STATISTIČKE I MATEMATIČKE METODE ZA REŠAVANJE PROBLEMA KLASTEROVANJA POŠTANSKIH PODATAKA KADA SU ONI NEPOTPUNI
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1 XXXV Simpozium o novim tehnologiama u poštanskom i telekomunikacionom saobraćau PosTel 2017, Beograd, 5. i 6. decembar STATISTIČKE I MATEMATIČKE METODE ZA REŠAVANJE PROBLEMA KLASTEROVANJA POŠTANSKIH PODATAKA KADA SU ONI NEPOTPUNI Nataša Glišović 1,2, Tatana Davidović 2, Neboša Boović 3, Nikola Knežević 3 1 Državni Univerzitet u Novom Pazaru, Departman za matematičke nauke 2 Matematički Institut SANU 3 Univerzitet u Beogradu, Saobraćani fakultet nglisovic@np.ac.rs, tanad@mi.sanu.ac.rs, nb.boovic@sf.bg.ac.rs, n.knezevic@sf.bg.ac.rs Sadrža: Kvalitet podataka e presudni faktor od koeg zavisi uspešnost obrade podataka. Značanu ulogu u kvalitetu podataka osim izvora imau i postupci pretprocesirana podataka. Podaci u izvornom obliku mogu biti nekompletni, atributi mogu imati nedostauće vrednosti ili može postoati nedostatak atributa. Cil ovog istraživana e da pokažemo matematičke i statističke metode koe rešavau probleme svrtstavana podataka u grupe kao i rada sa nedostaućim podacima. Metode su testirane na bazi podataka Evropske komisie koa sadrži podatke vezane za poštanski saobraća. Klučne reči: poštanski saobraća, nedostaući podaci, problem p-mediane, metaheuristike. 1. Uvod Poštanski saobraća i usluge predstvalau edan od servisa od opšteg društvenog interesa. Praćene i analiza učinka poštanskih kompania imau klučnu ulogu u podizanu nivoa kvaliteta poštanske usluge. Statistike i pokazateli koi karakterišu poštanski saobraća su broni i često e nihovo prikuplane i formirane baza podataka ograničeno nihovom dostupnošću i efikasnošću samog sistema. Primena modernih statističkih i matematičkih metoda u evaluacii poslovana poštanskih kompania omogućue sveobuhvatnu analizu koa uklučue veliki bro pokazatela, kao i veliku količinu podataka. Podaci u izvornom obliku mogu biti nekompletni, atributi mogu imati nedostauće vrednosti, ili može postoati nedostatak atributa. Isto tako može se poaviti nekonzistentnost unutar samih podataka kao posledica nedoslednosti u označavanu poedinih kategoria ili grupa. Kada se u podacima naiđe na nedostauću vrednost, tada se u procesu analize podataka koriste metode za predviđane nedostaućih vrednosti npr. neuronske mreže, regresione metode, linearna interpolacia, Baesove mreže, stabla odlučivana i slično [10]. U ovom radu ta problem prevaziđen e korišćenem rastoana
2 koe se može primeniti i u slučau nedostaućih vrednosti [1], tako da su vrednosti ostavlene u izvornom obliku onakvim kakve esu, t. nismo aproksimirane nedostauće vrednosti u bazi. Pored problema nedostaućih podataka, drugi aspekt obuhvata i heterogenost samih pokazatela i podataka koi karakterišu poslovane poštanskih kompania. Klasterovane predstvala pristup koim se uspešno može tretirati heterogenost podataka. Cil ovog rada e analiza dostupnih baza podataka o poštanskom saobraćau koom se podaci grupišu u klastere čii bro e dobien statističkim pretprocesiranem. Rad e podelen u nekoliko odelaka. Metodologia predložena za rešavane problema data e u narednom odelku. Opis podataka kao i rezultati istraživana dati su u Odelku 3. Posledni odelak sadži zaklučna razmatrana. 2. Metodologia Nedostaući podaci Problem nedostaućih podataka sve češće se avla pri analizi savremenih baza podataka. Podaci mogu nedostaati iz više razloga. Neki od nih su: podaci nisu raspoloživi, došlo e do grešaka u radu sa opremom, nekonzistentnosti sa drugim podacima, pa su zato izbrisani, nisu unešeni zbog nerazumevana, nisu smatrani bitnim u trenutku unosa itd. Bitna e odluka šta raditi sa nedostaućim podacima. Neke od mogućnosti su [3]: Izbrisati elemente kod koih se avlau nedostaući podaci-što nie preporučlivo posebno kod klasifikacie, a naročito ako nedostauće vrednosti variau od elementa do elementa, t. nedostau različiti elementi kod različitih obekata (vektora). Ručno popunavane nedostaućih vrednosti koe e zamorno i često neizvodlivo. Automatsko popunavane: nekom opštom konstantom, srednom vrednosti elemenata za sve obekte (vektore) koi pripadau isto klasi. Naverovatnia vrednost-zaklučak se donosi na osnovu Baesove formule ili prema stablu odlučivana. Kako ni edna od navedenih mogućnosti ne obezbeđue zadovolavauću transformaciu polazne baze, u ovom istraživanu problem e rešen korišćenem rastoana koe se koristi u slučaevima kada podaci nedostau, predloženom od strane Glišović i Rašković [1]. Ovo rastoane zasnovano e na logičkim formulama i ne zahteva popunavane nedostaućih podataka niti brisane nekih od atributa što e negova osnovna prednost Pretprocesirane Normirane podataka se često primenue kada e potrebno izbeći veliki utica poedine promenlive koa gravitira ka visokim apsolutnim vrednostima kod reševana problema klasterovana. Od metoda normirana podataka koe se naviše koriste u obradi podataka su: Min-max normirane Z-sklairane Decimalno skalirane
3 S obzirom na prirodu podataka koi se koriste u ovom istraživanu (postoe nedostauće vrednosti, t. nisu nam poznate maximalna i minimalna vrednost niza) opredelili smo se za Z skalirane, kao metodu adekvatnu u ovom koraku obrade podataka. y =, (1) gde e y nova (normirana vrednost), y izvorna vrednost atributa, sredna vrednost, a standardna deviacia poznatih (postoećih) atributa. Sredna vrednost i standardna deviacia se računau na uobičaeni način: = = (2) (3) Klasterovane Klasterovane podrazumeva da se slični podaci (u odnosu na odgovarauće atribute) grupišu zaedno u grupe koe nazivamo klasteri. Dok su elementi unutar klastera slični, klasteri se među sobom razlikuu. Klasterovane e vid nenadgledanog učena (engl. unsupervised learning) er klasteri nisu određeni pre ispitivana podataka. Postoi nekoliko formulacia problema klasterovana zavisno od funkcie cila koa se optimizue, a u ovom istraživalu e korišćena formulacia preko problema p- mediane [8]. Neka su x i i x y i t. d. y binarne promenlive definisane na sledeći način: 1, ako se obekat i nalazi u klasteru, = 0, inače. 1, ako obekat reprezentue odgovaraući klaster, = 0, inače. min dx i i (4) i x i = 1 za svako i (5) xi y za svako i, (6) y = p (7) { } x, y 0,1 (8) i
4 2. 4. Osnovna metoda promenlivih okolina Metoda promenlivih okolina e metaheuristika koa e predstavlena devedesetih godina prošlog veka [7][9] nakon čega e doživela mnogo promena i ekstenzia [5][6], kao i uspešnih primena [4]. Osnovna metoda promenlivih okolina (BVNS) e narasprostranenia varianta metode promenlivih okolina er obezbeđue više preduslova za dobiane kvalitetniih konačnih rešena. Kod BVNS metode osnovni koraci sadržani su u petli u koo menamo indeks okoline i, određuemo slučano rešene iz te okoline (korak razmrdavana), izvršavamo proceduru lokalnog pretraživana počev od tog slučanog rešena i proveravamo kvalitet dobienog lokalnog minimuma u odnosu na trenutno nabole rešene. Ove korake ponavlamo dok ne bude zadovolen neki od kriteriuma zaustavlana. Uloga koraka razmrdavana e da obezbedi diversifikaciu pretraživana. Prilikom svakog odabira okoline početna rešena generišemo na slučaan način kako bi obezbedili pretraživane različitih regiona kod svakog sledećeg razmrdavana u okolini i. Lokalnim pretraživanem rešena dobienog razmrdavanem intenzivira se pretraga rešena u negovo okolini. Adekvatan balans između intenzifikacie i diversifikacie obezbeđue se pravilnim izborom vrednosti k, osnovnog parametara BVNS metode. Okoline koe se koriste u BVNS metodi razlikuemo po brou transformacia (rastoanu) ili po vrsti transformacia (metrici). Napominemo da okoline za izbor slučanog rešena (razmrdavane) i lokalno pretraživane ne morau biti istog tipa. Pseudokod BVNS metode dat e na slici 1. Slika 1. Pseudokod BVNS metode BVNS metoda za kalsterovane podataka vezanih za poštanski saobraća implementirana e po uzoru na rad Glišović N., Davidović T. i Rašković M. [2]
5 3. Rezultati istraživana Podaci koi su korišćeni u ovom istraživanu nalaze se u bazama Evropske komisie, i dostupne su sa adrese Postoi šest baza koe za svaku od zemala sadrže sledeće podatke: Ukupan obim poštanskih usluga u unutrašnem i međunarodnom saobraćau, Prihod od poštanskih usluga u unutrašnem saobraćau Cene poštanskih usluga u unutrašnem saobraćau Kvalitet i rokovi prenosa u unutrašnem i međunarodnom saobraćau Dostupnost poštanske mreže Bro zaposlenih u nacionalnim poštanskim operatorima Obim pismonosnih usluga u unutrašnem i međunarodnom saobraćau Posmatrane baze podataka obuhvatau podatke za period godina za 31 bro zemala: Austria, Belgia, Bugarska, Hrvatska, Kipar, Češka, Danska, Estonia, Finska, Bivša Jugoslovenska Republika Makedonia, Francuska, Nemačka, Grčka, Mađarska, Island, Irska, Italia, Latvia, Litvania, Luksemburg, Malta, Holandia, Norveška, Polska, Portugal, Rumunia, Slovačka, Slovenia, Špania, Švedska, Velika Britania. Sve korišćene baze karakterišu se nedostaućim podacima. Važno e napomenuti da postoe zemle za koe nisu navedeni podaci ni za ednu godinu. Te zemle isklučene su iz analize u odgovaraućo bazi i podrazumevano e da su one svrstane u edan zaednički klaster. Nakon isklučivana zemala kod koih nisu postoali podaci ni za ednu godinu određen e procenat nedostaućih podataka. Karakteristike svake od baza sumirane su u tabeli 1. Tabela 1. Opis svake baze koa e analizirana iskazane kroz bro zemala koe se nalaze u bazi, bro Zemala kod koih nema podataka ni za ednu godinu (bro isklučenih zemala), kao i procenat nedostaućih podataka. Naziv baze Ukupan obim poštanskih usluga u unutrašnem i međunarodnom saobraćau Prihod od poštanskih usluga u unutrašnem saobraćau Cene poštanskih usluga u unutrašnem saobraćau Kvalitet i rokovi prenosa u unutrašnem i međunarodnom saobraćau Bro zemala Bro isklučenih zemala Procenat nedostaućih podataka % % % % Dostupnost poštanske mreže % Bro zaposlenih u nacionalnim poštanskim operatorima Obim pismonosnih usluga u unutrašnem i međunarodnom saobraćau % %
6 Prvo e izvršeno pretprocesirane, zatim analiza kako podataka tako i nihove prirode formirane su grupe sličnih zemala sa sličnim vrednostima atributa (određeni su intervali vrednosti u svako grupi). Zatim e primenen BVNS algoritam za klasterovane u okviru svake baze podataka na onoliko klastera koliko e određeno u pretprocesiranu podataka. BVNS metoda implementirana e u C# programskom eziku na računaru HP-15- d055, pod operativnim sistemom Windows 10 Pro. U pretprocesiranu za svaku od baza, na osnovu analiza stručnaka, oceneno e da su razvrstavna u tri grupe naadakvatnia. S obzirom na stohastičku prirodu metoda vršeno e 100 restartovana. Nabola rerešena, kao i bro puta koliko su dostignuta, zaedno sa prosečnim vremenom potrebnim za nalažene nabolih rešena dati su u tabeli 2. Nabole rešene karakterisano e vrednošću funkcie cila, t. zbirom rastoana unutar klastera. Metoda e uzvršavan 100 puta i pokazala e veliku stabilnost za svaku bazu. Tabela 2. Rezultati rada BVNS-a za svaku bazu podataka data po optimalnim rešenima, uspešnosti i vremenu dolaska do optimalnih rešena. Vreme Bro dolaska dostignutih BVNS primenen Nabole PROSEČNO naboliih na bazama rešene do optimalnog rešena rešena (uspešnost) (sekundama) Ukupan obim poštanskih usluga u unutrašnem i međunarodnom saobraćau Prihod od poštanskih usluga u unutrašnem saobraćau Cene poštanskih usluga u unutrašnem saobraćau Kvalitet i rokovi prenosa u unutrašnem i međunarodnom saobraćau Dostupnost poštanske mreže Bro zaposlenih u nacionalnim poštanskim operatorima Obim pismonosnih usluga u unutrašnem i međunarodnom saobraćau Za svaku od ovih baza daemo pregled klastera koi su dobieni gde su različitim boama predstavleni različiti klasteri (Tabela 3.)
7 Tabela 3. Klasteri dobieni primenom metode BVNS.. Ukupan Kvalitet i Prihod od Cene obim pošt. rokovi poštanskih poštanskih Dostupnost usluga u prenosa u usluga u usluga u poštanske unutraš. i unutraš.i unutraš. unutraš. mreže međun. međun. saobraćau saobraćau saobraćau. saobraćau Bro zaposlenih u NPO Obim pismonos. usluga u unutraš. i međun. saobraćau Bulgaria Belgium Belgium Belgium Belgium Belgium Bulgaria Czech R. Bulgaria Bulgaria Bulgaria Bulgaria Bulgaria Czech R. Denmark Czech R. Czech R. Czech R. Czech R. Czech R. Denmark Estonia Denmark Denmark Denmark Denmark Denmark Germany Greece Germany Germany Germany Germany Germany Estonia Spain Estonia Estonia Estonia Estonia Estonia Ireland Croatia Ireland Ireland Ireland Ireland Ireland Greece Italy Greece Greece Greece Greece Greece Spain Cyprus Spain Spain Spain Spain Spain Croatia Latvia France France France Croatia France Italy Lithuania Croatia Croatia Croatia Cyprus Croatia Cyprus Luxembourg Italy Italy Italy Latvia Italy Latvia Hungary Cyprus Cyprus Cyprus Luxembourg Cyprus Lithuania Austria Latvia Latvia Latvia Hungary Latvia Luxembourg Poland Lithuania Lithuania Lithuania Malta Lithuania Hungary Romania Luxembourg Luxembourg Luxembourg Netherlands Luxembourg Malta Slovenia Hungary Hungary Hungary Austria Hungary Netherlands Slovakia Malta Malta Malta Poland Malta Austria Finland Netherlands Netherlands Netherlands Portugal Netherlands Poland Sweden Austria Austria Austria Romania Austria Portugal Iceland Poland Poland Poland Slovenia Poland Romania Norway Portugal Portugal Portugal Slovakia Portugal Slovenia FYRM Romania Romania Romania Finland Romania Slovakia Slovenia Slovenia Slovenia Sweden Slovenia Finland Slovakia Slovakia Slovakia UK Slovakia Sweden Finland Finland Finland Iceland Finland Iceland Sweden Sweden Sweden Norway Sweden Norway UK UK UK FYRM UK FYRM Iceland Iceland Iceland Iceland Norway Norway Norway Norway FYRM FYRM FYRM
8 4. Zaklučak U radu e izvršena analiza baza podataka vezanih za poštanski saobraća u evropskim zemlama. Na osnovu te analize podaci su grupisani u odgovaraući bro klastera koi sadrže slične obekte. Za klasterovane e korišćena osnovna metoda promenlivih okolina koa koristi rastoane pogodno za obekte u bazama kod koih nedostau podaci. Predložena metoda pokazala e veliku stabilnost i uspešnost u klasifikacii podataka poštanskih servisa i usluga. Tako gurpisani podaci mogu se dale koristiti u cilu što bole analize zemala čii su se poštanski servisi i usluge razmatrani. Predloženi pristup može omogućiti primenu benčmarking i drugih alata u cilu analize poštanskih servisa kada su raspoloživi podaci nekompletni. Zahvalnica Ova rad e rezultat istraživana na proektu III "Razvo novih informaciono - komunikacionih tehnologia, korišćenem naprednih matematičkih metoda, sa primenama u medicini, energetici, telekomunikaciama, e-upravi i zaštiti nacionalne baštine" i proektu TR "Upravlane kritičnom infrastrukturom za održivi razvo u poštanskom, komunikacionom i železničkom sektoru Republike Srbie" koe finansira Ministarstvo prosvete, nauke i tehnološkog razvoa Republike Srbie. Literatura [1] Glišović, N., Rašković. M., Optimization for Classifying the Patients Using the Logic Measures for Missing Data, Scientific publications of the State University of Novi Pazar Ser. a: Appl. Math. Inform. and Mech. vol. 9, 1, , [2] Glišović, N., Davidović, T., and Rašković, M., Klasterovane kada podaci nedostau korišćenem metode promenlivih okolina, SYM-OP-IS, Zlatibor, septembra, pp , [3] Graham, J. W., Missing Data: Analysis and Design. Springer Science and Business Media, New York, [4] Hansen, P. and Mladenović, N., Variable neighborhood search. In Search methodologies (pp ). Springer US, [5] Hansen, P., Mladenović, N. and Pérez, J. A. M., Variable neighborhood search: methods and applications. Annals of Operations Research, 175(1), , [6] Hansen, P., Mladenović, N., Brimberg, J. and Perez, J. A. M., Variable neighborhood search Handbook of Metaheuristics ser. International Series in Operations Research & Management Science, 146, 61-86, [7] Mladenović, N., A Variable neighborhood algorithm a new metaheuristic for combinatorial optimization, Abstracts of papers presented at Optimization Days, Montreal, p. 112, [8] Mladenović, N., Brimberg, J., Hansen, P., Moreno-Perez JA, The p-median problem: a survey of metaheuristic approaches. European Journal of Operational Research 179: , [9] Mladenović, N., Hansen, P., Variable neighborhood search. Computers and Operations Research; 24(11); , [10] P. D. Allison, Missing data, Sage University papers series on quantitative applications in the social sciences, series Thousand Oaks, CA: Sage, Abstract: Data quality is a crucial factor that depends on the success of data processing. A significant role in the quality of data other than sources has a process of data preprocessing. The data in the original form may be incomplete, the attributes may have missing values, or there may be a lack of attributes. The aim of this research is to show the mathematical and statistical methods that solve the problem of grouping into groups as well as working with missing data. The methods were tested on the European Commission database, the database statistics database. Key words: postal traffic, missing data, p-median problem, metaheuristics. STATISTICAL AND MATHEMATICAL METHODS FOR SOLVING THE POSTAL DATA CLASSIFICATION PROBLEMS WHEN ARE MISSING DATA Nataša Glišović, Tatana Davidović, Neboša Boović, Nikola Knežević
KLASTEROVANJE KADA PODACI NEDOSTAJU KORIŠĆENJEM METODE PROMENLJIVIH OKOLINA CLUSTERING WHEN MISSING DATA BY USING THE VARIABLE NEIGHBORHOOD SEARCH
KLASTEROVANJE KADA PODACI NEDOSTAJU KORIŠĆENJEM METODE PROMENLJIVIH OKOLINA CLUSTERING WHEN MISSING DATA BY USING THE VARIABLE NEIGHBORHOOD SEARCH NATAŠA GLIŠOVIĆ 1, TATJANA DAVIDOVIC, MIODRAG RAŠKOVIĆ
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