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$ {,{,re,re t3 a- rronzero v"-to,r c Srrch H^a{ X c = o ,
l.tue cotuvr,uqS of fi,re- c.l.estgrn w.aFrti X q/e- sard lb be
co (t rvreac -l[ne,n X lrrors l"ar,t[< less lt^An p awd Xr X
I'ras t^o rr!^rque- iurue,rse.. RetoF"d oJi$authes av.use'tF
Llrte cptu,nnvrs oF X are- of *osF cott ,yrea" .
Th"e 'nnogF af.aiShl{or*.runo( go(u,.ho,n ts l-../ropcol"tr"r,ns f"t* X u,r Fit Hn* esh, ioF..s ,.re_ b*Fle-n
b"tr^oved.
A nnof e glst-ennahr qp? roa,ch l" clealrn' wilh wea[<
JestSr,r l^ncrhaeE ts nr dq s {eq(aSStot :eJ
bj re-Nr, hng *{ne ort3rv..q( yrnode{
+ 3 1p" r X,p,* e X, ,n x[p.))
s t-a" F
q,5
* = lp" + zY r e Z ^ *1g,-,-1
where, L' 7 1 o aurcl llne ot*.ago,,ral of ZrZ
12. 2,
cor'-\SrSlS "$ Y1 ' 5 '
Wq oLch tetie, t"[,.r.s lo .1 ce. vr l. .,^3 eo.rh co\ "r.* ,r of X r
b1 I ,-r\a Frurh *g il-s avera3e , ll,u^ diutd.t,n3 by its
s\a,*Ac.rd "levrhho,rr 1a.^d yu,.r[l-tp(yrhg by JA.
en., h* teast- S1*uve9 rsh**F,'g cr/e
AP, 1
t =€z)-'7r
W, +o re-pLuce- zrz- by
ztz + ,ro-, ,
,^.^dCcr"n.t. I )
wltou^g 7, >
the- Cofres poncl.rru1 ushntal'eS
io : (=" + orp_, )-' z'+are b,*tJ uvrtess h = o.
L-cr.n3e i/at*e-s ofi A lv\c-fe-rLss iLte- l,rto.s by slni,nkl,1E
\{re_ esh rnales |pqptc&S ttog DrrX{^ , b *F l{nis
17,3
Jecrecses l!.eir fcrftavr c9 .
T1^e vatre o[ n tE.[^"oseu'
o F c\ cni[e".brt sv.ch &9 l-tae
4v*ee cv ca) = 3- li;d't u
.*p tiicat(y to'-1
crogg - Va lr iq h$o
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ft fUo( vo l, e Sor t; p,reot..LF.J
r^unJe.t obhunel w[e-n l-tre-
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{*
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1e',terd\r.--J cvegs- rlaltdg.ho,n f.svvt o4 SJ r.r&reS :.t
Gcvn: t (+i -4, )'i=' g'
*!,.ur *- Hn , p-' 1^ l^t *- z (ztz + z rp_, )-'zt
rs iln"{- h"F ilnaMx c-orng)pgvrct'vtg fO H4- ,riAge
a,'1d lton- veCc,r of fr ftrl vct h.teg 1 - Ao utregrgggto r,r 7
Aepu^As o n Z
12" L+
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a, c$il^ t,nO /1 faLue , [4o ra Y = O r /Kctt{ citso \p. de ntveo(
\ B"iastan agu^g"Is '
(S.. AcU 9S:T.Z Sn ft;rlt.er olrscr;ssbn ,,o^d.hopl"rrrov\ Br^\ es\t\v1 rnnoolel.t . )
E rannptr' Cerruenrt A"fq ( AcD p . ss+ )
Example 8.3 (cement data) Table g.l conrains dara on the relationship betweenthe heat evolved in tlre setting of cement ancr its chemical compositiool uuru nnheat evolved, y' for each of n : 13 independent sarnples are av4irabrg, .no i"i .*nsample the percenrage weight in clinkers of four chemicals, x1,3cao.Ar7ot, xz,3 C a A . S i Oz, 4, 4C a O . Al 2 O 3. F e703, and .ra, 2C a O . S i O z,is recorded.
Figure 8.1 shows thaf although the response y depends on each ofthe covariatesx1, ...,.ra, lhe degrees and directions ofrhe dependences differ.
In this case we might {it the model
-r.r = do * hvi * fizxzt r- S3.r.r; * fcx+i * t'i.
where Figure 8.1 suggests rhat pl and F2 are positive. and that f3 and f4 a.e negarive.The design matri;r.has dimension 13 x 5, and is
tl 7 26 6 60r{r r 2s 1.5 r. Ix:1. : lrl: '. : : I\t ro 68 t nJthevectorsl,andahaledimension 13 x l andf hasdimension5 x L I
-7 7.5
trble8.l Cem"ntdatailvmds sJ al., 1932): y isheat evolved in raloriespergm ofement, and.{1, xl} tf , md & &p€rcertage \*eight ofclinkers, with .r1 ,3CaO.Al7O1x2.lCaO.SiAyx7,4CaA.Al2O1.F e2oz,md x+, 2CaO."9iO:.
figm8.1 Plct*ol{trent data. Ther{iables arc hmt evolv€dir raloris p€r gram.
'.per€entage weigbl intiinkets of -ti.-r{cO.Al2O3, x;,lfaO-Sj t7:, xr.4d a0 -AlzAz,FetQt,sxl xa.?CaO.SiQz.
Ca"se It J-l
I1
3456789
l011tz13
7 26 6 60 18.5| 29 15 52 .74.3
11 56 8 20 1i14.3lt 31 I 4? 87.61 52 6 33 95.9lt 55 I 22 109.?3 71 1't 6 n2.71 3i 22 44 72.5? 54 18 22 93.I
?l 47 4 ?6 115.9r 40 23 34 83.8
11 66 I 12 i13.310 68 I 12 109.4
€Qou
frs6)I
.:o6oT
I
j.t
,
I
ia
30 40
i. I:'-l5101520
Percenlage weight in clinkor€, x'l
-tI
5101520Percentage tveight in clinkers, x3
.---""--.--.----'-....-^...^.-----..-......+
6050
6ooo^6Ho=
io,
Ii
litolri . .I>l
-ol'5 01zol:i;81olII _t
Percentage weight in clinksrs, x2
.t.t10 28 30 40 50 60
P€rcentage weighl in clinkers, x4
-72.c
Table8.l2 Paemeter.Jdmates md thei{*andudenon futhefull:odel md a nduced,,odel fitt€d to the cement:tuIl.
Full model Reduced modsl
Parameler Estimate Standard error Esrimate Standirdr:rn:r
ftfit8z11z
Fq
62.41L550.510.l0
-0.l4
70.o74.74
0.750.71
"i 1.641.43Q.42
*0.:4
14.1,10. t:f).1 ?
Exarnple 8.28 {Cemeni dats) The astute reader will have realized that if the middlefour colurnns of Tabte 8. I are percentages, they may surn to 100. ln fact they sum to(99,97 ,95. 97, 98, 97 ,97 ,98,96. 98, 98, 93. 98). As there is a column nf r:nes in thedesign matrix for the full model, its colurnns are nearly dependent: estimadon of f veparameters is ahnosi impossible. This is reflected by the standard errors in Tatrle 8. i 2,The standard crror for p0 is vastly inflated by inclusion $fr3 because fp is almostimpossible to es:imate, whereas the other esrimates are less badiy affected.
The residual sum of squares tbr model rvithout.ti is 47 "97 , only slightly larger thanthat for the f'ull rnodel, 47.86. Thus inclusion of .r3 changes tbe fit of the rnodel verylittle, but has a drastic effect on the precision of paramrter estimation.
The eigenvalues of XrX with ail live columns of X are 44676,5965.4, 810.0,105.4 and 0.00012. The conditjon number oi 5056 indicates strong ill-conditioning.andfd;l : 821 seems very large.
The left panel of Figure 8.7 shows how the parameter estimates fi depend cnthe ridge parameter tr. All change fairly sharply as i increases from zero, and aremore stable for i > 0.2. The right panel shows that GCV{I) decreases sharplv when
lambda
t.u La
lambda
.), increases from zera, and is minimized when )" - 0.3. The dotted lines show thatwhen x3 is dropped both rhe D and GCV{I) depend mueh less on i, consistent witil
dlfJt\eF.
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Ftgure8.7 Ridgeregression analysis cfcfment data. Left:variation of elements of/; c a functicn of ),, furnrodels with ali l'ourcovariates {nqlid) md wilk.ri , Jt, md ;4 only {dols}.Rigbt: generaliz*dcross-vaiidation criterionGCV(}) lbr these trodois.
the discussioi above
s6. t
Pr.ltM (r\&ry ,n o le F, t\t;3 l{,te- tvrorn rnto/gt .
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too* a Fn.,tE rnodE\ 3 C ?) , b'",.{- r!e-
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t''
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A * Np(gq, T(*r)-'Kes)r[eg\-') (7)
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a,,rd ff tgt\ nAdy he e.sh^,taFed Jy
nt = i > to3 {(.*; ;d) > tqs +(1., ) A)
i'' )e- bg-r
aurd /\ y1 a \t : t )L log + q,, a) . il... s[ 5et-.]eo{
whe'-s K{oo1- n f }tsl F_i?ie) l!3fJ.rig) $tr)d?u ) ag-- >g'
arnol
If- ttg) ' f t1 '' e) 5o lko F rtn e ass uu(e d ctenrst Ly ri corr€eF,
fl,rer, 03 rs ltnu. ho,,re I 1 a,nol K@g ) = fg (&s J= -f (er) r
l'{ne tvur,,,tHvuricrFe ) e*pecl-€d r^forwuhbrn , autol (s;
pgj.r.ceS tb l'iue uStrat as y wr p b t a qP P
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f='=ilF- at vr tctrwrahonf f,*alt'if .
s6.t.*
We- *o\ cc^Asi**cf csnkJe^ce rntler*ra\s ,fr,r &: \,rsr^g
f - tt i-'A tPno x rv'^. q{-e d tslr\b..r,ho^ 'F L' lcetihood Ro ho stufisfie '
G1 I o tr 9e r\eS ,tPf no X I /v\aho 4 3 t{es
2\!to) -a(oe) \-- (o -s:)' I"- 2 r't'l tl+ t&)}td -*rlj=, as ber ) -
a r"\ L6 _eg). ,l (pu )td -er;
Nof,rrra\,\ oF I r,nn"p[tes $.al- fi,re- LtR. stohsh.,
h.q,s a. o\.sh;botho^ l|l^ pt"t-ruvrsl h" n', , b*l- u,i'l{n tvean
f" t I. cq)-' K (es) iT7@; = KtAs) r Jtviur3f l ll,.e wrodel \s co,.re.t )
V\tt C-c\\n p
