brillinger 1975 b

7/21/2019 Brillinger 1975 b

1/45

s ta t i s t i ca l Inference

fo r

Stationary Point.

Processes

by David R.

B r i l l i n g e r

The

University

of C a lifo rn ia , Berkeley

I n t

rOd ct

ion

This

work i s

divided into

t h r e e

p r i n c i p a l

sections

which a l s o

correspond to th e

t h r e e

l e c t u r e s

given a t Bloomington.

The t o p i c s cO ver, some u s e f u l

point process

parameters and the i r p r o p e r t i e s ,

estimation

o f time

domain parameters

and th e

estimation

o f

f r e ~ ~ e n e

domain

parameters.

The

work

may

be

viewed

as

an

extension o f

some

o f

th e

r e s u l t s in

Cox and Lewis

19.66, 1972) to apply ~

vector-vall1ed

processes and

to higher order

parameters. t w i l l proceed

a t a

h eu ris tic le v el

r a t h e r than formal.

A

fo rm al ap pro ach

may

be

f o ~ n

in

Da.ley

and Vere-Jones 1972) fo r example.

The

notation J f w i l l be used

fo r J

f x ) d ~ x , U being

Lebesgue

measure. A

general

lemma

concernin g

th e

e ~ -

istence of

c o n s i s t e n t

estimates

is

given in Section

IV.

Point Process

Parameters

Consider i so l a t e d points of

r

d i f f e r e n t types

randomly

d i st r i b u t e d along th e

r e a l l ine

R.

Prepared

while

th e

alxthor was a M i l l e r

Research

Professor

and

with

th e

support

o f

N.S.F.

Grant

GP-

31411.

55


2/45

D VID R. RILLINGER

x ~ p l e s that w

have

in mind include,

the

times

of

heart

beats or

earthqua.kes

in

the

case

r

=

1,

the

times

of nerve

pulses

released by a network of r

nerve

eel1s

the

case of general r . Let

Na A

denote

the

number of points

of

type

a fa l l ing

in

the

inter val A R and

le t

Na(t)

=

Na(O,tJ for

a

=

l , , r .

1 .

Suppose

Pr ob

[point

of type a in t , t +h]} lp a t h

as

h 0 Pa(t)

provides

a

measure of

the intensity

with which

points

of type a occur near t .

can

often conclude that

2.

Suppose,

for t

1

f

t

2

.

Prob

[point of

type

a

in

t

1

,

t

l

+ h

l

J and point

of

type

b

in

t

2

,

t

2

+

h

2

J}

as h

l

, h

2

IOPab(t

1

, t

2

)

provides ameas ure

of the

intensi ty with w aich points of

type

a occur near

t

l

and

simultaneously points of type b occur

near

t

2

A related useful measure is provided by

Prob[point

of type a

in t1 , t l+h

J I point

of

type

b

a t

t

2

}

56


3/45

fo r

a

b

STATISTICAL INTERFERENCE

as h 0

0

The r a t i o

P ab t

1

, t

2

) / P b t

2

)

i s seen to

provide a measure of

the

i n t e n s i t y with which type

a

point,s

occ ur:

near

t

1

, given t h a t

t h e r e

i s

a type

b

point a t

t

2

In th e

case t h a t type a

points are

distributed

independently of

type

b

point

s ,

Pa b t

1

, t

2

) =

P a t

1

)P b t

2

) ,

and

th e r a t i o e o ~ e s

Pa t

1

) , th e f i r s t order i n t e n s i t y .

The

function

Pa b t

1

, t

2

) i s

l i k e the

second

order

moment runction

of

ordinary time

s e r i e s ; however in

p r a c t i se i t

seelns

t o be

ml ch more u s e f u l as

i t

has a

f u r t h e r

i n t e r p r e t a t i o n as

a

p r o b a b i l i t y .

Often

i t

i s

true

t h a t

t t

= J J

P a b t l , t2 )d tld t2

o 0

t t

= fa f o Pa b t l , t 2 ) d t l d t 2

t

J

Pa t)

d t

fo r a

=

b

o

3. Suppose next t h a t ,

f o r

t1,

, t

k

d i s t i n c t an d

v 1 , . o . , v

r

non-negative i n t e g e r s wit.h

S urn

k

Prob

[type

a

p o i n t

in

each

of

t . ,

t

h . ] ,

J J J

j = L v 1 , , ~ v and a = l , ,r }

b


4/45

D VID RILLINGER

Prob{type a . point in t . , t .+h . l j = l , ,k}

J

J J

- PaI ak(tI , , tk) hI h

k

as

h1,

,h

k

0;

k

=

1,2 .0

The function

P (\)1) (\)r)

is

ca.lled pr.o.d-uct d E l ~ 1 - - i : t - y of or4er k.

Such

a function was

introduced by

S.

O Rice in

a

pa rt icu la r s it ua ti on and by A. ~ k r i s h n n in a

general s i t u a t i o ~ ,

see

Srinivasan

1974 . No claim

i s

made that

the

probabili ty in

(1) always .depends

on

hI

. ,h

k

in such

a direct

manner. Rather i t is

the

claim tha t

this

happens

for

an

interest ing

class

of e x a ~ p l e s . B r ~ l l i n g e r 1972 gives an expression

for

4.

The Erobabili ty g e ~ ~ r a t i n g f u n c t i o ~ a l of the

process

~ t = [Nl(t) , ,Nr(t)} i s defined

by

E[exp[J log

Sl(t) dNl(t)

J log Sr(t) dNr(t)}]

for

suitable

functions

Sl

o ~ r .

Writing

i t

as

r

er

{I

I; ( f)-I)}]

a:=l type a

point

a

and

expanding,

we can see

that i t

is given by

58


5/45


where we define

0

v

~

t

t

v -

This

fUnctional

is of use in computing probabi l i t ies

of

int r st for

the

process. For example

sett ing

~ a t

=

z

for

t

E A

a

=

for

t

and

deterlnining

the coefficient

of

j l

j r

z l

we

see

that

v1-j l

v jo

-1 r r

2

e m y l ikewise determine condit ional product

densit ies such as

59


6/45

E

\ ~ j

r r

D VID

R

RILLINGER

Yl) Yr)

p t

l

, t

k

N

l

A = j l

,Nr A) =

jr )

1

0)/ 2)

These

c onditiona l

product densities

are

u s e f u l

in

s ta t i s t i ca l inference. They provide likelihood

functions and also

o ~ the

i n v e s t i g a t i o n

of the

,distribution

of

s ta t is t ics

c onditiona lly the

observed

number

of

p o i n t s .

Were

N A

=

0 ,

one

wouldn t want

to

claim much.)

The integrated product

densities

give the

factoria l moments o f

the

process. For e.xaJ.ilple,

if N v) =

N N-l)

N - v + l ) , then

\)1) \ r)

E Nl A)

)

oN

A) ) =

\ 1 r \)r k

A

Also

of

use are

the c umulant d e n s i t i e s ,

\)l) \)r)

q t1 ,

, t

k

)

given by

3

60


7/45

ST TISTIC L INTERFERENCE

They measure

the degree of

dependence

of

increments

of the process a t d if fe r e n t t

j

Certain o th er c on di ti on al

product

d e n s i t i e s

are

of us e . We mention

Prob{type a point in each of

( t j , t j+h jJ ,

j

=

vb

vb and a = 1,.g. ,rtNl{O} l} /

b a

bsa

and f o r rrl, ,rr

k

Prob{type

1 point in

t , t + h ] \

po in ts of

type

1 , v

2

p oints of type 2 , a t

1

,

2

,

,

k

respectively}/h

l

+l v

2

v

- p r ( t ,

1 1

'1 2 ' ' ' . ' r

k

vI)g v

r

P rr1, ,rr

k

I f a l l points up to t a re in clud ed, this becomes the

complete i n t e n s i t y

lim Prob{type 1 point in t , t + h ] , (u) , u

s

~

5.

Certain

p r o b a b i l i t i e s and mOlnents

a re of

s p e c i a l

in teres t . We

l i s t

some o f

t h e s e .

61


8/45

DAVID R BRILLINGER

i)

th e renewal fUnctions

Uab. t

=

E{Na t)

Nb

{

=

I}

fo r

t

>

0

t

J Pab u,O) du / Pb O a b=l r

o

The renewal

density is Pab t,O)/pb O)

i i th e forward

recurrence

time d is trib u tio n

is

given by

Prob[event

before or

a t

t}

= Prob[time of

next event

from

0 is

t}

1 - Prob[N t) O}

1 _ ~ l lV r

p v)

\ J ~

\

6 t]\J

i i i

th e survivor function or d is trib u tio n of

l ifetime)

Prob[time of

next event

from 0 is >

t

N[O}

l}

Prob{N t) 0 \ N{O} I}

= p O -l r ~ ~ V S p V+l O,o.o

~ O,t]\J

1 - F t) say.

iv) the hazard function

or force

of mortali ty

~ t

=

f t / l

- F t)).

Prob {point in t , t+h ) t N

{O}

N t)

=

O}/h

where

F t)

is

given

in

i i i

and

f t is

i t s

derivative.

62


9/45

STATISTICAL

INTERFERENCE

(v) the variance time curve

var

N(t)

= E/N(t)(N(t) - 1) + E N(t) - E

N(t))2

t t

(2)

t

=

J J

p t

l

, t

2

)dt

l

dt

2

+ J p(t)

dt

a a t a

p

t

dt)2

a

(vi) the Palm functions

Ql(j1,

, j r

; t

=

prob{N

1

( t)

=

j1, o,N

r

(t)

=

j r

I

NI{a}

=

I}

vl-j l+ o+v

- j

= 1 (-1) r r

j1

J

j r P1(6)v (v

l

-3

1

(vr-J

r

J

1 1 r r

1

+1 v

2

v

r

p r

J

v

1

+ +v

r

(0 0

(0,

6.

We

next

indicate the

values of

a few

of

these

parameters for some examples

of in teres t .

Example

1 .

The Poisson process with mean intensity

p(t)oThe

numbers

of

points in dis joint intervals

I

1

,o

, I

k

are independent Poisson variates with

means P I l , .o . ,P I

k

respectively where

P(I) =

J

p(t) dt.

Here

I

and so

G[E]

= exp[ (s(t)

- 1

p(t)

dt}

Prob

{N A

j} =

P(A)j

exp{-P(A)}

J

k

I

_

jJ

t l , . oo , t

k

I

N A

=

J

= Z j k l ~ o

63


10/45

DAVID R BRILLINGER

I f

pet)

=

stp( t)

dt and N (s),

s E R is

a

o

Poisson

process

with mean

intensity

1, then the

general

process

may be represented

as

N t = N (P(t))

Example

2.

The

doubly

stochastic

Poisson process

o

Suppose

[ x l t ) , ~

, x r t ) ~ t E R+, is a process with

non-negative sample

paths,

moments

m v1 vr t1 , , t

k

=

E{Xl(tl) Xl(tvl)

X2(tvl+l) Xr(tk)}

and moment generating functional

M[Sl,o

,9

r

J = E[exp[J8

l

( t )x

1

( t )dt

jer(t)Xr(t)d t})

Suppose af ter a real izat ion

of

th is

process is

obtained, independent Poissons with mean intensi t ies

x1(t ) , ,xr t a re genera ted . Then

v1 o.o v

r

v1 o. v

r

p t

1

,

, t

k

m (t1,o , t

k

G [ ~ l , . o ~ r J

=

M[Sl-1 S r-

1

]

= E[exp{ (Sl(t)-1)x1(t)dt+

}]

I f Xa(t) = xa(t)dt,

and

Nl s ,

, N ~ s

are

independent Poissons with mean intensi t ies

1,

then

th is

process may

be

represented as

64


11/45

ST TISTI L

INTERFEREN E

N:L .Xl t))

8 , N ~ X r t ) )

This

process

seems

to

be

u s e fu l fo r

checking out

general

formulas

t h a t

have b ee n d ev el op ed , such

as

2)

an d (3),

among

other things.

EXaJ;llple 3: The c l u s t e r

process.

Suppose N

t) , ,

N ~ t )

i s

a primary process of c llls te r

centers

with

p r o b a b i l i t y

generating

functional

G [ ~ l , o ~ r J .

Suppose t h a t

secondary points are gener.ated

in

independent

c l u st e r s

centered

a t

th e

points

of

Suppose t h a t th e p g f

o

fo r c l u s t e r points o f

type

a

centered a t t

is

G a [ s l t J .

Then

th e p g f

o

of the o v e r a l l process is

G [ ~ l

, l ; r J

= E r n l ; l [ c r ~

J ~ k J n E ; r [ c r ~ + J ~ k J }

j k

j k

=

E { ~ G l [ g l l c r ~ J

~

G r [ g r l c r ~ J }

J J

= [ l [ S l \ J r [ ~ r \ J J

I f r =

2 ,

and

th e f i r s t

component

is th e primary

process and

th e

second component

corresponds

to

c lu s te rs of one member, then we have a process of

the

character

of the

G G oo

queue.

Example 4. The renewal process.

Here

th e

points

correspond to th e

par t ia l

sums of a random walk

with

p os itiv e s te ps .

Suppose r = 1 , t

l

< t

2

t

k

, then

65


12/45

DAVID R. BRILLINGER

As

the

process

has stat ionary

increments,

i t has a

spectral representat ion

and i f SU

denotes

the shi f t transformation,

S U ~ t ~ t + u ,

then

Pa

Pab t

l

- t

2

\}l \}r

p .

t l - t

k

t k_ l - t

k

Pa t

Pab t

l

t

2

\}l \}r

p

tl t

k

p t

... t

1 .

which

the process

is

stationary,

tha t i s probabil-

i ty dis tr ibut ions

are

invariant under t rans la t ions

of t . This means for example,

p 1 t

1

p 2 t

2

t

1

p 2 t

3

t

2

p 1 t

1

p 1 t

2

P 2 t

k

, t

k

_

1

p l t

k - l

where p l

and p 2

sat is fy renewal

eq :uations, see

p. 5 in Srinivasan

197

4

.

Example 5. Zero cross ing processes . Expressions

may

be

se t down

for

the

product

densit ies

of point

processes corresponding to the zeros of random

d b _ e t t e : : r 1 ~ 7 2 .

7. We now turn

to

a

consideration

of

the

case in


13/45

ST TISTI L INTERFEREN E

Na t = JC exp{itA}-l / iA ] dZa A

_co

~ o r a

l r

We

may

define

cumulant spectra of

order k by

V1 V

r

o Al+.o.+Ak

f Al,

,Ak_l dAl

dAk

=

cum{dZl Al ,G,dZl AVl ,,dZr Ak }

with o e

the

Dirac delt.a. ~ u n t i o n Alternately,

making use

of

product densi t ies , we might d e ~ i n e the

power

spectra by

OO


14/45

DAVID BRILLtNGER

mixing condition,

.Assumption

~ t ,

t E R,

i s

an

r

vector-valued

stat ionary

point

process sat isfying

1), whose

c tunulant

densi t ies

of 3 sat isfy

The second-order

spectra

of the

process,

fab A , possess

many of the same propert ies as the

spectra

of

ordinary

time

ser ies .

There

are

however

some

differences,

we mention

that

for mixing point

processes instead of

the

l imi t

for ffilxlng ordinary time seriea.

The spectral

representation ~

be used to

relate the point process

to the

associated ordinary

time series

h

f. t) =

h

t - ~ , t + ~ = exp[iAt}[ sin h

A

2

hA/2 J d ~ A

t This shows, for example,

tha t the

cross-

spectrum

of

the a-th

and b-th cOlnponents of

. ~ t

i s

8. A

key indicator of

the appearance of the process

of

points

of

type

s a y ~

is

provided

by


15/45


h small

the

empirical

intensi ty with

which

points of type

1

are

seen

to occur

near

to Models

for

the

process

may

usefully involve models for th is variate.

A

simple

statement says

Prob[point of type 1

in (t , t+h]}

~ Plh

for

h

small.

A

more

complicated statement i s

Prob[point

of

type 1

in

( t , t+h]

\

point

of

type

a at

1 }

..

PI

(t-1 )h/p

a a

In the

case

that the process 1,

near

t is

independent of the

process a,

near 1 th is las t

is

~ l h the marginal intensi ty . This

happens

often

as It-1 l

00 .

An even more

complicated

statement

involves

Prob[point of

type

1 in

( t , t+h]

vI

points of type

v

2

points of

type 2

a t

1 1

-2 ,

1 k

respectively}

(v

+l)(V

2

)(v

r

)

p t - 1 k

1-

1 k , 1 k_l- 1 k)h/

v

r

p (rr

l

-1 k,,1 k_l-1 k)

Suppose r ;

2.

A useful simple model here is ;

69


16/45

DAVID R BRILLINGER

Prob{point

of

type

1 in ( t , t+h]

N

2

(U),

C X < U ~ J

{fJ.

a(t-u)

d

2

(U)}h

4

l:

a(t- T .)}h

j

J

where

the

T .

are

the times

of the

events of

the

J

second process.

This

model

allows

the intensi ty ,

near t of

points

of

type

1

to

be affected in a

direct

manner

by

points

of

type

2.

f

the

system

is c ~ l s l then a(u) = 0, u < O The second

process

may excite or

inhibi t

the f i r s t process depending

on the sign of a(u) .

The model

implies,

for

example,

5

showing that ~

may be

i n t e r p r ~ t e d as

the intensi ty

with

which type

1 points would occ ur

where

P2

=

o.

Also

f

A A) = J a(u) exp{-iAu}du

then 5

and

6 lead to

70


17/45


suggesting

how the

para.meters ~ A A might be

ident if ied

o

I f P22 u

is constant,

as in the

Poisson

case,

then 6

leads

to

and

a t may

be

measured direct ly .

As an example of

the model

4 we mention

the

G/G/oo

queue

with

N

l

referrin g to the

process

of

exi t

t imes,

N

2

to the

process

of entry times, a -u

referring to the density of service times and

~ =

O.

Clearly, here

r o ~ { c u s t o m e r leaves in

the interval

t, t h] t

N

U ,

_ X < t }

,.. [ t a t-rr . }h

j

An

interest ing

problem

is

that of measuring the

degree

of

association of two point

processes.

A

measure

suggested

by

the

preceding

model

is

the

o ~ r n

see Bri l l inger 1974a . This

parameter

also

appears

as a measure

of the

degree of l inear

predictabi l i ty

of

the

proeess

N

l

by

the

process

N

2

I t

sa t i s f ies

a R

I2

A) 1

2

s 1.

Other

measures

of

association

could

be

based on the

nearness of

the

function

P12 u PIP2 to O.

We

mention next the self-exci t ing processes

introduced by

Hawkes see Hawkes

1972 .

For

71


18/45

DAVID R BRILLINGER

r

= 1, these sat isfy

Prob{point

in

t , t+h] ,

N u ,

u

~

t}

t

+ a t-u dN u }h

_ 0 0

~ + l a t-rr. }h

.

~ t

J

J

I f we have more

than

one

p rocess, then w

could also

set

up multivariate l inear models and

define

par t ia l

parameters.

As

another

extension,

we

could consider non-linear models such

as

Prob{point

of type

I

in t , t+h]

I N

2

U , _ o o ~ u o o }

-faa

+

J a1 t-u dN

2

U +uU

a

2

t-u,t-v dN

2

U

dN

2

V }h

More

detai ls concerning such extensions may be

found

in B rillin ge r

197

4b

9

We

end by mentioning

that

some, possibly

unexpected, relationships exist between certain of

the

parameters

that

have been defined. These are

the

Palm-Khinchin

relat ions,

00

Prob{N t

S

j}

= p Prob{N u = j I

N{ol

=

l ldu

t t

= l p

o

Prob N u = j t N{O} =l}du

Prob{N t >

j

N{O}

= I} =

1 + D+{p-l

j+l-k

j=O

Prob [N t .k:}}

EtN t N t -l

N t - k } =

72


19/45


k+l P J t E{N u) N u) - 1

N u) - k +

1

I

o

{ }

= I}

du

Such relat ionships are discussed in Cramer,

Leadbetter and

Serfl ing

1971).

In

th is

f i r s t

section

of

the paper we

have

sought to

provide

a framework

within

which

stat ionary

point processes may be handled when the

only

element of s ta t i s t ica l independence is

asymptotic.

I I . Estimation of

Time Domain

Parameters

for

Stationary

Processes

We consider

th e estimation of certain time

domain parameters g iven a realizat ion of a

process

t ) over the interval O,T], i . e . given

the

observed

times of events

in

O,TJ. We

begin

with the

f i r s t

order mean in tens i t ies p ,

a

a

z l , , r .

1.

Obvious estimates of the P

a

, a = 1 , , r , are

the

a l , r In connection with these we have,

Theorem 1.

Suppose

the

process sa t i s f ies

Assumption I . Then [Pl,

,PrJ is asymptotically

as T .. CX .

73


20/45

DAVID BRILLINGER

T his theorem , as

are

those given la te r , is

proved in th e f ina l sect i on

of

th e

paper.

The

e stim ate s a re a sy mp to tic ally

normal.

The

s ~ n p t o t i

variance o f

p i s 2n

T - l f

0). Were increments of

a aa

- 1

th e

process uncorrel at ed, th is wO uld be T Pa. We

w i l l

see

how to estimate f A)

next sect i on.

Were

aa

T

l a r g e , we might

s e t T = J U

an d

take

The r a t i o

2nf

O)/p is

u s e f u l

in describing

aa a

c e r t a i n asp ec ts o f th e process N

a

I f i t is

g r e a t e r than 1 , th e process is

said

to be cl ust ered

o r u n d e r d i s p e r s e d .

I f i t is l e s s than 1 , th e

process is c a lle d

overdispersed.

2 .

In th e

second

order

case

we are

in te re ste d in

estimating

Pab u) ~ r o b {type a in t+u,t+u+h

l

J and type b i n

t , t + h

2

]}/ h

l

h

2

fo r u fO and

Pab U)/Pb

Prob[type

a in t+u,t+u+h] 1 type b

a t

t }/h

fo r

u f

I t seems n a t u r a l

to

base

e stim ates o f the s e

on

J;b U)

=

[ j ,k) such tha t u -

S

t ; i

=

1,

, N(T) - 1 }

(iv)

Next

consid er th e estimation

of

the forward

recurrence

time dis t r ibu t ion

G(t)

1 - Prob{N(t) = O}

P (1 - F (

u))

du t

P[

(1

- F (

t ) ) t

J

P

J

U

dF

(

U )

where we use

a

Palm-Khinchin relat ion from

Section

1.9

and in tegra te

by

par ts . The l as t re la t ion

suggests the

estimate

82


29/45


G t) = ~ [ l - F t t J + p }: ( 1 i+1- 1 i) / N T )- l )

-i+l- -i ~ t

,..

t

[

i +1 - i > t }IT + L i +1- rr i IT

i+l- -i ~ t

j = 0 , .

,

J - l

xp[-iAt} dN t

a

I I I .

Estimation

of Frequency

Domain Parameters

1. We begin with a discussion of

f i r s t

order

s ta t i s t i c s . Suppose T = JU, J an

in teger .

Set

j+l)U

d ~ O j jJ

~

jU

brillinger 1975 b

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