ÇOK DEĞİŞKENLİ MARKOV ZİNCİRİ MODELİ VE BİR UYGULAMA

Marmara Üniversitesi øø%)'HUJLVL YIL 2010&ø/7;;,;, SAYI II, S. 577-590 d2.'(öøù.(1/ø0$5.29=ø1&ø5ø02'(/ø9(%ø5 UYGULAMA Ersoy ÖZ* Semra ERPOLAT** Özet 0DUNRY ]LQFLUOHUL SHN oRN X\JXODPD DODQÕQD VDKLS RODQ VWRNDVWLN süreçlerdir. Markov zincirlerinde incelenen sisteme ait veriler tek kaynaktan JHOPHNWHGLU dRN GH÷LúNHQOL 0DUNRY ]LQFLUL PRGHOL LVH D\QÕ ND\QDN YH\D EHQ]HU kaynaklar WDUDIÕQGDQUHWLOHQoRNOXNDWHJRULNYHULGL]LOHULQLQGDYUDQÕúÕQÕJ|VWHUPHN DPDFÕ LOH NXOODQÕODQ ELU PRGHOGLU %X oDOÕúPDGD 0DUNRY ]LQFLUOHUL ]HULQH NXUXOX RODQ oRN GH÷LúNHQOL 0DUNRY ]LQFLUL PRGHOL WHRULN DoÕGDQ GHWD\OÕ ELU ELoLPGH DQODWÕOPÕúWÕU 8\JXODPD RODUDN LVH GRODU NXUX DOÕú IL\DWODUÕQGD ROXúDQ D\OÕN GH÷LúLPOHU LOH ø0.% 8OXVDO  (QGHNVL GH÷HUOHULQGH ROXúDQ D\OÕN GH÷LúLPOHU LNL NDWHJRULNGL]LRODUDNHOHDOÕQÕSEXGL]LOHULQELUELUOHULQLQH RUDQGDHWNLOHGLNOHULoRN GH÷LúNHQOL0DUNRY]LQFLULPRGHOLLOH RUWD\DNRQPXúWXU Anahtar Kelimeler: Markov zinciri, Kategorik Veri Dizileri, Çok DH÷LúNHQOL0DUNRYZinciri. MULTIVARIATE MARKOV CHAIN MODEL AND AN APPLICATION Abstract Markov chains are stochastic processes with a wide range of application areas. In Markov chains, the examined data that belong to the system comes from a single source. Multivariate Markov chain model is used to show the behavior of the multi categorical data sequences that were produced by the same source or by similar sources. In this study, multivariate Markov chain model, which is based on Markov chains, is theoretically explained in detail. As the application, the monthly changes of the US Dollar buying rates and the monthly changes of the ISE National 100 Index values are taken into consideration as two categorical sequences and it is revealed with multivariate Markov chain model to what degree these sequences affect each other. * g÷U*|U'U<ÕOGÕ]7HNQLNhQLYHUVLWHVL0HVOHN<NVHNRNXOX7HNQLN3URJUDPODU%|OP e-mail: ** Yard. Doç. Dr. Mimar Sinan Güzel Sanatlar ÜniverVLWHVL)HQ(GHEL\DW)DNOWHVLøVWDWLVWLN Bölümü, e-mail: 577 g÷U Gör. Dr. Ersoy ÖZ*Yard. Doç. Dr. Semra ÖZPOLAT Key Words: Markov Chain, Categorical Data Sequences, Multivariate Markov Chain. *LULú Markov süreçleri LOHULGH RUWD\D oÕNPDVÕ RODVÕ GXUXPODUÕQ JHUoHNOHúPH RODVÕOÕNODUÕQÕQJHoPLúYHULOHUGHQGH÷LOúXDQNLYHULOHUGHQ\DUDUODQDUDNKHVDSODQGÕ÷Õ süreçlerdir1. %LU 0DUNRY VUHFL NRúXOOX RODVÕOÕN IRQNVL\RQX ³0DUNRY\HQ g]HOOLN YDUVD\ÕP ´ÕVD÷OD\DQELUVWRNDVWLNsüreçtir2. 0DUNRY VUHoOHULQLQ JHQHO WHRULVL  YH ¶OÕ \ÕOODUGD JHOLúWLULOPLúWLU Markov zincirleri hukuk (Stander vH GL÷HUOHUL   SD]DUODPD 'XUD (2006)), VD÷OÕN KL]PHWOHUL 5RPDJQXROR   JHOLU GD÷ÕOÕPÕ 'DUGDQRQL (1995)), finans (Aytemi]YHùHQJ|QO  5]JDU  ELOJLVD\DU\D]ÕOÕPÕVHoLPL 3RRUHYH GL÷erleri   :KLWWDNHU YH 3RRUH   YHUL PDGHQFLOL÷L *XLGLFL YH &DVWHOR   UHWLP *HYUHN YH ùHQJOOHU   6LPNLQ   J|o 1LHOVHQ YH Wakeley (2001)), meteRURORML .RoDN YH ùHQ   EDúWD ROPDN ]HUH ELU oRN DODQGDX\JXODQPDNWDYH\D\JÕQRODUDNNXOODQÕOPDNWDGÕU 0DUNRY]LQFLULPRGHOOHULQLQNXOODQÕOGÕ÷ÕWPX\JXODPDODUGD, veri tipine ve NXUXODQ PRGHOH ED÷OÕ RODUDN JHOHFHNWH ROXúDFDN GXUXP GL]LVL X]XQ G|nem denge) RODVÕOÕNODUÕ \XWXFX 0DUNRY ]LQFLUL DQDOL]OHUL YH\D PDUNDODUÕQ WHUFLK RODVÕOÕNODUÕ LOH LOJLOL VRQXoODU HOGH HGLOPHNWHGLU (OGH HGLOHQ EX VRQXoODUÕQ WP LoLQ NXOODQÕODQ YHULOHU WHN ND\QDNWDQ JHOPHNWHGLU gUQH÷LQ, tüketicilerin meyve suyu marka tHUFLKOHULLOHLOJLOLELUoDOÕúPDVDGHFHWNHWLFLOHULQúXDQGDWHUFLKHWWLNOHULPH\YHVX\X PDUNDODUÕ YH ELU VRQUDNL WHUFLK HGHFHNOHUL PH\YH VX\X PDUNDODUÕ GLNNDWH DOÕQPDNWDGÕUdRNGH÷LúNHQOL0DUNRY]LQFLULPRGHOLQGHLVHELUELUOHULLOHLOLúNLOLYHUL dizilerL GLNNDWH DOÕQPDNWDGÕU <DQL ELUELUOHUL LOH ELU NRUHODV\RQD VDKLS YHUL GL]LOHUL NXOODQÕOPDNWDGÕU %|\OHFH JHUoH÷H GDKD \DNÕQ WDKPLQOHU RUWD\D oÕNPDNWDGÕU <XNDUÕGDNL|UQHNLoLQNXOODQÕFÕODUÕQPH\YHVX\XWHUFLKOHULQGHGL÷HULoHFHNOHULQ de (soda, gazoz, ayUDQ YV  URO R\QDGÕ÷Õ GLNNDWH DOÕQGÕ÷ÕQGD GDKD JHUoHNoL VRQXoODU RUWD\DoÕNDFD÷ÕDúLNâUGÕUøúWHEXQRNWDGDoRNGH÷LúNHQOL 0DUNRY]LQFLUL PRGHOLQLQ NXOODQÕOPDVÕIDUNOÕELUEDNÕúDoÕVÕROXúWXUPDNWDGÕU dRN GH÷LúNHQOL 0DUNRY ]LQFLUL PRGHOLQLQ WHRULVL VRQ RQ \ÕO LoHULVLQGH JHOLúWLULOPLúROXSX\JXODPDDODQÕQGDNLoDOÕúPDODUÕQVD\ÕVÕROGXNoDD]GÕU %XoDOÕúPDQÕQLNLQFLE|OPQGH|QFHOLNOH0DUNRY]LQFLULWDQÕPODQDUDNoRN GH÷LúNHQOL 0DUNRY ]LQFLUL PRGHOL GHWD\OÕ ELU ELoLPGH WHRULN DoÕGDQ HOH DOÕQDFDNWÕU Üçüncü E|OPGH NDWHJRULN GL]L RODUDN GRODU NXUX DOÕú IL\DWODUÕQGD ROXúDQ D\OÕN GH÷LúLPOHU YH ø0.% 8OXVDO  (QGHNVL GH÷HUOHULQGH ROXúDQ D\OÕN GH÷LúLPOHU NXOODQÕODUDNoRNGH÷LúNHQOL0DUNRY]LQFLULPRGHOLROXúWXUXODFDNWÕU0RGHOLROXúWXUDQ LNL GL]LQLQ D\OÕN RODUDN L]OHQHQ GH÷LúLP GH÷HUOHULQLQ ELUELUOHULQL EHOOL ELU RUDQGD 1 R.I. Levin ve dL÷HUOHULQuantitave Approaches to Management, Fifth Edition, Tokyo, Mc-Graw-Hill, 1982, s.47 2 Sheldon M. Ross, Introduction to Probability Models, 10th Edition, United States of America, Academic Press, 2009, s.192 578 HWNLOH\HFH÷L YDUVD\ÕPÕ LOH HWNLOHPH RUDQODUÕ EHOLUOHQHFHNWLU  6RQ E|OPGH LVH HOGH HGLOHQoRNGH÷LúNHQOL0DUNRY]LQFLULPRGHOLLOHLOJLOL\RUXPODU\DSÕODFDNWÕU dRN'H÷LúNHQOL0DUNRY=LQFLri 2.1. Markov Zinciri %LUWHVDGILGHQH\LQEWQPPNQVRQXoODUÕQÕLoHUHQNPH\H|UQHNX]D\ denir3 %LU |UQHN X]D\ÕQGDNL KHU ELU ROD\Õ UHHO VD\ÕODUD WDVYLU HGHQ IRQNVL\RQD WHVDGILGH÷LúNHQDGÕYHULOLU4. .HVLN ]DPDQOÕ ELU VWRNDVWLN VUHo t  N {0,1, 2,...} parametreleri ile LQGHNVOHQHQ YH YHULOHQ ELU RODVÕOÕN X]D\Õ ]HULQGH WDQÕPOÕ WHVDGIL GH÷LúNHQOHULQ { X t , t  N } bir ailesidir. 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