Quickest detection
and
the
problem
of
two-sided
alternatives
Olympia Hadjiliadis
Outline
 The change detection problem
 Overview of existing results
 Lorden’s criterion & the CUSUM stopping time
 The Brownian motion model with multiple alternatives
 A modified Lorden criterion
 Optimality of the CUSUM rule for 1-sided alternatives
 Two sided 2-CUSUM test & the HMR
 Further optimality issues for 2-sided alternatives
& Optimality of Equalizer Rules
 Discussion and Open Problems
The change detection problem
We are observing sequentially a process  t with the
following characteristics:
 Change time
: deterministic (but unknown) or random
 In control distribution is known
 Out of control can be known or unknown
Detect the change “as soon as possible”
Applications include: systems monitoring, quality control,
financial decision making, remote sensing (radar, sonar,
seismology), occurrence of industrial accidents,
epidemiology etc
The observation process »t becomes available
sequentially;
This can be expressed through the filtration:
For detecting the change we are interested in
sequential schemes
Any sequential detection scheme can be represented
by a stopping time T (the time we stop & declare the
change)
The stopping time T is adapted to
In other words, at every instant t, we perform a test
(whether to stop or continue sampling) using only
available information up to time t
Overview of Existing Results
: the probability measure induced by , when the
change takes place at time
: the corresponding expectation
: all data under nominal (in-control) regime
: all data under alternative (out-of-control) regime
Optimality Criteria
They are basically a trade-off of two parts:
 The first is the detection delay
(out-of-control A.R.L.)
 The second is the frequency of false alarms
(in-control A.R.L.)
Possible approaches are Bayesian & Min-Max
Bayesian Approach (Shiryaev):
is random and exponentially distributed
The Shiryaev test consists of computing the statistics
and stop when
is optimum (Shiryaev 1978):
In discrete time: when
are i.i.d. before and after the
change
In continuous time: when
is a Brownian Motion with
constant drift before and after the change
Min-Max Approach (Shiryayev-Roberts-Pollak)
is deterministic and unknown
subject to
Optimality results exist only for discrete time when
are i.i.d before and after the change. Specifically if we
define the statistics
where
,
the common pdf of the data before and
after the change then (Yakir 1997) the stopping time
is optimum
Lorden’s Criterion
An alternative min-max approach: (Lorden 1971)
And solve the min-max problem
subject to
The test closely related to Lorden’s criterion and being
the most popular one used in practice is the Cumulative
Sum (CUSUM) test
Remark: In seeking solutions need only
The CUSUM statistic process
Define the CUSUM statistics
as follows:
;
The CUSUM stopping time (Page 1954):
where the threshold º is selected so that
Optimality results
Discrete time: when
are i.i.d. before and after the
change (Moustakides 1986, Ritov 1990)
Continuous time: when
is a Brownian Motion with
constant drift before and after the change (Shiryayev
1996, Beibel 1996)
0.4
0.0
ut
-0.4
mt
0.2
0.8
0.0
0.4
0.6
0.8
1.0
0.6
0.8
1.0
0.0
0.4
yt
0.0
0.2
0.4
TC
WHY THE CUSUM?
X1, X2,…,Xt i.i.d.
-Composite hypothesis testing
H0 : no change vs
H1 : change at time 1
or
H2 : change at time 2
or ...
Ht : change at time t
The Brownian motion model with one
alternative
We observe sequentially the process
following dynamics:
where the drift ¹ is known.
with the
Lorden’s criterion & the CUSUM
The Brownian motion model with
two/multiple alternatives
We observe sequentially the process
following dynamics:
where the
with the
are known
Remark: We are not interested in identifying which of
the
occurs, just in detecting the change time .
Due to the symmetry of the BM, it suffices to consider the
following cases:
Case 1:
Case 2 (Symmetric):
Case 3 (Asymmetric):
A modified Lorden Criterion
One possible way to extend Lorden’s criterion, to include
multiple changes, is in the following way:
We want to solve the min-max problem with the following
optimality criterion:
subject to
Remark: We are interested solely in the detection
problem. No special care is taken for estimating the
type of change (i.e. identifying i).
Optimality Issues
Case 1:
We can show that the 1-sided CUSUM test, run for
is optimal (in a sense detecting
is the worst case
scenario.
Key point in the proof is the following inequality:
6
5
4
3
2
1
0
T2
0.0
T2
T1
m
0.2
T1
0.4
0.6
0.8
1.0
Change-point detection in B.M. model
The process



is:
is observed sequentially
deterministic: (Min-Max approach)
and
: are known
Extended Lorden’s criterion
subject to
The optimal solution has to satisfy…
?
Let S be a s.t.
&
Let U be s.t. declares an alarm at the same time as S but when “-”
observations are received
Construct a randomized rule V : Flip an unbiased coin.
If Heads then follow U. If Tails then follow S.
The classical 2-CUSUM rule
And the corresponding CUSUM stopping times:
Then the 2-sided CUSUM stopping time is:
Notice
The best 2-CUSUM
OBJECTIVE:
Look in the 2-CUSUM class and find the
best rule (
) in both:

Symmetric case

Non-symmetric case
 Harmonic Mean 2-CUSUM rules
 Non-harmonic mean 2-CUSUM rules
0.0
0.2
0.4
0.6
0.8
1.0
Harmonic Mean 2-CUSUM rules
0 .0
0 .2
0 .4
0 .6
HMR:
Side
results
,
0 .8
1 .0
The best 2-CUSUM
Symmetric case
Pick
i.e. a Harmonic mean 2-CUSUM rule
Hence the best 2-CUSUM rules is a HMR
Is it unique?
Non-symmetric case
?
Non Harmonic Mean 2-CUSUM rules
0.0
0.2
0.4
0.6
0.8
1.0
Case
0 .0
0 .2
0 .4
Tc
1
0 .6
Tc
2
0 .8
1 .0
?
Non-Harmonic Mean 2-CUSUM (cont.)
Suppose that
for
big.
Therefore, the event
conditioned upon
For general
?
is
Non-Harmonic mean 2-CUSUM (cont’d)
change is
0
Andersen
(1960)
-2
-1
upl
1
2
no change
The first moment
change is
0
20
40
60
80
100
x
The FIRST MOMENT of a 2-CUSUM rule is
Non-harmonic mean 2-CUSUM (cont.)
<
Upper and lower bounds
<
Taylor (1975)
Lehoczky (1977)
no change
change is
change is
Non-harmonic mean 2-CUSUM
?
?
under
; change is
under
; no change
under
; change is
change is
no change
change is
Non-harmonic mean 2-CUSUM
The first moment of a general 2-CUSUM (
Under
Under
Under
)
Modified H.M. Eq. 2-CUSUM rules
And
thetwo
corresponding
CUSUMfor
stopping
Define
CUSUM processes
drifts times:
;
EqR.:
Case 2 (symmetric)
For small
the suggested rule is better than classical 2-CUSUM
Case 3 (asymmetric)
Select a modified 2-CUSUM HMR rule with
Case 3 (Asymmetric):
Define the class of Equalizer Rules:
Theorem
Proof:
LHS of (2) = RHS of (1)
Hence:
LHS of (1) > RHS of (2)
Equalizer Rules
2
1.5
2(µ2-µ1)
1
0.5
1
λ1’ λ1
2
3
λ1+2µ2
λ1’+2µ2
λ2’
4
5
6
λ2
We can rewrite (1) as follows:
7
A comparison
-mod. H.M. Eq. 2-CUSUM rules &
-classical Eq. 2-CUSUM rules
Non-symmetric case (cont.)
Mod. H.M. 2-CUSUM
EqR.:
The best modified H.M. Eq.R 2-CUSUM
OR
Classical 2-CUSUM
EqR.:
Pick
such that
RESULT: moderate or big values of
Classical better than Modified
Symmetric case
m=0.5
3
2.5
Classical 2-CUSUM
2
Modified drift λ 2-CUSUM
1.5
CS
MS
1
0.5
0
0
0.5
1
1.5
2
gamma
2.5
3
3.5
4
Non-symmetric case
m2=0.5,m1/m2=2
4
Modified 2-CUSUM HMR λ2=μ2 , λ1=2μ1–μ2
3.5
3
Modified 2-CUSUM
HMR λ2-λ1=2(μ2–μ1)
2.5
C
Classical 2-CUSUM ν1 > ν2
2
1.5
1
0.5
0.5
1.5
2.5
3.5
gamma
4.5
5.5
6.5
M
MO
The difference of the two detection delays tends to the
constant:
. This proves asymptotic optimality.
True drifts: ¹1=1, ¹2= 1.2;
Drifts used in 2-CUSUM: ¸ 1=1, ¸ 2= 1.4;
True drifts: ¹1=1, ¹2= 1.5
Drifts used in 2-CUSUM: ¸ 1=1, ¸ 2= 2
Publications




"Optimal and Asymptotically Optimal CUSUM rules for change
point detection in the Brownian Motion model with multiple
alternatives"
O. Hadjiliadis and G. V. Moustakides
Theory of Probability and its Applications, 50(1), 2006.
"Optimality of the 2-CUSUM drift equalizer rules for detecting
two-sided alternatives in the Brownian motion model"
O. Hadjiliadis
Journal of Applied Probability, 42(4), 2005.
'‘On the existence and uniqueness of the best 2-CUSUM rules
for quickest detection of two-sided alternatives in a Brownian
motion model''
O. Hadjiliadis and H. V. Poor
[Theory of Probability and its Applications, 53(3), 2008]
'‘A comparison of the best 2-CUSUM rules for quickest
detection of two-sided alternatives in a Brownian motion model''
O. Hadjiliadis G. Hernandez-del-Valle, I.Stamos.
[Sequential Analysis, 28(1), 2009]