Chapter 4 Desg of Expermets 4. Itroducto I Chapter 3 we have cosdered the locato of the data pots fxed ad studed how to pass a good respose surface through the gve data. However, the choce of pots where expermets (whether umercal or phscal) are performed has ver large effect o the accurac of the respose surface, ad ths chapter we wll explore methods for selectg a good set of pots for carrg out expermets. The selecto of these pots s kow as desg of expermets. Desg of expermets s heretl a mult-objectve optmzato problem. We would lke to select pots so that we maxmze the accurac of the formato that we get from the expermets. We usuall also would lke to mmze the umber of expermets, because these are expesve. I some cases the objectve of the expermets s to estmate some phscal characterstcs, ad these cases, we would lke to maxmze the accurac of these characterstcs. However, the desg applcatos, whch are of prmar terest to us, we would lke to costruct a respose surface that could be used to predct the performace of other desgs. I ths case, our prmar goal s to choose the pots for the expermets so as to maxmze the predctve capablt of the model. A lot of work has bee doe o expermetal desgs regular desg domas. Such domas occur whe each desg varable s bouded b smple lower ad upper lmts, so that the desg doma s box lke. Occasoall, sphercal domas are also cosdered. Sometmes each desg varable ca take ol two or three values, ofte called levels. These levels are termed low, omal ad hgh. I other cases, the desg space s approxmatel box lke, but t s possble to carr expermets wth the desg varables takg values outsde the box for the purpose of mprovg the propertes of the respose surface. I the ext secto we wll summarze brefl some of the propertes of expermetal desg box-lke domas, ad preset some of the more popular expermetal desgs such domas. For desg optmzato, however, t s commo for us to tr ad create respose surfaces rregularl shaped domas. I that case, we have to create our ow expermetal desg. Secto 4.3 wll dscuss several techques avalable for fdg good desgs a rregular shaped doma. 48
4. Desg of Expermets Boxlke Domas I ths case the rego of terest s defed b smple lower ad upper lmts o each of the desg varables ' x x x,,...,, (4..) l u where x l, ad x u are the lower ad upper lmts, respectvel, o the desg varable x '. The prme dcates that the desg varable has ot bee ormalzed. For coveece we ormalze the desg varable as x ' xl x x, (4..) xu xl The ormalzed varables are the all boud the cube x, (4..3) 4.. Iterpolato, extrapolato ad predcto varace The smplest expermetal desg for the cube s oe expermet at each oe of the vertces (Matlab ff). Ths desg s called a -level full factoral desg, where the word `factoral' refers to 'factor', a som for desg varable, rather tha the factoral fucto. For a small umber of desg varables, ma be a maageable umber of expermets. More geeral full-factoral desgs ma have dfferet umber of levels dfferet drectos (Matlab fullfact). However, for larger values of, we usuall caot afford eve the two-level full factoral desg. For example, for 0, we get 04. For hgh values of we ma cosder fractoal factoral desgs, whch do ot clude all the vertces (see Matlab s fracfact). If we wat to ft a lear polomal to the data, t certal appears that we wll ot eed awhere ear pots for a good ft. For example, for = 0, we have coeffcets to ft, ad usg 04 expermets to ft these coeffcets ma appear excessve eve f we could afford that ma expermets. However, wth fewer expermets we lose a mportat propert of usg the respose surface as a terpolato tool rather tha as a tool for extrapolato. To uderstad that we wll frst defe what we mea b terpolato ad extrapolato. Itutvel, we sa that our respose surface wll terpolate the data at a pot, f that pot s `completel surrouded' b data pots. I oe dmesoal space, ths meas that there s a pot to the rght of the terpolated pot, as well as a pot to the left of t. I two-dmesoal space we would lke the pot to be surrouded b at least 3 pots, so that t falls sde the tragle defed b these 3 pots. Smlarl, threedmesoal space, we would lke the pot to be surrouded b at least 4 pots, that s le sde the tetrahedro defed b these 4 pots. I -dmesoal space, we would lke the pot to be surrouded b + data pots, or other words, ft sde the smplex defed b the + pots. A smplex s the geeralzato of a tragle ad a 49
tetrahedro; a shape dmesoal space defed b + learl depedet pots. Gve a set of + pots -dmesoal space, x, x,..., x, the smplex defed b these pots cludes all the pots that ca be obtaed b a covex sum of these pots. That s, t cludes a pot x, whch ma be wrtte as x x, (4..4) Wth, ad 0,,,,,,,, (4..5) Gve a set of data pots, the set of pots where the respose surface performs terpolato s the uo of all of these smplexes, whch s also called the covex hull of the data pots. Oe measure that we ca use to estmate the loss of predcto accurac curred whe we use extrapolato s the predcto varace. Recall that the lear regresso model that we use ma be wrtte as ˆ b ( x ), (4..6) ( m) Defg the vector x b ( m) x ( x ), (4..7) we ca wrte Eq. 4..6 as ( mt ) ˆ x b, (4..8) Wth ose the data, b has some radomess to t, whle x ( m) s determstc. Usg Eq. 3..9, t s eas to show that the varace (square of the stadard devato) of ŷ s gve as X X Var ˆ x x x x x (4..9) ( mt ) ( m) ( mt ) T ( m) [ ( )] b, wth the stadard devato of ˆ( x) beg the square root of ths expresso. If we use the estmate ˆ stead of, we get the stadard error s of ŷ s X X ( mt ) T ˆ ( m ) x x (4..0) The stadard error gves us a estmate of the sestvt of the respose surface predcto at dfferet pots to errors the data. We would lke to select pots so as to make ths error as small as possble everwhere the doma where we would lke to estmate the respose. Itutvel, t appears that ths would be helped f the stadard error dd ot var much from oe pot to aother. Ths propert s called stablt. The followg example demostrates the effect of usg extrapolato o the stablt of the stadard error. 50
Example 4.. Cosder the problem of fttg a lear polomal b bx b3 x to data the square x, x. Compare the maxmum value of the predcto varace for two cases; (a) Full factoral desg (pots at all 4 vertces). (b) A fractoal desg cludg 3 vertces, obtaed b omttg the vertex (, ). T T T T x,, x,, x,, x,. 3 4 Full factoral desg: We umber the pots as For ths case we have Also 0 0, T X X X 4 0 0 0 0 x x ( ) x m ( m) T T ( m), x X X x 0.5( x x ),, (4..) (4..) Usg Eq. 4..0, we see that the stadard error of the respose vares betwee s ˆ at the org ad s 3ˆ at the vertces. Ths case represets a farl stable varato of the error of ol 3 betwee the smallest ad hghest value the doma of terest. Fractoal factoral desg: Ths tme we have 3 - T X, 3 - X X, (4..3) - 3 ad T X X 0.5 ( mt ) T ( m), x X X x 0.5 xx x x xx, (4..4) At the org the stadard error s stll s ˆ, at the three vertces we get s ˆ. Ths s expected, as wth ol 3 data pots the respose surface passes through the data, so that the error at the data pots should be measuremet error. However, at the fourth vertex (,), where the respose surface represets extrapolato, we get s 3 ˆ. B settg the dervatves of the predcto error to zero, we easl fd that the mmum error s at the cetrod of the three data pots, at ( 3, 3). At the cetrod s ˆ 3.Now the rato betwee the smallest ad hghest stadard errors s 5., wth the hghest errors the rego of extrapolato. 5
It ca be checked that whe we use a full factoral desg for a lear polomal T varables, we get X X I,, where I s a ut of order +. Ths wll gve us ˆ s x x... x (4..5) so that the maxmum predcto error (acheved at a vertex) s ˆ /. That s the qualt of the ft becomes ver good wth creasg. Ths reflects the fact that we use pots to calculate + coeffcets, so that we average out the effect of ose. Ths estmate s msleadg actual stuatos, because rarel do we have a true lear model. Whe the respose we measure s ot lear, we wll have modelg errors, also called bas errors whch are ot averaged out. If o the other had, we take ol eough measuremets to calculate the coeffcets, that s a saturated desg wth + coeffcets, the error gets progressvel bgger, because the porto of desg space covered b the smplex cotag the data pots becomes progressvel smaller. For example, for Example 4.. the three pots used for the saturated desg form a tragle coverg half of the desg doma. For a three dmesoal cube, 4 vertces wll spa a tetrahedro wth a volume of oe sxth of the volume of the eclosg cube. For the dmesoal case, the full-factoral desg s the vertces of a cube of volume, whle + vertces obtaed b perturbg oe varable at a tme from oe vertex spa a smplex of volume!. For example, for three dmesos, the maxmum predcto error wth a full-factoral desg s 0.5, whle the maxmum predcto error for the saturated 4-pot fractoal factoral desg s 7 (Exercse ). 4.. Desgs for lear respose surfaces For fttg lear respose surfaces, we tpcall use desgs wth ol two levels for each desg varable, ad the most popular fractoal desgs are the so called orthogoal desgs. A orthogoal desg s oe where the matrx XTX s dagoal, ad the populart of these desgs s partl based o the followg theorem (see Mers ad Motgomer, 995 p. 84): For the frst-order model (lear polomal) ad a fxed sample sze. If all varables le betwee - ad, the the varace of the coeffcets s mmzed f the desg s orthogoal, ad all the varables are at ther outer postve or egatve lmts (.e., - or +). It s eas to check that the full factoral desg s orthogoal, but t s ot trval to produce orthogoal desgs wth smaller umber of measuremets. Varous orthogoal desgs ca be foud books o desg of expermets (see Mers ad Motgomer, 995). To demostrate the beefcal propertes of orthogoal desgs we wll cosder the two-dmesoal case that we have studed the prevous example. I that example we ftted a two-varable lear polomal frst wth a full factoral desg ad the wth ol 3 pots. Fttg a lear polomal varables o the bass of + pots requres the pots to be learl depedet, so that the form a smplex. There s o redudac the desg, that the umber of pots s equal to the umber of coeffcets, ad ths s called a saturated desg. I order to get a orthogoal desg ths case, we have to gve up o havg the varables be ol at the levels, ad stead opt for a perfect smplex, wth the dstace betwee all pots beg the same. 5
Example 4.. Cosder the equlateral tragle whch results a scalar matrx (a scalar matrx s a scalar multple of the ut matrx) 3, ), ( 3, ), X T X. It cludes the pots ( (0, ). Check for the stablt of the predcto varace ad ts maxmum value the ut square for the lear model b bx b3 x. We have 3-3 0 0 X - 3 -, X T X 0 3 0. (4..6) 0 0 0 3 Also ( m) ( m) T T ( m) x x, x X X x x x. (4..7) 3 x Usg Eq. 4..0, we see that the stadard error of the respose vares betwee s ˆ 3 at the org ad s ˆ at the vertces. Ths s a mprovemet, both terms of stablt ad maxmum stadard error as compared to usg 3 vertces of the ut square. However, ths has come at the prce of dog the measuremets outsde the ut square. As we wll see later, ths reduces the varace error, but creases the so called bas error. Bas error s the error troduced whe the model that we tr to ft s dfferet from the true fucto. For example, f the model s lear ad the true fucto s quadratc, the smplex model that we have used ths example s lkel to crease the error rather tha decrease t. 4..3 Desgs for quadratc respose surfaces Quadratc polomals varables have (+)(+)/ coeffcets. To ft quadratc respose surfaces we eed at least that ma pots, ad at least three levels for each desg varable. For > 3 t s possble to have the requste umber of pots wth ol two levels. For example, a quadratc polomal 4 varables has 5 coeffcets, ad a full factoral desg two levels has 6 pots. However, f we let ol oe desg varable var at a tme, we ca easl check that we are left wth 3 coeffcets ad we eed three dfferet levels of that desg varable. We ca use a full-factoral 3-level desg for the quadratc respose surface, ad ths desg wll have3 pots. I most cases we caot afford such desgs eve for farl small values of. For example, for = 6, we requre3 6 79 expermets. A popular compromse whch reduces the umber of expermets to close to the -level full factoral desg s the cetral composte desg (CCD) (Matlab ccdesg). 53
The cetral composte desg s composed of the pots of the fullfactoral two-level desg, wth all the varables at ther extremes, plus a umber of repettos c of the omal desg, plus the pots obtaed b chagg oe desg varable at a tme b a amout. Fgures 4. ad 4. show the cetral composte desg for = ad = 3. The value of chose the fgures are such that all the pots outsde the org are of the same dstace from the org, so that we have a sphercal desg. Ths placemet of the pots s at the hgher ed of the tpcal choce for. A more popular choce s based o the cocept of rotatablt. The propert of rotatablt requres that the predcto varace be depedet ol o the dstace from the org ad ot o the oretato wth respect to the coordate axes. 54
It ca be show that for the cetral composte desg the rotatablt requremet wll be satsfed for 4 (4..8) Ths equato gves for =, whch s the same as the sphercal desg, however, for = 3 we get. 68, whch s slghtl smaller tha 3. Wth ether a sphercal desg or a rotatable oe, we fd that we eed a umber of replcate ceter pots (pots at the org) to obta good predcto varace stablt. Fgures 4.3 ad 4.4 show cotours of predcto varace wth oe cetral pot ad fve cetral pots, respectvel. It s obvous that the stablt of the latter s much better tha the former. Ths eed of cetral pots wth rotatable cetral composte desgs s a possble problem wth umercal expermets that gve exactl the same aswer whe the expermet s repeated at the same pot. Fortuatel, however, wth there s o eed for repeated cetral pots. The case of, whch s called the face cetered cetral composte desg (FCCCD), s ver attractve a lot of applcatos, because t does ot requre usg a other levels except (-,0,). Fgure 4.5 shows cotours of the predcto varace for the ut square wth a sgle cetral pot. It s see that whle the desg s ot rotatable, the stablt s qute good. 55
56
For hgh dmesoal spaces, the cetral composte desg s o loger practcal, because the umber of pots creases too fast. Oe possble soluto s to keep the rus whch perturb a sgle varable, but to have a fractoal factoral desg to replace the vertces. However, whle the umber of vertces creases as, the umber of polomal coeffcets creases as ( + )( + )/. Cosequetl, the tpe of fractoal desg used has to be modfed as creases. Istead there s a block desg (Matlab bbdesg), frst troduced b Box ad Behke (960), where the umber of expermets creases at the same rate as the umber of polomal coeffcets. The two-varable block desg s based o perturbg ol two varables from the omal value. That s, at each pot, we have a par (I, j), such hat x, x j, ad xk 0 for all k, k j. The two varables are perturbed all four combato of. For example, for = 3 (the lowest dmeso for whch ths desg makes sese) we wll have the followg desg pots 57
x x x3 - - 0-0 - 0 0-0 - - 0 (4..9) 0-0 0 - - 0-0 - 0 0 0 0 where the last pot s the cetral pot, whch ma be repeated. Fgure 4.6 shows ths desg. For the geeral case, we ca select varables out of ( ) was. For each such combato of two varables we have four desg pots, wth each oe of the varables takg the values of. The total umber of pots ths block desg s c ( ). For large values of ths teds asmptotcall to 4 tmes larger tha the umber of coeffcets that we eed to ft. However, ths happes for ver large values of. For example, for = 0 the umber of coeffcets s 66, whle the umber of pots the block desg wth oe ceter pot s 8 (for comparso, the umber of pots the cetral composte desg s 045). 58
. Block desgs are sphercal, that all the pots are at the same dstace from the org. For example, all the two-varable block desg pots are at a dstace of from the org. For large values of, ths dstace ca be much smaller tha the dstace of the vertces (whch s ). Therefore, extrapolatg to the vertces o the bass of the block desg ma be rsk. 4.3 Optmal Pot Selecto The specal expermetal desgs that we cosdered so far, as well as other expermetal desgs avalable the lterature wll satsf our eeds most of the tme whe we have a box-lke doma. Ofte, however, we wll ot operate a box-lke doma because of varous desg costrats. I fact, t s desrable to voke as ma costrats as we ca to reduce the volume of the desg doma, because ths tpcall creases the accurac of the respose surface. Wth a rregular desg doma, the stadard expermetal desgs are of o use to us. I fact, eve wth a box-lke doma we ma wat to perform a umber of expermets that does ot correspod to a of the stadard desgs. I these cases we have to create our ow expermetal desg, that s select a optmum set of desg pots. 4.3. Mmum varace desgs The commo approach to optmum pot selecto treats t as a combatoral problem. We start b creatg a pool of caddate pots where we could possbl evaluate the respose. We wll assume that we ca afford evaluatos of the desg, so that out of the pool we would lke to select the `best' pots. There are varous crtera as to what s the best set of pots, ad the most popular oes attempt to select the pots so as to mmze the varace of the coeffcets of the respose surface. A ke to that varace s the momet matrx T X X M (4.3.) The determat of the momet matrx T X X M (4.3.) ca be show to be drectl related to cofdece rego of the coeffcets of the respose surface (a cofdece rego for a coeffcet s the rego where the coeffcet wll le wth a gve probablt). I fact, t s versel proportoal to the square of the volume of the cofdece rego of the coeffcets. So maxmzg the determat creases our cofdece the coeffcets. Ths crtero s called D-optmalt, ad t s mplemeted several software packages. Fdg the D-optmal set of pots from a gve set of pots s ofte a dffcult combatoral problem. For example, f we eed to fd 0 D-optmal pots out of 50, we 0 0 ca have 0 possble combatos. So the umber of combatos becomes 50 huge eve for moderate sze problems. Soluto algorthms ca rarel fd the D-optmal 59
set, ad usuall settle o a suboptmal but good set. Some soluto algorthms are based o replacg oe pot at a tme ad takg advatage of expesve expressos for updatg the determat whe oe pots s chaged. Geetc algorthms have also bee used to fd a good desg based o D-optmalt. Matlab has several fuctos that ca be used for geeratg D-optmal desgs. For box-lke doma ad lear ad quadratc polomals, oe ca use cordexch (ame comes from usg a coordate exchage algorthm for searchg for the D-optmal desg). Example 4.3.: ------------------------------------------------------------------------------------------------------------------- We wat to ft a quadratc polomal varables (6 coeffcets) ad we wat to see the effect of the umber of pots o the desg ad the determat of the momet matrx. Usg cordexch we start wth the default 3 levels ad compare usg the mmum umber of sx pots to usg 0 pots. Wth sx pots: >> =6;beta=6; >> [dce,x]=cordexch(,,'quadratc'); >> dce' as = - - 0 - - - 0 >> det(x'*x)/^beta as = 0.0055 Wth pots: >> =; >> [dce,x]=cordexch(,,'quadratc'); >> dce' as = - - 0 0-0 - - - - - - 0 0 0 >> det(x'*x)/^beta as =0.00 We ote that the pots are selected o the boudar ad that for pots wth ol three levels, evtabl some pots are duplcated. We ca crease the umber of levels to tr ad avod duplcato. Icreasg the umber of levels to fve, we sert addtoal pots wthout elmatg a of the prevous levels. >> [dce,x]=cordexch(,,'quadratc','levels',[5 5]); >> dce' as = - - 0 - - - 0 0 0 - - - 0-0 0 0 >> det(x'*x)/^beta as =0.0087 The route stll selected ol the prevous 3 levels, but because the problem of optmzg the desg became more complex, a feror soluto was foud. Ths ca be remeded b lettg the algorthm multple tres from radom starts. >> [dce,x]=cordexch(,,'quadratc','levels',[5 5],'tres',0); >> dce' as = 60
- 0 0-0 - - - - - - 0 0 0 >> det(x'*x)/^beta as =0.00 Wth sx levels we ca force t to use other pots >> [dce,x]=cordexch(,,'quadratc','levels',[6 6],'tres',0); >> dce' as = -.0.0.0 -.0.0.0 -.0.0-0. 0. -.0-0. -.0 -.0.0 0. -.0 0. -.0.0 -.0-0..0.0 >> det(x'*x)/^beta as = 0.0094 But we sacrfce accurac. --------------------------------------------------------------------------------------------------------------------- Whe the desg doma s ot box-lke, we ca produce caddate pots wth a method we lke ad the choose amog them a D-optmal subset (Matlab fucto cadexch ). For example, we ma produce a full-factoral desg a box that cotas our desg doma, ad the prue a pots that fall outsde. Of course, the umber of levels eeds to be suffcetl hgh to esure that we are left wth suffcet umber of pots. Aother crtero s called A-optmalt. It s based o the fact that the dvdual varaces of the coeffcets of the respose surface are proportoal to the dagoal elemets of M. A-optmalt seeks to mmze the sum of these elemets, that s the trace of M. Istead of focusg o mmzg the varace of the coeffcets, t ma be more reasoable to mmze the predcto varace. We frst defe the scaled predcto varace ^ Var ( x) mt T m ( x) x ( X X ) x (4.3.3) A crtero that seeks to mmze the maxmum value of (x) the doma defed b thedata s called G-optmalt. It ca be show that uder the stadard statstcal assumptos about the error that the maxmum predcto varace the doma defed b the data pots s alwas larger or equal to the umber of terms the respose surface,, (see Mers ad Motogemer, 995, p. 367). Therefore, wth G-optmalt we have a target to shoot for. We would lke to seek a set of pots that wll brg the maxmum close to. Ths value s acheved b a two-level full-factoral desg for a lear model, see Example 4... 4.3. Mmum bas desgs Varace based optmalt crtera do ot cater to errors due to the fact that s ot possble to ft accuratel the true respose b the model used the respose surface. Ths error s called modelg error b egeers ad bas error b statstcas. To cosder ths errors we eed to troduce the cocepts of desg momets. Deote b R To avod havg duplcate pots, Matlab suggests to fd the commet Fd maxmum chage the determat ad sert mmedatel after dd(rowlst(rowlst>0)) = -If; 6
the rego of terest to us terms of predctg the respose, ad deote b the frst momets of the doma, that s x dr,,,,,,, (4.3.4) V R where V s the volume of the doma, that s V dr, (4.3.5) R Smlarl, we deote secod momets as j where j x x jdr, (4.3.6) V R ad so o. Repeated dces are used to defe powers, so that, for example x x jdr,,,,,,, j,,,,,,, (4.3.7) V R We ca smlarl defe the momets assocated wth the data pots. Deotg b x k the th coordate of the kth data pot, we defe m xk,,,,,,,, (4.3.8) k wth smlar defto for hgher momets. For example 3 m x k xk, (4.3.9) k Now cosder the possblt that we use oe model, but suspect that the respose ma be descrbed b a hgher order model. Deote the vectors x m assocated wth the frst model ad secod models b x m ad x m, respectvel. The the matrx x m x mt defes a matrx of products of the fuctos (usuall moomals used the two models). The tegral of ths matrx s a matrx of momets m mt M x x dr, V R Smlarl, the matrx ca be averaged over the data pots m mt M x x, k k (4.3.0) (4.3.) For a mmum bas desg we eed for the matrces to be the same (Mers ad Motgomer, p. 4). That s M M ad M M, (4.3.), I cotrast to the mmum varace desgs that ted to put pots the perpher of the desg doma, mmum bas desg ted to brg them closer to the cetrod. 6
Mmum bas desgs ofte also have low predcto varace, but the reverse s ot true. That s, mmum varace desgs ted to have large bas errors. Compromse desgs, whch M ad M are slghtl larger tha M, ad M, respectvel, are occasoall used. Example 4.3. We wat to ft a lear model to a fucto the ut square based o four measuremets, whle we kow that the exact fucto ma be quadratc. Costruct a 4-pot mmum bas desg ad compare t to the full factoral desg for fttg the fucto x x.. For the lear model ad quadratc model we have m T m T x, x, x, x, x, x, x, xx, x, (4.3.3) so that x x x xx x m mt 3 xx x x xx x xx xx, (4.3.4) 3 x xx x xx xx x For the ut square we have V dx dx 4, (4.3.5) Ad 0 0 0 3 3 m mt M 0 0 0 0 0 x x dx dx, (4.3.6) V 3 0 0 0 0 0 3 Note that because of the smmetr of the doma, all the tegrals wth odd powers of ether x or x are zero. Because x m m s a subset of x, we do ot eed to take care of M, ad t s eough to satsf the codto M M. If we pck 4 pots that are smmetrc wth respect to both the x axs ad the x axs, the the sums volvg odd powers wll vash, so that all the zeros M wll match the zeroes M. There are two possble choces: Oe s the set of pots r,0, 0,r where r s a costat. The secod s the set r, r, wth a dfferet r tha the frst set. The value of r s calculated b settg the ozero tegrals M equal to the correspodg sums. That s, 4, x. 3 4 3 4 4 For the frst set these equatos gves us x (4.3.7) r r, 4 3 or r 0.865. (4.3.8) For the secod set we get stead 63
r r r, or r 0.5774. r (4.3.9) 4 3 The two possble mmum-bas desgs are 0.865,0, 0, 0.865or 0.5774, 0.5774. Now let us compare these two sets to the full factoral desg, for fttg the fucto 3 so that the x x. For ether set, the value of the fucto at each pot s ^ respose surface s also 3. For the full factoral desg the value of the fucto at each ^ pot s, so that. Obvousl the mmum bas desg gves a better ft. To apprecate that the ft s optmal, cosder fttg the fucto b a geeral lear polomal. Because of the double smmetr of the fucto, all the lear terms should vash, so b that the respose surface should deed be of the form. The mea square error over the rego s the 8 4 e rms x x b dxdx b b. (4.3.0) 4 45 3 Dfferetatg the error wth respect to b ad settg to zero, cofrms the fact that b = /3 gves the mmum error, e 8 45. I cotrast, b = gves e 88 45. rms If ths example appears mpressve, ote that we dd ot have a ose at all the fucto, ad the example was selected to make the mmum bas desg look good. I other cases the results ma be less dramatc, ad a compromse betwee mmum bas ad mmum varace ma be called for. 4.4 Space Fllg Desgs Varace mmzg desgs are targeted at problems where ose the data s the ma problem. Whe bas errors are of ma cocer, there s a tutve appeal to desgs that leave the smallest holes the desg space. There are several popular methods that attempt to acheve ths goal. Lat hpercube samplg (LHS) desgs start wth the prcple that f we have data pots, the we should strve to have each varable have levels. Ths ca be doe b dvdg the rage of each varable to equal tervals ad requrg that the varable has a level each. Alteratel, we ca dvde the rage to - tervals ad requre that the varable has a value at the boudares. Ths ca stll leave out large holes the desg space. So ormall, LHS desg s accompaed b a procedure that wll optmze t to avod large holes. Fgure 4.4.4 compares three desgs wth 9 pots. ^ rms 64
Fgure 4.4. Desgs wth 9 pots. The leftmost desg s full factoral desg wth three levels. The mddle oe s a radom LHS desg, ad the rghtmost oe s a LHS desg optmzed for maxmum mmum dstace betwee pots. The full factoral desg has ol three levels for each varable, the radom LHS desg has substatal empt regos o the top rght ad bottom left areas whle the optmzed LHS desg has more uform coverage. It s ot clear that for ths partcular case the LHS desg s better tha the full factoral desg, because wth a small rotato of the coordate axes the full-factoral desg would look better. Cosequetl, ths s stll a matter of cotrovers. However, optmal LHS desgs allow us to specf a umber of pots, stead of beg lmted to specal umbers assocated wth full-factoral or most of the other desgs dscussed earler. Matlab s lhsdesg fucto produces LHS desg wth choce of two optmzato strateges. Oe maxmzes the mmum dstace ad the other mmzes a measure of the correlato betwee the varables. Example 4.4.: Geerate two 7-pot LHS desgs two dmesos usg the two crtera ad compare. >> x=lhsdesg(7,,'crtero','correlato','teratos',000) x=x(:,); >> x=x(:,); >> plot(x,x,'r+') >> x=lhsdesg(7,,'crtero','maxm','teratos',000); >> hold o >> xb=x(:,); >> xb=x(:,); >> plot(xb,xb,'o') 65
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0. 0. 0 0 0. 0. 0.3 0.4 0.5 0.6 0.7 0.8 0.9 Fgure 4.4.: LHS desgs. Plus smbols deote mmzed correlato desg ad crcles deote maxmum mmum dstace desg. It ca be see from the fgure that there s ot a great deal of dfferece betwee the desgs. The mmum dstace betwee the plus smbols (mmzed correlato) s obvousl somewhat smaller tha betwee the crcles (maxmzed mmum dstace). The chage correlato betwee x ad x s less obvous. It s -0.074 for the pluses ad -0.0777 for the crcles. 4.5 Exercses. Fd the maxmum predcto varace the ut cube for a lear polomal, whe the data s gve the four pots (-,-,-), (-,-,), (-,,-), (,-,-).. Fd the three pots the ut square that wll mmze the maxmum predcto varace the ut square for a lear respose surface. 3. For Example 4.3., fd the maxmum predcto varace for the mmum bas desgs ad compare t to that of the full factoral desg. 4(*). Costruct a mmum-bas cetral composte desg. You ma eed to replace the cetral pot wth 4 pots ear the org. 66