Partition Optimization for a Random Process Realization to Estimate its Expected Value

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1 SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol 4, No, Octoer 7, -4 UDC: 698:49]:59 DOI: htts://doorg/98/sjee7m Partto Otzato for a Rado Process Realzato to Estate ts Exected Value Vladr Marchu, Igor Shrafel, Dtry Cheryshov, Alexader Maev, Stea Buryaov Astract: The aer rovdes a aalytcal roof the otal uer of arttos of a o-statoary rado rocess realzato, whch s ecessary for estatg ts exected value whe usg the estato reroducto ethod for sgal rocessg Ths ethod allows to rocess sgal wth a lted volue of ror forato aout the desred sgal fucto ad statstcal characterstcs of the addtve ose cooet Keywords: Desred sgal, Nose, Sgal-to-ose rato, Exected value, Estato, Rado rocess Itroducto Processg of o-statoary rado rocess requres effcecy reservato codtos of lted aout of the ror forato, oth o the fucto of the useful sgal ad o the statstcal characterstcs of the addtve ose cooet The estato reroducto (ER) ethod, was descred [ 4], t shows good results such codtos of a ror deteracy I wors [5 ] ths ethod was fully descred, ut we dd ot cosder the questo of otal artto uer of sgal realzato I ths aer the a atteto wll e ayed to otzato of arttos uer for a o-statoary rado rocess ters of estatg exected value The Matheatcal Model of Method Let x() t a() t () t e the su of the deterstc sgal at () ad ts ose cooet () t at stat t Suose () t s cetered rado rocess wth arwse ucorrelated cross sectos ad a costat varace D () t D Cosder orgal sgal x() t equally dstat oets of te t t ( ) h ad ut x xt ( ), a at ( ),,, Susequetly M a x, D D x Isttute Shahty of DSTU, Shevcheo str 47, Shahty, Rostov rego, 465, Russa; E-als: archu@sssuru; shrafel7@alru; dcher@oxru; ayaev@galco; uryaov@galco

2 V Marchu, I Shrafel, D Cheryshov, A Maev, S Buryaov ad rado values x,, x are ucorrelated arwse Hereafter, the sae otato wll e used for a rado rocess ad ts leetato Let assue that soe realzato X ( x,, x ) was receved We choose rado teger uers ad ( ) P of,, that elog to eleetary suset suset Let,,, e eleets of P (at P ), that are arraged ascedg order, ; Set P geerates a artto of the set,, to eces T Hereafter, we shall call the set P a artto,,, Let us fd OLS estatos wth olyoals of degree zero a for fragets ( x ) T of realzato X at every ad wth T : y x,,, () Ths ethod of estatg the exected value of a rado rocess s used the estato reroducto ethod [ 4] As t s ow the crtero of the qualty of estato ( ) Y y s ea varace D ( ) M y a Soetes, to ehasze the deedece of the ea -varace o the artto P, we shall deote t y D ( P ) Let us call artto P ts corresodg ea varace ( ) D D P otal for ay artto P D D ( P ) I ths cotext, we derve the forula for the ea varace: Dy M y My My a My a Dy My a D M y a M y y a a Note that due to the arwse ucorrelated rado varales D Dx,, () x, Moreover, at the sae,, My a, where a s deoted as a Therefore 4

3 Partto Otzato for a Rado Process Realzato to Estate ts Exected Value the D D a a D a a 5, () (4) D D a a I ths aer we fd otal uer of arttos ad ea varace for the lear atheatcal exectato a t We ota where a t h a,,,,, a t h h;,, that eas that a, are costat The a a a a (5) a,,, We susttute ths exresso the last su of exresso (4): a a a 4 a a 4 (6) a a a a Now we ca fd: a a a a aa 6 (7)

4 V Marchu, I Shrafel, D Cheryshov, A Maev, S Buryaov Susttute ths exresso (4): a D D a a a 6 (8) D, where Fro the otaed exresso for the ea varace, artcular, t follows that ay erutato of eleets of sequece, that corresods to the otal artto, t also geerates a otal artto P Let us cosder the case searately I ths case the sallest value D D D s reached whe, ad equals D Now let us set Let us show that f P s a otal artto, the for all are, Assue that ths s ot so That eas that for soe ar of uers, For defteess, we assue that (the case s used a slar way) Set N at, ad N other cases ; Q N, M N N,,, The Q s artto M for all,,, M, M The: D Q D P M ( ) ( ) M M ( ) ( ) ( ) ( ) ( )( ( )) ( )( ) Ths cotradcts the otalty of the artto P Hece, for a otal artto P P a sequece of uers,, there are, at ost, two dfferet values If we deote saller uer as, tha gger oe wll e equal to If uer aears sequece tes, the ( )( ), t leads to ( ) ; ( ) Sce 6 (9)

5 Partto Otzato for a Rado Process Realzato to Estate ts Exected Value, the, (here ad after, the syolst ad t deote, resectvely, the teger ad fractoal arts of the uer t ) Wth A D, we get: D ( ) ( ) 4 A ( ( ) ) ( ) 4 A ( ) ( )(( ) ) () 4 A ( ) ( ) ( ) 4A ( ) ( 4 A) We ehasze that ths exresso s equal D oly at values, that ze t Let s fd all We wll call t otal values ad deote as Let f ( ) e the fucto wth cotuous arguet,, that s set y exresso (), where We show ext that fucto f ( ) s ecewse lear ad cotuous o the dcated terval We have,, At all, the codtos of equalty are fulflled, whece ; f ( ) ( ) ( 4 A), e fucto f ( ) s lear o, To rove ts cotuty o, t s suffcet to estalsh cotuty fro the rght f ( ) at the ots,,, For the dcated, we have: 7

6 V Marchu, I Shrafel, D Cheryshov, A Maev, S Buryaov whch cocdes wth l f ( ) ( ) ( 4 A) 4A 4A, f A ( )( ) (( ) ( ) 4 ) 4A Thus, the fucto f ( ) s cotuous o, Let us cosder the fucto g creases o, Moreover: ( ) 4 () () A It s cotuous ad g() 6 4 A, () Wth g( ) ( ) ( ) 4A 4 A A g ( ) g(),,, f ( ) decreases o, f( ) f( ), e (4) A D f( ) ( (4 A)) D, ( ) Now let us cosder a case where A We have 4 g ( ) g ( ),,, ad fucto f ( ) creases o,, f( ) f(), 8

7 Partto Otzato for a Rado Process Realzato to Estate ts Exected Value A D D f() ( ( ) 4 A) ( ) ( ) Fally wth A fucto g( ) o, has oe ad 4 oly oe root, that s foud y eas of Kardao forula: Put Moreover, f (e A A A A (5) At g( ), ad at g( ) s teger) g ( ), ad case g ( ) Fro the research results of the fucto f ( ), the cocluso follows that t decreases o, ad creases o, (whe s teger t creases o,, ut reas costat o, ) Cosder the case whe s o-teger The ad fucto f ( ) reaches ts lowest value o, a sgle ot, Now two cases are ossle If s teger ad, D f 4A D (6) If s o-teger, the for fdg t s ecessary to coare f ad f + If f < f + the, f f > + ; ad f f = f +, the oth values ad are otal f + the + 9

8 V Marchu, I Shrafel, D Cheryshov, A Maev, S Buryaov tegers If s teger the at, uers, otal values wll e all If the we choose as oe of the for whch the value f s less ( ) f f oth values ad are otal Let us suarze the results I case Theore Let x,, x e arwse ucorrelated rado values wth exected value M a X ad varace D D x, where,,, D, ad a, D are costat We troduce two ore costats: A (f we cosder A ) ad A A A A (f A ) Also we troduce fucto: f 4 A (7) Let us defe the set S as set of tegers as follows: f A ut S ; f ( ) A 4, the S ; I case ( ) A t creates rachg; 4 f -s teger, we deote q, q The: f q q the S q, q,, q ; f q q ut, q 4

9 Partto Otzato for a Rado Process Realzato to Estate ts Exected Value f f f the S ; f f f the S ; f f f, we assue S, Let The: f f e o-teger Put 4 ;, s teger, the S ; s o-teger, we deote ; f f < f + we assue S ; f f > f +, the S ; fally f f f + the S, For artto P to e otal t s ecessary ad suffcet so that for S sequece ( ) a uer was reset tes Ad f s o-teger the uer has to e reset tes Otal ea dserso for all etoed cases ca e foud wth forula D f I suaragrahs, ad D s calculated usg slfed equatos D D D D, D ( ) и D resectvely Cocluso I ths aer, the otal value of the artto ad the value of the ea varace were foud for the lear exected value Mea varace characterzes the qualty of the otaed estato 4 Acowledget The reorted study was suorted y the Russa Foudato for Basc research (RFBR), research roect

10 V Marchu, I Shrafel, D Cheryshov, A Maev, S Buryaov 5 Refereces [] V Marchu: Ital Measureet Results Data Processg wth Lted Aout of A Pror Iforato, Tagarog State Uversty of Radoegeerg, Tagarog, Russa, (I Russa) [] V Marchu, K Ruatsev: A New Way to Icrease the Measureet Results Relalty Sace Rocet Research, Aerosace Istruetato, No, 4, 5 55 (I Russa) [] V Marchu, K Ruatsev, A Sherstotov: Low-frequecy Processes Flterg wth a Lted Aout of Measureet Results, Radotecha, No 9, 6, 7 (I Russa) [4] V Marchu, I Shrafel: Methods for Detecto the Useful Cooet wth a Pror Ucertaty ad a Lted Aout of Measureet Results, South-Russa State Uversty of Ecoocs ad Servce, Shahty, Russa, 8 (I Russa) [5] V Marchu: Error Estato the Aroxato of the Useful Cooet whe the Realzato of the Measureet Results s Dvded to Itervals, Telecoucatos, No 8,, 6 (I Russa) [6] V Marchu, V Voro, A Sherstotov: Error Estato the Detecto of a Useful Sgal Durg Processg Codtos of a Lted Volue of a Pror Iforato, Radotecha, No 9,, 75 8 (I Russa) [7] V Marchu, A Sherstotov, S Grd, I Toorov: Techques for Irovg the Relalty of Searato of useful Sgal uder a Pror Ucertaty, The Successes of Moder Rado Electrocs, No 5,, 6 (I Russa) [8] V Marchu: Useful Cooet Aroxato Error Estato whe the Realzato of the Measureet Results s Dvded to Itervals, Moder Iforato Techologes, No 9, 4, 5 59 (I Russa) [9] V Marchu, S Maov, D Tofeev, M Psesova, A Fsuov: A Method of Sgal Estato Error Reducto a Pror Ideteracy, Telecoucatos Foru TELFOR, Belgrade, Sera, 4-6 Nov [] V Marchu, S Maov, A Maev, S Stradacheo: Studyg Accuracy of the New Desred Sgal Extracto Method y Resduals a Prory Ideteracy Codtos, 4th IEEE East-West Desg ad Test Syosu, Yereva, Area, 4-7 Oct 6 [] V Marchu, S Maov, A Maev, V Voro, D Cheryshov: Reovg of Systeatc Measureet Errors Caused y Asyetrc Dstruto Law of the Nose Cooet, IEEE East-West Desg ad Test Syosu EWDTS 6, Yereva, Area, 4-7 Octoer, 6 4