MATH2999 Directed Studies in Mathematics Matrix Theory and Its Applications

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1 MATH999 Drected Studes Mthemtcs Mtr Theory d Its Applctos Reserch Topc Sttory Probblty Vector of Hgher-order Mrkov Ch By Zhg Sho Supervsors: Prof. L Ch-Kwog d Dr. Ch Jor-Tg

2 Cotets Abstrct. Itroducto: Bckgroud. Fudmetl Cocepts d Results the Theory of No-egtve Mtrces. Dscrete Tme-Homogeeous Mrkov Chs.3 The Perro-Frobeus Theorem for Prmtve Mtrces. Hgher-order Mrkov Chs. Deftos d Cocepts. Codtos for Ech Pot the Smple Beg Sttory Vector.3 Applctos DNA Sequece Predcto 3. Cocluso

3 Abstrct I ths proect, our focus wll be how to determe the sttory vector of hgherorder Mrkov Ch. Specfclly, we mly focus o the tertve method proposed by L, Ng d Ye (0). Rther th the computto method tself, our problem s wht kd of codtos of the prmeters wll gve out ftely my solutos, multple solutos or uque soluto to the hgher-order Mrkov Ch, whch mes we wt to study the umber of sttory probblty vectors the soluto set. Ths s reltvely ew topc whch my led to more future lyss.. Itroducto: Bckgroud. Fudmetl Cocepts d Results the Theory of No-egtve Mtrces Defto A squre mtr T s clled o-egtve f ll ts etres re oegtve rel vlues. We wrte T 0 to represet such mtrces. Mewhle T s squre mtr whch s clled prmtve f there ests postve teger k such tht T k >0. Frst ote tht o-egtve mtr s ot suffcet to be prmtve mtr. A smple emple s 0 d 0 re both prmtve mtrces sce 0 d prmtve. 0 whle 0 0 s o-egtve but ot. Dscrete Tme-Homogeeous Mrkov Chs A dscrete-tme Mrkov ch s stochstc process Xt, t 0,, wth dscrete fte stte spce S such tht wth tme depedet probblty Pr X X, X, X,,, X, X Pr X X p t t t t t t 0 0 t t holds for ll,, 0,, t. The ut sum vector X s sd to be sttory probblty vector or dstrbuto of fte Mrkov Ch f XP=X where P p,, S. The followg theorem gurtees the estece d uqueess of the sttory probblty vector of dscrete-tme Mrkov Ch.

4 .3 The Perro-Frobeus Theorem for Prmtve Mtrces (Wthout Proof) Suppose T s by o-egtve prmtve mtr. The there ests egevlue r such tht r s rel postve smple root of the chrcterstc equto of T, r> λ for y egevlue λ r d the egevectors ssocted wth r re uque to costt multples. If 0 B T d β s egevlue of B, the β r. Moreover, β =r mples B=T. We hve to dmt tht the Perro-Frobeus Theorem s the most fudmetl theorem for o-egtve mtrces, d t hghlghts d gurtees the so-clled ture of fte Mrkov Chs, whch s the covergece of rreducble fte Mrkov Ch to ts sttory probblty dstrbuto.. Hgher-order Mrkov Chs. Deftos d Cocepts t, 0,, We cosder stochstc process wth sequece of rdom vrbles, X t, whch tkes o fte set S 0,,,, the process. clled the stte set of Defto. Suppose the probblty depedet of tme stsfyg Pr X X, X, X,,, X, X t t t t 3 t 0 t X X X X X p Pr,,,, t t t t 3 t m m,,,, m where,,,, m S, the t s clled m-th order Mrkov Ch, other words, the curret stte of the process depeds o m pst sttes. Observed tht p,,,, m. Whe m=, t s ust the regulr stdrd Mrkov Ch. Defto. Wrte 3 to be three-order -dmesol tesor, where 3 d,, 3, defe -dmesol colum vector X 3 gve 3, 3 X Wrg: ths three-order hs othg to do wth the m-th order the prevous Defto.

5 Defto.3 A three-order -dmesol tesor s clled reducble f there ests o-empty proper de subset I {,,, } such tht, f 0, I,, I 3 3 s ot reducble, we cll t rreducble. 3 I fct, f P s rreducble o-egtve three-order -dmesol tesor of hghorder Mrkov Ch, L, Ng d Ye (0) hs proved tht order to obt the sttory probblty vector X of hgh-order Mrkov Ch, we ust eed to solve X X. 3. Codtos for Ech Pot the Smple Beg Sttory Vector For smplcty, we rewrte the bove equto A A A where mtrces wth ll etres re rel umbers. X X for tesor s A s re colum stochstc Theorem. Proposto bout umber of the sttory vectors for cse Now we re cosderg, where ll,, b, b 0,, 0, b b b b The oe of the followg holds () If, b 0, b, the we must hve ftely my solutos, mely, every wth 0, s soluto to the bove equto. () If, b <, the we must hve two solutos or b b b to the bove equto. (3) Otherwse, we must hve uque soluto wth the codto tht b If b b 0, ecludg the codto (), the b b If b b 0, the b b b b gve 3

6 b b b b b b b Proof: The settg s s bove, wrte b b f b b b b b b b We wt to solve Observed tht f we set f to determe the soluto. g f b b b b b 0 g 0 b 0 d g 0, hece by the Itermedte Vlue Theorem, there must est t lest oe 0 0,, such tht g 0 0 Let b b b b b b b If b b 0, the qudrtc equto reduced to b b b 0, the f b b 0,.e., b 0, b, there re ftely my solutos; otherwse f b b 0, the the Itermedte Vlue Theorem b gurtees tht the uque soluto s b b If b b 0, there re two solutos to the qudrtc equto, whch re b b b b d b b b b Whe 0, o mtter b b < 0 or b b > 0, we get oly oe soluto b b b b Itermedte Vlue Theorem. Whe 0 d, t must be the uque soluto we wt by the >, ote tht b b 4b b b b b b b b 4 b b If b b < 0, of course b 0 sce we set > 0, hece b b <0 < b b<, 4

7 the b b < 0, the we coclude there s oly oe stsfed b b soluto whch s If b b 0 d b b b b >, the b b b b b 4 0 b b b b b 4 b b 0 (), b 0, the b b b >0 >, b b b b b b hece we coclude tht s the uque soluto b b we requre. (), b 0 d b, the the stuto s the sme s (), ust plug b 0 (), b 0 d b, the b, b b b b d b b b, hece we coclude the uque soluto we requre s (v), b 0 d b<, the there re two solutos, whch re b d b b 0< < b b b, ust the sme stuto wth () ecept tht (v), b 0 d b, drectly plug the vlue d solve the equto we get d 0 The we wt to eted the codto for ftely my solutos for cse 5

8 Theorem. For A A A, ech elemet the set, 0,,,, f d oly f for would be soluto to the equto A, A 3 3,, A,,,,,, A,, Proof: " ", trvl, by drectly checkg row by row. " ", frst ote tht Theorem., we hve proved tht for cse, f we hve ftely my solutos, the two mtrces must be of the form 0 A, A, hece, for 3 3 cse 0 6

9 3 b b b3 c c c3 3 X b b3 X 3 c c c3 X X, f we set the b b b c c c thrd etry of the sttory vector X to be 0, the we c hve ftely my solutos s X wth 0,, 0, f d oly f the sub-mtrces d b b b b of A d A must be wth the form of 0 d 0, whch uquely determe the etres of d b b b b Smlrly, we set the secod etry of X to be 0, we c uquely determe the sub-. mtrces d c c c 3 c 3 33 of A d A 3, whch must be wth the form of d 3 0 ; lso, we set the frst etry of X to be 0, we c 3 uquely determe the sub-mtrces d c c c 3 c 3 33 of A d A 3, whch must be wth the form of re uquely determed by d 3 0, therefore, the three mtrces A 0 0, A 3, A Smlrly, we use ths method d result for 3 3 cse to determe the four mtrces of 4 4 cse. Iductvely, we use the wth. A,,, A 3 3,,,, 7

10 8,,,,,, A,,,,,,,, A for =,,, (-) Whe we set the -th etry of X 0 for =,, respectvely, the we c determe the (-) sub-mtrces whch re ectly of the form,,,, B 3,, 3,, B

11 9,,,,,,,,,, B,,,,,,,,,, B,,,,,, B By combg ll the sub-mtrces obted wth steps, use the sme s from bove, hece, we get the mtrces A,, A beg the form we requre. Remrk: Observed tht f we pck up the -th row of ech A to form ew mtr M, we c esly see tht

12 3 M 3,,, whch s symmetrc mtr wth ll etres of the dgol equl to. L, Ng d Ye (0) stte tht gve P s rreducble o-egtve tesor of order p d dmeso, f s ot the egevlue of DT(), the Jcob mtr of T, for ll \, the X s uque where p T :, T X PX. I fct, f the -th colum of A s e, whch s -th colum of the detty mtr, d ll the other etres equl to, the there re solutos whch re e, e,, e. Ad there must be o commo zero k blocks wth these mtrces. Flly, we stte out depedet cocluso descrbg the ture of umber of solutos. Theorem.3 Gve y two solutos lyg o the teror of -dmesol fce of the boudry of the smple, the the whole -dmesol fce must be set of collecto of solutos to the bove equto. Proof: Observed tht, for cse, f we re gve two solutos of the form X d X where, 0, there re ftely my solutos X, 0, oly two solutos, oe of them must be Therefore, gve two soluto of the form X th th d X th th wth, the we must coclude tht sce f we hve two d X, whch s cotrdcto. 0,,, 0,,,, the the two sub-mtrces, whch re obted by pckg up the -th d -th rows d -th d -th colums from T T 0

13 the A d A must be of the form d 0 0, whch wll gve out ftely my solutos..e. ll pots le o the whole -dmesol fce wll be soluto. From ths perspectve, we coclude tht we could ot observe three pots whch two of them le o the sme -dmesol fce d sgle pot outsde the fce. We coecture tht gve y k+ solutos lyg the teror of the k-dmesol fce of the smple, d y q of them (q < k) do ot le o the sme (q-)- dmesol fce, the y pot lyg the whole k-dmesol fce, cludg the vertees d boudres, wll be soluto to the equto. We leve t for reders to prove or dsprove the result..3 Applctos DNA Sequece Predcto Hgher-order Mrkov Chs re ofte used to descrbe the flow drecto of sequeces of rdom vrbles. Oe mportt pplcto predctg the DNA sequece rses up recet yers. I the book wrtte by Chg d Ng (006), they lso hghlght ths spect by cosderg the mouse αa-crystll gee (Rftery d Tvre 994). The m de s to rewrte the model to the followg mthemtcl form: t+ s the stte vector t tme (t + ) d t+ depeds o t+ ( =,,..., ), the f Q s rreducble, λ > 0, the the model hs sttory dstrbuto, where s the uque soluto of the ler system (Zhu d Chg 0): Here we gore the detls sce we re cosderg the umber of sttory probblty vectors to our three-order -dmesol tesor. But deed, they hve some uderlyg coectos. For detls, you my refer to (Chg d Ng 006) d (Zhu d Chg 0).

14 3. Cocluso I ths pper, we strt wth the results proposed by L, Ng d Ye (0) d try to fgure out the ssumpto coecture they rsed ther pper. Orglly, they re cosderg the geerl soluto to p-order -dmesol tesor, but due to our uderstdg bout the tesor tself, we re ot cosderg my stutos. But evetully we ed up wth some beutful smll theorems descrbg the ture of ftely my solutos over the whole smple for three-order cse. My other corollres could be deduced from wht we stte Theorem., we leve t for reders to rech out some more fluetl coclusos.

15 Refereces. L. Ng. d Ye. Fdg Sttory Probblty Vector of Trsto Probblty Tesor Arsg from Hgher-order Mrkov Ch. Deprtmet of Computer Scece, Shezhe Grdute School, Hrb Isttute of Techology. Deprtmet of Mthemtcs, The Hog Kog Bptst Uversty, Hog Kog. 8 Februry 0.. Sek, T. A Elemetry Proof of Brouwer's Fed Pot Theorem. Tokyo College of Scece. Receved December 5, Morrso, C. d Styes, M. A Itutve Proof of Brouwer's Fed Pot Theorem R. Mthemtcs Mgze, Vol. 56, No., pp Mthemtcl Assocto of Amerc [Ole]. Avlble: Accessed: 0/03/0 08: Adke, R. d Deshmukh, R. Lmt Dstrbuto of Hgh Order Mrkov Ch. Jourl of the Royl Sttstcl Socety. Seres B (Methodologcl), Vol. 50, No., pp Blckwell Publshg for the Royl Sttstcl Socety [Ole]. Avlble: Accessed: 9/03/0 08: Zhu, D. d Chg, W. A Note o the Sttory Property of Hgh-dmesol Mrkov Ch Models. Itertol Jourl of Pure d Appled Mthemtcs. Volume 66, No. 3, [Ole]. Avlble: Accessed: 0/05/0 3:3. 6. Chg, W. d Ng, K. Mrkov Chs: Models, Algorthms d Applctos. Itertol Seres Opertos Reserch & Mgemet Scece. Volume Seet, E. No-egtve Mtrces d Mrkov Chs. New York: Sprger. [d ed]