Time Seies, Sochasic Pocesses and Comleeness of Quanum Theoy Maian Kuczynsi Deamen of Mahemaics and Saisics, Univesiy of Oawa, 585,av. King- Edwad,Oawa.On.KN 6N5 and Déaemen de l Infomaique, UQO, Case osale 50,succusale Hull, Gaineau. Quebec, Canada J8X 3X 7 Absac. Mos of hysical exeimens ae usually descibed as eeaed measuemens of some andom vaiables. Exeimenal daa egiseed by on-line comues fom ime seies of oucomes. The fequencies of diffeen oucomes ae comaed wih he obabiliies ovided by he algoihms of quanum heoy (QT). In sie of saisical edicions of QT a claim was made ha i ovided he mos comlee desciion of he daa and of he undelying hysical henomena. This claim could be easily ejeced if some fine sucues, aveaged ou in he sandad desciive saisical analysis, wee found in ime seies of exeimenal daa. To seach fo hese sucues one has o use moe suble saisical ools which wee develoed o sudy ime seies oduced by vaious sochasic ocesses. In his al we eview some of hese ools. As an examle we show how he sandad desciive saisical analysis of he daa is unable o eveal a fine sucue in a simulaed samle of AR () sochasic ocess. We emhasize once again ha he violaion of Bell inequaliies gives no infomaion on he comleeness o he non localiy of QT. The aoiae way o es he comleeness of quanum heoy is o seach fo fine sucues in ime seies of he exeimenal daa by means of he uiy ess o by sudying he auocoelaion and aial auocoelaion funcions. Keywods: foundaions of quanum mechanics, saisical and conexual ineeaion, comleeness of quanum heoy, non sandad daa analysis, and visualizaion., ime seies analysis, sochasic ocesses, quanum flucuaions, quanum infomaion, Bell inequaliies. PACS: 03.65. Ta, 03.67.Ud, 03.67.-a, 05.40-a, 07.05 Kf, 0.7.05 Rm, 4.50Lc, 4.50Xa INTRODUCTION The algoihms ovided by quanum heoy (QT) allow finding he obabiliy disibuions of exeimenal daa. In some exeimens beams of idenical hysical sysems ae eaed and hei ineacion wih a measuing aaaus oduces vaious ime seies of oucomes. In ohe exeimens a single hysical sysem laced in a a is obed by a lase beam and he measuemens ae made. The sae of he sysem in he a is ese o he iniial sae and he ocedue is eeaed. Any single exeimenal oucome is no edicable only he saisical egulaiies ae obseved and comaed wih he edicions of QT. I is well nown since he exlanaion of Beand s aadox [-3] ha a obabiliy disibuion is neihe a oey of a coin no a oey of a fliing device. I is only he chaaceisic of he whole andom exeimen: fliing his aicula coin wih ha aicula fliing device. Theefoe he claim ha QT ovides he comlee
desciion of individual hysical sysems can no be coec and by no means he quanum sae veco can be eaed as an aibue of an individual hysical sysem. Fo Einsein and Blohinsev QT was only a saisical heoy. Boh wisely insised on he wholeness of hysical exeimens undelying he eisemological, algoihmic and conexual chaace of he heoy. A conemoay saisical ineeaion of QT is conexual as i should be. The agumens in favo of his ineeaion and many ohe efeences may be found in Balleine [4, 5], Accadi [6-8], Khenniov [9-] and in [3, -6]. Saisical desciion ovided by QT leaves oen a quesion whehe i is ossible o find a deeminisic sub-quanum desciion of he henomena in which he unceainy of he individual oucomes would esul only fom he lac of conol of some hidden aamees descibing he hysical sysems and he measuing devices. Even if esonally I am no an advocae of hidden vaiable models because of hei ad hoc chaace and esiced alicabiliy I was imessed by he even based couscula model [7] esened duing his confeence by Hans de Raed. This model gives a unified desciion of all ineesing oical exeimens. I is a nice examle of a conexual model which even afe even builds he saisical disibuion of couns consisen wih he edicions of QT wihou using he Maxwell heoy o he quanum mechanics. I gives an addiional jusificaion fo he seach of fine sucues in he hysical daa as well as fo he seach of new alenaive moe deailed and non sandad models of hysical henomena. Many yeas ago we obseved ha if he hidden vaiables exised hen all ue quanum ensembles would be mixed saisical ensembles wih esec o hese vaiables [8]. Since mixed saisical ensembles diffe fom ue saisical ensembles he diffeence could be discoveed wih hel of he uiy ess [9, 0]. In coule of ecen aes we wen a se fuhe and we noiced ha even he edicable comleeness of QT has no been esed caefully enough [3,4,]. Namely we oined ou ha any hyoheical fine sucue in he ime seies of he exeimenal daa, if i exised, i would be aveaged ou when emiical hisogams wee consuced fom he daa and comaed wih he obabiliy disibuions ovided by QT. If some eoducible unexeced fine sucues wee discoveed in he daa i would be a significan discovey and a decisive oof of incomleeness of QT. In his ae which is a coninuaion of he ae [] we eview some saisical ools which ae used o deec auoegessive sucues in a ime seies of daa []. TIME SERIES THEORY A ime seies i is a family of andom vaiables { } whee =0,.. fo simliciy. Time seies is saionay if E( )=µ, va( )=σ and auo covaiance funcion γ() a lag does no deend on whee γ() = cov(, + ) = E( -µ, + -µ). A whie noise i is a ime seies {a } whee a ae nomal indeenden and idenically disibued (i.i.d) andom vaiables wih zeo mean and vaiance σ. The auocoelaion funcion () a lag is defined as ()= γ() / γ(0) and i is easy o see ha (0)=, () =(-) and () =0,±,..
I is useful o inoduce he following oeaos: B = - and = I-B. The imoan models of ime seies, which wee sudied exensively [], ae so called auoegessive inegaed moving aveage models ARIMA (,d,q): Ф(B)(I-B) d = Θ 0 + Θ(B)a whee Ф(B)=I-Ф B- Ф B - -Ф B and Θ(B)=I- Θ B- Θ B - - ΘqB q. In his ae we concenae on simle auoegessive models AR()= ARIMA(,0,0) wih µ=0 given by he equaion: Φ Φ Φ = a () A geneal soluion of his diffeence equaion is a sum of homogeneous and aicula soluions: = h () + (). Guessing h () =G we find ha allowed values G i of G can be found as he oos of he following olynomial equaion : Ф(G - )=0 If G i < hen AR () is saionay and if he olynomial equaion has diffeen comlex oos G i we find: h ( ) = C G + CG + + C G () Thus h () decays o zeo as a sum of exonenials and/o damed sine funcions and () if gows. Simila behavio has he auocoelaion funcion () which is usually denoed in saisical acages as ACF. Since E( +,a )=0 we ge fo >0 he following homogeneous diffeence equaion simila o Eq. : ) Φ ( ) Φ ( ) Φ ( ) 0 (3) ( = Thus he geneal soluion of his equaion is given by: ) = C G + C G + + C G (4) ( and ()=ACF is a quicly decaying funcion when inceases. Fom Eq, 3 one obains so called Yule-Wale maix equaions: 3 Φ Φ = Φ which allow o find he coefficiens Фi of AR() if ACF is nown. Anohe imoan funcion is so called aial auocoelaion funcion PAC which measues he imoance of he -h lag in he model AR (). I is found by solving: 3 whee φ =PAC. Fom Eq. 5 we see ha fo = φ i =Ф i and ha PAC=0 fo >. These wo auocoelaion funcions lay an imoan ole heling o find ou whehe a ime seies of some daa is a samle of some ARIMA ocess and in aicula of some saionay AR() ocess. φ φ = φ (5) (6)
EMPIRICAL AUTOCORRELATION FUNCTIONS. Le us conside a samle S= {z,, z n } of some ime-seies. The auocoelaion funcion = () can only be esimaed fom he daa by : = n = whee z-ba is a sandad samle mean.. If he unnown ime seies is a saionay AR() hen emiical ACF= should decease quicly when gows. To find he value of we have o sudy a family of AR() whee ϕ ϕ ϕ = ( z z)( z + z) (7) a (8) whee φ i ae unnown and may be only esimaed by using he Yule- Wale Eq. 6 in which i ae elaced by i. n = ( z z) 3 ˆ φ ˆ φ = ˆ φ (9) The emiical PAC= ) φ and is no exacly zeo fo > bu i should have a clea «cu off» a =. I means ha i should be equal o zeo wihin a sandad eo of he ode of n -0.5 whee n is a samle size. If we ge a samle of some ime seies we canno assume ha i is a samle of some saionay AR (). In ode o discove which ARIMA model if any can fi he daa we have o exloe ou samle wih hel of saisical sofwae we have a ou disosal. Fo a sudy of ime seies he ecommended acages ae S+ o R bu even a oula Miniab can do. In aicula we have o find: Hisogam. Nomal scoes lo. Simle ime seies lo (z, ). Lagged scae los ((z, z + ). Emiical ACF and PAC los. Residuals afe fiing los. If a ime seies is no a simle AR () i is no an easy as o deemine exacly is sucue bu in many cases i can be done wih high ecision []. In he nex secion we will aly he emiical ACF and PAC funcion in ode o deec a fine auocoelaion sucue in some simulaed daa.
A STUDY OF AN EMPIRICAL TIME SERIES In ode o ove ou oin ha a sandad daa analysis maes imossible he discovey of a fine sucue in ime seies of he daa we simulaed a samle of size 500 of AR (): 0.5 0. 5 = a (0) whee a wee nomal i.i.d. wih a uni vaiance. A sandad desciive analysis: summay, hisogam, and nomal scoes showed ha he daa can be viewed as a samle dawn fom some nomally disibued oulaion. FIGURE. A Hisogam FIGURE. Nomal Scoes Plo
Only he deailed analysis allowed discoveing he fine sucue in he sudied samle wha can be seen fom he figue below: FIGURE 3. Emiical PAC funcion The emiical ACF was decaying and he emiical PAC funcion had a clea <<cu off>> a lag hus we could conclude ha he samle was dawn fom a saionay AR(). The saisical acage S + using he Eq. 6 esimaed he coefficiens in he Eq. 0 o be: 0.43 and 0.487 vey close o he ue values 0.5 and 0.5 esecively. A moe deailed sudy of diffeen simulaed ime seies will be ublished elsewhee. CONCLUSIONS Sandad analysis of he daa is unable o discove ossible fine sochasic sucues in he daa. Suble saisical analysis of he ime seies of he daa using he ools descibed above and in [, ] is an aoiae mehod o es he comleeness and he limiaions of QT. The saemens ha he violaion of Bell inequaliies oves he non localiy of QT and/o is comleeness ae simly no ue [,3,7,9,3,4,6,3-5] and i is eally sange how ofen hey have been eeaed in he as and even duing his confeence.
ACKNOWLEDGMENTS I would lie o han Andei Khenniov fo he wam hosialiy exended o me duing his ineesing confeence. I am indebed also o Mahmoud aeou fom Univesiy of Oawa who iniiaed me o S + and heled me wih he daa analysis. I would lie o han also he Univesiy of Oawa fo a financial suo. REFERENCES. B.V. Gnedeno, The Theoy of Pobabiliy, New Yo: Chelsea, 96,.40.. M. Kuczynsi, Beand's aadox and Bell's inequaliies, Phys.Le.A, 05-07(987) 3. M. Kuczynsi, EPR aadox,localiy and comleeness of quanum heoy, in Quanum Theoy Reconsideaion of Foundaions-4, edied by G.Adenie e al, AIP Conf.Poc. 96, NY:Melville 007,.74-85 (axiv:070.350) 4.L.E.Ballenine, The saisical ineeaion of quanum mechanics, Rev.Mod.Phys. 4, 358(970) 5. L..E.Ballenine, Quanum Mechanics: A Moden Develomen, Singaoe:Wold Scienific, 998. 6. L.Accadi, Toics in quanum obabiliy, Phys.Re.77, 69-9 (98). 7. L.Accadi and M. Regoli, Localiy and Bell's inequaliy,in: QP-XIII, Foundaions of Pobabiliy and Physics, ed. A.Khenniov, Singaoe: Wold Scienific,00,. 8 8. L.Accadi and S.Uchiyama, Univesaliy of he EPR-chameleon model, in Quanum Theoy Reconsideaion of Foundaions-4, edied by G.Adenie e al, AIP Conf.Poc. 96, NY:Melville, 007,.5-7. 9. A.Yu. Khenniov, Bell s inequaliy: nonlocaliy, deah of ealiy, o incomaibiliy of andom vaiables?, in Quanum Theoy Reconsideaion of Foundaions-4, edied by G.Adenie e al, AIP Conf.Poc. 96, NY:Melville,007,.-7 0. A.Yu..Khenniov, Conexual Aoach o Quanum Fomalism, Fundamenal Theoies of Physics 60, Doech:Singe, 009.. A.Yu.Khenniov, Ubiquious Quanum Sucue, Belin: Singe, 00. M. Kuczynsi, Is Hilbe sace language oo ich, In.J.Theo.Phys.79, 39(973), eined in: Physical Theoy as Logico-Oeaional Sucue, ed. C.A.Hooe, Dodech: Reidel, 978,.89 3. M.Kuczynsi, On he comleeness of quanum mechanics, axiv:quan-h 0806, 00 4.. M. Kuczynsi, Seveny yeas of he EPR aadox in Albe Einsein Cenuy Inenaional Confeence edied by J-M Alimi and A Funzfa,AIP Conf.Poc. 86, NY:Melville,006,.56-53 (axiv:070.350) 5 A.S.Holevo, Saisical Sucue of Quanum Theoy, Belin:Singe, 00. 6. M.Kuczynsi, Enanglemen and Bell inequaliies, J.Russ.Lase Reseach 6, 54-3(005) (quan-hys 040799) 7 K.Michielsen, F.Jin, H.de Raed, Even-based couscula model fo quanum oics exeimens, axiv:006.78[quan-hys], 00 8. M..Kuczynsi, New ess of comleeness of quanum mechanics, Phys.Le.A,.5-53, 987. 9. M.Kuczynsi, On some imoan saisical ess, Riv.Nuovo Cimeno 7, 5-7(977) 0..J. Gajewsi and M. Kuczynsi, Puiy ess fo π - d chage muliliciy disibuions, Le.Nuovo Cimeno 6, 8-87(979). M. Kuczynsi, Is he quaum heoy edicably comlee?, Phys.Sc.T35, 04005 (009) (axiv :080.59). G.E.P Box, G.M Jenins and G.C Reinsel, Time Seies Analysis Foecasing and Conol, Hoboen: Wiley, 008 3. A.Yu Khenniov and I.V Volovich, Quanum non-localiy, EPR model and Bell's heoem, in 3d Inenaional Sahaov Confeence on Physics. Poceedings, edied. By A. Semihaov e al., Singaoe:Wold Scienific,003, 60-67. 4. W de Baee W, On condiional Bell inequaliies, Le.Nuovo Cimeno 40, 488 (984) 5 H. de Raed, K.Hess and K.Micielsen, Exended Boole-Bell inequaliies, axiv:090.546v