face-Centered Cube The face-centered cube a CCD wth α = ; conequently, t a degn on a cube, t not a phercal degn. Th degn can be run wth a few a n c = center pont. The predcton tandard devaton for the cae = 3 hown below: DESIGN-EXPERT Plot ctual Factor: X = Y = ctual Contant: C = StdErr of Ev aluaton Notce that depte the abence of center pont, the predcton tandard devaton relatvely contant n the center of the regon of exploraton. Note alo that the contour of contant predcton tandard devaton are not concentrc crcle, becaue th not a rotatable degn. Whle th degn wll certanly wor wth no center pont, th uually not a good choce. Two or three center pont generally gve good reult. elow a plot of the predcton tandard devaton for a face-centered cube wth two center pont. Th choce wor very well.

5 DESIGN-EXPERT Plot ctual Factor: X = Y = ctual Contant: C = StdErr of Ev aluaton S-5. Note on Rotatablty Rotatablty a property of the predcton varance n a repone urface degn. If a degn rotatable, the predcton varance contant at all pont that are equdtant from the center of the degn. What not wdely nown that rotatablty depend on both the degn and the model. For example, f we have run a rotatable CCD and ft a reduced econd-order model, the varance contour are no longer phercal. To llutrate, below we how the tandardzed tandard devaton of predcton for a rotatable CCD wth =, but we have ft a reduced quadratc (one of the pure quadratc term mng). DESIGN-EXPERT Plot ctual Factor: X = Y = StdErr of Ev aluaton

6 Notce that the contour of predcton tandard devaton are not crcular, even though a rotatable degn wa ued.