Sampling Procedure of the Sum of two Binary Markov Process Realizations

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1 Samplng Procedure of he Sum of wo Bnary Markov Process Realzaons YURY GORITSKIY Dep. of Mahemacal Modelng of Moscow Power Insue (Techncal Unversy), Moscow, RUSSIA, E-mal: VLADIMIR KAZAKOV Dep. of Telecommuncaons, ESIME-Zacaenco, SEPI, Naonal Polyechnc Insue of Mexco, MÉXICO E-mal: Absrac: The sum of wo bnary Markov processes s consdered. Resul of he sum can have four or hree saes. In he frs case here s a Markov process, n he second case he process s non Markov. We nvesgae samplng procedure of any realzaon of he sum process. The nvesgaon mehod akes no accoun he probably of he sae omsson and probably denses funcons of sayng mes n each sae. In resul, we oban he algorhm for choosng of samplng nervals for each sae. They are dfferen. One non rval example s gven. Key-Words: - Samplng Algorhm, Bnary Markov Process, Error reconsrucon.. Inroducon The problem of samplng - reconsrucon procedure (SRP) of realzaons of random processes s exensvely examned n leraure. Majory of publcaons are devoed o he SRP nvesgaon of realzaons of connuous processes. SRP of realzaons of dsconnuous processes s weakly consdered. Here we menon some papers conneced wh he dscussed problem [, ]. In [] samplng of realzaons of he Bnary Markov Process (BMP) s nvesgaed n order o descrbe new Bnary processes. In [] SRP of realzaons of BMP s analyzed. Bu hs analyss s based on he covarance approach. There are some auhors publcaons [ -5] devoed o SRP of dsconnuous process realzaons. Invesgaons n [4, 5] are based on he Condonal mean Rule. In resuls we obaned he sascal descrpon of samplng procedure and o esmae he qualy of he reconsrucon. I s necessary o menon ha here s a grea dfference beween he SRP mehod for connuous [6 ] and dsconnuous [4, 5, ] process realzaon. In he presen paper we consder SRP of realzaons of dsconnuous process whch s he sum ς ( ) of wo ndependen BMP processes. Such a problem s neresng n some conrol and elemery sysems. There are wo dfferen cases: ) he process ς ( ) has four dfferen saes; ) he process ς ( ) has hree dfferen saes. The frs varan can be consdered by he Markov mehod suggesed n [4, 5]. The nvesgaon of he second case s more dffcul n comparson of he frs one because he process under consderaon becomes non Markov. Here a new aspec of SRP problem s occurred. In he frs me, we need o nvesgae he deermnaon of he nsan momen of he nex sample, when he locaons of prevous samples are known. Ths problem mus be solved akng no accoun he gven probably of a sae omsson. Ths SRP problem s nvesgaed n deal. Fnally, here s one non rval example. ISBN:

2 . The General Varan ξ ( ), ξ ( ) are ndependen BMP wh Le correspondng saes: {x, x } и {y, y }, x < x, y < y, and wh ranson denses (or wh he nenses) {λ, λ } and {µ, µ }. These denses have a known sense, for nsance, λ : P{x x, } = λ + o( ) ec. Le us nroduce a wo dmensonal Markov process { ( ), ( ) } ξ ξ wh four saes. The process graph s presened n Fg.. Here one can see saes wh all possble ransons and correspondng denses. We consder a sum ς ( ) = ξ ( ) + ξ ( ). Fg.. The graph of he wo dmensonal process { ξ ( ), ξ ( ) } The process ς ( ) s observable. We have o oban a samplng algorhm for any realzaon of hs process wh he prncpal condon: he probably of a sae omsson mus be no more of a gven value γ. Le us consder wo dfferen cases. The frs case When (x, y ) λ λ (x, y ) µ µ µ µ λ (x, y ) (x, y ) λ x x - x y y - y ; () he process ζ() has four saes z, z, z, z : z = x + y, z = x + y, z = x + y, z = x + y Snce z z, akng no accoun () we have z - z = ς s (x - x ) (y - y ). Then he process Markov. The graph of hs process s he same as presens n Fg.. In hs case he probably densy funcon (pdf) of sayng me ν n he saes (,,,) E α wh parameers: η E(α = λ + µ ), η E(α = λ + µ ), z = s exponenal η E(α = λ + µ ), η E(α = λ + µ ) () The ranson probables are deermned by he followng formulas: µ λ P =, P =, µ + λ µ + λ P = λ µ, P = λ + µ λ + µ () In () we gve some requred expressons only. In order o descrbe SRP of such ype of he process, s necessary o use he mehod [4, 5]. The second case When x x - x = y y - y ; (4) he process ζ() has hree saes z, u, z : z = x + y, u = x + y,= x + y, z = x + y The sae u s formed by he unon of wo saes z and z of he process n he frs case. The graph of he dscussed process s shown n Fg.. z u z Fg.. The graph of he process ζ() n he second case The pdf of me sayng η and η n he saes z and z are he same as n (),.e. hey are exponenal ISBN:

3 wh he parameers α and α correspondngly. The suaon wh pdf of sayng me η u n he sae u s anoher. The pdf of he me ηu depends on he prevous sae z (=; ). If he realzaon comes no he sae u from z, hen he pdf wll be E( α ), (=; ). Then pdf of ηu s deermned by he formula of he oal probably. I means ha pdf s no exponenal. In hs case s necessary o apply analyss whch s vald for non Markov processes, or for an arbrary pdf.. The Esmaon of he Samplng Inerval Le us nroduce some desgnaons:, j, k he pas, he presence, and he fuure saes. ζ j a random nsan me of an oupu from j. Ths momen has s begnnng from me of he las sample of he sae. θ - an nerval from L unl a curren presen me. η k a sayng me n he nex sae k. ζ θj a remander of wang me for oupu from j f ζ j > θ. The samplng nerval T j can be chosen akng no accoun he requred demand for he probably of he omsson of he nex sae k: max P{ζ θj + η k < T j } = F Σ (T j ) =γ, (5) k where F Σ (;j,k) s he probably dsrbuon funcon of he sum Σ ζ θj + η k. a) If a curren sae j s equal o z or z (hey have he exponenal fdp), hen he remander of wang saes ζ θj, j =, have he same pdf and hey do no depend on θ. There are f j () = E(, α j ), j = ;. The sayng me ηk = ηu depends on j. Is pdf s deermned by he combnaon of wo exponenal pdf, correspondng o saes z и z. Pdf for η u s descrbed by he expresson: f u ( j) = p jse( ; αs ), j =,. (6) The probably dsrbuon funcon of he sum Σ can be smply found by he convoluon of wo exponenal dsrbuons. Le us desgnae he dsrbuon funcon of he sum of wo ndependen random varables wh he exponenal pdf by F(; α, β) (here α and β are parameers of pdf). Then he dsrbuon funcon of he sum n (5) can wre n he form: F Σ (; j, u) = Σ, P js F(; α j, α s ), j =, b) Le us assume he curren sae j s he sae u. We deermne pdf of he ranson momen τ zu from z no u. Takng no accoun resuls [4, 5] and he expresson (6) one can oban he formula for pdf w u () as a lneal combnaon of wo runcaed (by he value Т ) exponenal pdf E T (, µ): w u () = pset ( ; µ s ), =,, (7) µ s = α - α s, (8) µ µ T µ e ( e ), µ, where E T (, µ) = (9) T, µ = On he base of w u () one can deermned he condonal mean and varance. They are he prncpal characerscs of SRP. We deermne he samplng nerval followng n. (a), f a curren sae j s equal o z or z. In he case he curren sae j s he sae u he dsrbuon of he remander ζ θj = ζ θu of he oupu wang me depends on he pas (z или z ) and on he pas me θ. The pdf h u ( ) of he oupu momen ζ u (ζ u = τ u + η u ) from he sae u can be calculaed by he negral akng no accoun (b). We have: mn(, T ) h u ( ) = fu ( x ) wu ( x) dx = α q sαse s, < <. () ISBN:

4 We do no wre he expressons for he coeffcens q s because hey have a cumbersome form. The pdf of he random varable ζ θu (f ζ u > θ) can be wre usng () by he formula h θu ( ) =c h u (θ + ), < < () where c s a normalzed consan. We noe ha he dsrbuon funcon F Σ (T;u,k,θ ) depends on he pas sae, me θ and he nex sae k. Ths funcon s deermned by he negral: F Σ (; u,k,θ ) = P{ζ θu + η k < } = = P{ ηk < x} hθu ( x ) dx () We do no gve he resul of he negraon because s raher complex. The fnal expresson for he samplng nerval T u T u (θ,) can be obaned by a numercal soluon of he equaon (5). Ths nerval depends on he level γ, he prevous sae and me θ. The pdf of he jump momen τ uk from u no k s deermned by he formula: α θ µ w uk () = = с qsαse s e sk, () s = < < T u, T u T u (θ, ). µ sk = α s - α k,,k =,, On he base of () one can oban he condonal mean of he jump momen (hs s he esmaon of he jump momen) and he condonal varance (hs s he qualy of he esmaon). 4. Example In order o llusrae he nfluence of sayng nervals on he SRP, would be beer o choose dfferen values of average nervals of nal flows (or denses). Namely, we pu: T = 5.6, T = 6., T =, T =8. ffu(, ) u z fu(, ) Fg.. Graphs of he condonal pdf f ( ν u z) for wo values z (=; ). Takng no accoun he dependence of sayng nervals ν u from dfferen saes z (=; ), we mark he average nervals n he sae u n such manner: T z. Then he average nervals of Tu( z ) and u( ) he process ς ( ) are characerzed by he followng values: T = 5.6, T ( z ) = 8.6, u u T z = 7.4, T =8. In Fg., graphs of he condonal pdf f ( u z) ν for wo values z (=; ). The connuous lne s corresponded o he sae z and he doed lne o he sae z. Le us choose he requred probably of a sae omsson as γ =.. The nvesgaed realzaon ς ( ) s he consequence of saes z =, u=.5, z =, u=.5, z =. We pu he nal value ς = =. The concree correspondng sayng mes are equal: 5.; 5;.; 5.. Calculaons of he samplng nervals and are characerzed by he followng resuls: ) T( z ) =.; T z =.. ) T( u z ) =.,.5,.8,.,.,.5,.6; T( u z ) =.4,.5,.6. ISBN:

5 4 ξ ( ) ζ() ξ n 4 5, n Fg. 4. One realzaon of he sum process and he se of samples In Fg 4, one realzaon of he process ς( ) s presened. Besdes hs, here are some samples desgnaed by crosses. As one can see, he samplng nervals T z and T z are consan and hey don changed n me because boh pdf of he sayng mes are exponenal. Bu he nervals are dfferen, because hey depend on pdf of sayng mes n consdered and fuure saes. The nervals n he sae u are dfferen as well and hey are changed n me. However hs dependence s raher small. Ths effec s deermned by prevous sae z or z and by he condonal pdf of he sayng mes. 5. Conclusons The deermnaon of he requred samplng nerval s a very mporan procedure n he SRP sascal descrpon of he random processes wh jumps. References: [] B.J. Redman, D.G. Lampard. Sochasc Samplng of a Bnary Random Process. IEEE Transacons on Crcu Theory. March 96, pp [] B.Lacaze. Perodc b-samplng of saonary processes. Sgnal Processng, V. 68 (998), pp.8-9. [] V. Kazakov. Paen No USSR, (98);. Paen No 5, USSR, (98). [4] V.Kazakov, Y. Gorsky, Samplng- Reconsrucon of Bnary Markov Process.- Radelecroncs and Communcaon Sysems, Vol. 49, No, 6, pp. 6-. [5] Y. A. Gorsky, V.A.Kazakov. Samplng and Reconsrucon of Markov Processes wh lmed se of saes. Journal of Compuer and Sysem Scences Inernaonal., No, pp. 6-. [6] V. Kazakov. The samplng-reconsrucon Procedure wh a Lmed Number of Samples of Sochasc Processes and Felds on he Bass of he Condonal Mean Rule, Elecromagnec Waves and Elecronc Sysems. Vol., # -, 5, pp [7] V. Kazakov, D. Rodrguez. Samplng Reconsrucon Procedure of Gaussan Processes wh Jer Characerzed by Bea Dsrbuon. IEEE Transacons on Insrumenaon and Measuremen. Vol. 56, 7, #5, Ocober, pp [8] V.Kazakov. The reconsrucon of Gaussan processes realzaons wh an arbrary se of samples. Recen Researches n Telecommuncaons, Informacs, Elecroncs & Sgnal Processng, -h WSEAS Inernaonal Conference on Sgnal Processng, (SIP ), Lanzaroe, Canary Islands, Span, May 7-9,, pp.-5 [9] V.Kazakov, Y.Olvera. Samplng-reconsrucon procedures of non Gaussan processes by wo algorhms. Inernaonal Journal of Crcus, Sysems and Sgnal Processng, (Norh Alanc Unversy Unon - NAUN), Issue 5, Vol. 5,, pp [] V. Kazakov. General problems of he Samplng- Reconsrucon Procedures of Random Process Realzaons. Recen Researches n Appled Informacs & Remoe Sensng. Proceedngs of he h WSEAS Inernaonal Conference on Appled Compuer Scence (ASC-). Penang, Malaysa, Ocober -5,, pp. 98. ISBN: