Detecting changes in the rate Poisson process

Detecting changes in
the rate
of a
Poisson process
George V. Moustakides

Outline
Overview of the change detection
problem
CUSUM test and Lorden’s criterion
The Poisson disorder problem
CUSUM average run length for Poisson
processes
CUSUM optimality in the sense of Lorden
G.V. Moustakides: Detecting changes in the rate of a Poisson process 2

Change detection - Overview
Available sequentially an observation process
{ξ t } with the following statistics:
ξ t ~ P ∞ for 0 6 t 6 τ
~ P 0 for τ

We are interested in sequential schemes.
With every new observation the test must decide
Stop and issue an alarm
Continue sampling
Decision at time t uses available information
F t = σ{ξ s :06 s 6 t}.
up to time t.
Sequential test stopping time T adapted to
the filtration {F t }.
G.V. Moustakides: Detecting changes in the rate of a Poisson process 4

0
P ∞ P 0
τ
t
P τ
: the probability measure induced, when
change takes place at time τ
E τ [.]: the corresponding expectation
P ∞
P 0
: all data under nominal regime
: all data under alternative regime
Parameters to be considered
The detection delay T - τ
Frequency of false alarms
G.V. Moustakides: Detecting changes in the rate of a Poisson process 5

Bayesian approach (Shiryayev 1978)
Change time τ random with exponential prior.
J(T ) = c E[ (T - τ) + ] + P[ T ν }
Discrete time: i.i.d. observations
(Shiryayev 1978, Poor 1998)
Continuous time: Brownian Motion
(Shiryayev 1978, Beibel 2000, Karatzas 2003)
G.V. Moustakides: Detecting changes in the rate of a Poisson process 6

Non-Bayesian setup (Pollak 1985)
The change time τ is deterministic & unknown.
J(T ) = sup τ E τ [ (T - τ) | T >τ ]
Optimization problem: inf T J(T )
subject to: E ∞ [ T ] > γ
Discrete time: i.i.d. detect change in the pdf
from f ∞ (ξ) to f 0 (ξ). Roberts (1966) proposed
f 0 (ξ t )
S t = (S t-1 +1)
f ∞ (ξ t )
T SRP = inf t { t: S t > ν }
(Mei 2006)
G.V. Moustakides: Detecting changes in the rate of a Poisson process 7

CUSUM test and Lorden’s criterion
Discrete time, i.i.d. observations.
Pdf before and after the change: f ∞ (ξ n ), f 0 (ξ n )
Since change time τ is unknown
sup 06τ 6t
n=τ+1
t
Σn=1
t
f 0 (ξ n )
Σlog( )
f ∞ (ξ n )
-inf 06τ6t Σn=1
> ν
f 0 (ξ n )
log( )
f 0 (ξ n )
f ∞ (ξ n )
f ∞ (ξ n )
τ
log( )
> ν
u t
– m t
> ν
G.V. Moustakides: Detecting changes in the rate of a Poisson process 8

dP 0
u t = log( (F
dP t ))
∞
m t = inf 06s 6t u s
CUSUM process: y t = u t – m t > 0
The CUSUM stopping time (Page 1954):
T C = inf t { t: y t > ν }
G.V. Moustakides: Detecting changes in the rate of a Poisson process 9

ν
ν
ν
u t
m t
ML estimate of τ
T C
G.V. Moustakides: Detecting changes in the rate of a Poisson process 10

Non-Bayesian setup (Lorden 1971).
Change time τ is deterministic and unknown.
J(T ) = sup τ essup E τ [ (T - τ) + | F τ ]
Optimization problem: inf T J(T )
subject to: E ∞ [ T ] > γ
Discrete time: i.i.d. observations
(Moustakides 1986, Ritov 1990, Poor 1998)
Continuous time: BM (Shiryayev 1996, Beibel
1996); Ito processes (Moustakides 2004)
G.V. Moustakides: Detecting changes in the rate of a Poisson process 11

The Poisson disorder problem
Let {N t } homogeneous Poisson, with rate λ
satisfying:
λ = {
Bayesian Approach
λ ∞ , 0 6 t 6 τ
λ 0 , τ

CUSUM & average run length
u t = (λ ∞ - λ 0 )t + log(λ 0 /λ ∞ )N t
m t = inf 06s 6t u s
y t = u t – m t
T C = inf t { t: y t > ν }
We are interested in computing E[ T C ] when
N t is Poisson with rate λ.
Existing formula (Taylor 1975) for:
u t = at + bW t ; W t is standard Wiener
G.V. Moustakides: Detecting changes in the rate of a Poisson process 13

a>0; b

G.V. Moustakides: Detecting changes in the rate of a Poisson process 15

We end up with a DDE and the following
boundary conditions:
af 0 (y) + λ[f(y + b) - f(y)] = -1; y∈[0,ν)
f(y) = f(0) for y 6 0;
f(y 0 ) = E[ T C ]
u t
f(ν)=0
m t
T C
ν
G.V. Moustakides: Detecting changes in the rate of a Poisson process 16

(
Existing formula (Taylor 1975) for:
u t = at + bW t ; W t standard Wiener
b 2 f 00 (y) + af 0 (y) = -1; y∈[0,ν ]
2
f 0 (0)= 0; f(ν)=0
G.V. Moustakides: Detecting changes in the rate of a Poisson process 17
)

af 0 (y) + λ[f(y +b) - f(y)] = -1; y∈[0,ν)
f(y) = f(0) for y 6 0; f(ν)=0
Because b

a0
af 0 (y) + λ[f(y + b) - f(y)] = -1; y∈[0,ν)
f 0 (0)= 0; f(y)=0 for y > ν
Backward DDE
G.V. Moustakides: Detecting changes in the rate of a Poisson process 19

where p is defined as
with
G.V. Moustakides: Detecting changes in the rate of a Poisson process 20

Average over 10000 repetitions:
λ ∞ = 2, λ 0 = 1, (a=1, b=-log2), ν = 5.5
Formula
Simulation
E 0 [ T C ] 15.3832 15.3605
E ∞ [ T C ] 779.9669 771.1219
λ ∞ = 1, λ 0 = 2, (a=-1, b=log2), ν = 5.5
Formula
Simulation
E 0 [ T C ] 12.2885 12.2673
E ∞ [ T C ] 981.9811 986.7159
G.V. Moustakides: Detecting changes in the rate of a Poisson process 21

Optimality of CUSUM
J(T ) = sup τ essup E τ [ (T - τ) + | F τ ]
inf T J(T );
subject to: E ∞ [ T ] > γ
If T is such that
E ∞ [ T ] > E ∞ [ T C ] = γ
then
J(T ) > J(T C )
G.V. Moustakides: Detecting changes in the rate of a Poisson process 22

h(y) = E ∞ [ T C | y 0 =y]
g(y) = E 0 [ T C | y 0 =y]
essupE τ [(T C - τ) + | F τ ]=sup y g(y)=g(0)
T C is an equilizer rule therefore
J(T C ) = g(0)
For the false alarm we have
E ∞ [ T C ] = h(0)
G.V. Moustakides: Detecting changes in the rate of a Poisson process 23

We would like to show:
If E ∞ [T ] > E ∞ [T C ]
then J(T ) > J(T C )
Lemma
Sufficient:
If E ∞ [T ] > h(0)
then J(T ) > g(0)
G.V. Moustakides: Detecting changes in the rate of a Poisson process 24

We will show that this is true for any T
G.V. Moustakides: Detecting changes in the rate of a Poisson process 25

Consider the function f(y) defined as follows
then
f(y)=e y [g(0) - g(y)] - [h(0) - h(y)]
> 0
> 0 ¥
G.V. Moustakides: Detecting changes in the rate of a Poisson process 26

Conclusion
We considered the Poisson disorder problem
of detecting changes in the rate of a
homogeneous Poisson process, in the sense
of Lorden.
We obtained closed form expressions for the
average run length of the CUSUM stopping
time.
We used these formulas to prove optimality of
the CUSUM test in the sense of Lorden.
G.V. Moustakides: Detecting changes in the rate of a Poisson process 27

EnD
G.V. Moustakides: Detecting changes in the rate of a Poisson process 28