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CSUBSTANTIATION OF A STOCHASTIC MODEL OF

RHYTHMIC PHENOMENA

by

Ya. P. Dragan

93-03846

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FASTC-ID (RS)T-0562-92

HUMAN TRANSLATIONFASTC-ID(RS)T-0562-92 4 February 1993

cqC OO(3 14

MICROFICHE NR:

SUBSTANTIATION OF A STOCHASTIC MODEL OF RHYTHMICPHENOMENA

By: Ya. P. Dragan

English pages: 11

Source: Otbor i Peredacha Informatsii, Nr. 31, 1972;pp. 21-27

Country of origin: USSRTranslated by: Charles T. Ostertag, Jr.Requester: FASTC/TATE/1Lt Douglas E. CoolApproved for public release; Distribution unlimited.

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THIS TRANSLATION IS A RENDITION OF THE ORIGINAL PREPARED BY:FOREIGN TEXT WITHOUT ANY ANALYTICAL OR EDITO-RIAL COMMENT STATEMENTS OR THEORIES ADVO- TRANSLATION DIVISIONCATED OR IMPLIED ARE THOSE OF THE SOURCE AND FOREIGN AEROSPACE SCIENCE AND

DO NOT NECESSARILY REFLECT THE POSITION OR TECHNOLOGY CENTEROPINION OF THE FOREIGN AEROSPACE SCIENCE AND WPAFB, OHIOTECHNOLOGY CENTER.

FASTC- iD(Rs)T-0562-92 Date 4 Fs-hritiry

U. S. BOARD ON GEOGRAPHIC NAMES TRANSLITERATION SYSTEM

Block Italic Transliteration , Block Italic Transliteration"A a A A, a P P P P R9 r

56 5 6 B b CC Cc Ss

8 5 V V, V Jr To t

r r r * , g y y Y U, uA a 0 * D, d * 't 0 F, f

E * E # Ye, ye; E, e' X x X X I(h, kh?K At X Zh, zh U UI Ts, ts

3 3 3 9 Z z '4 '4 1 Ch, cht 4 H u I, i Wl m 1 w Sh, sh

SA ~ R Y, y IN 'U W w Shch, shch

KX K ,kx K-k It

1a JI A L, 1 bf m W Y, yMx M A Mm b b ,

H m H N, n 3 3 E, e0o 0 e 0, o 0 io A) Yu, yu

nl n /7 ,p p Rn R N Ya, ya*ye initially, after vowels, and after =, b; e elsewhere.When written as 4 in Russian, transliterate as y* or 6.

RUSSIAN AND ENGLISH TRIGONOMETRIC FUNCTIONS

Russian English Russian English Russian English

sin sin sh sinh arc sh sinh- 1

cos cos ch cosh arc ch cosh- 1

tg tan th tanh arc th tanh-1ctg cot cth coth arc cth coth-1sec sec sch sech arc sch sech-1

cosec csc csch csch arc csch csch-1

Russian English

rot curllg log

GRAPHICS DISCLAIMER

All figures, graphics, tables, equations, etc.merged into this translation were extractedfrom the best quality copy available.

I

SUBSTANTIATION OF A STOCHASTIC MODEL OFRHYTHMIC PHENOMENA

Dragan, Ya.P.

L'vovSubmitted 25 Jun 70

In a number of areas of science and technology it is necessary todeal with rhythmic processes, such as the parasitic modulation of shotfluctuations in radio-electronic equipment [8,9], changes in the levelsof stability of oscillatory parameters of biological systems [12],modulation of an acoustic field in a layer of water due to heaving ofits surface [7, 13], change in meteorological parameters under theaction of tidal forces [14], etc. Many investigators [1, 11, 12] notethe advisability of using methods of the theory of stationary randomprocesses for analyzing such essentially nonstationary phenomena asbiological, etc. rhythms, and stress the necessity of developingstatistical methods, based directly on the properties of this type ofnonstationary processes and having an explicit interpretation. Thefrequently used additive model - the determinate periodic functionf(t) on a background of stationary noise V(t)- f(t)+ V(t), analyzable

by methods of the theory of processes with stationary pulsations, doesnot describe the main property of rhythmic phenomena - change in theparameters of the process, analogous to pulsations (of modulation). Amultiplicative model of the type of modulation - ý(t) f•t), whereI (t) - stationary process, although it describes this feature it isnot general. Therefore the need arose for developing a generalprobability model of rhythmic phenomena and on this base to developeffective methods for analyzing them. Although in the literature [11]the important role of a study of periodic nonstationary randomprocesses is noted in connection with an analysis of rhythmicphenomena, this thesis is in no way substantiated, and the theory ofthis class of processes is found in the initial phase of development,to say nothing of its applications. In work [8] only the process of atype of modulation is studied. Taking this into consideration, theauthor together with K.S. Voychishin made an attempt at developing andsubstantiating a general model of rhythmic phenomena. The idea ofcreating such a model as a base for developing methods of analyzingthem, the use for this purpose of periodical)y correlated (PK) randomprocesses (SP)*, and the proof of the necessary theorems belong to theauthor, and for substantiation of the model the results from processing

S. . . . . .. m m m i | 1

the materials of observations cf the bioactivity of loaches andmeteorological elements which were obtained by K.S. Voychishin wereused considerably.

* Footnote. Here, just as in work [5], we will use the term"periodically correlated" in the sense of "periodically nonstatlonaryof the second order" (or in a wide sense, if an analogy with stationaryprocesses is used). This class of processes is sufficient for thetheory being developed. Periodic nonstationary in a narrow sense isdefined, as usual, through the periodic nature of n-dimensionaldistributions of probabilities [11]. [End of footnote]

For creating a meaningful theory, in contrast to mathematical, itis not sufficient just to select axioms, it is also necessary that theyrepresent significant sides of the phenomenon for which the theory isbeing constructed. Therefore in work [2] on the basis of an analysisof tracings of rhythmic phenomena it is postulated that if a phenomenontakes place in an unchanged mode, then it is reasonable to accept thefollowing as its significant characteristics: a) noise-like nature,i.e., the fundamental impossibility of their representation in the formof a complex determinate function; b) harmonizaDility - separabilityinto simple harmonic components; c) nonstationary state of the type ofmodulation, characterized by a periodic change in dispersion, and thismeans in covariation.

From these postulates it follows [6] that in the correlationtheory a periodically correlated random process (PKSP) should be ageneral model of a rhythmic phenomenon.

The purpose of this article, just like that of article [6],developing the ideas of work [5] and supplementing work [2], is afurther study of the structural properties of PKSP as a model ofrhythms, but in contrast to article (6] the mathematical expectation ofPKSP, generally speaking, is considered different from zero.

Definition 1. We will call the random process (SP) centered ifits mathematical expectation is equal to zero.

Then from a presentation of PKSP through stationary components(SK) [6]

and the uniqueness of the expansion into a Fourier series follows thevalidity of the theorem.

2

Theorem 1. A PKSP is centered whenf and only when, its stationarycomponents are centered.

Theorem 2. If the SK PKSP is uncorrelated and centered, with theexception maybe of zero, then it degenerates into into a stationary SP,if the SK are only uncorrelated, then into SP with stationarypulsations and a periodic mean.

Actually, from correlation [5]

(where rk,j(.) - mutual covariation of k- and j-th SK under thecondition of noncorrelation of SK rk,j(u)=rk(u)S j,k, from whichrl,ll-k(u)=rl(u) Sk,O) we obtain

bt( U ) ri () e BO (u).

where B 0 (-) - covariation of a stationary approximation to PKSP [5],i.e., SP ý (t) is stationary relative to covariation. And for themathematical expectation from formula (1) we find

Lh•t

where mk=E T'k(t) - in a general case (for all mkRO) a periodicfunction with a period of correlation.

Theorem 3. The non-correlatedness of odd SK PKSP with even andpaired together is sufficient so that its covariation would possesshalf a period.

Actually, under the conditions of the theorem

{0, if (jVk) are oddrjk(u), if (JAk) are even,

and then in equality (2) Bh(U)= r rN.._(.)= -) is identically

3

equal to zero in the case of odd k.

Corollaryj. Under the conditions of the theorem a FKSP possessesa half a period if its odd SK are centered.

Corollary 2. The absence of SK is sufficient so that the PKSPwould possess half a period.

Corollary 3. Centered SK, non-correlated pairwise and with allthe remainder, can be related to zero.

When all the SK are linear transforms of one of them (for example,this will happen when the PKSP is obtained by conversion of astationary SP by a system with periodically changing parameters):

h(A = () =e dZ

where 1/( -) - frequency characteristic of the corresponding invariantfilter Ak, then

~~(t4. T) ¶h(kjd(kT)

and the correlation components (KK)

COs O

and the covariations of components

CM

rhit (u) = e'f (X.) iF; (X) dS (X).

From here it is evident that the components will be non-correlated if

WhQ) *(%)=O.

In the particular case under consideration it is directly evidentthat a situation is possible when Bk(O)=O and Bk(u)#O0 in the case ofu•-O. For this it is sufficient to assume that the j-th SK is the k-thHilbert transform , i.e., 1 ( )=l, and 'i?, (9)=-i sgn 2 since

.l(u) - 2 ý sin (

4

Example. Assume

(1) = t(I)EIAI+ (i)-' 4 .

where V - denotes the conversion to the Hilbert transform, then

taking '•{U)r"(u)/,|.m -, (-'i) intc account we obtain

bq((tu) ' 2 r1(u)cos.% u+- B1 (u))2lt , wheI-e B(u)R= r )eiAm -r r (--u)-UAu,, f rom here

B2=0. The dispersion of this process oa(t)=2 (O)=2o•const If the

process S (t) is noncentered and E 5(t)=m, then m (t)=2m cos At.

In the case when the converter is a differential cperator wlthN

periodic coefficients (a (i).-I-=T/•.) , then *k a)= i a:(11 , where 4/I-0

the k-th coefficient of the Fourier function al(•). When only theactual process in the steady-state mode are considered, then odd 1 areabsent and the period of correlatedness of the process will be equal tohalf of the period of change of the parameters of the converter. Then,having designated the product of the series

Im-.=oS flm-- = r--rnL- -rn. Jm

for the covariation SK we find

'hj (") (kS e"P, (.) ds (L).

If we designate _( an even part of spectral density and

take into account the evenness of functions a•(.) , then

'r2.2i (0) 2 S P2k.21 ()L) s+ (X) d),.0

since the integral with an odd part is equal to zero. From here the

initial value of KK according to formula (2) B.A(O)=SQh(hLs)s+(X)d;A , whereN 0

Q2(0)=2'- p2 , ()...2) For example, for operators of the second and1-0

5

fourth orders Q•kA(0)= 4

_ 4 ;.6 ja 4 2 4 2 ~ 2-2 (12 = X$ ja2 (1_* la2-) Q'2 (,.k)02 1 ,+ ;L a .a-2(I

It follows from this that the values of KK in zero only inexceptional cases will be zeros, i.e., for steady-state processes,representing transforms of stationary processes with the help ofdifferential operators with periodic coefficients, as a rule, thedispersion and mathematical expectation (if the processes are notcentered) possess a precise period, equal to the period ofcorrelatedness and are half of the period of change of the operatorcoefficients. Thus the PKSP, the result of a stationary disturbance ofa system with periodically changing parameters, will possess a periodwhich is half the period of change of its parameters.

The null covariation component describes the distribution ofenergy by harmonics which make up the PKSP [6], and therefore its rolein the characteristics of the process is not at all different from therole of other KK, which is also evident from the following theorems.

Theorem 4. In order that the zero KK be identically equal to zeroit is necessary and sufficient that it be equal to zero in zero.

This follows directly from the fact that BO(-) is a covariation ofa stationary approximation to PKSP, and the covariation of thestationary process satisfies the condition r 1 (u).4r (0)= c , which inthis case gives the inequality IBo(u)/.,Bo(u).

From formula (2) we find that B*(O)= r (O) and since allSlI--.-

rt 1402 > 0 as dispersions of stationary processes, then the equality

Bo(O)=O is possible only in the case when all 0 =0. But since from

this same formula (2) IBA), ( IrjL.j_,kU)I and the mutual covariation of

stationary processes satisfies the inequality Irlk(u)1'<r,(O)rl(O , then

from here follows the conclusion, which, in view of its importance, wewill formulate as a lemma.

Lemma 1. If Bo(O)=O, then all KK are identically equal to zero.

6

Actually, since when B0 (O)=O pulsations are ansent, the processdegenerates into a determinate function. Thus the following thee crmtakes place.

Theorem 5. Equality to zero of the value in zero of a zero KK isnecessary and sufficient so that the PKSP would degenerate into adeterminate function (equal to its average).

The sufficiency of the condition follows from formula (8), lemra 1

and the definition of dispersio 2 o(t=bt(t,.0). And the necessity is

evident from the fact that the dispersion cf pulsations of a

determinate function is equal to zero. Then from the property of

covariation Ib t'u)1<a0(t+U)aM( it follows that lb (tu)j10 , and

since , then Bk(u)=O in the case of all k and u.0

Definition 2. As normalization of SP we will name, in an analogywith normalization of random variables, the transition from processS(t) to process

M, (1) (3)i

Its covariation

b- (t,u) b_ b(t. u) (4)

and the average and dispersion, naturally, are defined as rn-(t) =0,0-(t)0=1t t

in the case of all t. Therefore the unique characteristic of rhythm ofa normalized PKSP can only be covariation. Since formulas (3) and (4)have meaning, when OQ' (t)v.rO for all t, then from formula (2) it follows

t',at the function at(t+u) a(t) =- V V_ Bm_ (0)B(0)e

is periodic with respect to t with a period T and is non-vanishing.

Then based on the Wiener theorem [3] the function [a(t Vu +) at (O)F

is expanded into an absolutely convergent trigonometric series of thesame order.

7

Since the product of the periodic functions of one period is a periodicfunction of the same period, then this proves the validity of thetheorem.

Theorem 6. The periodic correlatedness of an SP is invariant withrespect to its normalization.

In particular, the covariation of a normalized PKSP, possessing aconstant dispersion (evidently 6 (t)=B0 (0)), will be

±18 a (u) eo pocesebB61(t.uA)- .. 7 . t is advisable that normalization of processes

by applied in the case of a comparative study of the rhythm ofdiverse-scale physical and other parameters.

Assume now that 1 (t) - a stationary SP with the covariation

R1(-), and f(f• -edF(4, where F() - certain complex measure of

limited variation, then the process

E W = u(Of(M (5)

is harmonizable [10] and its function of covariation

b. (t.u) = Rj (i.) (t + u) f (t), (6)

and dispersion

atan I WI US= u'.l f(r) (It

Then the normalized function of covariation

b & (t, u) - Q, (u) et m *• t]

Rw e (u)where 0()=a-•

8

Definition --3. A process of the type (5 we will call a prccess ofthe type of simple modulation.

If the function f(-) is periodic and - - :ts Fouriercoefficients, then the KK of the process g (t) will beBi, M = R, (U) C,,Ce

The following theorems stem from the previous formulas.

Theorem 7. A process of the type of simple modulation with anactual periodic modulating function is a PKSP which has been normalizedrto stationary.

Actually under the conditions of the theorem by(t, u)= ¾(u).

Theorem 8. A symmetry of the third kind of a modulating functionof an SP of the type of simple modulation is a necessary and sufficientcondition of the fact that the period of its dispersion would be halfthe period of the correlated state.

The condition of symmetry of the third kind is fulfillment of the

equality I t+-L- ) -f(t)in the case of all t, therefore the function

possesses a period -- and confirmation of the theorem followsfrom formula (7). 2

Theorem 9. Simple modulation is necessary and sufficient for theconversion of white noise into nonstationary white noise, and if themodulating function is periodic, then this is PK noise [6).

The necessity follows from the definition of nonstationary whitenoise and the fact that the negative function is a square of themodulus of the function g(t)=If(t)/ 2 ,O, and the sufficiency is foundfrom formula (6) and the property of the S-function:

&(x-a) f (x)= S (x-a) f (a).

From the analysis made of the PKSP it is evident that the preciseperiod of dispersion depends on the values in the zero of the KK, andthe precise period of the average - on the centrality of the SK. Sincein a general case these properties are independent, then the period ofdispersion and the period of the average can be used for estimating theprecise period of correlatedness of the PKSP. In this case thedispersion is more significant than the average, which frequently isconsidered equal to zero [4]. The most significant characteristic isthe function of covariation, describing the correlatedness of theharmonic components of the PKSP, and its period is determined by theKK.

"9

In particular, for a normalized PKSP the function of covariation is theunique characteristic of rhythm.

It is evident from formula (1) that when all SK are proportionalto one of them, which can be considered as zero, i.e., whengk(t)=Ock ; 0 (t), where { Ck} - complex numbers, then the PKSP turns

into a process of the type of simple modulation

(t)~~ ~ fe=% t (t).am

If all the SK are non-correlated and non-centered, then the PKSP turnsinto an additive model - a periodic determinate function on abackground of stationary noise:

k--,

where 1 (t) - stationary SP with a null average and the covariation

(U) Y3 rl "U) e t -

It is evident from the considerations presented that the idea ofisolation of the determinate component of the rhythmic phenomenon andthe study of the stationary residual turns out to be inconsistent,since only in an additive model can it have direct meaning. In thiscase the stationary residual cannot describe the rhythmic nature, andin other models the residual is not stationary.

The theorems proved above jointly with the theorems in works [5,6]serve as substantiation of the model of rhythmic phenomenon as PKSP andreveal a series of its significant properties, and also show thatneither additive nor multiplicative models can be considered as themost general for rhythmic phenomena, biological in particular, whichboth by isolation of mathematical expectation and by normalization arenot reduced to stationary processes.

10

REFERENCES

a. BP pTi e T 7 M. C. Bseaeftwe 8 TeOPHnO C.1Y'~aAHbx npou~eccoa. ;JA, M., 1958.B. oBo itWItWHH K. C.. .11pa r aH 351. [1.- B Kit.: OT60P H nepeaa'ia 1114OpmaUHHll. 29.' Hzi)Koso AyM:Ka*, K., 1971.

3. r eC L. 1 bi a Hl .M. A P 3 A K 0 8 A. lA., WU H o 03 r. E. KoNhimy1aTiisitbe HOPM3IfPOaH-Hfie IK0.1bUa1. 4DH3MaTrh13. M.. 1960.

4- 1".1 a .1 u wi enB E. r.- TeopIIR BePaHTHocTefi H ee flpNimetielfiI. 1963. Vill, 2.53. '1 p a r a it $. l.- B Ki. OT6op H nepeaama HHoopmaLumi, 22. '(HayKosa aymKav. K.,

1969.6. 3: P a r a it H. n1.- B Kit,: OT60P it nepeia'Ia )mqopmautisi, 30. 'Ha)'Kona AYMKai), K..

1971.7. A~pa rai q . nl.. MH x a A .10 B CK it B. H.. 351 Bo p c x t H. H.- B KH'.: OT6op it

nepeaam.a impop3Eaums, 28. cHayKosa ayxas, K.. 1971.8. K 0.%e., C. MK-B Kit.: Tpyam MOTH, IV, M., 1959.9. A ao. a x o a A. H. (DAyX~yautmi B aftTOoxoe6aTe.,bHb1lx CIICTeNMax. tHayxav, M., 1968.to. nl p ox o p oas 10. B., P 03 a Hi 0 9 10. A. Teoplits BepORT14ocTeri. cHayxav, M., 1967.11. Pma ueaiicH xo A. (b.. Ce p r es r . A . Bonpocbt npiii.iaaioro alla.1113- ciy'IAuIbix

flpoueccoo. cCoa. poaiio*. M., 1968.12. C 0.1 b66 e p r e D A.- B icx.: Btionorittemxie 'iacu. cMiipi. M., 1964.uI T o.1c To n H.. Ea e A K. C. AKYCTmcKa o1(e34a. 41MHlP:, M., 1969.14. Br i er G. XV.- Monthly weather Keview, 1965, 93, 2.

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