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1980-03

A mixed exponential time series model,

NMEARMA (p,q)

Lawrance, A. J.

Monterey, California. Naval Postgraduate School

http://hdl.handle.net/10945/29977

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* MIXED EXPONENTIAL TIME SERIES MODEL,

A

P

NMEARMA(p,q)

by

. J . Lawrance

and

. A. W. Lewis

March 3 980

Approved for public release; distribution unlimited,

Prepared for:

Naval Postgraduate SchoolMonterey, Ca. 93940

Q f\ A d A nw —

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1ZCURITV CLASSIFICATION OF THIS PACE (Whan Dm* EnfaraO

4. JjJTJ_E_fa«»d bwbHif) ... .„._ -.

' S "j* Mixed Exponential Time Series Model, )

f* ^NMEARMA (p.q^ ,J

<2s

REPORT DOCUMENTATION PAGE2. OOVT ACCESSION NO.

u--4

;TYPE^>F REPORT ft FCMM COVERF.O

Technical

?. AUTHORf«J

r\ A. J. /Lawrance -and P. A. W.yXewisl

• • PERFORMING OROANIIATION NAME ANO AOORESS

Naval Postgraduate SchoolMonterey, CA. 93940 •

II. CONTROLLING OFFICE NAME ANO AOORESS

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It. SUPPLEMENTARY NOTES

l». KEY WOROS fCc*ic/nua or, >*r»rt« aid* tl MMHM* and lamilty by block numnbtr)

Stochastic difference equation; Random coefficients; Mixed exponential;Mixed exponential! prccccc; Time scries; &£MA models; SKEARMA(p,q) ;

NEAR(l); Autogressive process; NMEAR(l) , Modelling Complex systems.

t\lO-^iABSTRACT fContlnoa on r*w tide H n*caaa«rr and Irfanflftr ftp alae* numb*)

A first-order stochastic difference equation with random coefficients is

shown to have a solution which ..takes the marginal distribution of thestationary sequence generated by the equation a convex mixture of twoexponential distributions. This Markovian process should be broadlyapplicable in stochastic modelling in operations analysis. Moreover it can

be extended quite simply to a mixed exponential process with mixed pth-orderautoregressive and qth-order moving average correlation structure. Coupling

DD ,'JSi, 1473**».

COITION OP 1 NOV CS IS OBSOLETES/N 103-014-660 1

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UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAOC (than Dim rmarad}

maaasBSSBESBaiS^^

UHCLASSIPIXDiacuwtv classification or this f»AGcrvh»n c«« JCa«m4}

20. ABSTRACT Coat.

— \ of the processes to model multivariate situations is also discussed.

t

SECURITY CLASSIFICATION OF THIS PAGEf»ln»«i D««« Enr.r»d)

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cfararcwaasraaa rss^ass^HTi^sammjm— it* g!**W ^awmnjsH

A MIXED EXPONENTIAL TIME SERIES MODSL, NMEARMA(p.q)

by

A. J. LavranceUniversity of BirminghamBirmingham, England

P. A. W. LewisNaval Postgraduate SchoolMonterey, California, USA

ABSTRACT

A first-order stochastic difference equation with random

coefficients is shown to have a solution which makes the marginal distribution

of the stationary sequence generated by the equation a convex mixture of

two exponential distributions. This Markovian process should be broadly

applicable in stochastic modelling in operations analysis. Moreover it

Is extended quite simply to a mixed exponential process with mixed pth-order

autoregressive and qth-order moving average correlation structure. Coupling

of the processes to model multivariate situations is also discussed.

Acct'-,jion For

fNTIS Sixinil

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1, Introduction

Sequences of independent, exponentially distributed random variables

{E } play a central role in stochastic modelling in operations analysis,

either to characterize the times-between-events in homogeneous Poisson arrival

processes or to characterize, for example, time series of successive service

times in a queue or successive response times at a computer termina 1 Un-

fortunately both the assumption of independence and/or the assumption of an

exponential distribution can be tenuous in practice. The distributional

assumption has long been relaxed, as in the renewal model for arrival processes,

but independence has been found difficult to relax in simple tractable models.

For Instance, the Box-Jenkins or ARMA models, veil known in time series analysis,

are mainly suitable for modelling time series with marginal Gaussian distri-

butions and are inappropriate for positive variables, such as response times.

However, a tractable and flexible time series model has recently been intro-

duced by Lawrance (1980), Lawrence and Lewis (1981), but developed only for

time series with exponential marginal distributions. The development here is

to extend this new model into mixed exponential variables. Mixed exponential

variables are overdispersed relative to exponential variables while retaining

their nonmodal distributional aspect. Further, they are often used when data

suggests that exponential assumptions are incorrect.

2. The Model to be Developed

A time series of equally spaced obc**rv»t"ions will be represented

by the random variables {X , n 0, +1, +2,...}. We shall be concerned

with the following general model,

X - e +n n

8X , w.p. a

(2.1)

w.p. 1-a

c + BVX , (2.2)n n n-i

XT

t * ii- i.>**.,r.Jtmtm^:f,\ u,„,<^Mm* J

"^i^'iCTp»*kW??^'

1• i.

where the V are i.l.d. binary variables with P(V - 1) - 1 - P(V =0) » a,

and {e } are assumed to be an i.i.d. sequence. This was the two-parameter

model developed in Lawrence and Lewis (19S1) for stationary X *s with

exponential marginal distributions, and there called the NLAR(l) model (new

exponential autoregressive of first order); here we develop the special

X - c + V X .n n n n-1

(2.3)

for {X } having a mixed exponential marginal distribution. The first

and crucial step is to determine the distribution of e in terms of then

marginal distribution of {X ); the model (2.1) or (2.3) provides no

guarantee that this can be achieved. The model (2.3) turns out to be very

tractable, while still preserving a geometric a correlation structure;

however, it does restrict sample path behavior to a preference for runs

of rising values. Developments of the model in Section 5, 6 and 7 to

both higher order autoregressive and higher order moving average dependency

overcome this effect to a considerable degree; the more general model (2.2) with

mixed exponential marginals has been found to be very limited by tractability.

The marginal distribution function for the mixed exponential vari-

ables {X ) will be denoted by Fy(x) where

-x/w, -x/u.

1 - Fx(x) - itje + it

2e (2.4)

and » , w- > 0, •+•»«« 1 and p., p« 2 "» Pi ^ U2

; since "*» v ? > ° and

V\ ¥ Vj we are considering only a convex mixed exponential and not including the

ordinary exponential.

Equations (2.3) and (2. A) constitute a model which will be called

INMEAR(l),

synonymous for 'new mixed-exponential first-order autoregressive

process". The necessary choice for the distribution of cn

is given in Section 3.

'; r'\

/,

i ,

.ui»i toii»l

\

- *- ,--. e*vrp •—

!

A mixed exponential random variable with distribution function

given by (2. A) will be denoted by ME(tt .u^i^.u^; it can be usefully

written as

X - K E'n n n

(2.5)

where P(Kn

- yij) - 1 - P(Kn

« p2> - t^ and E

qis a unit exponential

random variable. The customary measure of its dispersion relative to an

exponential variable is the coefficient of variation C(X) and

C(X) - sdLeooJ

=

2 2 2 1/2sd(X)

{Vl + "2W2+ 1t

l1,

2(urp

2) }

E(X) ^1M1+ P

21T 2^

(2.6)

This is always greater than one, its value in the exponential case.

3. The Innovation Process for the NMEAR(l) Process

To establish the existence and distributional form of the innovation

sequence c in (2.3) we prove the following result:

Theorem . For <^ a < 1 the stationary Markovian sequence (X } defined by

(2.3) has a convex mixed exponential marginal distribution (2. A) if and only

if the l.i.d. sequence {e } has the convex mixed exponential distribution

function F (x) given byc

-x/y, -x/y2

1 - Fc(x) = i^e + n

2e

where n,, n2

> 0, r\

1+ n

2- 1, Y^i Y

2> and

V TJjPj + ir

2u2

;

(x > ) , (3.1)

(3.2)

1 2Yl*

Y2 *2 * p l

+ M2

" °P - '^vl

+ p2

~ap^ " A ( 1_0 ) u

ip2^ ; (3,3 *

YQ " Vl + V2 :

nl

" (y1-y )/(y

1-y

2); n

2- (y

2-y )/(y

2-y

1) .

To prove the Theorem we need the following preliminary result:

(3.A)

(3.5)

• 1 MMSMPloMttAaiitaai t.^i**wj«^a ~*i** ,.:*&-.^k «*»W*i«H i« >*«Anu»iAriuk»i , J -

.^ ..--. -, i f«-" --y-

|^Mi^ 1ih'r-f^l^^r, *^^J^^''' "' - J— *~ '™

*.- "ll "I

L«mm«. For < a < 1 and < v. < 1.

_~~ '-' - '"JL'lL.:'- '

'

'•' i"T -~~~~

-

"

(yx+ y

2- ay)

2- 4(1-0^2 > t*

x+ ^

2" (2"a)u) 2 ° • (3.6)

Ue have

(VXP2-(2-o)y} - (y

x+ P

2- ay) - 4(l-o)(y. + V

2- oy)u + 4(l-a)

2y2

and cubs ti tut ing this into the left-hand side of (5.6) and cancelling like

terms, the inequality (3.6) is seen to be equivalent to

y(y, + y„) > y.p + y or1 ' "2' ' -1-2

Using (3.2) for y we get

w(pl+ v

2- y) > v^.

[*1y1+ (l-ir

1)y

2][y

1+ y

£- (it^ + (l-ir^)] > y^

[w^ + (l-ir1)y

2ll(l-ir

1)y

1+ ir^] > p^ ,

A.

-

or

or

[(yx - y

2)*

x + lJ2^ (lJ2"Ml)lT

l+ y

l*> y

ly 2 *

2 2 2- (^^2^ *i

+ *pl~y 2* *1 + yly2 * y

lu2

< it

x< 1

which is true by assumption. Thus the Lemma is proved.

Proof of the Theorem

For the stationary sequence given by (2.3) we define

x(t) - E{exp(-tX)}, $_( t ) - E{exp(-te)},

so that transforming both sides of (2.3) gives

e(t) - $ x

(t)/{a4>x(t) + (1-a)} .

VJ. It

(3.8)

With *x(t) - (1 + Y()t)/{(1 + y lt )(l + y

2t)} from (2.4), (3.8) yields

i^in-W..^ » il».»» >-~. — In .

.

' - jlilrirfn-'-^fir^'T —»*»' » «ill«"^" 1"

.

» :

Xt-i MGSi.T i| i „ — »...„,. a

- \

c(t) - (1 + Y t)/(1 + (y

x+ V

z- au)t + ri-c)!!^'). (3.9)

The reciprocal roots of the quadratic denominator are given by (3.3) and are

real by the Lemma; furthermore th.-.y are distinct and nonnegative since

A(l-a)u-y2

> 0. A partial fraction expansion of (3.9) then gives

£(t) - njU + Y^r1

+ n2(l + YjO*"

1(3.10)

This inverts to the required mixed exponential distribution provided that

the velghts n. and n?

are both positive and sum to one; the latter is

clearly true from (3.5). Now r\ > requires, from (3.5), that y, "> YQ»

and t\~ > requires that y2* Y « The first of these is equivalent

to

or

1 7

2*P1

+ ^2 " aU + ^^i + y2" 'xy) " *^1"«^»,

1V2]} > W

2 + W2

"

Aivx * »2- au)

z- 4<l-a)v

2ii2

] > vx+ p

2- (2-a)p (3.11)

and the second to

Aivx+ V

2- op) - A(l-a)

yiy2} > - { U] + v

2- (2-a)y) . (3.12)

Squaring (3.11) and (3.12) show that the condition for y > y and

Y2< Yq is just the condition proved in the Lemma. The key requirement

is that is. should be a non-zero probability.

-:*

I / /

i

.'

.

J-

<v

A. Utility of the Theorem

(i) Note that the required distribution for the i.i.d. e 's is

again the convex mixed exponential; the c 's can then be generated as

e - L En n n

(4.1)

where P(L - y,) 1 - P(L yJ) - n. and E is a unit-niean exponential.u l n z i n

Thus the model (2.3) gives a means of probabilistically transforming an i.i.d.

exponential sequence into a first order autoregresslve Markov mixed exponential

i\

s.

A

fac-quencc with autocorrelations p - ar

> 0. If a - 0, {X } are l.i.d.r — n

H2(w1,M

1;»

2,y

2) random variables. Since the process Is Karkovian It Is easy

to ehow that setting X^ to be a ME^.y, ;ir2,y

2) random variable independent

of el,c2*"* makes the Pr°ces8 {X

q , n - 1,2,...} stationary. Finally it

la easy to see that the regression of X on X . is linear: from (2.3)n n—

1

n n-l n

(11) The theorem also holds when the marginal distribution of X

Is the zeio-jump exponential. In our notation, let v. - and y~ * u',

so there is a probability ir. that X - 0. This structure is related to the

queuing situation where the waiting time can be either zeio when no customer

is being served, or an exponential length of time when the server is already

busy. The theorem gives the following form of c ;

(1-aiOy'Ei n

w.p. (l-a)Tr2/(l-air

2)

(4.2)

) w.p. *1/(l-a^

2)

Thus the c variable also has a zero-Jump exponential distribution.

(iii) There are two cases in which a mixed exponential marginal

distribution ME^.,u.ji^.iO of {X } can reduce to a single exponential

distribution; these are when y, « v 2and when elther n^ or *

2is zero.

Neither case is covered by the theorem, but each can be treated directly

from the model (2.3). With y as the mean of the exponential {X }

sequence, {c ) is an i.i.d. exponential sequence with mean (l-a)y. This

situation is the TEAR(i) process studied in Lawrance and Levis (1981).

5. Mixed Exponential Moving Average Processes

It is possible to think of (2.3) and its mixed exponential solution

as a way of combining independent ME(r\ y ;r\ y ) and ME^.w^ji^.y.^)

variables into another ME(Tr.,y ;ir2,y

2) variable. This key result allows

the constructing (cf. Lawrance and Lewis, 1977) of a first order moving

average process

i , .. -

•

Xn " L

nEn+ V

*K»-1E«-1 • • " 0. 1. ± 2, ... , (5.1)

where Vq , K

qand L

qare defined at (2.2), (2.5) and (4.1), respectively.

The dependency parameter Is a. 3y the results of Section 3, X will haven

the M2(*1,p.;ir

2,y

2) distribution (2.4), but the dependency will be that of

a first order moving average process; the model will be denoted as NHEMA(l).

The moving average aspect Is clear because X and X . . have just En n+l n

In common while X and X . (r > 1) will have no E 's in common and thusn n+r n

be Independent. However, the NMEMA(l) process has a very limited range of

correlations; corr(X ,X ) Is easily derived asn n+i

Px

- a(l-a)(*1u1+ »

2w 2

)2/[*

1Vj + *

2U2 + *

1»2(p

1-y

2)

2] (5.2)

- a(l-a)/C2(X) . (5.3)

The a(l-a) is always less than one-fourth and the coefficient of variation

C(X) of the mixed exponential distribution is always greater than on.;; thus

an overall bound below one-fourth is Implied. Note that the oaxiuum allowable

value of p. decreases as C(X) increases. I.e. as the dispersion of X

relative to an exponential random variable for which C(X) - 1, incteasec A

stationary sequence is clearly obtained for n 1,2,... by starting the

moving average over EL, E. , . . . .

6. Higher-Order Autoregreosive Mixed Exponential Models

Alt'^'Ugh it is possible to construct higher-order analogs of the

autoregressive equation (2.1), as in Lawrance and Lewis (1980), a simple

p-th order extension of the model (2.3) is obtained by letting X , be a

probability mixture of X ,,..., X . If the X *s are marginallyn-i n-p n

MEdr. ,u. ;7r.,ij-) then this mixture will be ME(ir ,u.;*2,u

2) also, and the

error sequence c in the autoregressive extension

IMrflirtatfWi'hrtiri^- - * -J k "*^*****™-<*********^^ ^KnrMv^^imifcMmm

n n

n-1

n-2

w.p. 1 - (« +...+ a )1 P

w.p. a,

w.p. a.(6.1)

XI

n-p w.p. a

1. given „ at (3.!) to (3.5) to be HEt,^,,^, with „ repUce<J by

"1 * "2+-+ », »« autocorrelation now foUov pth order linear, conatant

coefficient difference equation* paralleling those of the standard auto-regressive models.

The second-order situation NMEAKtf) is probably aa Mgh „ oncwould want to go in Celling data, in particular to take car. of situationswhere the marginal distribution is elxed exponential but either the processis not first order Markov or the sample paths do not tend to run

up. In this case simple computations give

PX" a

1/(l-a

2) >

0l , P2- a + o-p, .ri (6.2)

The range of Pj and Pj combinations is restricted by < a,. a2

< 1, and

•l + «2 < 1. In particular any first autocorrelation pj between zero and one

is attainable for value*? of « „•,,* ^ ,-values of 0j and «2

satisfying a2

= i-^/p*. Slnce th±sis linear. ^ ranges between and p* as «

2,anges from 1 to ,

where a2 - gives the NMEAR(l) case. Note thai. «, - imp lies that

{xj» yy ...} are independent of {x X * , a ^J lV V"-> and that the odd and the evensequences are governed by separate but ideocica! firs t order processes. The^fect. practical. of balng able to choo=e ,^ ^ ^ _^ ^will be to adjust run-up behavior.

1

7. Mixed Autoregressive-Moving Average Extensions

It is possible to combine the autoregressive structure (6.1)

and the moving average structure (5.1), as was done by Jacobs and Lewis (1977)

and Lavrance and Lewis (1980) for exponential variables, to obtain a complete

NMEARMA(p,q) process with mixed pth order autoregressive-qth order moving

average correlation structure. This yields a much richer process in terms

of sample-path behavior but since the process is not Karkovian it is diffi-

cult to do estimation for the parameters. Thus for the sake of completeness

we consider only the (1,1) model, extending the allowable range of marginal

behavior of the EARMA(1,1) process of Jacobs and Lewis (1977); higher-order

extensions are obvious. The NMEARMA(1,1) process is defined by the equations

X - L (o)E + V (a)Y . ,n n n n n-1

Yn- L'(a')E + V»(a ')Y ,

n n n n n—

l

n - 0, + 1, + 2, .... (7.1)

where the independent i.i.d. sequences V (a) and V'(cx') have

P{V (a) - 1) - a, P{V'(a')- 1) - a'. Also L (a) and L'(a*) are inde-n n n n

pendent with distributions given at. (4.1). By the Theorem of Section 3 the

{X } sequence has a ME(n. ,y. ;ir_,y.) marginal distribution and a correlationn i i. z z

structure typical of standard ARMA(l.l) models. Other properties of the

model go through In complete analogy with the EARMA(l.l) process. In fact

they will be simpler sii.ce that process is based on the EAR(l) model of

Gaver and Lewis (1980) whose zero-defect gives most of the problems in

the limit theorems of Jacobs and Lewis (1977).

8. Modelling and Multivariate Extensions

The mixed exponential time series generated by (2.3) and written,

using (4.1) and (2.5), as

X LE + V X ,,n n n n-1' °. ± 1. ± 2,...

Is a simple, random linear combination of random variables. As with the

ordinary ARMA models, this makes it ideal for modelling complex systems andfor extension to multivariate time series. The use of EARMA models in

modelling queues has been discussed in Lewis and Shedler (1977) and Jacobs

(1980); the NEAK(l) structure used here is actually better for this purposesince, unlike the EAk(l) structure, the error term does not disappear.

These applications of the NMEAR(l) process will be discussed elsewhere.

Here we only discuss direct multivariate extensions.

Thus let (E;,EJJ)

be a bivariate pair with unit exponential

marginals and let X* and X" be

n n n n n-1 '

X" - V'E" + V"X", ,n nun n n-1 *

n "0. ±1. ±2.... (8#1)

where L* is chosen to make X' be ME{«* u'*v* u*l ,„j T » < ul ». n nB^w2»»*£« u 2 ,v2 chosen to

make X£ be ME(»J,pJ ;ir»,uJ)

, The i. 1#d . squences { L »,l") and {V'.V"}

could be dependent within pairs. The sequences {X'} and {X"} will haven nnegative cross-correlation if th*> /f' v"\ ^otv^n xr tne tE

n'V Pairs are negatively correlated.

A cross-coupled version of this process, as in Gaver and Lewis (1980), will

create alternating and posibly negative autorcorrelation in X' and X"n n*Many other multivariate extensions are possible.

9. Discussion

This paper has focussed on replacing the 'Poisson process' assumptions ofexponentiality and independence which are often made in modelling sequences

of identically distributed positive random variables. The exponential

assumption has been replaced by a mixed exponential marginal distribution,

allowing more variable behavior, while the independence has been replaced

10

by autogressive and moving average dependence. Utility of the model has beenreferred to in queuing and multivariate situations. Simulation of the model

18 possible from easily obtainable i.i.d. exponential sequences.

Various aspects of the model are currently being explored or are opento development. Extensions incorporating negative dependency can be treatedby the method of cross-coupling used in Gaver and Lewis (1980) and Lawrence-d Levis (1981). Estimation via the likelihood is available in the pure auto-recessive models, but the standard assumptions are not satisfied because of

«.<< rontinuities in the likelihood. This aspecr is investigated in Raftery

<1980a.b) under the assumption of an exponential marginal distribution.

Extensions to nonconvex mixed exponential solutions may be useful when the

marginal distribution is less dispensed relative to the exponential distri-

bution; one such case is from the sum of two independent non-identical expo-

nentials. However, the model has no Gamma solution. A more flexible two-

parameter autoregressive model is available, but its development is limited

by mathematical tractability.

Acknowledgments

Work of P. A. W. Lewis was supported by the United States Office

of Naval Research (Grant NR-42-284) and by the Naval Postgraduate School

Foundation. Work of A. J. Lawrance was supported by the Naval Postgraduate

School Foundation and the United Kingdom Science Research Council.

11

rjma*-'

REFERENCES

Gaver, D. P. and Lewis, P.A.W. (1980). First order autoregressive gammasequences and point processes, J. Appl. Prob . 17, to appear.

Jacobs, P. A. (1980). Heavy traffic results for single server queues

with dependent (EARMA) service and interarrival tines,

J. Appl. Prob . 17 , to appear.

Jacobs, P. A. and Lewis, P. A. W. (1977). A mixed autoregressive movingaverage exponential sequence and point process, EARMA(1,1). Adv.Appl. Prob . 9, 87-104.

Lawrance, A. J. (1980). Some autoregressive models for point processes.Colloquia Mathematics Societatis Janos Bolvai . 25, Point Processesand Queuing Problems, Keszthely (Hungary) 1978.

Lawrance. A. J. and Lewis, P. A. W. (1977). A moving average exponential

point process, EMA(l) . J. Appl. Prob. 14. 98-113.

Lawrance, A. J. and Lewis, P. A. W. (1980). The exponential autoregressive

moving average EARMA(p.q) process. J. R. Statist. Soc. B. 42 . to appear.

Lawrance, A. J, and Lewis, P. A. W. (1981). A new autoregressive time

series model in exponential variables, NEAR(l). Naval Postgraduate SchoolTechnical Report NPS55-80-011.

Lewis, P. A. W. and Shedler, G. S. (1977). Analysis and modelling of pointprocesses in computer systems. Bull. Int. Sta«- t««.- vt Vt,m>

193-210.

Raftery, A.E. (1980a). Un processus autoregressif a loi marginale exponentielle:proprietes asymptoiques et estimation de maximum de vraisemblance. AnnalesScientifiques de l'Universite de Cle^n.ont . to appear.

Raftery, A.E. (1980b). Estimation efficace pour un processus autoregressifexponentiel a densite discontinue. Publ. Inst. Stat. Univ. Paris . 25,64-90.

12

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