Some signals have no waveform. Consider the measurement of when
lightning strikes occur within some region; the random process
is the sequence of event times, which has no intrinsic waveform.
Such processes are termed point processes, and have
been shown (see Snyder) to have
simple mathematical structure. Define some quantities first.
Let N t
N t
be the number of events that have
occurred up to time t t
(observations are by convention assumed to start at
t=0
t
0
). This quantity is termed the counting process, and
has the shape of a staircase function: The counting function
consists of a series of plateaus always equal to an integer,
with jumps between plateaus occuring when events occur.
The increment
N
t
1
,
t
2
=
N
t
2
−
N
t
1
N
t
1
,
t
2
N
t
2
N
t
1
corresponds to the number of events in the
interval
t
1
t
2
t
1
t
2
.
Consequently,
N
t
=
N
0
,
t
N
t
N
0
,
t
.
The
event times comprise the random vector WW; the dimension of this vector is
N t
N t
, the number of events that have
occured. The occurrence of events is governed by a quantity
known as the intensity
λ
(
t
;
N
t
;
W
)
λ
(
t
;
N
t
;
W
)
of the point process through the probability law
Pr
N
t
,
t
+
Δ
t
=1|
N
t
;
W
=
λ
(
t
;
N
t
;
W
)
Δt
N
t
;
W
N
t
,
t
+
Δ
t
1
λ
(
t
;
N
t
;
W
)
Δ
t
for sufficiently small
Δt
Δ
t
. Note that this probability is a conditional
probability; it can depend on how many events occurred
previously and when they occurred. The intensity can also vary
with time to describe non-stationary point processes. The
intensity has units of events, and it can be viewed as the
instantaneous rate at which events occur.
The simplest point process from a structural viewpoint, the
Poisson process, has no dependence on process history. A
stationary Poisson process results when the intensity equals a
constant:
λ
(
t
;
N
t
;
W
)
=
λ
0
λ
(
t
;
N
t
;
W
)
λ
0
. Thus, in a Poisson process, a coin is flipped every
Δt
Δ
t
seconds, with a constant probability of heads (an
event) occuring that equals
λ
0
Δt
λ
0
Δ
t
and is independent of the occurrence of past (and
future) events. When this probability varies with time, the
intensity equals
λt
λ
t
, a non-negative signal, and a nonstationary Poison
process results.
From the Poisson process's definition, we can derive the
probability laws that govern event occurrence. These fall into
two categories: the count statistics
Pr
N
t
1
,
t
2
=n
N
t
1
,
t
2
n
, the probability of obtaining
nn events in an interval
t
1
t
2
t
1
t
2
, and the time of occurrence statistics
p
W
n
w
p
W
n
w
, the joint distribution of the first
nn event times in the observation
interval. These times form the vector
W
n
W
n
, the occurrence time vector of dimension
nn. From these two probability
distributions, we can derive the sample function density.
We derive a differentio-difference equation that
Pr
N
t
1
,
t
2
=n
N
t
1
,
t
2
n
,
t
1
<
t
2
t
1
t
2
, must satisfy for event occurrence in an interval to
be regular and independent of event occurrences in disjoint
intervals. Let t
1 t
1 be fixed and
consider event occurrence in the intervals
t
1
t
2
t
1
t
2
and
t
2
t
2
+δ
t
2
t
2
δ
, and how these contribute to the occurrence of
nn events in the union of the two
intervals. If kk events occur in
t
1
t
2
t
1
t
2
, then
n−k
n
k
must occur in
t
2
t
2
+δ
t
2
t
2
δ
. Furthermore, the scenarios for different values of
kk are mutually exclusive.
Consequently,
Pr
N
t
1
,
t
2
+
δ
=n=∑k=0nPr
N
t
1
,
t
2
=k
N
t
2
,
t
2
+
δ
=n−k=Pr
N
t
2
,
t
2
+
δ
=0|
N
t
1
,
t
2
=nPr
N
t
1
,
t
2
=n+Pr
N
t
2
,
t
2
+
δ
=1|
N
t
1
,
t
2
=n−1Pr
N
t
1
,
t
2
=n−1+∑k=2nPr
N
t
2
,
t
2
+
δ
=k|
N
t
1
,
t
2
=n−kPr
N
t
1
,
t
2
=n−k
N
t
1
,
t
2
+
δ
n
k
0
n
N
t
1
,
t
2
k
N
t
2
,
t
2
+
δ
n
k
N
t
1
,
t
2
n
N
t
2
,
t
2
+
δ
0
N
t
1
,
t
2
n
N
t
1
,
t
2
n
1
N
t
2
,
t
2
+
δ
1
N
t
1
,
t
2
n
1
k
2
n
N
t
1
,
t
2
n
k
N
t
2
,
t
2
+
δ
k
N
t
1
,
t
2
n
k
(1)
Because of the independence of event occurrence in disjoint
intervals, the conditional probabilities in this expression
equal the unconditional ones. When
δδ is small, only the first
two will be significant to the first order in
δδ. Rearranging and taking
the obvious limit, we have the equation defining the count
statistics.
dd
t
2
Pr
N
t
1
,
t
2
=n=-λ
t
2
Pr
N
t
1
,
t
2
=n+λ
t
2
Pr
N
t
1
,
t
2
=n−1
t
2
N
t
1
,
t
2
n
λ
t
2
N
t
1
,
t
2
n
λ
t
2
N
t
1
,
t
2
n
1
To solve this equation, we apply a
zz-transform to both sides.
Defining the transform of
Pr
N
t
1
,
t
2
=n
N
t
1
,
t
2
n
to be
P
t
2
z
P
t
2
z
,
we have
∂∂
t
2
P
t
2
z=-λ
t
2
1−z-1P
t
2
z
t
2
P
t
2
z
λ
t
2
1
z
-1
P
t
2
z
Applying the boundary condition that
P
t
1
z=1
P
t
1
z
1
, this simple first-order differential equation has
the solution
P
t
2
z=ⅇ-1−z-1∫
t
1
t
2
λαdα
P
t
2
z
1
z
-1
α
t
1
t
2
λ
α
To evaluate the inverse
zz-transform, we simply exploit
the Taylor series expression for the exponential, and we find
that a Poisson probability mass function governs the count
statistics for a Poisson process.
Pr
N
t
1
,
t
2
=n=∫
t
1
t
2
λαdαnn!ⅇ-∫
t
1
t
2
λαdα
N
t
1
,
t
2
n
α
t
1
t
2
λ
α
n
n
α
t
1
t
2
λ
α
(2)
The integral of the intensity occurs frequently, and we
succinctly denote it by
Λ
t
1
t
2
Λ
t
1
t
2
. When the Poisson process is
stationary, the intensity equals a constant, and the count
statistics depend only on the difference
t
2
−
t
1
t
2
t
1
.
To derive the multivariate distribution of WW, we use the count statistics
and the independence properties of the Poisson process. The
density we seek satisfies
∫
w
1
w
1
+
δ
1
…∫
w
n
w
n
+
δ
n
p
W
n
vdvdv=Pr
W
1
∈
w
1
w
1
+
δ
1
…
W
n
∈
w
n
w
n
+
δ
n
v
w
1
w
1
δ
1
…
v
w
n
w
n
δ
n
p
W
n
v
W
1
w
1
w
1
δ
1
…
W
n
w
n
w
n
δ
n
The expression on the right equals the probability that no
events occur in
t
1
w
1
t
1
w
1
, one event in
w
1
w
1
+
δ
1
w
1
w
1
δ
1
, no event in
w
1
+
δ
1
w
2
w
1
δ
1
w
2
, etc. Because of the independence of event
occurrence in these disjoint intervals, we can multiply
together the probability of these event occurrences, each of
which is given by the count statistics.
Pr
W
1
∈
w
1
w
1
+
δ
1
…
W
n
∈
w
n
w
n
+
δ
n
=ⅇ-
Λ
t
1
w
1
Λ
w
1
w
1
+
δ
1
ⅇ-
Λ
w
1
w
1
+
δ
1
ⅇ-
Λ
w
1
+
δ
1
w
2
Λ
w
2
w
2
+
δ
2
ⅇ-
Λ
w
2
w
2
+
δ
2
…
Λ
w
n
w
n
+
δ
n
ⅇ-
Λ
w
n
w
n
+
δ
n
≈∏k=1nλ
w
k
δ
k
ⅇ-
Λ
t
1
w
n
W
1
w
1
w
1
δ
1
…
W
n
w
n
w
n
δ
n
Λ
t
1
w
1
Λ
w
1
w
1
+
δ
1
Λ
w
1
w
1
+
δ
1
Λ
w
1
+
δ
1
w
2
Λ
w
2
w
2
+
δ
2
Λ
w
2
w
2
+
δ
2
…
Λ
w
n
w
n
+
δ
n
Λ
w
n
w
n
+
δ
n
k
1
n
λ
w
k
δ
k
Λ
t
1
w
n
for small δ
k δ
k . From this
approximation, we find that the joint distribution of
the first nn event times
equals
pW
n
w=∏k=1nλ
w
k
ⅇ-∫
t
1
w
n
λαdαif
t
1
≤
w
1
≤
w
2
≤…≤
w
n
0otherwise
p
W
n
w
k
1
n
λ
w
k
α
t
1
w
n
λ
α
t
1
w
1
w
2
…
w
n
0
(3)
For Poisson processes, the sample function density describes
the joint distribution of counts and event times within a
specified time interval. Thus, it can be written as
Pr
N
t
|
t
1
≤t<
t
2
=Pr
N
t
1
,
t
2
=n|
W
1
=
w
1
…
W
n
=
w
n
pW
n
w
t
1
t
t
2
N
t
W
1
w
1
…
W
n
w
n
N
t
1
,
t
2
n
p
W
n
w
The second term in the product equals the distribution derived
previously for the time of occurrence statistics. The
conditional probability equals the probability that no events
occur between w
n w
n and
t 2
t 2 ; from the Poisson process's count statistics, this
probability equals
ⅇ-
Λ
w
n
t
2
Λ
w
n
t
2
. Consequently, the sample function density for the
Poisson process, be it stationary or not, equals
Pr
N
t
|
t
1
≤t<
t
2
=∏k=1nλ
w
k
ⅇ-∫
t
1
t
2
λαdα
t
1
t
t
2
N
t
k
1
n
λ
w
k
α
t
1
t
2
λ
α
(4)
From the probability distributions derived on the previous
pages, we can discern many structural properties of the
Poisson process. These properties set the stage for
delineating other point processes from the Poisson. They, as
described subsequently, have much more structure and are much
more difficult to handle analytically.
The counting process
N
t
N
t
is an independent increment process.
For a Poisson process, the number of events in
disjoint intervals are statistically independent of each
other, meaning that we have an independent increment
process. When the Poisson process is stationary, increments
taken over equi-duration intervals are identically
distributed as well as being statistically independent. Two
important results obtain from this property. First, the
counting process's covariance function
K
N
tu
K
N
t
u
equals
σ2min{tu}
σ
2
t
u
. This close relation to the Wiener waveform
process indicates the fundamental nature of the Poisson
process in the world of point processes. Note, however,
that the Poisson counting process is
not continuous almost surely. Second,
the sequence of counts forms an ergodic process, meaning we
can estimate the intensity parameter from observations.
The mean and variance of the number of events in an interval
can be easily calculated from the Poisson distribution.
Alternatively, we can calculate the characteristic function
and evaluate its derivatives. The characteristic function
of an increment equals
Φ
N
t
1
,
t
2
v=ⅇⅇⅈv−1
Λ
t
1
t
2
Φ
N
t
1
,
t
2
v
v
1
Λ
t
1
t
2
The first two moments and variance of an increment of the
Poisson process, be it stationary or not, equal
E
N
t
1
,
t
2
=
Λ
t
1
t
2
N
t
1
,
t
2
Λ
t
1
t
2
(5)
E
N
t
1
,
t
2
2=
Λ
t
1
t
2
+
Λ
t
1
t
2
2
N
t
1
,
t
2
2
Λ
t
1
t
2
Λ
t
1
t
2
2
σ
N
t
1
,
t
2
2=
Λ
t
1
t
2
N
t
1
,
t
2
Λ
t
1
t
2
Note that the mean equals the variance here, a trademark of
the Poisson process.
Consider the conditional density
p
W
n
|
W
n
-
1
,
…
,
W
1
w
n
|
w
n
-
1
,
…
,
w
1
p
W
n
|
W
n
-
1
,
…
,
W
1
w
n
|
w
n
-
1
,
…
,
w
1
.
This density equals the ratio of the event time densities
for the nn- and (
n−1
n
1
)-dimensional event time vectors. Simple
substitution yields
∀
w
n
,
w
n
≥
w
n
-
1
:
p
W
n
|
W
n
-
1
,
…
,
W
1
w
n
|
w
n
-
1
,
…
,
w
1
=λ
w
n
ⅇ-∫
w
n
-
1
w
n
λαdα
w
n
w
n
w
n
-
1
p
W
n
|
W
n
-
1
,
…
,
W
1
w
n
|
w
n
-
1
,
…
,
w
1
λ
w
n
α
w
n
-
1
w
n
λ
α
(6)
Thus the
n
th
n
th
event time depends only on when the
n−1th
n
1
th
event occurs, meaning that we have a Markov
process. Note that event times are ordered: the
n th
n th
event must occur after the
n−1th
n
1
th
, etc. Thus, the values of this Markov process
keep increasing, meaning that from this viewpoint, the event
times form a
nonstationary Markovian sequence.
When the process is stationary, the evolutionary density is
exponential. It is this special form of event occurence
time density that defines a Poisson process.
Exploiting the previous property, the duration of the
n
th
n
th
interval
τ
n
=
w
n
−
w
n
-
1
τ
n
w
n
w
n
-
1
does not depend on the lengths of previous (or
future) intervals. Consequently, the sequence of interevent
intervals forms a "white" sequence. The sequence may not be
identically distributed unless the process is stationary.
In the stationary case, interevent intervals are truly
white - they form an IID sequence - and have an exponential
distribution.
∀τ,τ≥0:p
τ
n
τ=
λ
0
ⅇ-
λ
0
τ
τ
τ
0
p
τ
n
τ
λ
0
λ
0
τ
(7)
To show that the exponential density for a white sequence
corresponds to the most "random" distribution,
Parzen proved that the
ordered times of
nn events sprinkled
independently and uniformly over a given interval form a
stationary Poisson process. If the density of event
sprinkling is not uniform, the resulting ordered times
constitute a nonstationary Poisson process with an intensity
proportional to the sprinkling density.
Here, the intensity
λt
λ
t
equals a sample function drawn from some waveform
process. In waveform processes, the analogous concept does
not have nearly the impact it does here. Because intensity
waveforms must be non-negative, the intensity process
must be nonzero mean and non-Gaussian.
The authors shall assume throughout that the intensity
process is stationary for simplicity. This model arises in
those situations in which the event occurrence rate clearly
varies unpredictably with time. Such processes have the
property that the variance-to-mean ratio of the number of
events in any interval exceeds one. In the process of
deriving this last property, we illustrate the typical way
of analyzing doubly stochastic processes: Condition on the
intensity equaling a particular sample function, use the
statistical characteristics of nonstationary Poisson
processes, then "average" with respect to the intensity
process. To calculate the expected number
N
t
1
,
t
2
N
t
1
,
t
2
of events in an interval, we
use conditional expected values:
E
N
t
1
,
t
2
=EE
N
t
1
,
t
2
|∀λt,
t
1
≤t<
t
2
=E∫
t
1
t
2
λαdα=
t
2
−
t
1
Eλt
N
t
1
,
t
2
λ
t
t
1
t
t
2
N
t
1
,
t
2
α
t
1
t
2
λ
α
t
2
t
1
λ
t
(8)
This result can also be written as the expected value of the
integrated intensity:
E
N
t
1
,
t
2
=E
Λ
t
1
t
2
N
t
1
,
t
2
Λ
t
1
t
2
. Similar calculations yield the increment's second
moment and variance.
E
N
t
1
,
t
2
2=E
Λ
t
1
t
2
+E
Λ
t
1
t
2
2
N
t
1
,
t
2
2
Λ
t
1
t
2
Λ
t
1
t
2
2
σ
N
t
1
,
t
2
2=E
Λ
t
1
t
2
+σ
Λ
t
1
t
2
2
N
t
1
,
t
2
Λ
t
1
t
2
Λ
t
1
t
2
Using the last result, we find that the variance-to-mean ratio
in a doubly stochastic process always exceeds unity, equaling
one plus the variance-to-mean ratio of the intensity process.
The approach of sample-function conditioning can also be used
to derive the density of the number of events occurring in an
interval for a doubly stochastic Poisson process. Conditioned
on the occurrence of a sample function, the probability of
nn events occurring in the
interval
t
1
t
2
t
1
t
2
equals (Equation 1)
Pr
N
t
1
,
t
2
=n|∀λt,
t
1
≤t<
t
2
=
Λ
t
1
t
2
nn!ⅇ-
Λ
t
1
t
2
λ
t
t
1
t
t
2
N
t
1
,
t
2
n
Λ
t
1
t
2
n
n
Λ
t
1
t
2
Because
Λ
t
1
t
2
Λ
t
1
t
2
is a random variable, the unconditional
distribution equals this conditional probability averaged
with respect to this random variable's density. This
average is known as the Poisson Transform of the random
variable's density.
Pr
N
t
1
,
t
2
=n=∫0∞αnn!ⅇ-αp
Λ
t
1
t
2
αdα
N
t
1
,
t
2
n
α
0
α
n
n
α
p
Λ
t
1
t
2
α
(9)
-
D. L. Snyder. (1975). Random Point Processes. New York: Wiley.
-
E. Parzen. (1962). Stochastic Processes. San Francisco: Holden-Day.
