Point processes with correlated gamma interarrival times

Statistics & Probability North-Holland

Letters

28 September

15 (1992) 135-141

Point processes with correlated interarrival times

1992

gamma

C.H. Sim Department of Mathematics, lJnir?ersity of Malaya, Kuala Lumpur, Malaysia Received December 1990 Revised January 1992

Abstract: This paper presents two point processes identically distributed gamma variables. Keywords: Point process;

gamma

process;

where the intervals

autoregressive

and moving

between

average

successive

correlation

events form a sequence

of correlated

and

structure.

1. Introduction In this paper we shall discuss two point processes that can be used for the modelling of non-Poisson series of events. For the first point process, the sequence of random intervals (X,) between successive events is assumed to follow the gamma AR(l) model of Sim (1990). The random intervals X, are taken to be identically distributed GammaCat -p>, V) random variables (r.v.‘s) and are constructed according to the autoregressive representations X,, =

NW,_,) c y+

E,

(1)

j=l

where (i) the E, are i.i.d. Gamma(a, V) r.v.‘s with cz, v > 0; (ii> the y. are i.i.d. Exponental r.v.‘s; (iii) for each fixed positive value of x, N(x) is a Poisson r.v. with parameter pax, and 0

(2)

where (i) the Z, are i.i.d. Gamma(A, v + 1) r.v.‘s with A, v > 0; (ii) the U,, are i.i.d. random coefficients defined on the interval F,(U) = 24”; (iii> the U,, and Z, are mutually independent, and p > 0. Correspondence

to: Prof. C.H. Sim, Department

0167-7152/92/$05.00

0 1992 - El sevier Science

of Mathematics,

Publishers

University

of Malaya,

B.V. All rights reserved

[0, 1) with

59100 Kuala

distribution

Lumpur,

function

Malaysia.

135

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STATISTICS&PROBABILITY

Let 4,, be the Laplace we have 4,(s)

= uE[exp(

transform

LETTERS

(LT) of the probability

function

(p.d.f.)

1992

of X,. From (2)

-spZ,)]~lU”plE[exp( -suZ,+~)ldu

which demonstrates that the X, are marginally distributed variables Gamma(h, V> and Gamma(A/p, v + 1). The a.c.f. of the mixed-gamma MA(l) process is given by Corr(X,+j,

density

28 September

Xn) =

r41/[v+(v+l)P2],

as the sum of two independent

gamma

j=I,

i 0,

j>

where the first-order serial correlation is non-negative X n + , given X, = x is nonlinear and is given by

vpx

I,

and bounded

by i. The conditional

moment

of

,F,[v+1;2V+2;(h_h,)x]

2v+l

,F,[v+l;2v+l;(A-A,)x]’

where A, = A/P and ,F,[a; 6; z] = Cz=, [(a),z”/(b),n!l is the confluent hypergeometric function with (a), = T(a + n)/T(cY). In the sequel, we shall denote Tk =X, + . . . +Xk as the time of the kth event in a point process starting from an arbitrary event with (i) = To < T, < T2 < . . . and (ii) Tk - 03 as k + 03. Note that the first condition prohibits coincident events, while the second ensures that every finite time interval contains only a finite number of events. By knowing the distribution of Tk, the distribution of the random number of events occuring in the time interval (0, t), N(t), which starts from an arbitrary event can then be obtained by using the fundamental relationships between counts of events and times between events (McFadden, 1962). The distribution of Tk for each of the processes (1) and (2) is given respectively in Section 2 and 3.

2. Point process with gamma AR(l) For the gamma is X r,...,Xn

AR(l)

&l(S r,...,s,)

process

interval

sequence

of (l), it has been

=E[exp(-six,--

...

where 1, is the identity matrix of order n; . V, is the positive definite matrix $1, sz,...,s,, p); and 1A 1 is the determinant of the square The LT of the p.d.f. of Tk =X, + . . . +Xk f&(S)

=&(s,...,s)

and may be expressed

= lzk+esl/k

-s,X,)]

by Sim (1990) that the LT of the joint p.d.f. of = II,+OS,V,

I-”

(3)

S,, is the n x II diagonal matrix with diagonal elements with elements uij =~l~-j”~, i, j = 1, 2,. . . , n; 0 = l/d1 matrix A. can be obtained from (3) as

I-”

in the form (Kotz and Adams,

4k(S) =
proved

1964)

(4)

STATISTICS&PROBABILITY

Volume 15, Number 2

roots of the where hkr, i = 1, 2,. . . , k, are the characteristic interpreted as the sum of k independent but non-identically (4) is the subject of discussion by Kabe (1962) and Kotz and the p.d.f. of Tk which can be computed iteratively is given On applying the result in the Appendix with vi = v and 1

2

Pr( Tk G t) = r( kv) fi

(0h,,)”

b,(r)(kv

matrix V,. Thus the random sum Tk can be distributed gamma variables. Inversion of Adams (1964). An alternative expression for in the Appendix. p, = l/oh,, for i = 1, 2,. . . , k, we have

- ~)r

(kv),r!

r=”

28 September 1992

LETTERS

’ a’ / 0’

kv+rpl

exp( -u/Oh,,)

du

i=l

tk” exp( - t/BA,,)

T(kv+

5

l)fi(Bh,,)‘.

bk(r)(kV - v), (a,t)‘,F,[l, (ku + l),r!

r=”

kv + r + 1; t/8h,,]

i=l

where ‘k(k-1, ak

and hi(r),

=

-

‘kk

ehkkhk(k-l)

r = 0, 1,. . . , can be computed

bi(r)= I?bi-l(j) i

recursively

(iu-2u)j(-r)j (iu

_

u)

j=O

ci,

,j,

I

as

i=3

,...,

k,

.

with *k(i-l)-*k(r-2) ci

*k(i-1)

-

*k,

ii

*k(i-

1) 1

hki, i = 1, 2, . . . , k can be obtained

and the eigenvalues MATHEMATICA. The distribution

of N(t)

can then be obtained,

=k] =Pr(T,
Pr[N(t) Note that Gamma(B,,

*ki

=

-Pr(Tk+i

for large k, the p.d.f. vk) distribution where

kv

of

from some readily

avaliable

softwares

such as

for k 2 0, as Gt).

Tk can be approximated

by (Kotz

and

Neumann,

1963) the

1 IT, = ~

0h( k) ’

Vk=ho’ with

h(k)

= 1+ j?-$

- k;l-:I))’

Sometimes it is useful to examine is specified by the renewal density

m(t> =

the probability

of an event in the small time interval

(t, t + At). This

-&(r)] =k$&(T,
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STATISTICS&PROBABILITY

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0

4

Fig. 1. The renewal

density

m(t) for the gamma

AR(l)

I3

6 alpha

LETTERS

X

10

28 September

1992

12

t

process. The function p = 0.2, 0.5 and 0.8.

m(t)/a

is plotted

against

at for values of v = 2 and

The function m(t)/cu is plotted against cut in Figure 1 for values of v = 2, and p = 0.2, 0.5 and 0.8. The computation of m(t) was terminated at the Kth term of the summation if (d/dt) Pr(T, < t> is less than 10p6. The value of m(t)/a increases as p decreases for fixed values of cut. For fixed values of p, m(t) is initially zero and increases until the maximum value is obtained near t = pX; m(t) then decreases slowly to the limiting value l/pX, where pLx is the mean value of the Gamma(a(1 -p), v) variable. Finally we note that, for this gamma AR(l) process, the index of dispersion for intervals is

Thus the observed events exhibit some degree of clustering for p > (v - l>/(v + 1).

3. Point process with mixed-gamma

MA(l)

interval

sequence

For the mixed-gamma MA(l) process of (21, the distribution of Tk may be obtained as follows. We first note that the double LT of the joint p.d.f. of Tk and Zk+ 1 satisfies the recurrence relation

lk( P, s) = E[exp(-PT, - s-G+~)] l&1( P7 BP)@4

=

(&)( *+;+p)"k-l~P'

for k = 2, 3,. . . with

138

“-‘E{exp[ -(s

=

+pu)Z,+,]}

PP)

du

(5)

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By solving (5) recursively

28 September

LETTERS

1992

s = 0, the LT of the p.d.f. of Tk is then

and by setting

P=O,

(6)

I(&)(&J”l*‘; PZO,

where A, = A//3 and A, = A/(1 + p). From (6), for p # 0, the random variable Tk can be interpreted as the time of the kth event in a modified renewal process, in which the p.d.f. of the first time interval has LT fl(p) = [A/CA + interevents times are i.i.d. with LT f,(p) = [A,/(A, +p)l[A2/(A2 p)l”[A,/(A, + P#‘+’ and the subsequent +p)]“. By inverting (6) with respect to p, we obtained the p.d.f. of Tk as A( At)vk-’ fTk( t) =

[ A,( ht)“(

~,t)“+~-‘(

4$2)[vk-vy,

I

P=O,

exp( -At)/T(vk), ~~t)“~-~

y + k + vk)]

exp( -A,t)/r(

v; v+k+vk;

(A, -A2)t,

(Al-A)t],

P > 0,

and thus Pr(T,
[(ht)‘(~,t)“+~(h,t)“~~~exp(-A,i)/T(u+k+~k+l)] 4(23)[vk-~,

Y, 1; v+k+vk+l;

(A,-A,)t,

(A,-A)t,

A,t]

where @(;l)(b,, . . .) b,; c; x1,. . .) x,) n m”co =g&) =

~

(;2-l)(bp...

c m,mn!

E

. . . go

,b,-,;

c+m,,;

(“l~~~~~~,:~-.(~~l~~~~~~

,y n’

n

m,=O

x~,...,x,-I)

is the hypergeometric series of n variables (Erdelyi, 1953, Vol. I, p. 225) convergent for all finite values of Xi, x2,. . . , X”. By summing Ik(p, 0) of (6) over k, the LT of the renewal density of N(t) is then given by A*“+‘( A +p + pp)” m;(P)

= [(

A +P)(A

+Op)l’[(A

+PP)(~

+P+BP)‘-A~+‘]

’

(7)

However, except for the case when v = 1, inversion of (7) is complicated. Hence for v # 1, an alternative is to obtain the required renewal density m(t) by computing and summing fr,(t) over k. The function m(t) is plotted against t in Figure 2 for values of A = 1, /3 = 1, and v = 1, 2 and 3. The value of m(t) increases as v decreases for fixed values of t. However, for fixed values of V, m(t) increases from zero until the maximum value is obtained near t = pX; m(t) then decreases slowly to the limiting value l/pX, where pX is the mean value of the stationary mixed-gamma MA(l) process. For this process the index of dispersion for intervals is J=

[(l +p)*V+pq/[(l

+p)v+pJ2. 139

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6

28 September

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3

1992

15

t

Fig. 2. The renewal

density

m(r) for the mixed-gamma

MA(l) process. The function fi = 1, and Y = 1,2 and 3.

m(t)

is plotted

against

t for values of A = 1,

Thus the observed events exhibit some degree of clustering when p < (1 - v>/(l + V) and v < 1.

Appendix Theorem.

Let X,, . . . , X, be k independently distributed r.v.‘s and let the p.d.f. hj( x) = ,J’~xy~-’ exp( -j_~~.~)/r( vi),

The p.d.f.

of Xi be

x > 0.

of the sum Tk=X1+X2+

...

+Xk

is

m h(r)(a,-11,

t”kwl exp( -pu,t)

C r=O

(a,)/!

K/-h

-Pk-lPr

where i = 2,

1, bi( r) =

6 j=O

i = 3,. . . , k,

bi-,(j)(ai-*)j(-~)ic:, (ai-l)j.i!

for r = 0, 1, 2,. . . , and

cj= (~i-2-~i-l)/(ELi-cLI-l).

•l

This theorem can easily be proved by mathematical

induction.

References Cox, D.R. (1962), Renewal Theory (Methuen, London). COX, D.R. and V. Isham (19801, Point Processes (Chapman and Hall, London). 140

Erdelyi, A. et al. (19.531, Higher Transcendental Functions (McGraw-Hill, New York). Kabe, D.G. (1962), On the exact distribution of a class of

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multivariate test criteria, Ann. Math. Statist. 33, 11971200. Kotz, S. and J.W. Adams (19641, Distribution of a sum of identically distributed exponentially correlated gammavariables, Ann. Math. Statist. 35, 277-283. Kotz, S. and J. Neumann (1963), On distribution of precipitation amounts for the periods of increasing length, J. Geophys. Res. 68, 3635-3641.

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28 September

1992

McFadden, J.A. (19621, On the lengths of intervals in a stationary point process, J. Roy. Statzkt. Sec. Ser. B 24, 364-382. Sim, C.H. (19871, On the modelling of stationary non-Gaussian processes, Ph.D. Thesis, Dept. of Math. Univ. of Malaya (Kuala Lumpur, Malaysia). Sim, C.H. (19901, First-order autoregressive models for gamma and exponential processes, J. Appl. Probab. 21, 325-332.

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