HYPER-RANDOM PHENOMENA: DEFINITION AND DESCRIPTION. Igor Gorban

Transcription

1 nternatonal Journal "nformaton heore & Applcaton" Vol HYPER-RANDOM PHENOMENA: DEFNON AND DECRPON or Gorban Abtract: he paper dedcated to the theory whch decrbe phycal phenomena n non-contanattcal condton he theory a new drecton n probablty theory and mathematcal tattc that ve new poblte for preentaton of phycal world by hyper-random model hee model take nto conderaton the chann of object properte a well a uncertanty of tattcal condton Keyword: uncertanty random hyper-random phenomenon probablty tattc ntroducton he mot of phycal phenomena (electrcal electromanetc mechancal acoutc nuclear and other) are an ndetermnate type Uually dfferenochatc method are ued to decrbe them However poblte of uch method are lmted here are erou problem when the obervaton condton are chaned n pace or tme and t mpoble to determne the tattcal reularty even by a lare expermental ample ze he chaned condton are met everywhere t mpoble to mae any real event value proce or feld n abolutely fxed condton All ma meaure are led n the varable condton controlled only partly he fxed tattcal condton and the probablty meaure are lnked toether When t ad the fxed (contant) condton abou meanhahere the probablty meaure for every ample of the reearched et When a phycal phenomenon oberved n more or le nvarable condton there poblty to acheve the tattcal reular reult However f the condton chaned n wde bound the tattcal etmate are not table and t mpoble to obtan probable etmate o mae a depth of the problem let u apply to well known clac tak wth ton a con he tablty of head or tal ( A or B ) eentally depend from the tyle of ton [] n a fxed tattcal condton there are table event frequence pn ( A ) pn ( B ) whch tend to any probablte PA ( ) PB ( ) when the number of experment N tend to nfnty n cae of varable condton the frequence pn ( A ) and pn ( B ) are contnuouly chaned hey ocllate n any nterval and noend to any fxed probablte he condton tablty play the mportant role n the probablty theory that marked by a number of centt bennn from Jakob Bernoull [] R von Me propoed even to defne [3] the probablty concepton on the bae of the event frequency n fxed condton t not mple to determne correctly a probablty meaure for real phycal phenomenon h fact wa marked n many work for ntance n the artcle [4] Dffculte and often mpoblty to ue the probablty theory tmulate the developn of new theore uch a fuzzy loc [5] neural network [6] chaotc dynamcal ytem [7] and other he new theory of hyper-random phenomena the bae of whch are preented below may be ncluded to th lt he am of the paper to revew the ornal author reearche publhed n artcle [8 5] and eneralzed n the monoraph [6] he theory orented to decrpton of dfferenype uncertanty a a contnency when the probablty meaure ext a another one when the probablty meaure doe not ext n modern mathematc the random phenomena are defned by the probablty feld that aned by the trad ( Ω P ) where Ω repreenhe et of the mple event ω Ω the Borel feld P the probablty meaure of the ubet he hyper-random phenomena may be defned by the tetrad ( Ω GP ) [9] where Ω and are the et of mple event and the Borel feld (a n the cae of the probablty feld) G the et of the condton G and P the probablty dtrbuton for the condton

2 04 nternatonal Journal "nformaton heore & Applcaton" Vol5 008 Any hyper-random phenomena (event varable functon) may be rearded a a et (famly) of random ubet n th contructon every ubet aocated wth any fxed obervaton condton he probablty meaure are determned for the element of each ubet; however the meaure are not determned for the ubet of the et Hyper-random Event and Varable he hyper-random event A from the Borel feld cannot be decrbed by any probablty However the event A under the condton G may be preented by the probablty P ( A) P( A ) h probablty ocllate when condton chaned he rane of the ocllaton may be decrbed by the upremum P (A) and the nfmum P (A) of the event probablty defned a P ( A) up P( A ) P( A) nf P( A ) n the contant condton ( cont ) thee bound are conruent and the hyper-random event deenerate to the random one wth the probablty P( A) P( A) P( A) he bound P (A) P (A) are half-meaure here ha been obtaned the expreon that are mlar to the formula decrbn the product and the addton rule the Baye' and other theorem of the probablty theory o decrbe the calar hyper-random varable X a number of the charactertc have been propoed hey are mlar to the probablty charactertc of a random varable he man of them are the upremum F ( x ) and the nfmum F ( x ) bound of the dtrbuton functon and alo the probablty denty functon f ( x ) and f ( x ) of thee bound hey are determned by the follown expreon: F ( x) up P X x F ( x) nf P X x where P{ X x } { } { } df( x) df( x) f( x) f( x) dx dx the probablty of the nequalty X x for the condton t ha been found thahe bound of the dtrbuton functon and the probablty denty functon of the bound for hyper-random varable have the ame partcularte a accordn charactertc for a random varable and n addton F ( x) F ( x) Amon the bound of the dtrbuton functon there a zone of the ambuty (f ) For a random varable X t wdth Δ F( x) F ( x) F ( x) equal to zero for all x f the upremum F ( x ) of the dtrbuton functon tend for all x to unt and the nfmum F ( x ) to zero zone of the ambuty tend to maxmum n th cae the hyper-random varable approache to a chao one o decrbe the hyper-random varable the charactertc mlar to random varable one may be ued hey are the bound F he bound of the dtrbuton functon crude and the central moment determned for the hyper-random and the zone of the ambuty (the black-out part) varable X on the bae of the bound expectaton M [ φ( X )] M [ φ( X )] of the functon φ( X ) : M ϕ ( X ) ϕ( xf ) ( x)d x M ϕ ( X) ϕ( xf ) ( x)d x [ ] [ ]

3 nternatonal Journal "nformaton heore & Applcaton" Vol n partcular the bound mean m x x mx M X mx M X For the real hyper-random varable X the bound varance D x D x are Dx Μ ( X mx ) Dx Μ ( X mx ) he bound crude moment are determned by the expreon mx M X mx M X and the bound central moment by the one μ M x ( X m ) x μ xm ( X mx ) where the order of the moment o decrbe the hyper-random varable other type charactertc may be ued too hey are upremum and nfmum of the crude moment and the ame bound of the central moment hee charactertc are determned on the bae of the expectaton of the functon φ( X ) : m are [ ] [ ] M[φ( X )] up φ( x ) f( x )d x M [φ( X )] nf φ( x ) f( x )d x where f ( x ) the probablty denty functon n condton n partcular for the hyper-random varable X the upremum m x and the nfmum m x of the mean are mx M [ X] mx M [ X ] and the upremum D x and the nfmum D x of the varance are Dx M [( X mx ) ] Dx M [( X mx ) ] where m x repreenhe mean of the random varable X he crude moment bound m x and m x are decrbed by the expreon mx M X mx M X and the central moment bound μ x and μ x by the follown one μ M [( X m ) ] μ M [( X m ) ] x x x x n eneral the operator M [ ] M [ ] dffer from the operator M [ ] [ ] M and the bound moment dffer from the moment bound althouh n ome partcular cae they may be expreed by each other for ntance when the dtrbuton functon F( x ) for dfferent condton have not ntercepton pont hen f the varance D x raed wth ran the mean m x ( a type dtrbuton) there are the follown equalte: mx mx mx mx Dx Dx Dx Dx ; f the varance D x reduced wth ran the mean m x ( b type dtrbuton) there are the equalte: mx mx mx mx Dx Dx Dx Dx he reult were eneralzed to complex X and vector X hyper-random varable to real X () t complex X () t and vector X () t functon Hyper-random Functon he calar hyper-random proce X () t ha been preented a a famly of the random procee X () t determned for a et condton G he proce decrbed by the upremum F ( x; t ) and the nfmum F ( x; t ) of the dtrbuton functon probablty denty functon f ( xt ; ) f ( xt ; ) of thee bound the bound moment m x () t m x () t μ x () t μ x () t and the moment bound m x () t m x () t μ x () t μ x () t where ( ) the order vector of the moment the meaure of the dtrbuton hee charactertc are decrbed by expreon that are mlar to one for hyper-random varable: F ( x; t) up P{ X( t ) x X( t ) x } F ( x; t) nf P{ X( t ) x X( t ) x }

4 06 nternatonal Journal "nformaton heore & Applcaton" Vol5 008 F ( x; t) F ( x; t) f( x; t) f ( x; t) x x x x m () t M[ Xt ()] xf( xtdx ;) m() t M[ Xt ()] xf( xtdx ;) x x x m ( t t ) M [ X ( t ) X ( t )] x x f ( x x ; t t ) dx dx x μ x ( ) M [( ( ) ( )) ( ( ) ( )) ] X t mx t X t mx t m ( t t ) M [ X ( t ) X ( t )] x x f ( x x ; t t ) dx dx μ x ( ) M [( ( ) ( )) ( ( ) ( )) ] Xt mx t Xt mx t m () t M[ X()] t m () t M[ X()] t x m ( t t ) M [ X ( t ) X ( t )] m ( t t ) M [ X ( t ) X ( t )] x x μ x ( ) M [( ( ) ( )) ( ( ) ( )) ] Xt mx t Xt mx t μ x ( ) M [( ( ) ( )) ( ( ) ( )) ] Xt mx t Xt mx t he bound correlaton functon and the bound covarance functon are K ( ) M [ Xt ( ) Xt ( )] K ( ) M [ Xt ( ) Xt ( )] x x R ( t t ) M [( X( t ) m ( t ))( X( t ) m ( t ))] x x x Rx( ) M [( X( t) mx ( t))( X( t) mx( t))] and the correlaton functon bound and covarance functon bound are K ( ) M [ Xt ( ) Xt ( )] K( ) M [ Xt ( ) Xt ( )] x x R ( t t ) M [( X( t ) m ( t ))( X( t ) m ( t ))] x x x R ( t t ) M [( X( t ) m ( t ))( X( t ) m ( t ))] x x x x tatonary and Erodc Hyper-random Functon t ha been found that ome hyper-random functon have pecal tatonary and erodc properte A functon X () t ha been called a tatonary hyper-random one f the bound mean do not depend from tme and bound correlaton functon depend only from tme nterval τ t t : Kx( ) Kx (τ) Kx( ) Kx(τ) A functon X () t ha been called tatonary hyper-random one for all condton f the mean mx () t xf( x; t )dx doe not depend from tme t ( mx () t mx ) and the correlaton functon K ( t t ) x x f( x x ; t t )dxdx x depend only from the nterval τ and the condton : Kx ( ) Kx ( τ ) he bound correlaton functon K x (τ) K x(τ) are determned by bound pectral denty xx ( f ) ( f ) that lnked each other by the follown expreon: xx xx ( f) Kx(τ) exp( jπfτ)dτ xx ( f) Kx (τ)exp( jπfτ)dτ

5 nternatonal Journal "nformaton heore & Applcaton" Vol Kx(τ) xx( f)exp(jπfτ)d f Kx(τ) xx ( f)exp(jπfτ)d f where f a frequency he pectral denty bound are determned by expreon xx ( f) up xx ( f) ( f) nf ( f) where ( f ) the pectral denty for condton : xx xx xx xx ( f) Kx (τ) exp( jπfτ)dτ Kx (τ) xx ( f)exp(jπfτ)df For two hyper-random functon X () t Yt () tatonary lnked each other the bound correlaton functon are determned by the follown expreon: K (τ) ( f)exp(jπfτ)df K (τ) ( f)exp(jπfτ)d f xy xy where ( f) ( f ) are the bound pectral denty: ( f) K (τ)exp( jπfτ)dτ xy ( f) K (τ) exp( jπfτ)dτ xy xy xy he pectral denty bound are xy ( f) up xy ( f) xy ( f) nf xy ( ) f where ( f ) the pectral denty for condton : xy ( f) Kxy (τ) exp( jπfτ)dτ K (τ) ( f)exp(jπfτ)df xy xy t ha been determned the partcularte of thee charactertc and ntroduced a number of new concepton n partcular hyper-random whte noe ome hyper-random functon X () t may be preented a a et of the random functon determned on the djont nterval [ ( + )) wth lontude on thahe condton are not chaned ( 0 ± ± ) et X () t the part of the functon X () t accordn to nterval and reduced to nterval [ ) : X () t f t X ( t ( + 05)) 0 f t he functon X () t n a fxed condton 0 ± ± the random functon determned on the nterval t [ ) he et of thee functon n uncertanty condton a hyper-random functon Yt () { X () t 0 ± } A hyper-random functon any functon φ( Yt ( ) Yt ( )) too where t t [ ) A hyper-random functon X () t that tatonary for all condton and lm mφ( ) mφ ha been called an erodc one Here mφ( ) the ample mean: xy xy xy m( )M [φ( Yt ( ) Yt ( ))] φ( Yt ( + t) Yt ( + t))dt φ xy xy

6 08 nternatonal Journal "nformaton heore & Applcaton" Vol5 008 and mφ M[φ( Y( t) Y( t ))] the mean of the functon φ( Yt ( ) Yt ( )) A hyper-random erodc functon X () t may be preented by the follown erou: Xt () lm X ( t ( + 05)) When the mean bound and the correlaton and the covarance functon bound are decrbed by the follown expreon: mx up x ( ) d mx nf x ( ) d Kx (τ) up x ( τ) ( )d t+ x where mx ()d x Hyper-random Model Kx (τ) nf x ( τ) ( )d t+ x R (τ) up [ x ( t+ τ) m ][ x ( t) m ]d t x x x R (τ) nf [ x ( t+ τ) m ][ x ( t) m ]d t x x x Developed approache ve poblte to model dfferenype of real phycal object and ther etmate under uncertanty chann of object properte and tattcal obervaton condton t ha been propoed dfferent meaure model: determne hyper-random random hyper-random and hyper-random hyperrandom one n thahe object are preented by determne random and hyper-random model and ther etmate by hyper-random model n cae of determne hyper-random meaure model n the fxed condton the accuracy of vector etmaton Θ of parameter θ may be decrbed by the expectaton of error quare Δ Μ[ Θ θ θ ] where M expectaton operator For the ndefnte condton the accuracy characterzed by the nterval where the value Δ may be tuated he bound of th nterval are Δ mn mn[δ Δ ] Δ max max[δ Δ ] where Δ M[Θ θ θ] Δ M[Θ θ θ] are the bound quadratc etmate he accuracy of pont etmaton may be characterzed by bound of quadratc etmate: Δ up Μ[ Θ θ θ ] Δ nf Μ[ Θ θ θ ] n calar cae the volume Δ up[σ + ε ] 0 Δ Δ and Δ Δ nf[σ ε ] Δ may be preented a 0 Δ σ + ε Δ σ + ε and 0 0 σ Μ Θ θ + where ( m ) ( m ) are the varance of error bound ( m θ ) σ Μ Θ θ σ Μ Θ θ the error varance for condton ε 0 ( m θ) ε 0 ( m θ) are the ytematc error for etmaton dtrbuton bound and ε ( m θ) the ytematc error for condton (f ) 0 θ

7 nternatonal Journal "nformaton heore & Applcaton" Vol F (θ θ ) F (θ θ) ε 0 θ θ F (θ θ) ε 0 ε 0 θ k σ m m θ m θ kσ θ F he fan of dtrbuton functon F (θ θ ) (thn curve) for dfferent condton and upremum F (θ θ) and the nfmum F (θ θ) bound of the dtrbuton functon (bold curve) o characterze the error Δ Θ θ of calar parameter θ and t etmate Θ the nterval [ε0 kσ ε0+ kσ ] and [ m kσ m + kσ ] may be ued correpondently where σ σ are the bound error tandard devaton and k the contant (f ) f condtonal dtrbuton of random value Θ θ are not penetrate and the varance D x raed or reduced wth rn the mean m x the lat nterval determned by error mean bound m m and error tandard devaton bound σ σ For a type dtrbuton t may be preented a [ m kσ m + kσ ] and for b type dtrbuton a [ m kσ m + kσ ] A hyper-random etmate Θ of fxed parameter θ wa called content one f t convered n probablty to th parameter under all condton G : lm P{Θ θ > ε θ } 0 where N a ample N ze for every condton and ε > 0 he neceary condton thahe hyper-random etmate a contenype tha deenerate to random etmate when N o etmate are not content f they tay hyper-random type when N t wa made a hypothe (hyper-random hypothe) that all real phycal phenomena are exted n contnuouly chaned tattc condton and therefore all phycal phenomena uually condered a a random type really are the hyper-random type h partcularty ext not only n cae of fnte but nfnte nterval obervaton t followed from th that all real etmate are not content and t mpoble to acheve nfntely lare accuracy n any condton he bound of error quare expectaton Δ Δ formed on the bae of ample X ze N and bound D Θ θ D Θ θ of etmate varance D Θ θ are decrbed by the nequalte ε 0 ε 0 Δ D Θ θ up + J N Δ D Θ θ nf + J N θ θ where J N Fher ntrnc accuracy for random value Θ θ : ln fn( X θ ) ln fn( X θ ) J N M M θ θ

8 0 nternatonal Journal "nformaton heore & Applcaton" Vol5 008 f N ( xθ ) probablty denty functon of ample X θ he bound quadratc etmate nequalte Δ Δ and the bound varance D Θ θ are defned by D Θ θ ε 0 ε θ θ ln f ( θ) ln ( θ) M N X f M N X Δ D Θ θ Δ D Θ θ θ θ were fn ( x θ) fn ( x θ) are bound probablty denty functon of ample X θ Analoue reult were obtaned for hyper-random hyper-random meaure model too On the bae of hyper-random hypothe wa hown that n any cae accuracy of any real phycal meaurement lmted all real etmate are not content one and therefore all real phycal phenomena are hyper-random type Procen of Hyper-random nal Developed body of mathematc may be effectvely ued for nal procen he example llutratn uch poblte preent below et u look the meaure proce of the level noe n the producton area when there a lot of producton equpment whch tme to tme wtch on and wtch off and therefore the noe condton chaned n wdely boundare he meaurement done on the ba of the data obtaned for a lon tme h tak may be concretzed by dfferent manner f the noe n fxed condton and the rule of chann condton may be rearded a random procee the tak become a clac one that cont of etmaton of a random varable or ome random varable o olve th tak t requeted to know the dtrbuton functon type or at leat have nformaton that uch dtrbuton ext f t mpoble to propoe the chann condton may be decrbed by any dtrbuton the tak a hyperrandom type n th cae the recorded data a ample from a eneral populaton of the hyper-random functon X () t By the procen of th data t poble to obtan etmate of dfferent charactertc he mae of the recorded data and etmate of ome charactertc ve the f 3 4 t ha been propoed thahe proce an erodc type F 3 Current noe level n the producton area (old lne) etmate of the bound of the expected value m x m x (traht old bold lne) and the bound of the tandard devaton m ± σ m ± σ (dahed lne) x x x x F 4 he etmate of the dtrbuton functon F ( x ) (old lne) the etmate of the bound of the dtrbuton functon F ( x ) F ( x ) (old bold lne) and the etmate of the dtrbuton functon F ( x ) calculated n the hypothe thahe data are random type (bold dahed lne) t followed from the fure that preented parameter and functon ve a lot of ueful nformaton that eentally more nformatve than charactertc uually ued for decrbn of random procee

9 nternatonal Journal "nformaton heore & Applcaton" Vol5 008 Concluon Any hyper-random phenomena (event varable functon) may be rearded a a et (famly) of random ubet n th contructon every ubet aocated wth any fxed obervaton condton he probablty meaure are determned for the element of each ubet; however the meaure are not determned for the ubet of the et Hyper-random varable and functon may be decrbed by the upremum and the nfmum bound of the dtrbuton functon Amon the bound of the dtrbuton functon there a zone of the ambuty Random and chaotc phenomena are the deenerate hyper-random phenomena 3 n addton to the bound of the dtrbuton functon the man charactertc decrbe hyper-random varable and functon are bound crude and the central moment and alo crude and the central moment bound hey are n partcular bound mean bound varance and alo mean bound varance bound and o on 4 Etmaton of hyper-random varable and functon were reearched t wa pad attenton to all real tattcal condton were contnuouly chaned herefore all real phycal phenomena uually reard a random tape n really are hyper-random tape h partcularty occur not only n cae of fnte but alo n cae of nfnte nterval obervaton t follow from th that all etmaton of real varable and functon are not content and o t mpoble to acheve nfnte phycal meaurement accuracy n any real condton Bbloraphy [] JB Keller he probablty of head Am Math Monthly vol 93 P [] Jakob Bernoull he art of conjecturn 73 [3] R von Me Mathematcal theory of probablty and tattc Edted and complemented by H Gerner NY and ondon Acad Pre 964 [4] Yu Almov Yu A Kravtov probablty a normal phycal quantty? Phyc-Upekh vol 35 no 7 pp [5] A Zadeh and J Kacprzyk (Ed) Fuzzy loc for the manaement of uncertanty John Wley & on New York 99 [6] M Haan HB Demuth and MH Beale Neural network den Boton MA: PW Publhn 996 [7] RM Crownover ntroducton to fractal and chao Jone and Bartlett Pub nc Boton ondon 995 [8] Gorban Random hyper-random chao and uncertanty Стандартизація сертифікація якість no 3 pp [9] Gorban Hyper-random phenomena and ther decrpton Акустичний вісник vol 8 no pp [0] Gorban Hyper-random functon and ther decrpton Радиоэлектроника (Radoelectronc and Communcaton ytem) no pp [] Gorban Decrpton of hyper-random varable and functon Акустичний вісник vol 8 no 3 pp [] Gorban tatonary and erodc hyper-random functon Радиоэлектроника (Radoelectronc and Communcaton ytem) no pp [3] Gorban Etmaton of hyper-random varable Математические машины и системы no pp [4] Gorban Etmaton method for hyper-random varable Математические машины и системы no pp [5] Gorban man phycal phenomena by hyper-random model Математические машины и системы no pp [6] Gorban heory of hyper-random phenomena Kev: Natonal Academy of cence 007 Author' nformaton or Gorban Deputy Drector General n cence he tate Enterpre Ukranan centfc-reearch and rann Center PhDDrc Prof Kyv Ukrane e-mal: orban@ukrndncorua