TECHNICAL REPORT SECTIONNAV&L POSTGRADUATE SCHOOlMOM i EfvEY. CALirOstMIA 93940
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NAVAL POSTGRADUATE SCHOOL
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A MOVING AVERAGE EXPONENTIAL POINT PROCESS (EMA1)
-by
A. J . Lawrance
and
P. A. W. Lewis
June 1975
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.
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The work reported herein was supported in part by the Office of NavalResearch, the National Science Foundation and the United Kingdom ScienceResearch Council.
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Linear CombinationsPoisson ProcessMoving Average
Point ProcessRandom SequenceVariance Time Curve
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A construction is given for a stationary sequence of random variables{X.} which have exponential marginal distributions and are random linearcombinations of order one of an i.i.d. exponential sequence {e.}. Thejoint and trivariate exponential distributions of X > X. and X. -
are studied, as well as the intensity function, point spectrum and variancetime curve for the point process which has the {X.} sequence for successive
DD 1 JAN 73 1473 EDITION OF 1 NOV 65 IS OBSOLETES/N 0102-014-6601
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times between events. Initial conditions to make the point process count
stationary are given, and extensions to higher order moving averages and
Gamma point processes are discussed.
UNCLASSIFIEDSECURITY CLASSIFICATION OF THIS PAGEfWh.n D.f- Enf.r.*
A MOVING AVERAGE EXPONENTIAL POINT PROCESS (EMA1)
A. J. LawranceUniversity of Birmingham
England
and
P. A. W. LewisNaval Postgraduate SchoolMonterey, California
ABSTRACT
A construction is given for a stationary sequence of random variables
{X.} which have exponential marginal distributions and are random linear
combinations of order one of an i.i.d. exponential sequence {e.}. The joint
and trivariate exponential distributions of X. , , X. and X... are studied,i-l i l+l
as well as the intensity function, point spectrum and variance time curve for
the point process which has the {X.} sequence for successive times between
events. Initial conditions to make the point process count stationary are
given, and extensions to higher order moving averages and Gamma point processes
are discussed.
1. Introduction
In this paper we discuss the stationary sequence of random variables
{X.} which are formed from an independent and identically distributed
exponential sequence {e.} according to the linear model
*Support from the Office of Naval Research (Grant NR042-284) , the National
Science Foundation (Grant AG476) and the United Kingdom Science ResearchCouncil is gratefully acknowledged.
!e . with probability 3;1
(O£0£l,i=O,±l,±2,...). (1.1)
.th probability 1-3.'1_
)I Be. + z wilv l l+l
In fact, the {X.} form a sequence of exponential random variables, and it will
be seen from (1.1) that adjacent members will be correlated. Such a type of
first order moving average model arose out of the companion paper, Gaver and
Lewis (1975); there the first order autoregressive model
Xi
= pXi-l
+ Ei
(i = 0,±1,±2,...), (1.2)
00
" ^ P i-kk=0
X *
with exponential marginal distributions for the {X.} is investigated. It is
found there that the e! must be a mixture of a discrete component at zero and
an exponential variable. The motivation behind both models (1.1) and (1.2)
was three-fold: partly as an alternative to the normality theory of time
series, partly as a model for correlated positive random variables with expo-
nential marginal distributions but chiefly as a simple point process model with
which to analyze non-Poisson series of events and to study the power of Poisson
tests—particularly in situations where there is no obvious physically motivated
model.
In the present paper we give a fairly complete picture of the model (1.1),
which will be called EMA1 (exponential moving average of order 1), as a station-
ary point process. Distributions of the sums of the X. are obtained and lead
to counting properties of the process; the joint distributions of two and three
adjacent intervals X. are derived and appear to be new bivariate and trivar-
iate exponential distributions. The distributions are investigated through
their conditional means and variances, and computations of a conditional correlation
are given. Extensions of the model and estimation problems are briefly
discussed.
In developing the properties of the process we will also point out
similarities to a backward first order moving average which is defined as
[Be. + E. . wiiV 1 l-l
Be. with probability B,
(O£0£l;l-O t ±l,±2,...) . (1.3)
th probability 1 - B.
Properties of the processes are very similar, but those of the forward
model (1.1) have simpler derivations.
It should also be noted that the model (1.1) can be written as a very
special type of linear model with random coefficients:
X. = Be. + I.e (O^B^l, i = 0,±1,±2,...)
,
where the I. are i.i.d. Bernoulli random variables which are 1 withl
probability 1-B and with probability B. This characterization is not
very helpful for the first order model; the main point is that since the
random coefficient has a probability which is just the parameter B, many
of the theorems for linear processes are not applicable.
2. Some Basic Aspects of the EMA1 Model
The simplest aspect of the EMA1 model is the exponential marginal
distribution of the intervals (X.}; in point process terminology (see e.g.
Lawrance, 1972) this is the synchronous distribution of intervals and refers
to the distribution of the interval from an arbitrarily chosen event to the
next event. For the Laplace transform of its probability density function
(p.d.f.) f (x) , we writeA •
1
* -sX.
fx
(s) = E{eX
}
i
-s$e. -s8e.-se= E{e
1)B+ E(e
X X i}(l-B) (2.1)
using (1.1). Now the i.i.d. random variables e. have exponential distributions
with parameters X, say, and so their Laplace transform is X/(A+s). Thus
(2.1) becomes
fx.<
s) -d5r B + x3T»s-<1-,) -&i
This demonstrates that the X. have identical exponential distributions as
asserted. The parameter X is thus the number of events per unit time, or
the rate of the point process.
The correlation between X. and X. M is easily obtained on consid-l l+l
ering the product of X. from (1.1) with
xi+i
[$e with probability 6,
'3e.+1
+ e. +2with probability (1-3)
Thus, again using straightforward conditioning arguments,
CX.X.,.) = FJBe.r )B 2
UB^.e.^e.e.+2
)B(l-B)
Ef6 2 e.e. J_.4-Be2 )B(l-6)
E(^ e. Ei^ £
. ei+2^. +ie . +2+Bc21+1
)(l-B)2,
and simplification of this result leads to
±= corr(X
i,X.
+1 ) = B(l-B). (2.2)
By the construction of EMA1, the higher order serial correlations will be zero,
and thus the spectral density of intervals (Cox and Lewis, 1966, p. 70),
fM (ta) - — {1*2 I o cos(kaj)}, (O^u^tt),
71
k=lk
becon ~
f.(u) - - {1 + 2S(l-8)cos(0))}. (O^oxtt). (2.3)
The result (2.2) is the greatest limitation of the EMA1 model since it implies
that the first order serial correlation is non-negative and bounded by 1/4;
this may be compared with an ordinary MAI model assuming two-sided e . dis-
tributions of mean zero for which |p1
|£ 1/2. In both cases it can be
anticipated that the restrictions are a consequence of the linearity of the
models
.
A further simple aspect of the EMA1 model is that the {X.} sequencel
reduces tc the Poisscr process when B = or 1, and this gives checks on
most of our results. We mav also note that the moving average is taken in the
forward sense ; t:he backward model (1.3) could equally have been treated,
although producing different but similar results. This serves to emphasize
that there is no time-reversibility in the process, in the sense that
{Xn
, . . .,X } does not have the same joint probability distribution as
{X , , . . . ,X , } for all finite k, where k > 2
.
-1 -k —
3. Distributions of Sumr; and Counts jr. (x } Sequence
In the point process theory of the model, the distribution of the sums
T = X,-4-. . .+X are very useful; if these can be obtained then the distribu-r 1 r
tions of counts, both in the synchronous and asynchronous mode, can then be
derived. \s shown in Cox and Lewis (1966, Chapter 4) for instance, these
then .lead to the second order properties such as the intensity function, the
(Bartlett) soectrum of counts and the variance time curve. It is, therefore,
a particularly attractive feature of the EMA1 model that the distribution of
the T may be obtained, and we shall now give a simple derivation.
Define i>(s) as the Laplace transform of the p.d.f. of the e. distri-i
bution: except where otherwise remarked this distribution is exponential of
parameter X ar ' so il^(s) = X/(A+s). Define the rouble Laplace transform
I equivalently the joint moment generating function) of T and e.
-,> as
-s T -s_e4
T.(s
1,s ) = E(e
L r l r L} for r = 1,2,... . (3.1)
For : = 1, we have
-s X -s e -s Be -s e -s Be -(s +s )e
i.(s.,sj= E(eL X A 2
} = E{eL l l 2
}8 + E(eX 1 L l 2
}(1-B)1 z
= t!;(BSi )[BHi(s 2) + (l-B)^( Sl+s 2
)] (3.2)
and we shall write
JH'sr s2
) = BiJj(s2
) + (1-B)i|;(s +s ). (3.3)
This is the double Laolaco transform of a joint distribution in which the
first iriable has mass B at zero and with probability (1-B) is exponen-
tial distributed. We shall now relate <}> (s ,s ) and cj> .(s1,s„). Since
T = T . + Xr r-1 r
T , + Be with probability 3r-1 r
T , + Be + e , , with probability 1-3,r-1 r r+1
we have
-S..T , -s.pe -s_e -s.T ,-s.Be -(s.+s )e
r(8r 8
2) - E(e
lr"! !r 2r - 1
}3 + E{eX r"1
*r X 2 r+1
}(l-3)
= 4>
r_ 1(s
1,3s
1)iJ;(s
2)3 + 4.
r_ 1(s
1,3s
1)^(s
1+s
2)(l-B)
= [B*(a2) + (l-3)«(s
1+s
2)]«
r_1(8
1,3s
1). (3.4)
Solving (3.4) gives,
4>
r(s
1,s
2) = *(B8
1)[«(s 1> Bs1
)]r"1
*(slt a2),(3.5)
and setting s = 0, we have for the Laplace transform of the p.d.f. of T ,
*r(s) = [3<K3s) + (1-B)iKBsWs)][BiKBs) + (1-B)i|»((l+B)s) ]
r_1(3.6)
X(A+23s)X+s (a+23s){X+(l+3)s] } , r * 1, (3.7)
This is our required result; from (3.7) it will be observed that T is dis-
tributed like the sum of r independently distributed variables, such as in
a delayed renewal process, although these are not X variables. The structure
of (3.6) or (3.7) is explained by the fact that the number of intervals which
are of the 3e. form or Be. + e.,-, form are binomially distributed with1 x 1+1
parameter B or 1-3; further consideration of the adjacencies of the two
types of intervals than leads to the terms in the binomial expansion of (3.6) .
We now obtain the distribution of N , the synchronous counting process
of number of events occurring in the interval (0,t] beginning at an arbitrary
event; this is related to the distribution of T through the equivalence of
the events N < r and T > t for r i 1. Let F (t) denote the distri-t r r
burin- of T , and then sincer
Prob{N[f)
= r} = Fr(t) - F^Ct), r ^ 0, (3.8)
with F (t) = 1 for t 2> 0, we have for the p.d.f. of N ,
N °°
E{zt
} m * ( Z ;t) - I zr[F
r(t) - F (t)]
r=0
oo
- 1 + (z-1) I zr_1
F (t). (3.9)r=l
r
Inserting (3.7) in the Laplace transform of (3.9) gives
* , m B(l+B)s 2 + [-B(l-B)z-*-2B+l]As + X 2 .
"f;
(s-"-X)l6(l+B)s ;: + (l+2B-2Bz)As + (l-z)A 2]
U--"^
This result is required in Section A to follow.
4. The Intensity Function and Spectrum of Counts
The intensity function of a point process is the derivative with
respect to t of E{N } and will be denoted by m (t) . The (Bartlett)
spectrum of counts, the Fourier transform of the covariance density of the
differential counting process, then has the simple expression
8+(w) = ^ {1 + m*(iu>) + m*(-iu))}, (4.1)
where m (s) is the Laplace transform of mf (t); this expression for g,(w)
is derived in Cox and Lewis (1966, Section 4.5).
For the EMA1 process, the result from (3.10) is that
m*(*\ - X(A+gs){A+(l+g)sj[(L on
fvs;
' 8(l+B)s(X+s){s+X/(8 z+B)}' K ]
In inverting the Laplace transform (4.2) it will be noted that the
case 3+3=1, i.e. 3 - 0.6185, must be treated separately since there
will then be a factor (X+s) 2 in the denominator. Partial fraction expansions
and their inversion then give, for t ^ 0,
mf(t) = ^ +f^ {e"
Xt/(e2+B)- e~
Xt] (B*W1). (4.3)
= Atl + 33Xte"
Xt] (3
2+3=l). (4.4)
We see in both cases that the initial value of m (t) is X and that they
both increase until maximum values are obtained at t = X (B2^) x log[(3 2+3)/
(32+3~l) ] and at t = X respectively for (4.3) and (4.4); both functions
then decrease exponentially to X. There is no apparent reason for the
3+3=1 case. When 8=0 or 1 both functions are constant at X, as
is appropriate to the Poisson process.
10
The function m (t) is plotted in Figure 1 for several values of 8.
The spectrum of count? follows easilv by inserting (4.2) into (4.1), and has
the expressions
( \ Mt± -m? 6(1-6) f g2+B 1
~\\ , fl2xcin ( , ,,
gj>) - - ^1 -f 2A 2 ^^'L(e„6)y^2 " T^TtJ) (6
2+6*l) (4.5)
(62+6=l). (4.6)
We observe that both these are ratios of 4th order polynomials in u). Esti-
mation of both m (t) and g^(w) given an actual sequence of interevent times
is considered in Cox and Lewis (1966, Chapter 5); in practice these would then
be compared with our given theoretical functions which are graphed in Figure 2.
Note that unlike the 2nd order -joint moment functions p, and f,(w)— k +
for intervals, the second order moment functions for counts mJ.(t) and g, (co)t +
do discriminate between the cases where the parameter is 3 or (1-6)
.
tiowp ,,n.r the graphs in Figure 2 indicate that the count spectra of models with
6 in the ranee (0.25. 0.75) are fairly close to each other; therefore, the
spectrum will not be entirely suitable for discriminating between different
6 values for small sample sizes.
The variance time curve is considered in Section 7, along with the
stationary initial conditions for the process.
11
5. The Joint Distribution of X. and X. inx 1+1
We now discuss the joint distribution of X. and X. in which willJ 1 l+l
be a bivariate exponential distribution. Several authors have discussed bivar-
iate exponential distributions, including Downton (1970), who makes some compari-
sons with those of Gumbel, Moran and Marshall-Olkin . The distribution to be
discussed here does not appear to be one of the earlier ones, although it is
fair to say that in common with earlier ones, it is not the 'perfect' bivariate
exponential.
The double Laplace transform of the joint pdf of X. and X is
easily calculated using (1.1); the required expectation is
„."SlVS
2Xi+l, **
E{e } = £x.,x. <
sr s 2>1 1+1-3s..e.-3s „e -Bs.e.-s (Be +E )
=E{e lx 2l+1}3
2 +E{e lx 2 1+1 1+2}3(l-3)
-s- (3e.+£. .-J-as-e..,+E{e 1 i i+l 2 i+l
}B(1_g)
-s (3e.+e )-s (Be +e )
+E{e 1 i i+l 2 i+l i+2}(1_e)
2} (5#1)
which can be written
fX X
(s1,s
2) = ij;(3s
1)[3'P(3s
2)+(l-3)^(s
1+3s
2)][3+(l-3)^(s
2)] (5.2)
i' i+l
X 2 (X+3s1+3s
2)
(X+3s1)(X+s
2)(X+s
1+3s
2)
* (5.3)
We note that (5.3) is not symmetrical in s and s_, and this is to be
expected since the process is not time reversible; this is one feature which
distinguishes it from earlier bivariate exponentials. The backward moving
average model (1.3) corresponding to (1.1) has the joint interval distribution
which is specified by (4.3) with s and s interchanged.
12
An explicit form of the joint distribution (5.3) can be obtained
directly, rather than by inversion of the transform which is less
informative. By the structure of the model the joint distribution of (X..X.,,)i l+l
is a mixture of the joint distributions of (Be .,8e ) ,
(Be . ,Be +e )
,
(Be . -*«..- ,Be ._._) and (Be .-K Be +e ) with the corresponding probabili-1 l+l l+l l l+l l+l i+2 & r
ties B2
, 8(1-6), 6(1-8) and (1-6) 2. These joint pdf's can be listed in
an obvious notation as follows:
f _ (x,y) = (X/8)e-(>/S)x
(A/6)e-(A/6)^ (x,y>0)
fcs£ . ,fcsei i+1
fflp PF +F (x,y) - Xe-
Xx(l-B)-
1[Xe-
Xy-Xe-
(X/B)y], (x,y>0)
ec . , tie . _+e . _i i-1-! i+2
Be. -•-£.,., Be...i i+I i+I
(x.y) - a/8)e-<X/B)(*-y/6
>(X/B>e-(X/B)y
, (6x>y>0)
Cei i+l
,t5ei+l'
ei+2
(x.y) = <
2e-(X/6)x
[eX(l-6)y/8 2
_e-Xy
]/(1_ 6+62)) (gx>y>0)
JA2[e
-X(l-6)x_e-(X/6)x
]e-Xy
/(1_ e+62) (y>6x>0). (5.4)
We thus see that the joint pdf of X. ,X..
, will be continuous in both variablesl i+I
but will have different analytical expressions over the regions Qx > y and
Bx <_ v ; there appears to be no compact analytical form for f (x,y) .
x.,x.+1
This is unfortunate because it makes it difficult to derive maximum likelihood
estimates of the parameters \ and B in the model.
13
Different bivariate exponentials also can be compared through
their conditional properties and so we will derive these for the present
distribution. Conditional pdf's are not succinct enough, and so we
concentrate on conditional moments. These may be obtained from (5.3). For
instance, to obtain E(X. |X._=t) we differentiate with respect to s~, set
s_ = 0+, invert with respect to s and then divide by the marginal (exponen-
tial) density of X. The two conditional means are in this way found to be
E(X.|Xi_ 1
=t) = X'h^t + ^| +^ e-X(1-e)t/B
] (5.5)
and
E(X.|x.+1
=t) = X"1[l+B-e"
(1_e)Xt/B]. (5.6)
Thus, both regressions have exponential components; this property is shared by
the Marshall and Olkin bivariate distribution, although that distribution has
a singular component along X. = X . For the continuous distribution treated
by Downton both the conditional means are linear, as are the conditional
variances
.
Examining these regression functions more closely we see that E(X. |X =t)
is equal to A ' for 3=0 or 3=1; otherwise it increases exponentially
from 3X to the constant value (1+3)X " as t increases. The transient
is long for 3 close to 1, but very short when 3 is close to 0. Thus unlike
the serial correlation coefficient p, there is differentiation in this condi-
tional mean between the cases where the parameter is 3 and the case when it
has value 1-3.
The conditional mean (5.5) is more complex. It starts at t = with
value X ' and negative slope B - 1. There is a unique minimum at t =
-3 in 3/{X(l-3)} and the function eventually increases linearly with t. Since
we have for large t that
14
E[xJ xi-l
=t:] ~ x-1
Bt,
the rate of increase depends onlv on 8, not on X.
The conditional variances for the present bivariate exponential are
also exponential functions, and their explicit forms are given by
VarfX. X. = t) = A1
' l-l
-2 1-2^2^, 2B 2 (1+At) -(l-6)At/S
(1-6)^ l-B "(1-6)-2(l-B)Xt/(
(5.7)
and
Var(X. X. , =t) = Al ' l-fl
-2 l+S-!-B2-8 3
1-f_2/-A_ + At) -x(i-B)t
(1-8 B i
/B -2(1-6)A- e
t/B]. (5.8)
These conditional variances are quite different forms as shown in Figures 3
and A. In practice it is clear that conditional means and variance could only
be calculated for t in the more central regions of the marginal distributions
In these situations Var (X. X. ., ) is fairly constant, while Var(X. X #11 )i' i-l i' l+l
is reasonably linear in t. In all cases the asymptotic values are reached
much quicker for the lower value of S.
15
6. The Conditional Correlation of X , X given X±
We now wish to carry the study of dependence in the sequence of intervals
{X.} a step further, in particular to trivariate distributions. The dependence
in the EMA1 process has a very particular structure: X. is dependent on X. .. and
X but not on X._ , X , X._3> X , and so on. It thus appears that the
-joint distribution of X. ., X., X. in has some natural significance for thisJ l-l 1 l+l
process, and it will be a trivariate exponential distribution; we should note
however that in view of the coupling effect of the dependence, this trivariate
distribution is not enough to describe completely the dependence in the sequence
{X.}. In particular the sequence is certainly not Markovian since the distri-
bution of X. in |x.,X. n will depend on the value of X. ., .
l+l ' l l-l l-l
The process, by its structure, has the somewhat strange feature that
although X. n and X. in both depend on X., the variables X. .. and X., n6 l-l l+l r l l-l l+l
are independent. For this reason, it is felt that the joint distribution of
X. , and X.,- conditional on X. is of interest, and we shall give calcula-l-l l+l l
tions of the conditional correlation of X. n and X.., given X. = t. Thel-l l+l & l
other two pairwise conditional joint distributions may also of course be used,
but the corresponding unconditional joint distributions show that the intervals
concerned are not independent. We think of the conditional correlation, written
Corr(X , X. |x. = t) , as a descriptive function of the higher order dependence,
with the thought that it may be used comparatively with other trivariate
exponentials. The general properties of conditional correlations are not well
understood, but Lawrance (1975) has shown that it is equal to the corresponding
partial correlation only in very special cases, one of which is the trivariate
normal, and the present distribution is not one of these cases.
The triple Laplace transform of the joint p.d.f. of X. , , X., X... isJ r i-1 l l+l
calculated by a straightforward extension of the procedure used to obtain the
16
bivariate Laplace transform at (5.2). The result is the sum of eight expecta-
tion terms with their associated binomial probabilities, and can be cast in
the form
r
" SlXi-l~
S2Xi"
S3Xi+l^ *** , ,Ele
A\ = f (s s s )X._r X.,X.
+11 2 3
=iKBs ) (£^(es2
) + (1-6) iK S;L+6s
2)} {3^(6s
3) + (1-6) iKs
2+6s
3) *3 +U+B) iKs )}
(6.1)
This reduces to the appropriate bivariate distribtuions where one s is set to
zero. Before passing to the conditional moments, we may note that the
generalization of (6.1) to r adjacent intervals is
r r
E(exp[- Z s.X.]}= iKBs ) n IBK3s )+(l-B) iKs. ,+ s.)]Ig+(l-3) iKs)].i=l
xj=2 2 J J
(6.2)
When s, = s„ = ••• - s we recover the result for X. + X„ + • •• + X
1 2 r 1 2 r
given at (3.5).
We now return to CorrfX. , X |x. = t) which we shall denote as P~(t)i— 1 i+1 i L
the conditional correlation of X. , and X.,, given X. = t. This has thel-l l+l l
explicit expression
E(X.1,X,,.|x.=t) - E(X. . |x.=t) E(X, ,_ |X,=t)
,. 1-1 1+1' 1 1-1
' 1 1+1' 1 ,, ,v
P ?(t) = zrjz . (6.3)
[Var(X._1JX.=t) Var(X.
+1|x.=t)]
i^
In view of the results (5.5)-(5.S), there only remains to calculate
E(X. ,, X ]X.=t). This is obtained from (6.1) by inverting
17
2(x;At)_1 3i^ If
x1_rx i'
xi+i
(Sl,S2 ' S3)3(si=s
3=0) '
(6 ' 4)
as a function of s„, to recover the variable t. After subtraction of the
product of the conditional means, we have for the conditional covariance
Cov(xi-i-
xi+il
xi=t >
. - -^ + (tt-B)Xt -B> e-(1"e)Xt/B - JL .-2d-B)^t/B •
(6 . 5)1-P 1-P
Hence the expression for p 9(t) and the graphs given in Fig. 5. The conditional
correlation is far from constant in t, although in the range (0,2X), within
which it would be possible to estimate it in practice, the values are positive
and small.
18
7 . Stationary Initial Conditions
Up to this point we have dwelt on aspects of the process which involve
the intervals between the events, we have emphasized that these are a corre-
lated but stationary sequence of exponential variables. This situation is
typified by the choice of an arbitrary event for the initial point of a
sequence of intervals. We now consider the corresponding problem when the
initial point is chosen without knowledge of the event times; this is usually
called an arbitrary time and is of interest when stationarity in the counts
of events is suggested (Cox and Lewis, 1966, Chapter 4), as opposed to
stationarity in the intervals between events. However, for stationarity in
counts of events, the initial point of the interval of the counting must be
chosen in a particular probabilistic way. We shall now obtain the appropriate
initial conditions, using the approach and definition discussed in Lawrance
(1972) in which the process is considered at time t and t is then allowed
to tend to infinity. The sequence of intervals between events beginning with
the arbitrary time, usually called the asynchronous sequence, is not exponen-
tial or stationary, but the counting variable of this sequence has stationary
increments, although not Poisson distributed.
At time t in the process (after a start in any convenient way) it is
apparent that for the process to continue, we must specify:
(i) the time to the next event in the {X.} sequence, and
(ii) the random variable e.., which is associated with the end of thel+l
X. interval covering t. The first of these will be denoted by x anc* is
just the forward recurrence time of the EMAI process, and this is bound asymp-
totically to be exponential, but it will be dependent on the second, denoted
by e which will not be exponential, even asymptotically. It is their joint
distribution as t -> °° which gives the required initial conditions.
19
Suppose the process starts at t = in the synchronous mode, and
suppose that in (0,t] there are r-1 events. Let the joint pdf of T
and 3e be f„, Q (x,y). When the r— interval is of the Be form,r T ,pe r
r-1 r
then the joint pdf for (x = w, e = z) is
ft
fT Rc (x,t-x+w)dx *_ (z) ,
x=0 r-1 r
(7.1)
where i> (z) is the pdf of e . If the r— interval is of the Be + ce . r+1 r r+1
form, there are two similar expressions according as z < w or z > w; these
are
f Rc (x,t-x+w-z)dx i|> (z) (z<w)x=0 r-1 r
(7.2)
uid
t-(z-w)f
fi(x,t-x+w-z)dx ij> (z)
x=0 r-1 r
(z > w) (7.3)
The expressions become evident on considering the configuration of events
The joint pdf of x and e at time t may thus be written
00 , t
1 f „ (x,t-x+w)dx ty (z) with probabilityr=0 J x=0 r-l
,P£r
£
f (w,z;t) =X,e
oo -t
1=0 J x=0 r-1'
ooft
1
r=
00 ft-(z-w)
(x, t-x+w-z)dx \\> (z)>e er
oor t
(z<w) (7 4)
Iwith
|probability
] 1 — 8 .
f„ Q (x,t-x+w-z)dx ty (z) (z>w)r=0 J x=0 r-1 r
The r = and 1 terms here are really special cases, but will not contrib-
ute as t -* °° and do not need to be obtained explicitly. We shall now use
the result that
lim f (w,z;t) = lim sf*(w,z;s) = f (w,z)
t-*»X ' e
s-*0X,e
(7.5)
20
to obtain the limit distribution at an arbitrary time. Now for the Laplace
transform with respect to t of (7.4) we need the joint pdf of T _ and
3c , which by (3.4) is
f (x,y) = C , (x-u)k(u,y)du,r-1 r J u=0
(7.6)
and in terms of Laplace and double Laplace transforms
C*_1(s) = ^(3s)[^(s,3s)]
r 2, and k**( S;L ,s 2
) = W (Bs2
) + (1-3)^ ( Sl+Ss 2) . (7.7)
Hence the Laplace transform with respect to t of the first line of (7.4)
after ignoring the r = and r = 1 terms is
H3s)l-iKs,3s)
u=0 a=0e ' k(u,a+w)duda ty (z) (7.8)
Taking the limit as in (7.5) then gives
v^£(z)
u=0 a=0k(u,a+w)duda = vty (z) f (a+w)da = v^(w/3)4^(z) > (7.9)
a=0 M er
where v is the mean of the e distribution and ¥ (z) is its survivore
function. The limits of the other terms in (7.4) can similarly be obtained,
and give the final result as
v-1
y (w/3H (z)e e
-1
X»e
v "
^e(z)
;,w > with probability3
f (w,z) = < v "4* {(w-z)/3>^ (z) < z < w
( wi
< w < z
th probability1-3
(7.10)
21
The marginal distribution of e has pdf
fe(z) = B^(z) + (l-B)zi|»
e(z)/v. (7.11)
The marginal distribution of x is in general rather complicated, but in the
EMAI case is exponential with parameter X. From (7.11) we see that in the
EMAI case the distribution of the first e variable after an arbitrary time
(e) is the weighted sum of exponential and Erlang 2 distributions. This result
implies that the second asynchronous interval does not have the exponential
distribution, although all the following intervals do; the non-stationarity of
the asynchronous sequence of intervals is thus caused only by the second interval
The distribution for the number of events in (0,t] when t = is
an arbitrary time, that is in the stationary situation, may now be obtained
directly. As in the synchronous case of section 3 we need the distribution
t"L|
of the time to the r— event for r ^ 1. The function <}>
1(s ,s ) of section
3 is now the double Laplace transform of (7.10), and so
*1 (S1' S
2)
= vs"^^ -'Kl3 s1Mf^(s
2)+ (l-6)i|;(s
1+s
2)}]. (7.12)
Generally, for the double Laplace transform of the pdf of T and e
measured from an arbitrary event, we have as at (3.2),
*r(s
l'S2)
= <K sr s2H<Ks
1,es
1)]
r~<J,
1(s
1,6s
1) (r£2). (7.13)
This leads, using (3.9) to the Laplace transform of the pgf of N(t) as
* v S(l+3)s 2 + [-g(l-(B)z+ 28+ l]Xs+ [1+ B(l-B)z(l-z)]A 2
9 KZ ' S) (s+A){B(l+B)s z + (l+2B-28z)As+ (l-z)X z} '
K ''^ }
22
Setting 6=0 or 1 reduces this to the Poisson process result and reminds
us that the distribution of N(t) here can be considered as a generalization
of the Poisson distribution appropriate to counting events in a correlated
exponential sequence. The customary differentiations and inversions of (7.14)
give
E{N(t)} = At
and
Var{N(t)} = [l+26(l-6)]At-26(l-6)(l+3+B 2) - l^il'V [(6
2+3) 2e~At/(B +3) -e
"Xt] (7.15)
P TP-l
when 32 + 3 ^ 1; there is an individual expression for (7.15) when 6+6=1.
We notice that the distribution is asymptotically over dispersed as compared
to the Poisson distribution. The results (7.14), (7.15) may also be obtained
from general theory and the previous synchronous results, but the initial
conditions have much wider applicability.
We have then been able to explicitly obtain the main probabilistic prop-
erties of the EMA1 process in respect of stationary intervals and stationary
counts; the process is thus unusually tractable, and this is of considerable
merit as compared with many other models.
23
8. Conclusions and Extensions
There are several extensions to both the first order autoregressive
and moving average point porcesses and sequences which will be considered
subsequently:
(i) By replacing e in (1.1) with Ye.,., with probability Y
and with Y£.,, + e -,i we obtain a second order moving averagel+l i+2
process. This may be extended to any order; like the present
model the serial correlations are restricted to lie between
and 1/4.
(ii) The autoregressive and moving average structures can be combined
to give what appears to be a much richer class of processes,
(iii) In Gaver and Lewis (1975) it is shown that is the X. is taken
to be Gamma distributed (K,\) , then the solution to (1.2) shows
that e! has Laplace transform { (pX+s)/ (X+s) } and this is the
Laplace transform of an infinitely divisible distribution. Thus
autoregressive, moving average and mixed Gamma processes can be
constructed. Their properties are much more complex than the
corresponding exponential processes, but are tractable.
The EMA1 and EMAp processes are easily simulated, as are the Gamma
processes for integer k. Estimation problems remain to be considered; they
are treated for the first-order autoregressive processes in Gaver and Lewis
(1975). The use of the EMA1 sequence and point process in cluster processes,
congestion models and computer systems models will be discussed elsewhere.
24
o
ofO
ocsi
0)
T-
O) *»"
CVJ
ro
C7»
U=M X I
! X) JDA
bii
C\J o m(0 IO CVJ
O o ou n H
oa. oa. ca.
(^=|X!
XI !X) JDA
ID
k.
{>= ! XI,X!
Xt, -
! x}jJO0 = (^)2cy
BIBLIOGRAPHY
Downton, F. (1970). Bivariate exponential distributions of reliabilitytheory. J. R. Statist. Soc. B_ j32 1 408-417.
Cox, D. R. and Lewis, P. A. W. (1966). The Statistical Analysis of Seriesof Events . Methuen, London and Wiley, New York.
iver, D. P. and Lewis, P. A. W. (1975). First order autoregressive Gammasequences and point processes. To appear.
Lawrance, A. J. (1972). Some models for stationary series of univariateevents. In Stochastic Point Processes (P. A. W. Lewis, ed.) Wiley,New York, 199-256.
Lawrance, A. J. (1975). On conditional and partial correlation. To appear,
25
Figure Captions
Figure 1. The intensity function m (t) for the EMA1 process. The functions
is plotted for values 3 = 0.1, 0.3, 0.5, 0.7 and 0.9 and A = 1. The
deviation from the constant, Poisson process value A = 1 is small. Unlike
the serial correlations for intervals this function does discriminate between
the cases 3 and 1-3.
Figure 2. The spectrum of counts g,(w) for the EMA1 process. The spectrum
is flat with value 1/tt for the Poisson process (3=1 or 3 = 0). Unlike
the spectrum of intervals it does discriminate between the cases 3 and 1-6.
Figure 3. The conditional variance of X., given X = t, for the bivariate
exponential distribution (A = 1) arising in the EMA1 process.
Figure 4. The conditional variance of X , given X. _ = t, for the bivariate
exponential distribution (A = 1) arising in the EMA1 process.
Figure 5. The conditional correlation p~(t) for intervals X. , and X.,,,2 l-l i+I
given X. = t, for the EMA1 process. The joint distribution of X. . , X.,1 l-l i
X is a trivariate exponential distribution. Again there is differentiation
between the cases 3 and (1-3)
.
26
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