A Simplified Higher-Order Markov Chain Model

Size: px

Start display at page:

Download "A Simplified Higher-Order Markov Chain Model"

Julianna Bruce
6 years ago
Views:

1 Wold Acadey o Scece Egeeg ad ecology Ieaoal Joual o Maeacal ad Copuaoal Scece Vol:7 No: A Spled Hge-Ode Maov Ca Model Cao Wag g-zu Huag Ce Ja Ieaoal Scece Ide Maeacal ad Copuaoal Scece Vol:7 No: waeog/ublcao/99967 Abac I pape we pee a pled ge-ode Maov ca odel o ulple caegocal daa equece alo called a pled ge-ode ulvaae Maov ca odel e ube o e paaee o e ew odel oly O wc le a O e ge-ode ulvaae Maov ca odel Nuecal epee lluae e bee o ou ew odel Keywod Hge-ode ulvaae Maov ca odel Caegocal daa equece Mulvaae Maov ca I INRODUCION ARKOV ca ae o ee a wde age o Mapplcao o eaple elecoucao ye auacug ye ad veoy ye ee o ace [6] ad e eeece ee I ece yea e pedco o caegocal daa equece [4] a becoe oe ad oe ueul ay eal wold applcao uc a ale dead pedco []-[] DNA equecg [] ad ced daa odelg [8]-[] Ipovg e odel o eplog ee elaop aog e gve caegocal daa equece a poa eeac aea ee yea Dee odel ave bee popoed o ulple caegocal daa equece pedco A ulvaae Maov ca odel a bee peeed [] ey couced a ew a by ea o e ao pobably ace aog dee equece Fo eecvee a poved ulvaae Maov ca odel a bee ebed o peed up e covege ae o copug e aoay o eady ae oluo I e poved odel ey copoaed e pove ad egave aocao pa [8] e ube o e paaee O A oe advaced aely ge-ode ulvaae Maov ca odel a bee uded [7] Moeove ee ae oe oe pape cobue o e ulvaae Maov ca odel eg [] []-[] [7] Howeve w e develope o cece ecologe ad e applcao e daa equece wll be loge e pedco eul eeded o be oe pece I evable a e copuao o a lage caegocal daa equece goup wll caue g copuaoal co Now e ge-ode ulvaae Maov ca odel peo e be e pedco o ulvaae dcee-e Maov ca u ueul o ply e ge-ode ulvaae Maov ca odel Fo e above pupoe we popoe a pled Cao Wag ad g-zu Huag ae w e Scool o Maeacal Scece Uvey o Elecoc Scece ad ecology o Ca Cegdu Scua 67 R Ca e-al: wagcao64@6co gzuuag@6co Ce Ja w Scool o Accoug Jl Uvey o Face ad Ecooc Cagcu Jl 67 R Ca e-al: cece98977@6co ge-ode ulvaae Maov ca odel pape e e o e pape ogazed a ollow I Seco II we evew e Maov ca odel [] e ulvaae Maov ca odel [] e ge-ode ulvaae Maov ca odel [7] ad wo lea [6] [8] I Seco III we pee a pled ge-ode ulvaae Maov ca odel o ulple caegocal daa equece Moeove oe popee o e pled ge-ode ulvaae Maov ca odel ae alo aalyzed Seco IV gve eao eod o e paaee o e pled ge-ode ulvaae Maov ca odel Nuecal epee o wo eaple deoae e bee o ou ew odel Seco V Fally cocludg ea ae gve Seco VI II A REVIEW ON HE MARKOV CHAIN MODELS I eco we bely oduce oe Maov ca odel ad eveal lea eg e Maov ca odel e ulvaae Maov ca odel e ge-ode ulvaae Maov ca odel ad e eo-fobeu eoe A e Maov Ca Model F we oduce oe deo o e Maov ca o [] [] Code e ae e o e caegocal daa equece be M { } e dcee-e Maov ca w ae ae e ollowg elaop: ob θ θ θ θ ob θ θ wee θ M { } e codoal pobable ob θ θ ae called e oe-ep ao pobable o e Maov ca I we ewe e ao pobable a p ob θ θ M e Maov ca odel ca be epeeed a ollow: wee [ p ] p p M ad e al pobably dbuo e ae pobably dbuo e Mulvaae Maov Ca Model A ulvaae Maov ca odel a bee popoed [] Ieaoal Scolaly ad Scec Reeac & Iovao 7 74 colawaeog/7-689/99967

2 Wold Acadey o Scece Egeeg ad ecology Ieaoal Joual o Maeacal ad Copuaoal Scece Vol:7 No: Ieaoal Scece Ide Maeacal ad Copuaoal Scece Vol:7 No: waeog/ublcao/99967 wee e ube o caegocal daa equece > ad a e ollowg o: wee { } { } { } 4 Hee e al pobably dbuo o e equece e ae pobably dbuo o e equece a e e ae pobably dbuo o e equece a e Hee e oe-ep ao pobably a o e ae e equece a e o e ae e equece a e I a o we ave wee ca be obaed decly o e caegocal daa equece ad e paaee ca be go by e lea pogag deal [] C e Hge-Ode Mulvaae Maov Ca Model Le e ube o caegocal daa equece be > be e ube o ae evey equece e ge-ode ulvaae Maov ca odel [7] ca be peeed a ollow: wee ae e al pobably dbuo ad ae 6 7 Hee e ae pobably dbuo o e equece a e e -ep ao pobably a o e ae e equece a e o e ae e equece a e e ae pobably dbuo o e equece a e Moeove a e ollowg o: R 8 I a o we ave wee ele I I I 9 Eac colu u o e eave a o eceay equal o oe wle eac e colu u o ao pobably ace equal o oe Fo aalyzg e popee o e pled ge-ode ulvaae Maov ca odel a wll be peeed e eco wo Lea ae gve a Lea eo-fobeu eoe [] Le A R be a o-egave ad educble a e; A a a pove eal egevalue equal o pecal adu e a A wee A deoe e egevalue o A o ee a egeveco z o ee beg eal ad pove uc a Az z a ple egevalue o A Ieaoal Scolaly ad Scec Reeac & Iovao 7 7 colawaeog/7-689/99967

3 Wold Acadey o Scece Egeeg ad ecology Ieaoal Joual o Maeacal ad Copuaoal Scece Vol:7 No: Ieaoal Scece Ide Maeacal ad Copuaoal Scece Vol:7 No: waeog/ublcao/99967 Lea [] Le be a eave a o ulvaae Maov ca odel ad be a ae dbuo a e I educble ad apeodc e ee a uque aoay dbuo π ayg π π ad l π III SIMLIFIED HIGHER-ORDER MULIVARIAE MARKOV CHAIN MODEL I eco o educg e ube o e paaee we popoe a pled ge-ode ulvaae Maov ca odel o caegocal equece ad bely dcu oe popee o e ew odel I ew odel e ae pobably dbuo o e equece a e deped o e ae pobably dbuo o all e equece a e Fo { } { } e pled ge-ode ulvaae Maov ca odel gve a ollow: wee ae e al pobably dbuo ave bee deed Seco IIC ad ae wee { } { } Aug a R wee I a o e pled ge-ode ulvaae Maov ca odel ca be peeed a wee ele I I I 4 Eac e colu u o ao a equal o oe e ube o e paaee o e popoed ew odel oly O wc le a O o e ge-ode ulvaae Maov ca odel Ne oe popee o e pled ge-ode ulvaae Maov ca odel ae dcued eoe : I e pled ge-ode ulvaae Maov ca odel o { } { } e e a a a egevalue equal o oe ad e odulu o all egevalue ae le a o equal o oe oo: Le wee ad { } 6 Ieaoal Scolaly ad Scec Reeac & Iovao 7 76 colawaeog/7-689/99967

4 Wold Acadey o Scece Egeeg ad ecology Ieaoal Joual o Maeacal ad Copuaoal Scece Vol:7 No: Ieaoal Scece Ide Maeacal ad Copuaoal Scece Vol:7 No: waeog/ublcao/99967 { } Fo e a o-egave ad eac colu u equal o oe Ug e popee o coeco educble Accodg o Lea ee e a pove veco uc a y y y y y y y ] Sce a [ y y ad y y 7 a pobably ao a o ad [ ] I clea a R 8 [ y y y y ] [ y y y y ] 9 uc a oe a egevalue o Subequely we pove e odulu o all e egevalue o ae le a o equal o oe Suppoe a D v dagv ad Ug 9 a v y D D v v v v e la o Fo 9 a we oba ρ ρ e cocluo o eoe ave bee poved o eep e educble o we ll e colu o w we e colu u o zeo I educble e alo educble Fo Lea ee e a uque pove veco uc a IV ESIMAING HE ARAMEERS OF SIMLIFIED HIGHER-ORDER MULIVARIAE MARKOV CHAIN MODEL I eco we wll eae e paaee o e ew ulvaae Maov ca odel Le' eae e ao ace I e daa equece ae gve ad e ae e M { } F equecy o e ae e equece a e o e ae e equece a e o e e ao equecy ace equece ca be couced a: F F o e daa ao pobably ace ca be obaed by oalzg e equecy ao ace a ollow: wee p oewe Subequely e way o eag e paaee wll be oduced Code beg a o aoay pobably dbuo ad ca be peeed a wee ad Oe would epec a Ieaoal Scolaly ad Scec Reeac & Iovao 7 77 colawaeog/7-689/99967

5 Cealy o-equaly ca be ewe a: wee > ad a all a poble Oe way o eag o ao o a zao poble a e ollowg o: ubeco Hee we cooe e o a e o ad u e above zao poble ca be peeed a: a ubeco wee [] e ey o e veco Ug e dea o [] e zao poble becoe a lea pogag poble a ollow: C C ubuco wee [ C ] ad V NUMERICAL EERIMENS I eco we epo o e uecal eul w dee ge-ode ulvaae Maov ca odel o wo eaple obaed w a Malab 7 pleeao o a Wdow w 9GHz 64-b poceo ad G eoy A Sple Eaple ee ae wo caegocal daa equece {44444} {44} S S I eaple we a a coucg a pled -ode ulvaae Maov ca odel Coug e lag ao equece we ave F F W e ae poce e ecod lag ao equece ae gve a ollow: F F Coug e e lag ao equece a F F Noalzg e ao equecy ace we oba e ao pobably ace: p 4 4 p Wold Acadey o Scece Egeeg ad ecology Ieaoal Joual o Maeacal ad Copuaoal Scece Vol:7 No: 78 Ieaoal Scolaly ad Scec Reeac & Iovao 7 colawaeog/7-689/99967 Ieaoal Scece Ide Maeacal ad Copuaoal Scece Vol:7 No: waeog/ublcao/99967

6 Wold Acadey o Scece Egeeg ad ecology Ieaoal Joual o Maeacal ad Copuaoal Scece Vol:7 No: Ieaoal Scece Ide Maeacal ad Copuaoal Scece Vol:7 No: waeog/ublcao/99967 p p p p e al ae pobably dbuo ca be obaed by copug e popoo o e occuece o eac ae eac o e equece ad ca be calculaed by lea pogag poble wc coepod o e pled ecod-ode ulvaae Maov ca odel o e wo caegocal daa equece W e eul o e pled -ode ulvaae Maov ca odel peeed a ollow: A Applcao o Sale Dead edco I pa e ale dead equece ae peeed o ow e eecvee o e pled ge-ode ulvaae Maov ca odel o ulvaae Maov ca Sce e equee o e ae lucuae eavly e poduco plag ad e veoy cool decly aec e eae co u udyg e eplay bewee e oage pace equee ad e oveall gowg ale dead a peg ue o e copay Hee ou goal o pedc e ale dead o e ae o zg e eae co Code e poduc ca be claed o poble ae 4 6 eg o ale volue vay low ale volue low ale volue 4 adad ale volue a ale volue 6 vay a ale volue e cuoe ale dead o ve poa poduc o e copay o a yea a bee gve [] We cooe e pled 8-ode ulvaae Maov ca odel o odel ve caegocal daa equece y copug e popoo o e occuece o eac ae eac o e equece e al pobably dbuo o e ve caegocal daa equece ae e ao pobably ace ca be obaed ae oalzg e ao equecy ace y olvg e coepodg lea pogag poble obaed ad e pled d-ode ulvaae Maov ca odel ca be peeed a ollow: Ne we ue e pled 8-ode ulvaae Maov ca odel o pedc e e ae o e equece a e wc ca be ae a e ae w e au pobably e [ ] [ ] Fo evaluag e peoace ad e eecvee o dee ulvaae Maov ca odel e pedco accuacy popoed [] ad deed a: δ % wee e leg o e daa equece ad θ δ oewe Fo copao e uecal eul o e 8-ode ulvaae Maov ca ae alo peeed able I wee we deoe e e copuaoal e p e ube o e paaee e odel H a ge-ode ulvaae Maov ca odel SH a pled ge-ode ulvaae Maov ca odel ad e pedco accuace o oduc A oduc oduc C oduc D oduc E ae deoed a A C D E epecvely Obevg e daa able I e pedco Ieaoal Scolaly ad Scec Reeac & Iovao 7 79 colawaeog/7-689/99967

7 Wold Acadey o Scece Egeeg ad ecology Ieaoal Joual o Maeacal ad Copuaoal Scece Vol:7 No: accuace o ou ew odel ae ealy e ae a e ge-ode ulvaae Maov ca odel I addo obvouly a e copuaoal e ad e ube o e paaee o ou ew odel ae uc le a oe o e ge-ode ulvaae Maov ca odel ALE I REDICION ERRORS OF HE SIMLIFIED HIGHER-ORDER MULIVARIAE MARKOV CHAIN MODEL AND HIGHER-ORDER MULIVARIAE MARKOV CHAIN MODEL IN SALES DEMAND REDICIONS A C D E e p H SH Ieaoal Scece Ide Maeacal ad Copuaoal Scece Vol:7 No: waeog/ublcao/99967 Subequely aoe pedco ceo o Maov ca odel oduced Noe a A e ube o e caegocal daa e equece ` a pedc pobably a e a ac value a e ad [ ] I e ac ae a caegocal daa equece e { } R We deoe e pedco eo e odel a pe wc ca be eaed by e equao: pe A 9 I able II we deoe a e pedco eo o e pled ge-ode ulvaae Maov ca odel pe e pedco eo o e ge-ode ulvaae Maov ca odel pe e ube o e paaee o e pled ge-ode ulvaae Maov ca odel p e ube o e paaee o e ge-ode ulvaae Maov ca odel p e copuaoal e o e pled ge-ode ulvaae Maov ca odel e e copuaoal e o e ge-ode ulvaae Maov ca odel e epecvely Sop ceo o e lea pogag poble ca be oud Malab ode lpog able II povde a e peoace o e pedco eo pled ge-ode ulvaae Maov ca odel wc copaable w e ge-ode ulvaae Maov ca odel able II lluae e bee o e pled ge-ode ulvaae Maov ca odel e o e copuaoal e ad e ube o paaee coollg ALE II REDICION ERRORS OF HE SIMLIFIED HIGHER-ORDER MULIVARIAE MARKOV CHAIN MODEL AND HIGHER-ORDER MULIVARIAE MARKOV CHAIN MODEL IN SALES DEMAND REDICIONS pe p e pe p e VI CONCLUSION I eal applcao evable a e copuao o a lage caegocal daa equece goup wc wll caue g copuaoal co Fo avg copuaoal co we popoe a pled ge-ode ulvaae Maov ca odel pape e ube o e paaee o e ew odel oly O wc le a O o e ge-ode ulvaae Maov ca odel e eul o e pedco accuacy ad e pedco eo ee wo odel ae copaable o ealy e ae Moeove uecal epee lluae e bee o ou ew odel e coug e ube o e paaee coollg ad e oage equee Cealy e pled ge-ode ulvaae Maov ca odel ca alo be appled ced ad oe eeac aea REFERENCES [] W Cg E Fug ad M Ng A ulvaae Maov ca odel o caegocal daa equece ad applcao dead pedco IMA Joual o Maagee Maeac [] W Cg E Fug ad M Ng A Hge-ode Maov odel o e ewboy poble J Opeaoal Reeac Socey [] W Cg E Fug ad M Ng Hge-ode Maov ca odel o caegocal daa equece Ie J Nav Re Log [4] W Cg M Ng ad E Fug O Couco o ocac geec ewo baed o gee epeo equece Ieaoal Joual o Neual Sye 97- [] W Cg ad M Ng Maov ca: odel algo ad applcao Ieaoal See o Opeao Reeac ad Maagee Scece 6 [6] W Cg uldg ge-ode Maov ca odel w ECEL Ie J Maeacal Educao Scece ad Egeeg [7] W Cg M Ng ad E Fug Hge-ode ulvaae Maov ca ad e applcao Lea Algeba ad Applcao Ieaoal Scolaly ad Scec Reeac & Iovao 7 8 colawaeog/7-689/99967

8 Wold Acadey o Scece Egeeg ad ecology Ieaoal Joual o Maeacal ad Copuaoal Scece Vol:7 No: Ieaoal Scece Ide Maeacal ad Copuaoal Scece Vol:7 No: waeog/ublcao/99967 [8] W Cg Su ad L L A poved paoou ulvaae Maov ca odel o ced Joual o Ced R 9 - [9] W Cg H Leug H Jag L Su ad Su A Maova ewo odel o deaul aagee Ieaoal Joual o Iellge Egeeg Ioac 4-4 [] V Cvaal Lea pogag Feea 98 [] M Dav ad V Lo Modelg deaul coelao bod poolo I Maeg R 4- [] R Ho ad C Joo Ma aaly Cabdge Uvey e [] M Ka K Koobaya ad E Suzu A ulvaae Maov odel o ulag coelaed deaul Joual o R 4 - [4] A Raey A odel o g-ode Maov ca J R Sa Soc [] Su W Cg M Ng ad E Fug O ulvaae cedbly appoac o poolo ced eauee Quaave Face 4-6 [6] S Ro Ioduco o pobably odel Acadec e Aeda oo ed [7] D Zu ad W Cg A oe o e aoay popey o gdeoal Maov ca odel Ieaoal Joual o ue ad Appled Maeac 66 - [8] ZY You LC Wag A cocep o olea bloc dagal doace J Copu Appl Ma Cao Wag wa bo 98 Jl Ca He eceved S 7 Appled Maeac o e Hua Uvey ad MS Scool o Maeacal Scece o Uvey o Elecoc Scece ad ecology o Ca He wog o D a e Uvey o Elecoc Scece ad ecology Cegdu Ca H cue ee clude ulvaae Maov Ca odel addle po poble ad lea copleeay poble g-zu Huag eceved e S MS ad D degee Copuaoal Maeac o e Depae o Maeac 'a Jaoog Uvey 'a Ca ad epecvely Dug e wa a vg cola Dep o Copue Scece Lougbooug Uvey o UK He cuely a Full oeo e Scool o Maeacal Scece UESC He cuely a edoal o Advace Nuecal Aaly Joual o ue ad Appled Maeac: Advace ad Applcao Joual o Elecoc Scece ad ecology o Ca He e auo o coauo o oe a eeac pape H cue eeac ee clude uecal lea algeba w applcao eave eod o lea ye ad addle po poble pecodog ecologe ad a aaly w applcao Ce Ja eceved e S degee o e Scool o Accoug Jl Uvey o Face ad Ecooc Cuagcu Ca 8 Se cuely a pogaduae e Scool o Accoug Jl Uvey o Face ad Ecooc Ieaoal Scolaly ad Scec Reeac & Iovao 7 8 colawaeog/7-689/99967

Modified Inverse Weibull Distribution

Modified Inverse Weibull Distribution J. Sa. Appl. Po. No. 5-5 NSP Moded Ivee Webull Dbuo Muhaad Shuab Kha ad Robe K School o Maheacal ad Phcal Scece he Uve o Newcale Callaha NSW 8 Auala Eal Adde huab.a@al.co obe.k@ewcale.edu.au Receved Apl