Çok Değişkenli Markov Zinciri Modeli ve Bir Uygulama

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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 WDUDIÕQGDQUHWLOHQoRNOXNDWHJRULNYHULGL]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%|OP} e-mail: ersoyoz@yildiz.edu.tr

**_{Yard. Doç. Dr. Mimar Sinan Güzel Sanatlar ÜniverVLWHVL)HQ(GHEL\DW)DNOWHVLøVWDWLVWLN} Bölümü, e-mail: serpolat@msu.edu.tr

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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.

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0DUNRY VUHoOHULQLQ 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|QO5]JDUELOJLVD\DU\D]ÕOÕPÕVHoLPL3RRUHYH GL÷erleri :KLWWDNHUYH3RRUHYHUL PDGHQFLOL÷L*XLGLFLYH&DVWHOR UHWLP *HYUHN YH ùHQJOOHU 6LPNLQ J|o 1LHOVHQ YH Wakeley (2001)), meteRURORML .RoDN YH ùHQ EDúWD ROPDN ]HUH ELU oRN DODQGDX\JXODQPDNWDYH\D\JÕQRODUDNNXOODQÕOPDNWDGÕU

0DUNRY]LQFLULPRGHOOHULQLQNXOODQÕOGÕ÷ÕWPX\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 WP LoLQ NXOODQÕODQ YHULOHU WHN ND\QDNWDQ JHOPHNWHGLU gUQH÷LQ, tüketicilerin meyve suyu marka tHUFLKOHULLOHLOJLOLELUoDOÕúPDVDGHFHWNHWLFLOHULQúXDQGDWHUFLKHWWLNOHULPH\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÷LúNHQOL0DUNRY]LQFLULPRGHOLQLQ NXOODQÕOPDVÕIDUNOÕELUEDNÕúDoÕVÕROXúWXUPDNWDGÕU

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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

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HWNLOH\HFH÷L YDUVD\ÕPÕ LOH HWNLOHPH RUDQODUÕ EHOLUOHQHFHNWLU 6RQ E|OPGH LVH HOGH HGLOHQoRNGH÷LúNHQOL0DUNRY]LQFLULPRGHOLLOHLOJLOL\RUXPODU\DSÕODFDNWÕU

dRN'H÷LúNHQOL0DUNRY=LQFLri

2.1. Markov Zinciri

%LUWHVDGILGHQH\LQEWQPPNQVRQXoODUÕQÕLoHUHQNPH\H|UQHNX]D\ denir3 %LU |UQHN X]D\ÕQGDNL KHU ELU ROD\Õ UHHO VD\ÕODUD WDVYLU HGHQ IRQNVL\RQD WHVDGILGH÷LúNHQDGÕYHULOLU4_.

.HVLN ]DPDQOÕ ELU VWRNDVWLN VUHo tN {0,1, 2,...} parametreleri ile LQGHNVOHQHQ YH YHULOHQ ELU RODVÕOÕN X]D\Õ ]HULQGH WDQÕPOÕ WHVDGIL GH÷LúNHQOHULQ

{X t_t, N} bir ailesidir. Bir Markov sUHFLELUVWRNDVWLNVUHoWLUYHNRúXOOXRODVÕOÕN GD÷ÕOÕP IRQNVL\RQX ³0DUNRY\HQ g]HOOLN´ RODUDN DGODQGÕUÕODQ |]HOOL÷L JHUoHNOHU Markov süreciQLQGXUXPX]D\ÕNHVLNOLLVH0arkov süreci NHVLN]DPDQOÕELUVWRNDVWLN süreç olur ve bir Markov Zinciri (MZ) olarak DGODQGÕUÕOÕU

Genel olarak kategorik bir veri dizisi x x x₁, ₂, ,...,₃ x_T vektörlerinin bir dizisi ile gösterilir. Bu ifadede yer alan T LQGLVL GL]LQLQ JHQLúOL÷LGLU (÷HU VUHo k durumunda ise x_i e_k¶GÕU ek Girdi için birim vektördür). m VD\ÕGD NHVLNOL

GXUXPOXELULQFLGHUHFHGHQNHVLN]DPDQOÕ0=DúD÷ÕGD\HUDODQED÷ÕQWÕ\ÕVD÷ODU 1 0 1 1 1 0 1 1 ( t x_t x , x,..., t x_t) ( t x_t t x_t), i . P x e x e x e x e P x e x e x M (1) 1 1 ( ) n n n x n x P x e x e (2)

LIDGHVLQGHNL NRúXOOX RODVÕOÕN 0=¶QLQ WHN DGÕP JHoLú RODVÕOÕNODUÕ RODUDN DGODQGÕUÕOÕU %X RODVÕOÕNODU ]DPDQ SDUDPHWUHVL n ’den n1¶H JHoWL÷LQGH i

durumundan j GXUXPXQD JHoLúLQ NRúXOOX RODVÕOÕNODUÕQÕ YHULU $\UÕFD EX RODVÕOÕNODU

n ]DPDQSDUDPHWUHVLQGHQED÷ÕPVÕ]GÕUYHDúD÷ÕGDNLELoLPGH\D]ÕOÕU 1 ( ), , . ij n i n j p P x e x e i jM (3)

3_{Tuncay Can, 6HNW|UOHU$UDVÕøOLúNLOHULQ0DUNRY=LQFLUOHULLOH$QDOL]LYH7DKPLQL}

7UNL\HgUQH÷LøVWDQEXO'HULQ<D\ÕQODUÕV

4

Sheldon M. Ross, Stochastic Processes, Second Edition, New York: Jhon Wiley & Sons Inc., 1996, s.7

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ij

p HOHPDQODUÕQGDQ ROXúDQ P PDWULVL JHoLú RODVÕOÕNODUÕ PDWULVLGLU YH

DúD÷ÕGD\HUDODQLNL|]HOOL÷LVD÷ODU i. 0d pijd1,i j, M (4) ii. 1 1, . m ij i p j M

¦

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Öneri 1: P PDWULVLQLQ ¶H HúLW ELU |]GH÷HUL YDUGÕU YH P’nin tüm |]GH÷HUOHUL¶HHúLWYH\DNoNHúLWWLU5_.

Öneri 2: A matrisi m dereceli ve indirgenemeyen bir kare matris olsun. %XGXUXPGDDúD÷ÕGD\HUDODQoLIDGHJHoHUOLGLU

i. A,

O

max_k

O

_k( )A RODFDN úHNLOGH SR]LWLI ELU UHHO |]GH÷HUH sahiptir. Burada Ok( )A , A¶QÕQ .k |]GH÷HULQLJ|VWHULU

ii. Az Oz olacak biçimde girdileri reel ve pozitif olan bir z vektörü YDUGÕU

iii. O, A¶QÕQEDVLWELU|]GH÷HULGLU

<XNDUÕGD\HUDODQLNL|QHULNXOODQÕODUDNPz z olacak biçimde pozitif bir

z YHNW|UQQROGX÷XV|\OHQHELOLUz YHNW|UVDELWGXUD÷DQRODVÕOÕNYHNW|URODUDN DGODQGÕUÕOÕU 1 1. m i i z

¦

(6) $\UÕFDzi, i GXUXPXQGDNLVLVWHPLQVDELWGXUXPRODVÕOÕ÷ÕGÕU

dRN'H÷LúNHQOL0DUNRY=LQFLUL0RGHOLQLQ<DSÕVÕ

dRNGH÷LúNHQOL0=PRGHOLD\QÕND\QDNYH\DEHQ]HUND\QDNODUWDUDIÕQGDQ UHWLOHQ oRNOX NDWHJRULN GL]LOHULQ GDYUDQÕúÕQÕ J|VWHUPHN LoLQ NXOODQÕODQ ELU modeldir6. Her biri m adet durum içeren s adet kategorik dizi ve ( )k

n x , n

5

Wai-.L&KLQJYHGL÷HUOHUL³$Multivariate Markov Chain Model for Categorical Data Sequences and Its Applications in Demand Predictions”, IMA Journal of Management Mathematics, Vol. 13, 2002, pp. 187-199.

6

Jun-ichi Maskawa, “Multivariate Markov Chain Modeling for Stock Markets”, Physica A, Vol. 324, 2003, pp. 317-322.

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]DPDQÕQGDDQÕQGD .k GL]LQLQGXUXPYHNW|UROVXQ(÷HU .k dizi n ]DPDQÕQGD j dXUXPXQGDLVHDúD÷ÕGDYHULOHQHúLWOLN\D]ÕODELOLU7: ( )k _{(0,..., 0,1, 0,..., 0) .}T n j x e (7) dRNGH÷LúNHQOL0=PRGHOLQGHDúD÷ÕGD\HUDODQED÷ÕQWÕODUJHoHUOLGLU ( ) ( ) ( ) 1 1 , 1, 2,..., s j jk k n jk n k x

¦

O P x j s (8) (8LIDGHVLQGH\HUDODQSDUDPHWUHOHUDúD÷ÕGDYHULOPLúWLU 0, 1 , jk j k s O t d d ve 1 1, 1, 2,..., . s jk k j s O

¦

(9) 1.

n DGÕPGD .k GL]LQLQ GXUXP RODVÕOÕN GD÷ÕOÕPÕ (jk) ( )k n

P x ¶QÕQ D÷ÕUOÕNOÕ

RUWDODPDVÕQD ED÷OÕGÕU %XUDGD _P(jk)_, _{k GL]LQLQ GXUXPODUÕQGDQ}_. _j_._dizinin

GXUXPODUÕQD JHoLú RODVÕOÕNODUÕGÕU YH ( )k n x , .n DGÕPGD .k GL]LQLQ GXUXP RODVÕOÕN GD÷ÕOÕPÕGÕU8_{0DWULVIRUPXQGDDúD÷ÕGDNLLIDGHOHU\D]ÕODELOLU} (1) (11) (12) (1 ) (1) 1 11 12 1 (2) (21) (22) (2 ) (2) 1 21 22 2 1 ( ) ( 1) ( 2) ( ) ( ) 1 1 2 s n s n s n s n n n s s s ss s n s s ss n x P P P x x P P P x x Qx x P P P x O O O O O O O O O ª º ª º ª º « » « » « » « » « » « » { { « » « » « » « » « » « » « » « » « » ¬ ¼ ¬ ¼ ¬ ¼ veya xn1 Qxn (10)

Öneri 3: 1d j k, ds LoLQH÷HUO_jk>0 ise Q PDWULVLGH÷HUOLELU|]GH÷HUH sahiptir ve Q¶QXQ|]GH÷HUOHUL¶HHúLWYH\DNoNNDWVD\ÕODUDVDKLSWLU

Öneri 4: 1d j k, ds için O_jk>0 ve (jk)

P ¶QÕQ LQGLUJHQHPH\HQ ROGX÷X YDUVD\ÕOVÕQ%XGXUXPGDx Qx ve

7_{Wai-Ki CKLQJYHGL÷HUOHUL, “A New Multivariate Markov Chain Model with Applications to} Sales Demand Forecasting”, International Conference on Industrial Engineering and Systems Management IESM 2007, Beijing – China, May 30-June 2-2007, pp. 1-8.

8_{Shu-Qin Zhang YHGL÷HUOHUL “Construction and Control of Genetic Regulatory Networks: A} Multivariate Markov Chain Approach”, J. Biomedical Science and Engineering, Vol. 1, 2008, pp. 15-21.

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( ) 1 [ ] 1,1 m j i i x d dj s

¦

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olacak biçimde bir _x _[_x(1)_,_x(2)_,_,_x( )s _]T_{YHNW|UYDUGÕU}

2.3. Model Parametrelerinin Tahminleri

Model parametrelerinin hesaplanabilmesi için her bir veri dizisinin JHoLú RODVÕOÕNODUÕ PDWULVL KHVDSODQPDOÕGÕU %XQXQ LoLQ HOH DOÕQDQ veri dizilerinde .k

dizideki durumlardan j. GL]LGHNL GXUXPODUD JHoLú IUHNDQVODUÕ VD\ÕOÕU %|\OHFH YHUL GL]LVLLoLQJHoLúIUHNDQVÕPDWULVLROXúWXUXOXU*HoLúIUHNDQVÕPDWULVLQHQRUPDOL]DV\RQ LúOHPLQLQX\JXODQPDVÕVRQXFXQGDLVHJHoLúRODVÕOÕNODUÕPDWULVLHOGHHGLOLU9_. ( ) j k jk i i f , _{ ( )k_} n x dizisindeki ik durumundan ( ) { j} n x dizisindeki i durumuna j JHoLúIUHNDQVÕQÕJ|VWHUPHN]HUHJHoLúIUHNDQVÕPDWULVLDúD÷ÕGDNLELoLPGH\D]ÕODELOLU ( ) ( ) ( ) 11 21 1 ( ) ( ) ( ) ( ) 12 22 2 ( ) ( ) ( ) 1 2 . jk jk jk m jk jk jk jk m jk jk jk m m mm f f f f f f F f f f ª º « » « » « » « » « » ¬ ¼ (12) *HoLúIUHNDQVÕPDWULVL (jk) F NXOODQÕODUDNJHoLúRODVÕOÕNODUÕPDWULVL (jk) P elde edilir. ( ) ( ) ( ) 11 21 1 ( ) ( ) ( ) ( ) 12 22 2 ( ) ( ) ( ) 1 2 ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ jk jk jk m jk jk jk jk m jk jk jk m m mm p p p p p p P p p p ª º « » « » « » « » « » ¬ ¼ (13) burada _ˆ( ) j k jk i i

p GH÷HUOHULQLQ KHVDSODQPDVÕ LoLQ DúD÷ÕGD \HU DODQ ED÷ÕQWÕ NXOODQÕOÕU

9

S-Qin Zhang YHGL÷HUOHUL, “A Simplified Multivariate Markov Chain Model for the Construction and Control of Genetic Regulatory Networks”, Proceedings of the 2nd International Conference on Bioinformatics and Biomedical Engineering, Shanghai, China, 2008, pp. 569-572.

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( ) ( ) ( ) ( ) 1 1 , 0 ˆ 0, ÷HU GXUXPODUGDdi j k j k k j k j k k jk _m i i _jk i i m jk jk i i i i i i f f p f z ° ° ® ° ° ¯

¦

(14) (jk)

P için tahminler bulunduktan sonra O_jk SDUDPHWUHOHULQLQKHVDSODQPDVÕ JHUHNLUdRNGH÷LúNHQOL0= x ile gösterilen bir VDELWGXUD÷DQRODVÕOÕNYHNW|Une sahiptir. x YHNW|U KHU ELU GL]LGHNL KHU ELU GXUXPXQ PH\GDQD JHOPH RUDQÕQÕQ KHVDSODQPDVÕLOHWDKPLQHGLOHELOLUYHDúD÷ÕGDNLELoLPGHJ|VWHULOLU (1) (2) ( ) ˆ (ˆ ,ˆ , ,ˆs ) .T x x x x (15) $\UÕFD O_jk parametreleri (jk)

P matrisleri ve x VDELW GXUD÷DQ RODVÕOÕN vektörü ile ilgiOLX]XQG|QHPGHQJHYHNW|ULOHLOJLOLRODUDNLIDGHVL\D]ÕODELOLU

(11) (12) (1 ) 11 12 1 (21) (22) (2 ) 21 22 2 ( 1) ( 2) ( ) 1 2 ˆ ˆ s s s s s s ss s s ss P P P P P P x x P P P O O O O O O O O O ª º « » « _{» |} « » « » « » ¬ ¼ (16)

(16) ifadesinden, O {O_jk} SDUDPHWUHOHULQLQ WDKPLQLQGH DúD÷ÕGD YHULOHQ optimizasyon probleminin çözümü bir yöntem olarak verilebilir10:

( ) ( ) ( ) 1 1 ˆ _ˆ _ˆ min max subject to 1, and 0, . m jk k j jk i _k i s jk jk k P x x k O O O O _ª _º ° _« _» ° ¬ ¼ ® ° _t ° ¯

¦

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Her bir j için (17) ile ifade edilen problem s VD\ÕGDlineer programlama SUREOHPLQHG|QúWUOU

( 1) (1) ( 2) (2) ( ) ( )

ˆ _ˆ ˆ _ˆ ˆ _ˆ

[ j j js s ]

B P x P x P x ROPDN ]HUH PRGHO DúD÷ÕGDNL ELoLPGH \D]ÕODELOLU11_:

10_{Dongmei Zhu- Wai-Ki Ching, “A New Estimation Method for Multivariate Markov Chain} Model with Application in Demand Predictions”, The 3rd International Conference on Business Intelligence and Financial Engineering (BIFE 2010), Hong Kong, August 13-15-2010, pp. 126-130.

11

Wai-Ki Ching- Michael K. Ng, Markov Chains: Models, Algorithms and Applications, United States of America, Springer Science+Business Media, Inc., 2006, s. 146

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Amaç fonksiyonu: minw _j (18) .ÕVÕWODU 1 1 2 2 ( ) ( ) 1 ˆ , ˆ 0 1, 0, . j j j j j j j j j j j js j js j s jk jk k w w w w x B x B w w w k O O O O O O O O § · § · § · § · °¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ °¨ ¸_t ¨ ¸ ¨ ¸_t ¨ ¸ °¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ °¨_¨ ¸_¸ ¨_¨ ¸ ¨_{¸ ¨} ¸_¸ ¨_¨ ¸_¸ °© ¹ © ¹ © ¹ © ¹ ® t ° ° ° _t ° ° ¯

¦

(19)

3. Uygulama

Markov ]LQFLUOHUL LOH X\JXODPDVÕ \DSÕODQ DODQODUÕQ ELU NÕVPÕ LoLQ oRN GH÷LúNHQOL 0= PRGHOL NXOODQÕODUDN GD X\JXODPDODU JHUoHNOHúWLULOHELOLU dRN GH÷LúNHQOL0=PRGHOLLOH\DSÕODQX\JXODPDODUÕQoRND]ROPDVÕKDWWDONHPL]GH\RN GHQHFHNNDGDUD]ROPDVÕEXDODQGDELUHNVLNOLNRODUDNJ|]HoDUSPDNWDGÕU

%XNÕVÕPGDYHULRODUDN7UNL\H¶GHGRODUNXUXDOÕúIL\DWGH÷HUOHULLOHø0.% 8OXVDO(QGHNVLGH÷HUOHULQLQ2FDNLOH$UDOÕNWDULKOHULDUDVÕQGDELU\ÕO ER\XQFD D\OÕN RODUDN ROXúDQ GH÷LúLP GH÷HUOHUL NXOODQÕOPÕúWÕU12_{'H÷LúLP GH÷HUOHUL} EHOLUOHQLUNHQ 0DUNRY ]LQFLUOHUL \DSÕVÕ JHUH÷L VDGHFH ELU |QFHNL D\ÕQ YHULVL GLNNDWH DOÕQDUDN YHUL GL]LOHUL ROXúWXUXOPXúWXU %|\OHFH LNL WDQH NDWHJRULN YHUL GL]LVL HOGH HGLOPLúWLU %X GL]LOHULQ VRQOX NHVLNOL GXUXPOX YH NHVLNOL ]DPDQOÕ ROGX÷X EDVLWoH söylenebilir.

'RODU NXUX DOÕú IL\DWODUÕ YH ø0.% 8OXVDO (QGHNVL GH÷HUOHULQLQ GH÷LúLPOHULLoLQRUWDODPDDUWPDYHRUWDODPDD]DOPDGH÷HUOHULEXOXQPXúWXU2UWDODPD DUWPD GH÷HULQLQ ]HULQGHNL DUWPD GH÷HUOHULD₁ RUWDODPD DUWPD GH÷HULQLQ DOWÕQGDNL DUWPDGH÷HUOHULD₂RUWDODPDD]DOPDGH÷HULQLQ ]HULQGHNLD]DOPDGH÷HUOHULD₃ ve RUWDODPD D]DOPD GH÷HULQLQ DOWÕQGDNL D]DOPD GH÷HUOHUL D4 ile semEROL]H HGLOPLúWLU

$UWPDYHD]DOPDGH÷HUOHULQHJ|UHYHULGL]LOHULS₁ ve S₂ LOHLIDGHHGLOPLúWLU 1 { ,1 1, 2, 3, 4, 4, 3, 2, 4, 2, 2, 2}, S D D D D D D D D D D D D 2 { 4, 3, 2, 1, 2, 2, 1, 1, 2, 4, 4, 1}. S D D D D D D D D D D D D 1

S YHUL GL]LVL GRODU NXUX DOÕú IL\DWODUÕQGD ROXúDQ D\OÕN GH÷LúLPOHUL YHS₂

YHULGL]LVLø0.%8OXVDO(QGHNVLGH÷HUOHULQLQGH÷LúLPOHULQLJ|VWHUPHNWHGLU%X

12

7UNL\H&XPKXUL\HWL0HUNH]%DQNDVÕ(OHNWURQLN9HUL'D÷ÕWÕP6LVWHPL http://evds.tcmb.gov.tr/cbt.html, (ULúLPWDULKL(02.08.2010).

(9)

GL]LOHUH J|UH oRN GH÷LúNHQOL 0= PRGHOLQLQ G|UW GXUXPOX ROGX÷X V|\OHQHELOLU %X YHULGL]LOHULQHJ|UHJHoLúIUHNDQVÕPDWULVOHULKHVDSODQPÕúWÕU (11) 1 0 0 0 1 2 1 1 0 1 0 1 0 1 1 1 F § · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ , (12) 0 0 0 1 1 1 1 2 0 2 0 0 2 1 0 0 F § · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ , (21) 0 2 1 1 1 1 1 1 1 0 0 0 0 1 0 1 F § · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ ve (22) 1 2 0 1 2 1 1 0 0 0 0 1 0 1 0 1 F § · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ . <XNDUÕGDHOGHHGLOHQJHoLúIUHNDQVÕPDWULVOHULQGHQ_F(11)_, 1 S dizisinin kendi GXUXPODUÕDUDVÕQGDNLJHoLúOHUGHQHOGHHGLOHQJHoLúIUHNDQVODUÕQÕ (12) F ise S2 dizisi

GXUXPODUÕQGDQ S₁ GL]LVL GXUXPODUÕQD JHoLú LOH HOGH HGLOHQ JHoLú IUHNDQVODUÕQÕ göstermektedir. _F(12)_{PDWULVLQLQ HOGH HGLOPHVL LoLQ DúD÷ÕGD YHULOHQ J|VWHULP}

NXOODQÕOPÕúWÕU 1 1 1 2 3 4 4 3 2 4 2 2 2 2 4 3 2 1 2 2 1 1 2 4 4 1 { , , , , , , , , , , , } { , , , , , , , , , , , } S D D D D D D D D D D D D S D D D D D D D D D D D D %HQ]HU úHNLOGH (22)

F , S₂ GL]LVLQLQ NHQGL GXUXPODUÕ DUDVÕQGDNL JHoLúOHUGHQ HOGH HGLOHQ JHoLú IUHNDQVODUÕQÕ _F(21)_ise

1 S GL]LVL GXUXPODUÕQGDQ S₂ dizisi GXUXPODUÕQDJHoLúLOHHOGHHGLOHQJHoLúIUHNDQVODUÕQÕJ|VWHUmektedir. *HoLúIUHNDQVÕPDWULVOHULQLQQRUPDOL]DV\RQXLOHJHoLúRODVÕOÕNODUÕPDWULVOHUL KHVDSODQPÕúWÕU (11) 1 0 0 0 2 1 1 1 1 2 2 2 3 ˆ 1 1 0 0 4 3 1 1 1 0 4 2 3 P § · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ , (12) 1 0 0 0 3 1 1 2 1 3 4 3 ˆ 1 0 0 0 2 2 1 0 0 3 4 P § · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ ,

(10)

(21) 1 1 1 0 2 2 3 1 1 1 1 2 4 2 3 ˆ 1 0 0 0 2 1 1 0 0 4 3 P § · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ ve (22) 1 1 1 0 3 2 3 2 1 1 0 3 4 ˆ 1 0 0 0 3 1 0 0 0 4 P § · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ . 1

S veri dizisi ile S₂ YHUL GL]LVLQGH \HU DODQ GXUXPODUÕQ PH\GDQD JHOPH RUDQODUÕNXOODQÕODUDNDúD÷ÕGDYHULOHQ x YHNW|UOHULHOGHHGLOPLúWLU 1 2 1 5 1 1 1 1 1 1 ˆ , , , ,ˆ , , , . 6 12 6 4 3 3 12 4 T T x §_¨ ·_¸ x §_¨ ·_¸ © ¹ © ¹

7HRULN DoÕNODPDODUGD m GH÷LúNHQL durum VD\ÕVÕQÕ J|VWHUGL÷LQGHQ GROD\Õ 4

m ve s GH÷LúNHQL kategorik GL]L VD\ÕVÕQÕ J|VWHUGL÷LQGHQ GROD\Õs 2’dir. Bu durumda j 1 ve j 2 LoLQD\UÕD\UÕo|]POHU\DSÕOPDOÕGÕU

(18) ve (19) ifadeleri ile verilen lineer programlama probleminin çözümü LoLQDúD÷ÕGDNLLúOHPOHUJHUoHNOHúWLULOPLúWLU

1

j için,

Amaç fonksiyonu: min w1

.ÕVÕWODU 1 1 1 (1) 11 1 (1) 11 1 12 1 12 1 1 ˆ , ˆ w w w w x B x B w w w w O O O O § · § · ¨ ¸ ¨ ¸ § · § · ¨ ¸_t _¨ _¸ ¨ ¸_t _¨ _¸ ¨ ¸ _© _¹ ¨ ¸ _© _¹ ¨ ¸ ¨ ¸ © ¹ © ¹ 1 2 1 11 12 11 12 1 0 1 1, , 0. k k w O O O O O t t

¦

(11)

(11) (1) (12) (2) 1 1 12 12 4 11 9 24 ˆ _ˆ ˆ _ˆ [ ] 3 1 16 6 13 11 48 36 B P x P x § · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ úHNOLQGHGLU 2 j için,

Amaç fonksiyonu: min w₂

.ÕVÕWODU 2 2 2 (2) 21 2 (2) 21 2 22 2 22 2 2 ˆ , ˆ w w w w x B x B w w w w O O O O § · § · ¨ ¸ ¨ ¸ § · § · ¨ ¸_t _¨ _¸ ¨ ¸_t _¨ _¸ ¨ ¸ _© _¹ ¨ ¸ _© _¹ ¨ ¸ ¨ ¸ © ¹ © ¹ 2 2 2 21 22 21 22 1 0 1 1, , 0. k k w O O O O O t t

¦

úHNOLQGHRODFDNWÕU.ÕVÕWODUGD\HUDODQB matrisi ise

(21) (1) (22) (2) 3 13 8 36 17 7 48 18 ˆ _ˆ ˆ _ˆ [ ] 1 1 12 12 3 1 16 12 B P x P x § · ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ ¨ ¸ © ¹ úHNOLQGHGLU

(18) ve (19) ifadeleri için j 1 ve j 2 DOÕQDUDN \DSÕODQ OLQHHU programlama problemlerinin çözümleri ile O GH÷HUOHULHOGHHGLOPLúWLU

O PDWULVL GH÷HUOHUL LIDGHVLQGH \HULQH \D]ÕOGÕ÷ÕQGD S1 ve S2 ile

J|VWHULOHQ NDWHJRULN YHUL GL]LOHUL LoLQ oRN GH÷LúNHQOL 0= PRGHOL DúD÷ÕGDNL ELoLPGH verilir:

(12)

(1) (11) (1) (12) (2) 1 0,332ˆ 0, 668ˆ n n n x P x P x (2) (21) (1) (22) (2) 1 0,5ˆ 0,5ˆ . n n n x P x P x

4. Sonuç ve Yorumlar

dRN GH÷LúNHQOL 0= PRGHOLQGH ELUELUOHUL LOH ELU NRUHODV\RQD VDKLS YHUL GL]LOHULNXOODQÕOPDNWDGÕU%XVD\HGH0=PRGHOLQGHHOHDOÕQDQWHNND\QDNOÕYHULOHULQ DQDOL]LQHJ|UHIDUNOÕEDNÕúDoÕODUÕYH\RUXPODUROXúPDNWDGÕU

dRN GH÷LúNHQOL 0= PRGHOL WHRULVLQLQ VRQ RQ \ÕOGD JHOLúWLULOPLú ROPDVÕ X\JXODPDDODQÕQGDNLoDOÕúPDODUÕQoRND]YHONHPL]GHEXNRQXLOHLOJLOLoDOÕúPDODUÕQ \RNGHQHFHNNDGDUD]ROPDVÕNRQX\XYHX\JXODPD\ÕLQFHOHQHELOLUKDOHJHWLUPHNWHGLU %XoDOÕúPDGDGRODUNXUXDOÕúIL\DWODUÕQGDROXúDQD\OÕNGH÷LúLPOHULOHø0.% 8OXVDO(QGHNVLGH÷HUOHULQGHROXúDQD\OÕNGH÷LúLPOHULNLNDWHJRULNGL]LRODUDNHOH DOÕQÕS EX GL]LOHULQ ELUELUOHULQL QH RUDQGD HWNLOHGLNOHUL üçüncü bölümde KHVDSODQPÕúWÕU(OGHHGLOHQVRQXoODUDúD÷ÕGD\HUDOPDNWDGÕU

(1) 1

n

x GH÷LúNHQL(n1). DGÕPGD GL]LQLQ \DQL GRODU NXUX DOÕú IL\DWODUÕQGD ROXúDQD\OÕNGH÷LúLPOHULQELU|QFHNLDGÕPGDLOHNHQGLVLQH (1)

n

x ) ve 0,668 ile 2.

GL]L RODQ ø0.% 8OXVDO (QGHNVL GH÷HUOHULQGH ROXúDQ D\OÕN GH÷LúLPOHUH (2)

n x ) ED÷OÕGÕU (2) 1 n x GH÷LúNHQL(n1). DGÕPGD GL]LQLQ \DQL ø0.% 8OXVDO (QGHNVL

GH÷HUOHULQGHROXúDQD\OÕNGH÷LúLPOHULQELU|QFHNLDGÕPda 0,5 ile kendisine( (2)

n x ) ve LOH GL]L RODQ GRODU NXUX DOÕú IL\DWODUÕQGD ROXúDQ D\OÕN GH÷LúLPOHUH (1)

n x )

ED÷OÕGÕU

%X oDOÕúPDGD oRN GH÷LúNHQOL 0= PRGHOL WHRULN DoÕGDQ RODELOGL÷LQFH DoÕN olarak ifade HGLOPLúYHELUX\JXODPDLOHWHRULN\DSÕDQODúÕODELOLUELUKDOHJHWLULOPLúWLU øOHULGH\DSÕODFDNoDOÕúPDODUGDNDWHJRULNYHULGL]LVLVD\ÕVÕQÕQDUWWÕUÕOPDVÕLOH ELUELULQL HWNLOH\HQ GL]LOHU LoLQ GDKD EHOLUJLQ VRQXoODU HOGH HGLOHELOLU $\UÕFD GL]L\L ROXúWXUDQ GXUXP VD\ÕVÕQÕQ DUWWÕUÕOPDVÕ YH J|]OHQHQ YHULOHULQ GDKD ID]OD ROPDVÕ PRGHOLQJHUoH÷HGDKD\DNÕQVRQXoODURUWD\DNR\DFD÷ÕGúQOPHNWHGLU

(13)

Kaynakça

$<7(0øZ Tevfik-ù(1*g1h/ Ahmet, “Markov Zincirlerinin Ekonomik Bir 3UREOHPH 8\JXODQPDVÕ 3HUDNHQGH $OÕúYHULúOHUGH %LUH\VHO 2ODUDN .XOODQÕODQ 0DGHQL 3DUD 6WUDWHMLOHULQLQ .DUúÕODúWÕUÕOPDOÕ $QDOL]L´ Dokuz Eylül Üniversitesi Sosyal Bilimler Enstitüsü Dergisi, Cilt. 6D\Õ. 4, 2004, s. 29-43.

CAN Tuncay, 6HNW|UOHU $UDVÕ øOLúNLOHULQ 0DUNRY =LQFLUOHUL LOH $QDOL]L YH 7DKPLQL7UNL\HgUQH÷LøVWDQEXO'HULQ<D\ÕQODUÕ

CHING Wai-Ki, FUNG Eric S. and NG Michael K., “A Multivariate Markov Chain Model for Categorical Data Sequences and Its Applications in Demand Predictions”, IMA Journal of Management Mathematics, Vol. 13, 2002, pp. 187-199.

CHING Wai-Ki, LI Li-Min, LI Tang and ZHANG Shu-Qin, “A New Multivariate Markov Chain Model with Applications to Sales Demand Forecasting”, International Conference on Industrial Engineering and Systems Management IESM 2007, Beijing – China, May 30-June 2-2007, pp. 1-8. CHING Wai-Ki-NG Michael K., Markov Chains: Models, Algorithms and

Applications, United States of America, Springer Science+Business Media, Inc., 2006.

DARDANONI Valentino, “Income Distribution Dynamics: Monotone Markov Chains Make Light Work”, Social Choice and Welfare, Vol. 12, No. 2, 1995, pp. 181-192.

DURA Codruta, “The Use of Markov Chainsin Marketing Forecasting”, Annals of the 8QLYHUVLW\3HWURúDQL(FRQRPLFV, Vol. 6, No. 1, 2006, pp. 69-76. GEVREK $OL øKVDQ-ù(1*h//(5 øONHU ³0DUNRY =LQFLUL $QDOL] <|QWHPLQLQ

/LQ\LWøoHUHQ=ÕUQDN)RUPDV\RQXQD3OL\RVH+ÕQÕV8\JXODQPDVÕ´Jeoloji 0KHQGLVOL÷L'HUJLVL6D\Õ. 41, 1992, s. 84-90.

GUIDICI Paolo-CASTELO Robert, “Improving Markov Chain Monte Carlo Model Search for Data Mining”, Machine Learning, Vol. 50, No. 1-2, 2003, pp.127-158.

.2d$..DVÕP-ù(1=HNDL³.XUDNYH<D÷ÕúOÕ*Q2OXúXPODUÕQÕQ0DUNRY=LQFLUL <DNODúÕPÕ LOH 8\JXODPDOÕ øQFHOHQHmsi”, Tr. J. Of Engineering and Environmental Science6D\Õ. 22, 1998, s. 479-487.

LEVIN R.I., KIRKPATRICK C.A. and RUBIN D.S., Quantitave Approaches to Management, Fifth Edition, Tokyo, Mc-Graw-Hill, 1982.

MASKAWA Jun-ichi, “Multivariate Markov Chain Modeling for Stock Markets”, Physica A, Vol. 324, 2003, pp. 317-322.

NIELSEN Rasmus-WAKELEY Jhon, “Distinguishing Migration From Isolation: A Markov Chain Monte Carlo Approach”, Genetics, Vol. 158, No. 1, 2001, pp. 885-896.

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POORE J.H., WALTON G.H.-WHITTAKER J.A., “A Constraint-Based Approach to the Repsentation of Software Usage Models”, Information and Software Technology, Vol. 42, No. 12, 2000, pp. 825-833.

ROMAGNUOLO J., MEIER M.A.-SADOWSKI D.C., “Medical or Surgical Therapy for Erosive Reflux Esophagitis: Cost-Utility Analysis Using a Markov Model”, Annals of Surgery, Vol. 236, No. 2, 2002, pp. 191-202. ROSS Sheldon M., Introduction to Probability Models, 10th Edition, United States

of America, Academic Press, 2009.

ROSS, Sheldon M., Stochastic Processes, Second Edition, New York, Jhon Wiley & Sons Inc., New York, 1996.

RÜZGAR Nursel S., ³%LU øúOHWPHQLQ gGHPHOHU 'HQJHVLQLQ 0DUNRY 6UHoOHUL <DUGÕPÕ\OD$QDOL]L´Dokuz Eylül Üniversitesi Sosyal Bilimler Enstitüsü Dergisi, Cilt. 6D\Õ. 1, 2003, s. 164-179.

SIMKIN Mark G., “Forecasting the Sale of Telephone Switchboard Equipment with an Interactive Computer Model and a Markov Chain”, Review of Business and Economic Research, Vol. 18, No. 1, 1982, pp. 27-36.

STANDER J., FARRINGTON D.P., HILL G.-Altham P.M.E., “Markov Chain Analysis and Specialization in Criminal Careers”, British Journal of Criminology, Delinquency and Deviant Social Behaviour, Vol. 29, No. 4, 1989, s. 317-335.

7h5.,<( &80+85,<(7, 0(5.(= %$1.$6, (OHNWURQLN 9HUL 'D÷ÕWÕP Sistemi, http://evds.tcmb.gov.tr/cbt.html, (02.08.2010).

WHITTAKER J.A.,-POORE J.H., “Markov Analysis of Software Specifications”, ACM Transaction on Software Engineering and Methodology, Vol. 2, No. 1, 1993, s. 93-106.

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H., “Construction and Control of Genetic Regulatory Networks: A Multivariate Markov Chain Approach”, J. Biomedical Science and Engineering, Vol. 1, 2008, s. 15-21.

ZHANG Shu-Qin, CHING Wai-Ki, JIAO Yue, WU Ling-Yun and CHAN Raymond H., “A Simplified Multivariate Markov Chain Model for the Construction and Control of Genetic Regulatory Networks”, Proceedings of the 2nd International Conference on Bioinformatics and Biomedical Engineering, Shanghai, China, 2008, s. 569-572.