O tehnica fuzzy de partitionare si inductie automata bazata pe extensia fuzzy a distantei c 2
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1 76 Revta Iformatca Ecoomca, r. (4 / 000 O tehca fuzzy de arttoare ducte automata bazata e etea fuzzy a dtate c Cof.dr. Vale GEORGESCU Uvertatea d Craova, vgeo@cetral.ucv.ro Lucrarea roue u tem de achzte automata a cuottelor ce clude mecame fuzzy de arttoare a domeulu varablelor cotue de geerare ductva a arborelu de decze. mbele rocee au la baza adatarea dtate? etru a oera î tabele de cot- geta fuzzy, î care frecvetele e defec r grade de aarteeta la elemetele uor artt fuzzy. Î coul uer de acord a uu tet robablt cu o decrere fuzzy a datelor -a trodu u crteru ce retrctoeaza alegerea chemelor de acoerre a domeulu ue varable cotue cu multm fuzzy. lcarea crterulu e ermte a terretam vectorul gradelor de aarteeta ataat fecare valor Dom( X î terme ue dtrbut de robabltat. Cu autorul codfcar fuzzy atfel defte, u algortm de dcretzare duce arttle fuzzy otmale ale tuturor redctorlor. rborele de decze fuzzy ete geerat ao recurv r electa otmala a redctorulu î raort cu care are loc ramfcarea î fecare od, electe ce aeleaza tot la dtata? adatata tabelelor de cotgeta fuzzy. Cuvte chee: fuzzy, ducte automata, clafcare, dcretzare, algortm, codfcare. lgortm de ducte clafcare automata Tehcle actuale de achzte a datelor roduc colect mave de format. Geerarea automata a cuottelor d date ete u roce comle ce aeleaza la mecame de ducte automata face obla traformarea formate amorfe, latete, îtr-o forma tructurata, telgbla dec "dgerabla" etru comutatea ttfca d domeul de eertza reectv. Stemele de îvatare automata ut reute de obce ub umele de algortm de ducte (ducer au dret co a geereze mecame de clafcare automata (clafer cu mare utere redctva, r etragerea, abtractzarea codfcarea formate cotute îtr-o baza de date. Î acet cotet, algortm ce duc arbor de decze rerezta trumete artculare deoebt de utle. E geereaza rocedur de clafcare ce utlzeaza tructura de arbore etru a electa regula de decze cea ma otrvta îtr-u cotet reczat. Cuoaterea atfel tructurata oate f medat traua îtr-u formalm de rerezetare comatbl cu temele feretale bazate e regul. vataul lor maor ete gradul îalt de telgbltate uurta cu care ot f roceate cuottele etru a fera o cocluz, decz, au redct. Î cotuare vom troduce câteva otat deft. Fe T u et de date detat îvatar regullor de decze (trag, ce cotau d ette (umte eemle, fecare avâd ataata o aumta etcheta. Orce etta etchetata are doua art: - o arte eetchetata, otata r (,, care ete u vector ale caru elemete S ut valor ale atrbutelor coreuzatoare ( X, m - u atrbut omal ecal, etcheta, otata r Y rerezetâd varabla tta (au decza, adca va rabla ce focalzeaza roceul de îvatare dere care dorm a facem redct (o valoare actuala y a etchete Y, culata cu vectorul atrbutelor, ecfca o clafcare cuocuta, adca o combate ce trebue folota etru a îvata regulle de decze. ;
2 Revta Iformatca Ecoomca, r. (4 / trbutele (X ut umte ueor redctor au u rol elcatv î clafcare. Fe Dom(X domeul redctorulu X. Fecare etta eetchetata ete u elemet al atulu ettelor eetchetate m X Dom(X... Dom(X, ude m ete umarul atrbutelor. Daca deemam r Y multmea valorlor oble ale etchetelor, atuc o etta etchetata ( X, y Y ete u elemet al atulu X Y. Sarca uu algortm de ducte I ete a roduca u mecam efcet de clafcare dtr-o multme de ette etchetate, adca a alce u et de date T detat îvatar, î multmea clafcarlor oble C. Mecamul de clafcare atfel geerat (bazat e aocerle ertete dtre artea eetchetata a ettelor o aumta etcheta oate f folot ao etru a clafca orce alta etta eetchetata X, r uerea a î coreodeta cu o etcheta y Y, care ete tta redcte. Preuuem ca etul de date ete alcatut d ette deedete detc dtrbute, motv etru care vom aoca atrbutelor o terretare couctva (artea codtoala a regul rerezta o coucte de reme mle. Î artcular, u mecam de clafcare cu tructura de arbore, cote arborele de decze du, care ue î coreodeta o etta eetchetata cu o claa r arcurgerea drumulu de la radaca la o fruza retureaza etcheta ce deemeaza aceata claa î fruza. Î cotuare, coul otru ete a vetgam câteva tehc etru geerarea automata a mecamelor de clafcare (clafcatoarelor fuzzy, folod algortm de ducte ecal modfcat, caabl a roduca regul de decze cu remze fuzzy. rbor de decze fuzzy rbor de decze fuzzy ot f rvt ca o geeralzare aturala a arborlor de decze tradtoal. Imlemetâd algortm de îvatare dcrmatv oerâd r arttoare recurva, dele lor de baza ut acelea: a arttoeze atul obervatlor î ubeatoae dua crter de electe due de date a rerezte arttle ub forma uu arbore. Mecamul feretal ete detul de aemaator: cuoaterea dua ete folota etru a roduce alte clafcar, r aalza coreodete dtre atrbutele uu o u e- emlu codtle îtâlte e traeele de la radaca la fruzele arborelu. Cu toate acetea, ut multe aecte î care cele doua abordar dfera. Î mod eetal, arbor de decze fuzzy foloec rerezetar fuzzy, care furzeaza u cadru mbolc etru achzta cuottelor î cotete mrece certe ermte îmbuatatrea metodologlor etete î cele doua comoete maore ale acetora: rocedurle de geerare a arborelu rocedurle feretale. coerrea fuzzy, comaratv cu arttoarea trcta a uu domeu cotuu Î temele de îvatare automata tradtoale, doua tur de dome e îtâlec î mod curet: cele coreuzâd varablelor mbolce (omale, defte lgvtc rtr-u umar mc de modaltat cele coreuzâd varablelor umerce (dcrete au cotue. Cu toate ca varablele cotue e îtâlec frecvet î ractca, umero algortm de ducte du doar de mecame de îvatare î atul atrbutelor omale. Petru a oera cu varable cotue, atfel de algortm utlzeaza î mod uzual tehc de arttoare a domeulu varablelor cotue, etru a roduce a- trbute omale. emeea tehc e ma umec rocedur de dcretzare (dcretzer. Data fd o artte a uu domeu real î tervale, u "dcretzor" ete o fucte de codfcare ce ue î coreodeta valorle ue varable reale cotue ce cad
3 78 îtr- u aumt terval, cu modaltatle (categorle aocate ale uu atrbut omal. tât arttle trcte cât acoerrle fuzzy trebue a fe comlete, adca fecare valoare a domeulu trebue a aarta cel ut uu terval (au multme fuzzy. Cotrar ue artt, care trebue a fe coteta, acoerrea fuzzy oate f coteta, datorta uu aumt grad de acoerre recroca a multmlor fuzzy (altfel u, o valoare a domeulu oate cadea î ma multe multm fuzzy, cu aumte grade de aarteeta. Terme lgvtc deft î mod gradual de elemetele ue cheme de acoerre fuzzy cofera abltatea de a modela detal de cuoatere foarte fe, recum obltatea de a trata date afectate de erturbat au avâd atrbute la. Utlzarea retrctlor fuzzy î arttoarea recurva a arborelu rbor de decze fuzzy dfera totodata de ce tradtoal r caactatea de a folo crter de ramfcare ce uorta u uma retrct trcte, c retrct fuzzy. Geerarea arborelu e bazeaza e o arttoare recurva. Radaca arborelu de decze rerezta îtregul atu de obervat atât tm cât u ut mue retrct. Î coecta, el cote toate ettele detate îvatar. Retrctle (fe trcte, fe fuzzy ut defte î raort cu terme lgvtc a fecaru atrbut X au forma: X ete, ude deota fe o categore mbolca, fe uul d elemetele ce acoera domeul (tervale, au multm fuzzy. Î coul ramfcar uu od î tmul roceulu de eadare recurva a arborelu, trebue a electam u atrbut X (ce u aare e drumul de la radaca catre, de obce cel care mamzeaza o aumta maura de mlartate, au câtg de formate, etru ettele detate îvatar rezete î od. cum, odul oate f ramfcat r arttoarea ettelor ale otrvt retrctlor "X ete " care ut Revta Iformatca Ecoomca, r. (4 / 000 folote ao etru a geera codt care a coduca re odurlor f (. ceea rocedura ete reetata recurv etru fecare od fu, âa e atge u crteru de to. Dua cum tm, arbor de decze clac oereaza cu varable cotue r dcretzarea domeulu lor î artt defte î mod trct, ale caror comoete ut multm clace (tervale, î cele ma multe cazur. D acet motv, valoarea uea dtre varablele ce decru o etta atface eact ua d oblele retrct ce e ramfca d odul reectv (aceea aocata multm clace a care fucte caractertca ete. Pr urmare, fecare etta cade eact îtr-u ubod atfel eact îtr-o fruza (cu eceta cazurlor câd lec valorle uor atrbute. Dmotrva, arbor de decze fuzzy ut caabl a roceeze varable cotue r geerarea de acoerr fuzzy (ecvete ordoate de multm fuzzy ce e urau artal. Petru ca fecare retrcte fuzzy ete defta de fucta de aarteeta a multm fuzzy, deve obl ca u eemlu dat a atfaca ma multe codt atfel a cada î ma multe odur fu. rbor de decze fuzzy ot rocea ette (eemle care cot date ermate atât r valor umerce cât r terme fuzzy. Petru a face fata acete tuat ete ufcet a duem de u oerator at a maoare gradul de cocordata (mlartate dtre ettat u uma de acela t, c de tur dferte (fe ele valor umerce, au terme fuzzy. De eemlu, fe X u atrbut, U Dom(X uverul au de dcur, ar o retrcte fuzzy. Î cazurle î care atrbutul a o valoare umerca, otata, o maura a gradulu î care atface retrcta fuzzy ete data r char fucta de aarteeta a lu, evaluata î : µ (.
4 Revta Iformatca Ecoomca, r. (4 / Câd atrbutul a o valoare fuzzy otata F, u dce de cocordata fuzzy oate f ermat, de eemlu, r maura de obltate: Π ( F, u µ (u u m( µ (u, µ (u u U F u U F Petru ca de obce multmle fuzzy e urau, ar dcele de cocordata ete u umar d tervalul [0, ], cea ma mare arte a ettelor atfac î acela tm ma multe retrct fuzzy (dar uma artal atfel cad î ma multe fruze, fecare dtre ele cu o odere cura î [0,]. Î odul radaca, toate cele ette au aceea odere: w root (,. Petru a atge u od, u aumt umar de retrct fuzzy trebue a fe atfacute. Î acet od, o etta data e are oderea w calculata îtr-u mod couctv, e baza dclor de cocordata ce deemeaza gradul î care ut atfacute retrctle fuzzy de-a lugul drumulu de la radaca la. Fe ful de rag al lu a reuuem ca e cuoate tuc, oderea î, otata r oate f evaluata recurv: w T w (, Π(e, w. w, ude T ete o T-orma (de obce m, au rodu, ar ete termeul fuzzy aocat cu retrcta fuzzy ce coduce la. Cumulâd oderle etru toate ettele ce cad î obtem frecveta de aarte a termeulu lgvtc : T ( w, Π(e, Î forma relatva, aceata frecveta etmeaza robabltatea ca o etta rezeta î a atfaca retrcta aocata termeulu lgvtc : T w T w (, Π(e, (, Π(e, alog, robabltatea ca o etta aflata î a rerezte claa lu Y aocata termeulu lgvtc B ete etmata de: y (, Π(e, B T w B T w y (, Π(e, B Dcretzare r codfcare fuzzy tadard. O terretare robablta Codfcarea fuzzy ete u trumet geeral care oate f adatat cu uurta etru dferte cour, dar ma ecfc el aare î dcretzarea varablelor aleatoare cotue e tervalele ue artt fte a lu R. Cocret, el cota î geerarea codfcar Drac (care atrbue fecare obervate uu terval r aocerea ue obervat (evetual la ma multe tervale, cu oder (grade de aarteeta ror fecaru terval. Fe Z o varabla aleatoare cotua, defta e atul de robabltate ( Ω,, µ. De obce, tattcaul u e ocua cu feomeul ubacet cocretzat r Z, c uma cu u eato de volum ( X( ω,..., X ( ω etra dtre valorle ue varable obervate X, care ar utea f o "aromare" a lu Z. Petru a furza u model al fuzzctat dtre X Z, utem folo o codfcare fuzzy a lu X, care a traforme eatoul orgal îtr-u eato codfcat, aocat ue varable aleatoare omale ordoata. Cocetul de codfcare fuzzy tadard e ermte a obtem o terretare robablta a rocedur de dcretzare: aceata îeama a aocem fecare obervat X( ω a ue varable reale aleatoare X, o dtrbute de robabltat P e R, îzetrata cu o σ-algebra fta, geerata de o artte a lu R î tervale. Î coul obter acete terretar robablte, oderle obervatlor refertoare la fecare terval trebue a fe umere oztve, a caror uma ete.
5 80 Î terme geeral, o codfcare fuzzy C c,..., de la tadard ete o alcate ( c R la [ 0, ], care atface codtle: ( C ete maurabla î raort cu σ-algebrele Borelee ale lu R [ 0, ] ; ( R, c (. Fuctle c,..., c ut umte fuct de codfcare fuzzy tadard.,..., a lu R î Câd o artte ( tervale ete data etru o varabla aleatoare reala, obtem o terretare a fuctlor de codfcare î terme uor robabltat de aarteeta ale ue obervat la dvere clae. Î acet caz, o dtrbute de robabltat P X e ( R, B, defta de PX ( c (, ete aocata fecare obervat X( ω, ude B ete σ -algebra,...,. geerata e R de artta ( Ma formal, o codfcare fuzzy tadard aocata ue artt (,..., ete be defta câd ete data o tructura tattca ( R, B, { PX; ( R, BR }. Fuctle maurable c,..., c d ( R, B R î ([ 0, ], B [0,] defte de c ( PX ( ut umte fuct de codfcare fuzzy tadard. Potrvt acete deft, P ( P X X R ete o dtrbute de robabltat de trazte e ( R, B R B, adca B B, P X (B ete o fucte maura-, X bla R P ete o dtrbute de robabltat e B. tfel, eatoulu tal ( X( ω,..., X ( ω e va aoca u eato codfcat ( PX (,..., PX ( ω ω rerezetâd o dtrbute de robabltat. Ma mult, etru orce varabla aleatoare reala X, varabla -dmeoala Xˆ c o X,..., c o X ete umta varabla ( codfcata fuzzy tadard. Î artcular, avem de a face cu o codfcare de t terval atuc câd utlzam codfcarea Drac PX δx, ude X Revta Iformatca Ecoomca, r. (4 / 000 δ ete maura Drac δ X ( c ( (X, ar ete fucta caractertca a tervalulu. Tabele de cotgeta etru atrbute fuzzy Fe ( X, Y u culu de doua varable aleatoare. Câd cel ut ua dtre acete varable ete cotua, o tabela de cotgeta u ar utea f cotruta fara ca varabla a fe ma îtâ dcretzata. Coderam doua cazur: Cazul I: O codfcare fuzzy tadard ete alcata lu X o codfcare Drac lu Y Preuuâd ca X ete cotua, decdem a o traformam îtr-o varabla fuzzyevaluata, folod o codfcare fuzzy tadard care geereaza o acoerre fuzzy. Preuuem, de aemeea, ca Y ete fe omala, fe cotua (î ultmul caz dcretzarea e va face r alcarea ue arttoar trcte î tervale.,..., o acoerre fuzzy a dome- Fe ( ulu lu X, (,..., B o artte trcta e B domeul lu Y (, y, u eato de volum. Petru a clafca cele valor B,..., a obte ale lu Y e artta ( B dtrbuta a de frecvete avem evoe doar de alcarea codfcar Drac: ude c (y δy (B B B (y ;, ete fucta caractertca a lu y B B : B (y, cu ( y 0 y B B. Petru a clafca cele valor ale lu X e acoerrea fuzzy (,..., a geera dtrbuta a de frecvete, utem folo o codfcare fuzzy tadard: c ( µ ( ;, ude fucta de codfcare fuzzy c ( ete defta î mod atural de fucta de aarteeta : µ (, uua codte de a avea o terretare robablta:
6 Revta Iformatca Ecoomca, r. (4 / c( µ ( P (, Dom(X D aceata codte (atfacuta ecluv de o codfcare fuzzy tadard utem deduce urmatoarea roretate: µ T µ ( ( µ Fe acum e (, y u eemlu etra dtr-u eato dat de volum. De fat, e deemeaza realzarea uu evemet comu. D acet motv, clafcarea acetu eemlu î raort cu acoerrea fuzzy (,..., artta ( B,..., B oate f terretata ca o coucte a doua fuct de codfcare. Deoarece modul obut de a rerezeta o coucte ete folorea ue T-orme (m rod ut cele ma oulare, avem: T Ma mult: ( µ (, ( y ;, ;, B ( µ (, B (y T( µ (, B (y ( Îtr-adevar, ca rezultat al fatulu ca o valoare data y cade eact îtr-u terval B, eta eact u terme î uma T ( µ (, B (y etru care B ( y. tuc, alcâd câteva roretat ale T- orme (adca T (u, 0 0 utem deduce ca: T T (u, u ( µ (, B (y T( µ (, ( µ Cazul II: O codfcare fuzzy tadard ete alcata atât lu X cât lu Y Cotrar cazulu recedet, u uma valorle atrbutulu X atfac ma multe retrct fuzzy, c uele valor y Dom(Y au clafcar multle. Putem vorb acum uma dere gradele de aarteeta ale acetor eemle la dvere clae lgvtce. D acet motv, umarul eemlelor e (, y ce atfac u culu (, B de retrct fuzzy crete emfcatv. Totu, frecvetele aocate tabelelulu de cotgeta ot f defte aalog: T ( µ (, µ B (y ;, ;, Prcala dfereta ete ca u oate f egala cu, datorta algebre T-ormelor, char daca utlzam o codfcare fuzzy tadard. Etea fuzzy a maurlor de mlartate, bazate e tattca tetulu Tetul de deedeta a doua varable aleatoare tetzate îtr-u tabel de cotgeta are ca obect vetgarea dferetelor î frecveta câd u eato de volum ete clafcat î clae dua rmul atrbut î clae dua al dolea. Frecvetele eemlelor î fecare clafcare ot f rerezetate mbolc ca î tabelul urmator:
7 8 Revta Iformatca Ecoomca, r. (4 / 000 Y Clae etru Y B... B... B X Clae etru X M M... M... M M M M... M... M M Stattca tetulu ete: ceata urmeaza o dtrbute cu (- (- grade de lbertate. Daca deaete valoarea crtca, atuc oteza ula coform carea cele atrbute ut deedete uul fata de celalalt ete rea. Icoveetul acetu tet tattc ete ca el dede de umarul gradelor de lbertate. Lmta ueroara ce oate f ata a fot tablta atfel: m (, Petru a faclta comaratle dtre atfel de valor ale tetulu etru tabele de cotgeta de dferte dmeu, au fot roue alte maur de mlartate, cu valor î [0, ]. Cele ma cuocute ut: coefcetul lu Tchurow: c Φ T, ude ( ( ( ( c Φ coefcetul de cotgeta al lu Cramer: C m(, coefcetul lu K. Pearo: Φ m(, P + Cea ma coveabla olute etru a agura comarabltatea dtre valorle tetulu calculate etru tabele de cotgeta de dferte dmeu ete a evaluam (r tregrare umerca maura de robabltate: P ( ν < e ν ν/ ν Γ 0 c o alta maura u dcrmeaza cu ma multa acuratete ître atrbutele aflate î comette, ma mortat, aceata maura e da obltatea de a cotrola robabltc roceul de ramfcare a arborelu. Petru a ue de acord atfel de tete robablte cu o decrere fuzzy a datelor, utem folo codfcarea fuzzy tadard etru a geera acoerr fuzzy coreuzatoare varablelor cotue etru a cotru tabele de cotgeta, aa cum am ugerat î ectuea recedeta. Se obte atfel u tet ma ut ebl la fluctuatle modelulu (alegerea tervalelor, fluctuatle de eatoare, etc.. Detal rvd mlemetarea algortmulu tetele umerce efectuate vor face o- bectul ue lucrar vtoare. Bblografe [] Brema, L., Fredma J.H., Ole, R.. & Stoe, C.J. (984. Clafcato ad Regreo Tree, Wadworth. [] Fayyad, U.M & Ira, K.B. (993. "Mult- terval dcretzato of cotuou- valued attrbute for clafcato learg", Proceedg of the 3th Iteratoal Jot Coferece o rtfcal Itellgece, Morga Kaufma, d
8 Revta Iformatca Ecoomca, r. (4 / [3] Georgecu, V. (998. Fleble etmato of cot fucto baed o fuzzy logc modelg aroach". Fuzzy Ecoomc Revew, Vol.III, o., [4] Georgecu, V. (998. ew etmato method fuzzy regreo aaly, baed o roecto theorem ad decoulg rcle. Fuzzy Ecoomc Revew, Vol.III, o.,. -38 [5] Georgecu, V. (996. fuzzy geeralzato of rcal comoet aaly ad automatc clafcato, baed o a ew metrc cocet: the dmlarty betwee fuzzy et. Proceedg of the Thrd Cogre of SIGEF, Bueo re, Paer.5 [6] Georgecu, V. (995. Eert ytem degg fuzzy logc ad obltc logc. Craova, Ed. Itarf [7] Haw, D.M. & Ka, G.V. (98, "utomatc Iteracto Detecto", Haw, D.M. (ed., Toc led Multvarate aly, , Cambrdge Uv. Pre: Cambrdge. [8] Qula, J.R. (986. "The Effect of oe o Cocet Learg" Mache Learg II, R. Mchal, J. Carboell & T. Mtchell (ed, Morga Kaufma. [9] Qula, J.R. (986. "Iducto o Deco Tree", Mache Learg, Vol., [0] Qula, J.R. (987 "Deco Tree a Probabltc Clafer'', Proceedg of the Fourth Iteratoal Worho o Mache Learg, [] Qula, J.R. (993. "Combg Itace Baed ad Model Baed Learg", Proceedg of Iteratoal Coferece o Mache Learg, Morga Kaufma, [] Qula, J.R. (993. C4.5: Program for Mache Learg, Morga Kaufma, Sa Mateo, C.
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