Global optimization of statistical functions with simulated annealing

35 Pages·2001·2.35 MB·English

lanruoJ fo scirtemonocE 06 )4991( .99-56 dnalloH-htroN
labolG noitazimitpo fo lacitsitats
snoitcnuf htiw detalumis
*gnilaenna
mailliW .L effoG
(/nirersiiy of Southrrn ,ipissisxiM Hattieshurg, MS 39406, USA
Gary D. reirreF
Unicrrsit? fo Arkansas, ,rllicettw~cF AR 72701, D’SA
nhoJ sregoR
Memphis Stute Unioersit>~, Memphis. NT 38152, USA
devieceR rebmevoN ,0991 lanif noisrev deviecer enuJ 2991
ynaM lacitsitats sdohtem yler no laciremun noitazimitpo ot etamitse a s’ledom .sretemarap -utrofnU
,yletan lanoitnevnoc smhtirogla semitemos .liaf nevE nehw yeht od .egrevnoc ereht si on ecnarussa
taht yeht evah dnuof eht ,labolg rehtar naht a ,lacol .mumitpo eW tset a wen noitazimitpo ,mhtirogla
detalumis ,gnilaenna no ruof cirtemonoce smelborp dna erapmoc ti ot eerht nommoc lanoitnevnoc
.smhtirogla toN ylno nac detalumis gnilaenna dnif eht labolg ,mumitpo ti si osla ssel ylekil ot liaf no
tluciffid snoitcnuf esuaceb ti si a yrev tsubor .mhtirogla ehT esimorp fo detalumis gnilaenna si
detartsnomed no eht ruof cirtemonoce .smelborp
wordys: eK detalumiS ;gnilaenna labolG ;noitazimitpo noitamitsE smhtirogla
ecnednopserroC :ot mailliW .L ,effaG tnemtrapeD fo scimonocE dna lanoitanretnI ,ssenisuB -inU
ytisrev fo nrehtuoS ,ipississiM ,grubseittaH MS ,60493 .ASU
*We dluow ekil ot ,knaht tuohtiw ,gnitacilpmi snaH ,nammA yraG ,niethsniraB divaD ,yelsleB
T. Randolph ,draeB namreH ,snereiB leahciM ,ytooB divaD ,grebdloG natS ,rreK divaD ,yveL
eissE ,imuosaaM leinaD ,yerffaCcM drahciR ,tdnauQ dna leahciM 1laeV rof lufpleh snoitseggus dna
stnemmoc no suoiverp .stfard owT suomynona seerefer dedivorp ynam elbaulav .snoitseggus treblA
regnaluoB saw yrev lufpleh htiw eht erutaretil .yevrus roF lanoitatupmoc ,pleh ew dluow ekil ot
knaht trA etnomleB dna avA htavroH fo .LSMI miJ llaH dna lliB yentruoC fo IBM yldnik dedivorp
a elpmas .noitatskrow haraS ,norraB naJ ,dryB treboR ,ssenkraH rehtuL ,releeK nhoJ ,ssepseR dna
soripS salleV fo eht ytisrevinU fo saxeT retneC rof hgiH ecnamrofreP gnitupmoC osla dedivorp
elbaulavni .pleh
ehT lanoitatupmoc krow rof siht repap saw demrofrep ta eht ytisrevinU fo saxeT retneC rof hgiH
ecnamrofreP .gnitupmoC
00.60$/49/6704-4030 )c reiveslE-4991 ecneicS srehsilbuP .V.B llA sthgir devreser
66 W.L. Gaffe et al., Global noitazimitpo of statistical snoitcnuf
.1 noitcudortnI
ynaM cirtemonoce ,sdohtem hcus as raenilnon tsael ,serauqs the dezilareneg
dohtem of ,stnemom dna the mumixam doohilekil ,dohtem yler nopu -azimitpo
noit ot etamitse ledom .sretemarap Not ylno do these sdohtem yal the -teroeht
laci noitadnuof for ynam ,srotamitse yeht are netfo desu ot yllaciremun etamitse
.sledom ,yletanutrofnU siht yam eb tluciffid esuaceb these smhtirogla are -emos
semit yhcuot dna yam .liaf roF the mumixam doohilekil ,dohtem remarC (1986,
p. 77) stsil a rebmun of tnasaelpnu‘ :’seitilibissop the mhtirogla yam ton -noc
egrev ni a elbanosaer rebmun of steps, ti yam head toward yletinifni egral
retemarap ,seulav or neve pool hguorht the emas tniop emit dna .niaga ,oslA the
mhtirogla yam evah ytluciffid htiw segdir dna .suaetalp nehW faced htiw hcus
,seitluciffid the researcher si netfo decuder ot gniyrt tnereffid gnitrats seulav
remarC[ (1986, p. 72) dna hcniF et .la (1989)]. ,yllaniF neve fi the mhtirogla
,segrevnoc there si on ecnarussa that ti lliw evah degrevnoc ot a ,labolg rather
naht a ,lacol mumitpo ecnis lanoitnevnoc smhtirogla tonnac hsiugnitsid -eb
neewt the two. nI ,mus there si a poor hctam neewteb the power of these
sdohtem dna the laciremun smhtirogla desu ot tnemelpmi .meht
A wen dna yrev tnereffid noitazimitpo ,mhtirogla detalumis ,gnilaenna -netop
yllait sevorpmi siht .hctam tI serolpxe the s’noitcnuf eritne ecafrus dna seirt ot
ezimitpo the noitcnuf elihw gnivom htob llihpu dna .llihnwod ,suhT ti si ylegral
tnednepedni of the gnitrats ,seulav netfo a lacitirc tupni ni lanoitnevnoc -ogla
.smhtir ,rehtruF ti nac escape morf lacol amitpo dna go no ot dnif the labolg
mumitpo yb the llihpu dna llihnwod .sevom detalumiS gnilaenna osla sekam
ssel tnegnirts snoitpmussa gnidrager the noitcnuf naht do lanoitnevnoc -ogla
smhtir ti( deen ton neve eb .)suounitnoc esuaceB of the dexaler ,snoitpmussa ti
nac erom ylisae laed htiw snoitcnuf that evah segdir dna .suaetalp ,yllaniF ti nac
ezimitpo snoitcnuf that are ton denifed for emos retemarap .seulav
sihT paper tsrif serapmoc the anaroC et .la (1987) noitatnemelpmi of
detalumis gnilaenna ot lanoitnevnoc smhtirogla no ruof cirtemonoce .sledom
,suhT these stluser yam pleh ediug other researchers. ehT tsrif ledom si na
elpmaxe morf the erutaretil htiw elpitlum aminim [Judge et .la (1985,
pp. ,])759-659 dna the dnoces si a lanoitar snoitatcepxe egnahcxe rate .ledom
ehT driht ledom si na ycneiciffe yduts of the gniknab yrtsudni gnisu a golsnart
cost reitnorf metsys dna the htruof stif a laruen krowten ot a citoahc emit .seires
esehT sledom yrav yltaerg ni .ezis ehT Judge ledom has ylno two ,sretemarap the
egnahcxe rate ledom has 14, the laruen krowten has 35, dna the cost yduts has
62. llA three elbairavitlum noitazimitpo smhtirogla ni the LSMI yrarbil tneserper
the lanoitnevnoc .smhtirogla ecniS yeht are evitisnes ot gnitrats ,seulav 100
tnereffid sets of meht are desu ni lla cases.
sihT paper neht secudortni snoisnetxe ot the anaroC .mhtirogla enO -ifidom
noitac checks that the labolg mumitpo si deedni ,deveihca elihw rehtona swolla
the researcher ot tcirtser noitazimitpo ot a tesbus of the retemarap space
.L.W @oG te ,.la labolG noitazimitpo fo snoitcnif,lacitsitats 16
siht( nac eb yrev lufesu for gnidnatsrednu the .)noitcnuf ehT driht noisnetxe
swolla the researcher ot enimreted a lacitirc laitini retemarap for the ,mhtirogla
elihw the lanif eno stcerid the noitceles of sretemarap that lortnoc the ssentsubor
of the .mhtirogla sihT swolla the researcher ot eziminim the noitucexe emit of
the .mhtirogla
eW dnif detalumis gnilaenna ot eb etiuq .lufrewop nO the Judge ,noitcnuf
detalumis gnilaenna syawla sdnif the labolg ,mumitpo elihw the LSMI senituor
do .ton roF the lanoitar snoitatcepxe ,ledom the lanoitnevnoc smhtirogla
tonnac ezimitpo dna do ton offer yna nosaer for rieht .eruliaf nehW desu as
a citsongaid ,loot detalumis gnilaenna senimreted the .melborp nevE nehw siht
melborp si ,detanimile the lanoitnevnoc smhtirogla ylno ylerar dnif the -itpo
,mum elihw detalumis gnilaenna does so .ylisae ehT lanoitnevnoc smhtirogla
tonnac ezimitpo the golsnart cost reitnorf ledom elihw detalumis gnilaenna does
so .ylisae roF the laruen ,krowten detalumis gnilaenna sdnif a hcum retteb
mumitpo naht yna of the lanoitnevnoc .smhtirogla ,revewoH ti si nwohs that the
erutan of siht noitcnuf sekam ti yllautriv elbissopmi for yna dohtem ot dnif the
labolg .mumitpo
,yllaitnessE detalumis gnilaenna sedivorp erom serutaef at the cost of na
esaercni ni noitucexe emit for a elgnis nur of the .mhtirogla ,revewoH nehw
derapmoc ot the elpitlum snur netfo desu htiw lanoitnevnoc smhtirogla ot test
tnereffid gnitrats ,seulav ti si .evititepmoc nI emos cases, detalumis gnilaenna si
neve evititepmoc htiw a elgnis nur of a lanoitnevnoc .mhtirogla sA nwohs ni
xidneppa ,A retupmoc power si gnisaercni yldipar dna the secruoser dedeen for
detalumis gnilaenna lliw ,noos fi ton ,ydaerla eb elbaliava ot tsom researchers
for neve the tsom tluciffid .smelborp
sihT paper si dezinagro as .swollof noitceS 2 sebircsed detalumis gnilaenna ni
liated dna serapmoc ti ot lanoitnevnoc ,smhtirogla elihw noitces 3 serapmoc sti
ecnamrofrep no the ruof test smelborp ot the LSMI .smhtirogla sihT noitces
osla secudortni ynam snoisnetxe ot detalumis .gnilaenna noitceS 4 ylfeirb
sweiver these snoisnetxe dna secudortni .rehtona A noisulcnoc sezirammus the
shtgnerts dna sessenkaew of detalumis .gnilaenna nA xidneppa senimaxe -na
other labolg noitazimitpo ,mhtirogla the citeneg ,mhtirogla dna sdnif that
detalumis gnilaenna si .roirepus
2. ehT smhtirogla
2.1. Conventional algorithms and their snoitatimil
lanoitnevnoC noitazimitpo ,smhtirogla hcus as noshpaR-notweN dna
,llewoP-rehctelF-nodivaD yrt ot evom llihpu noitazimixam( si demussa ni the
lareneg noissucsid of the )smhtirogla ni na evitareti .rennam nI yrev lareneg
,smret gnitrats morf a ,tniop yeht enimreted the tseb noitcerid dna step htgnel ot
68 .L.W Goffe et ,.la Global optimization of statistical functions
head .llihpu retfA gnivom there, siht process si repeated litnu emos gnippots
airetirc si .deveihca lacitsitatS packages hcus as ,SAS ,PST dna STAR esu these
.smhtirogla lareneG sweiver no meht nac eb dnuof ni rehtie Judge et .la (1985)
Press et .la (1986), or tdnauQ (1983), dna a erom suorogir sisylana si ni sinneD
dna .)389l(lebanhcS riehT noitarepo si ralimis ot a dnilb nam gniklaw pu a ,llih
whose ylno egdelwonk of the llih semoc morf what passes rednu sih feet. fI the
llih si elbatciderp ni emos ,noihsaf he lliw reach the top, tub ti si ysae ot enigami
gnidnuofnoc .niarret ynaM lanoitnevnoc smhtirogla ekam the noitcnuf -tciderp
elba yb gnimussa the noitcnuf si yletamixorppa .citardauq ,yletanutrofnU -ats
lacitsit snoitcnuf yam ton eb yletamixorppa .citardauq ticilpmI here si rehtona
noitpmussa nommoc ot lanoitnevnoc :smhtirogla the noitcnuf has eno -itpo
,mum so that yna lacol mumitpo si osla the labolg .mumitpo elihW lladneK dna
trautS (1978, p. 44) show yllacitotpmysa that doohilekil snoitcnuf evah eno
,mumitpo the llams elpmas seitreporp are .nwonknu nekaT together, ti si ysae
ot see yhw lanoitnevnoc smhtirogla yam evah ytluciffid gnidnif the labolg
.mumixam sA the txen noitces shows, detalumis gnilaenna takes a yrev tnereffid
approach ot gnizimitpo snoitcnuf that sevlovni yltnacifingis fewer .snoitatimil
2.2. Simulated annearing rof continuous variable problems
detalumiS s’gnilaenna roots are ni ,scimanydomreht where eno seiduts a -sys
s’met lamreht .ygrene A noitpircsed of the gnilooc of netlom latem setavitom siht
.mhtirogla retfA wols gnilooc ,)gnilaenna( the latem sevirra at a wol ygrene state.
tnerehnI modnar snoitautculf ni ygrene swolla the gnilaenna metsys ot escape
lacol ygrene aminim ot eveihca the labolg .muminim tuB fi delooc yrev ylkciuq
(or ,)’dehcneuq‘ ti thgim ton escape lacol ygrene aminim dna nehw ylluf delooc ti
yam niatnoc erom ygrene naht delaenna .latem detalumiS gnilaenna stpmetta ot
eziminim emos eugolana of ygrene ni a rennam ralimis ot gnilaenna ot dnif the
labolg .muminim sliateD nac eb dnuof ni Press et .la (1986, pp. .)433-623
ylraE detalumis gnilaenna smhtirogla deredisnoc lairotanibmoc ,smetsys
where the s’metsys state sdneped no the noitarugifnoc of .selbairav Perhaps the
tseb nwonk si the gnilevart namselas ,melborp ni hcihw eno seirt ot dnif the
muminim pirt ecnatsid gnitcennoc a rebmun of .seitic lairotanibmoC detalumis
gnilaenna has neeb desu yllufsseccus ni retupmoc dna tiucric ngised -kriK[
kcirtap et .la (1983) dna gnoW et .la (1988)], noitullop lortnoc tnewreD[
(1988)], a laiceps case of l-O gnimmargorp lxerD[ (1988)], laruen skrowten
namressaW[ dna Schwartz (1988)], noitcurtsnocer of enillatsyrcyllop serutcurts
ylleT[ et .la (1987)], dna egami gnissecorp ilavenraC[ et .la (1985)].
Other labolg noitazimitpo smhtirogla evah neeb decudortni ni tnecer .sraey
yehT edulcni evitpada modnar search otaznorP[ et .la (1984)], citeneg -ogla
smhtir grebdloG[ (1989)], the dellif noitcnuf dohtem upneR[ (1990)], -itlum
level sdohtem naK[ et .la (1987)], dna a dohtem gnisu citsahcots laitnereffid
.L.W Gofe et ,.la Global optimization ‘fo statistical functions 69
snoitauqe initneP-iffulA[ et .la (1988)]. htoB tlibrednaV dna eiuoL (1984) dna
yksvehcahoB et .la (1986) evah deifidom detalumis gnilaenna for suounitnoc
elbairav .smelborp
,revewoH the anaroC et .la noitatnemelpmi of detalumis gnilaenna for -noc
suounit elbairav smelborp appears ot offer the tseb noitanibmoc of ease of esu
dna ,ssentsubor so ti si desu ni siht yduts na[ ylrae dna yrev etelpmocni test nac
eb dnuof ni effoG et .la (1992)]. elihW a etelpmoc noitpircsed nac eb dnuof there,
a yrammus swollof dna the mhtirogla ni retupmoc-oduesp code si dedivorp ni
xidneppa .B ehT nartroF code of the mhtirogla si elbaliava morf the srohtua dna
ni drahciR s’tdnauQ 6TPOQG lacitsitats noitazimitpo package. ehT laitnesse
gnitrats sretemarap ot ezimixam the noitcnuf f(X) are r,, the laitini -repmet
;eruta X, the gnitrats rotcev of ;sretemarap dna V, the step htgnel for X. Note
that X dna V are htob srotcev of htgnel ,n the rebmun of sretemarap of the
.ledom reppU case refers ot srotcev dna rewol case ot sralacs htiw( the noitpecxe
of ,erutarepmet .)T
A noitcnuf noitaulave si edam at the gnitrats tniop X dna sti eulav f si
recorded. Next, a wen X, X’, si nesohc yb gniyrav tnemele i of X,
Xi = Xi + r ’ ,iV (1)
where r si a ylmrofinu detubirtsid modnar rebmun morf [ - 1, 1 1 dna iU si
tnemele of V. ehT noitcnuf ’feulav si neht .detupmoc ’ffI si greater ;fnaht X’ si
i
accepted, X si set ot X’, dna the mhtirogla sevom .llihpu fI siht si the ,ftsegral ti
dna X are recorded ecnis siht si the tseb tnerruc eulav of the .mumitpo
’”jfI si ssel naht or lauqe ot f, the siloporteM noiretirc sediced no ecnatpecca
scimanydomreht( setavitom siht .)noiretirc ehT eulav
p = e(S’-f)/T (2)
si detupmoc dna derapmoc ot p’, a ylmrofinu detubirtsid modnar rebmun
morf [0, 11. fI p si greater naht p’, the wen tniop si accepted, X si detadpu htiw
X’, dna the mhtirogla sevom .llihnwod ,esiwrehtO X’ si .detcejer owT factors
decrease the ytilibaborp of a llihnwod :evom rewol serutarepmet dna regral
secnereffid ni the s’noitcnuf .eulav oslA eton that the noisiced no llihnwod sevom
sniatnoc a modnar .tnemele
retfA N, steps hguorht lla stnemele of X lla( hcus ‘N’ sretemarap are set yb the
,)resu the step htgnel rotcev V si detsujda so that 50% of lla sevom are accepted.
ehT laog here si ot elpmas the noitcnuf .ylediw fI a greater egatnecrep of stniop
are accepted for ,iX neht the tnaveler tnemele of V si .degralne roF a nevig
,erutarepmet siht sesaercni the rebmun of snoitcejer dna decreases the -tnecrep
age of .secnatpecca retfA TN semit hguorht the evoba ,spool the ,erutarepmet T,
si .decuder ehT wen erutarepmet si nevig yb
T’=r,.T, . (3)
70 .L.W Gofi et ,.la Global optimization of statisticalfunctions
where Tr si neewteb 0 dna 1. A rewol erutarepmet sekam a nevig llihnwod evom
ssel ,ylekil so the rebmun of snoitcejer esaercni dna the step shtgnel .enilced nI
,noitidda the tsrif tniop deirt at the wen erutarepmet si the tnerruc .mumitpo
ehT rellams steps dna gnitrats at the tnerruc mumitpo sesucof noitnetta no the
tsom gnisimorp area.
ehT mhtirogla sdne yb gnirapmoc the tsal N, seulav of the tsegral noitcnuf
seulav morf the dne of each erutarepmet noitcuder htiw the tsom tnecer eno dna
the mumitpo noitcnuf .eulav fI lla these secnereffid are ssel naht E, the mhtirogla
.setanimret sihT noiretirc spleh erusne that the labolg mumixam si reached.
Note that detalumis gnilaenna tsrif sdliub pu a hguor weiv of the ecafrus yb
gnivom htiw egral step .shtgnel sA the erutarepmet sllaf dna the step htgnel
decreases, ti ylwols sesucof no the tsom gnisimorp area. llA the ,elihw the
mhtirogla nac escape morf lacol amixam hguorht llihnwod .sevom ,yllautnevE
the mhtirogla dluohs egrevnoc ot the s’noitcnuf labolg .mumixam
detalumiS gnilaenna has lareves laitnetop segatnavda revo lanoitnevnoc -ogla
.smhtir ,tsriF ti nac escape morf lacol .amixam nI cimanydomreht ,smret elihw
lanoitnevnoc smhtirogla hcneuq yb ylpmis gnidaeh pu the tnerruc llih tuohtiw
regard ot others, detalumis gnilaenna sevom htob llihpu dna .llihnwod ,oslA the
noitcnuf deen ton eb yletamixorppa .citardauq nI fact, ti deen ton neve eb
elbaitnereffid anaroC[ et .la (1987) yllufsseccus etartsnomed detalumis gnilaenna
no a cilobarap noitcnuf derutcnup htiw .]seloh rehtonA tifeneb si that the step
htgnel sevig the researcher elbaulav noitamrofni tuoba the .noitcnuf fI na tnemele
of V si yrev ,egral the noitcnuf si yrev talf ni that .retemarap ecniS ti si denimreted
yb noitcnuf snoitaulave at ynam ,stniop ti si a retteb llarevo erusaem of ssentalf
naht a tneidarg noitaulave at a elgnis .tniop ,yllaniF detalumis gnilaenna nac
yfitnedi renroc snoitulos esuaceb ti nac ’elgguns‘ pu ot a renroc for snoitcnuf that
t’nod tsixe ni emos .noiger ehT tsom tnatropmi egatnavda of detalumis gnilaenna
si that ti nac ezimixam snoitcnuf that are esiwrehto tluciffid or elbissopmi ot
.ezimitpo sihT si detartsnomed ni the txen noitces htiw the test .smelborp
ehT elos kcabward ot detalumis gnilaenna si the deriuqer lanoitatupmoc
power, tub siht melborp si gniraeppasid or has .deraeppasid xidneppA A -ed
sebircs the gniunitnoc stnemevorpmi gnirrucco ni hgih ecnamrofrep .gnitupmoc
,ylfeirB the power of the tsrif yarC retupmocrepus nac won ylisae eb tup no
a desktop, dna siht dnert shows on ngis of .gniwols ,suhT detalumis gnilaenna si
na evitcartta noitpo for tluciffid .snoitcnuf
3. detalumiS gnilaenna derapmoc to lanoitnevnoc smhtirogla
3.1. Comparison krowemarf
ehT suoiverp noitces detartsnomed that detalumis gnilaenna has esimorp for
gnizimitpo lacitsitats .snoitcnuf oT see fi siht esimorp ,sdloh detalumis gnilaenna
W.L. Gaffe et ,.la Global noitazimitpo fo lacitsitats functions 71
was derapmoc ot three lanoitnevnoc smhtirogla no ruof tnereffid lacitsitats
.smelborp ehT esoprup here si ot see fi detalumis gnilaenna works no these
,smelborp elihw noitces 4 sessucsid gnivorpmi the .mhtirogla
llA three etairavitlum noitazimitpo smhtirogla morf the LSMI yrarbiL/htaM
noitidE 10 were nesohc for nosirapmoc htiw detalumis .gnilaenna sihT yrarbil
was nesohc esuaceb of sti ytilauq dna .ytilibaliava ehT senituor are ,LOPMU
a xelpmis ,mhtirogla ,FGCMU a etagujnoc tneidarg mhtirogla htiw laciremun
,sevitavired dna ,FNIMU a notweN-isauq mhtirogla htiw laciremun .sevitavired
ehT laciremun evitavired snoisrev of FGCMU dna FNIMU were nesohc
esuaceb gnitupmoc the lacitylana sevitavired si yrev tluciffid ecnis two of
the ruof snoitcnuf were etiuq xelpmoc (the xelpmis mhtirogla does ton esu
.)sevitavired
ecniS lanoitnevnoc smhtirogla are evitisnes ot gnitrats ,seulav each elgnis test
desu 100 tnereffid ylmodnar detceles gnitrats .seulav sihT rebmun sevig the
mhtirogla a good ecnahc of gnidnif the .mumitpo ,sulP the lacipyt researcher si
ylekilnu ot esu .erom ehT egnar desu ot tceles meht dluohs ton eb ylbirret
lacitirc ecnis so ynam are .desu htiW detalumis ,gnilaenna 100 gnitrats seulav
were desu ni eno ,melborp tub retupmoc emit stniartsnoc detneverp gnisu
siht approach for the other three. nI these cases, the stluser were deifirev ni
rehtona .yaw
3.2. Test functions
Judge et .la (1985, pp. )759-659 edivorp the tsrif test .noitcnuf sihT raenilnon
tsael serauqs melborp si nettirw ni noitaziminim mrof dna has two lacol .aminim
tI has two sretemarap dna ytnewt .snoitavresbo elihW llams dna ,laicifitra ti
sevres as a tneinevnoc test morf the erutaretil for noitazimitpo smhtirogla
esuaceb of the two lacol .aminim sA lliw eb ,nees lanoitnevnoc smhtirogla
tonnac yb sevlesmeht hsiugnitsid neewteb the two, elihw detalumis gnilaenna
has on .ytluciffid Note that ti si dezimitpo ni noitaziminim .mrof
ehT dnoces noitcnuf sevired morf the lanoitar snoitatcepxe noisrev of the
yratenom yroeht of egnahcxe rate noitanimreted as detneserp yb ooW (1985).
ehT ledom stsisnoc of the gniwollof larutcurts :spihsnoitaler
h - m:) - (PC - ?+iP = ’1 + (CPY, - cp*Y:) + 4Wl - @- )1 - IR(r, - ):r
a,
- 4Pt - 1 - -+iP 1) + tE( -
r: +
rt = W,+ 4 I ,l,I
1 -
12 W.L. Gaffe et al., Global noitazimitpo of statistical snoitcnuf
where m, ,p e, dna y are the gol slevel of the yenom stock, ecirp ,level egnahcxe
rate, dna ,tuptuo ,ylevitcepser dna where star stpircsrepus etoned selbairav ni
the citsemod .ymonoce esehT snoitauqe tneserper the dradnats -tsujda-kcots
tnem yenom dnamed snoitauqe nehw the tseretni rate seiticitsale dna -tsujda
tnem stneiciffeoc are the emas ni htob .seirtnuoc gnisahcruP power ytirap dna
derevocnu tseretni ytirap .dloh ehT citsahcots noitacificeps of the ledom si
detelpmoc yb gnimussa that the suonegoxe selbairav of the ,metsys
M, = m, - m:, yr, dna :y evlove gnidrocca ot a elbats )l(RA noitatneserper
nevig yb
,x = v,,
1-,XB +
where X, E [M, y, ’]:y dna where B si a 3 x 3 retemarap xirtam hcus that the
roots of Z(ted - BL) eil nihtiw the tinu .elcric
ehT noitulos dna noitamitse erudecorp swollof that deniltuo yb imelaS (1986),
hcihw syolpme the elbats rotcev AMRA noitatneserper for the selbairav of the
metsys deilpmi yb the ledom .evoba oT evired siht ,noitatneserper tel
,Z = ,‘];X,e[ dna etirw the ledom evoba as
,1 + ,ZlZ + 1-Z.zC + ],II,+,Z[EO = ,,U
where CO, z‘, , ,2C dna 0 are 5 x 4 retemarap secirtam gniniatnoc the larutcurts
sretemarap M, /3, ,y q, dna cp*, as llew as the enin stnemele of ,B dna where ,U si
the rotcev of larutcurts .secnabrutsid A noitulos of the mrof
2, = Cc, + A(L)U, = oC + [A, + LIA + AzL2 +. . .]U,
neht stluser ni the elbats AMRA noitatneserper nevig yb
oC + [O + GIL + Z,L*]Z, = [A&J + ZL]U,,
where the stnemele of 0A evah neeb detcirtser so as ot yllacitylana lecnac the
elbatsnu root of the RA .laimonylop sA detacidni yb ,imelaS the setamitse of the
sretemarap of siht ledom yam neht eb deniatbo yb noitaziminim of
htiw respect ot the evif metsys sretemarap (c(, ,I/ ,y p, cp*) dna the enin RA
sretemarap ,b( 1, . , bj3), tcejbus ot the xelpmoc raenilnon snoitcirtser desopmi
across these sretemarap ni the stnemele of ,& [the noitisnart morf a mumixam
.L.W effaG et al., labolG noitazimitpo snoitcnuflacitsitatsfo 37
ledom ot a ledom desab nopu the noitaziminim of a mus of serauqs si
debircsed yb nosliW-effilcinnuT (1973)].
ehT driht melborp si morf a yduts of mrif noitcudorp ycneiciffe desab no
a metsys of a reitnorf cost noitcnuf dna sti tupni share snoitauqe [see reirreF dna
11evoL (1990)]. eerhT error smret no the cost noitauqe erutpac the effects of two
sepyt of ycneiciffeni dna modnar error. ehT error smret no the share snoitauqe
are deknil ot eno of the error smret no the cost noitauqe (errors ni tupni share
noitceles tluser ni ,ycneiciffeni suht gnisiar costs). retfA gnikam snoitpmussa
gninrecnoc the suoirav ecnabrutsid ,smret noitamitse of the metsys si deirrac tuo
yb gniziminim a noitcnuf devired morf the doohilekil .noitcnuf ehT noitamitse si
deirrac tuo no laer dlrow data no .S.U yrotisoped snoitutitsni dna sevlovni 62
sretemarap dna 575 .snoitavresbo
ehT htruof test noitcnuf semoc morf the laruen krowten .erutaretil elihW
laruen sten evah ynam cimonoce snoitacilppa [see muaB (1988) dna nauK dna
etihW (1991) for ,]syevrus the esu here swollof tnallaG dna etihW (1989). yehT
esu a laruen ten ot etamixorppa na nwonknu ,noitcnuf g(x), nevig yb
.
SKtxlB9 Y) = i: Pj’G i xi’Yi,j + YO,j
j=l ( i=l >
larueN skrowten are dezinagro ni ,sreyal morf the tupni reyal ot the tuptuo
.reyal nI siht laruen ,ten the x rotcev of htgnel r si the tupni .reyal ehT jY
,srotcev =j ,...,l ,K are sthgiew ot the txen reyal eton( the tpecretni or saib
mret ni each .)rotcev sihT reyal si -’neddih‘ ylno the tupni dna tuptuo sreyal are
.’nees‘ ecniS there si ylno eno neddih ,reyal siht si a elgnis neddih reyal laruen
.krowten ehT noitcnuf G si the neddih tinu noitavitca noitcnuf (a citsigol ni
tnallaG dna )etihW dna 0, a rotcev of htgnel K, si the neddih ot tuptuo reyal
.sthgiew
tnallaG dna etihW propose ot esu the laruen ten ot etamixorppa the -nu
nwonk noitcnuf that setareneg a citoahc .seires oT test siht approach, a citoahc
seires of htgnel n si detareneg yb a etercsid noisrev of the ssalG-yekcaM
:noitauqe
.
10.2.x,_, + ( xt-#O1
-rx,s-,x(g 1) = 1-tX + (10.5). r-,x.1.0-
[
sihT si a lufesu rotareneg of data for the yduts of citoahc cimonoce dna laicnanif
seires as ti setareneg data ralimis ot those dnuof ni laicnanif .stekram
ehT I/ dna the iy srotcev are nesohc ot eziminim
47 .L.W eSfoG te ,.la labolG noitazimitpo fo snoitcnuflacitsitats
,suhT the laruen ten sretemarap are nesohc so that ti setamixorppa the noitcnuf
that setareneg x, nevig x,_ ,1 . . , ,x _ .5 sihT si a tluciffid noitazimitpo melborp
esuaceb lanoitnevnoc modsiw has ti that siht noitcnuf has ynam lacol aminim
siht( noitressa si deifitnauq .)retal
nI siht ,sisylana Y = 5 siht( sdleiy xis stnemele for each jy rotcev ecnis the
tpecretni mret si ton dedulcni ni the ,)noitammus K = 5, dna n = 1000. esehT
seulav of Y dna K dleiy 35 sretemarap ot ’niart‘ or tif the laruen ten ot the
.noitcnuf esehT seulav were nesohc esuaceb tnallaG dna etihW dnuof yeht
mrofrep llew ni gnittif the noitcnuf ot siht seires htgnel dna ti si a melborp of
etaredom .ezis
3.3. Computing environment
llA gnitupmoc was enod at the ytisrevinU of saxeT retneC for hgiH -rofreP
ecnam gnitupmoC ni nitsuA no yarC 42/PM-X dna yarC 468/PM-Y -mocrepus
sretup lla( gnitupmoc was enod at eno etis neve hguoht siht level of ecnamrofrep
was ton syawla .)dedeen llA smargorp were nettirw ni nartroF 77 htiw emos
nartroF 90 hcihw( sedulcni ynam serutaef that yfilpmis gnitirw smargorp htiw
)syarra snoisnetxe gnisu the 77TFC ,relipmoc snoisrev 2.0, 3.0, dna 4.0. A tib-46
noitatneserper was desu for lla gnitaolf tniop ;srebmun elihw elgnis noisicerp no
a ,yarC siht stneserper elbuod noisicerp ni the Vax, MBI ,emarfniam dna CP
.sdlrow
ehT yarC detneserp eno .ytluciffid nehW a gnitaolf tniop error srucco no
,ti the margorp .setanimret sihT si tnereffid morf the MBI emarfniam dlrow
where a software noitcerroc netfo takes ecalp dna noitucexe .seunitnoc ,suhT ti
dluow appear that stluser detareneg no a yarC yam ton yrrac revo ot other
.sretupmoc ,revewoH the yarC seldnah na yllausunu ediw egnar of ,srebmun
morf 1O2466 ot 6642-O1 ,suhT gnitaolf tniop errors are ylno ylekil ot rucco fi the
mhtirogla si dnoyeb the ylekil noiger of a .noitulos nI these ,snoitautis a -moc
retup htiw a puxif for gnitaolf tniop errors si ylekilnu ot do .retteb ,suhT the
s’yarC kcal of a puxif for gnitaolf tniop errors dluohs ton timil the snossel
denrael here.
3.4. Results htiw Judge’s function
elbaT 1 shows the stluser of lla smhtirogla for the Judge .ledom llA smhtirogla
were nur 100 semit htiw tnereffid gnitrats seulav for the ledom sretemarap (the
emas seno were desu yb lla .)smhtirogla A ylmrofinu detubirtsid modnar
rebmun rotareneg detceles meht morf - 100 ot 100 ot etalumis the researcher’s
ytniatrecnu tuoba the tseb gnitrats .eulav roF the LSMI ,senituor rieht -seggus
snoit for mhtirogla stupni were desu where ,nevig except the mumixam rebmun
of noitcnuf snoitaulave was set ot a yrev egral eulav ot diova ylrae .noitanimret