George V. MoustakidesDepartment of Electrical & Computer Engineering

UNIVERSITY OF PATRAS, GREECE

Detecting Changesthat occur atObservable Random Times

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 2

Outline

The sequential change detection problem, Lorden’s performance measure and the CUSUM test

Variant of the change detection problem and modification of Lorden’s criterion

Extended CUSUM test and performance evaluation for the BM case

Optimality of ECUSUM test

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 3

Sequential change detectionWe are observing sequentially a process {ξt}with the following statistics:

ξt ~ P∞ for 0 6 t 6 τ

~ P0 for τ < t

Change time τ : deterministic (but unknown)Probability measures P∞ , P0 : known

Applications include: systems monitoring; quality control; financial decision making; remote sensing (radar, sonar, seismology);...

Detect the change as soon as possible

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 4

We are interested in sequential schemes.

With every new observation the test must decideStop and issue an alarmContinue sampling

Decision at time t uses available information Ft = σ{ξs : 0 6 s 6 t}.

up to time t.

Sequential test stopping time T adapted to the filtration {Ft}.

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 5

Lorden’s performance measure

τ t

P∞ P0

0

Pτ : the probability measure induced, whenchange takes place at time τ

Eτ[.]: the corresponding expectationP∞ : all data under nominal regimeP0 : all data under alternative regime

Lorden’s performance measure (1971):

J(T ) = supτ>0 essup Eτ[ (T - τ)+ | Fτ ]

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 6

Optimization problem: infT J(T )

subject to: E∞[ T ] > γ

The CUSUM rule (Page 1954):

Sν = inft>0 { t: yt > ν }

Optimality of CUSUM in continuous time: when {ξt} is a BM with constant drift (Shiryayev, Beibel 1996) and; when {ξt} is Ito process (Moustakides 2004).

mt = inf06s 6t us .dP0

dP∞(Ft)ut = log( )

CUSUM process: yt = ut – mt > 0

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 7

Sν

νν

ut

mt

ν

ML estimate of τ

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 8

Variant of change detection problemIn addition to the observation process {ξt} we have available (deterministic) sequence of times {tn} with tn ∞. We assume that for some n

ξt ~ P∞ for 0 6 t 6 tn~ P0 for tn < t

The change can occur only at some time instant tn from the known sequence.

J(T ) = supn>0 essup Et [ (T - tn)+ | Ft ]

n n

Lorden’s criterion must be modified as follows:

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 9

In addition to the observation process {ξt} we observe a sequence of random times {τn}

with τn ∞ a.s. We assume that for some n

ξt ~ P∞ for 0 6 t 6 τn~ P0 for τn < t

Earthquake damage detection in structures.

Ft = σ{ξs , Ns : 0 6 s 6 t}

Define Nt = supn{τn6 t}

then τn becomes a s.t. adapted to {Ft}

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 10

J(T ) = supn>0 essup Eτ [ (T - τn)+ | Fτ ]

n n

Optimization problem: infT J(T )

subject to: E∞[ T ] > γ

Extended CUSUM (ECUSUM)

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 11

Sν

ν

ut

mt

τ1

τ2τ3

τ4 τ5

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 12

Performance of ECUSUM test for BM

The ECUSUM test takes the form

Let

The number of occurrences {Nt} is independent from {ξt} and Poisson distributed with constant rate λ.

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 13

The expectation h(y)=E∞[Sν |y0=y] satisfies

The expectation g(y)=E0[Sν |y0=y] satisfies

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 14

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 15

Optimality of ECUSUMSelect ν to satisfy

h(0)=E∞[Sν]=γ

(the false alarm constraint with equality). Then the Lorden measure of ECUSUM is

J(Sν)=g(0)

For optimality, sufficient to show that for any Tsatisfying: E∞[ T ] > γ = h(0)

we have J(T) > g(0)

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 16

Define the function

then we can verify that p(y) 6 0.

6 0

This implies

¥¥

6 0

G.V. Moustakides, University of Patras, Greece. IWSM-2007. 17

EnDEnDThank you for your Thank you for your

attention!!attention!!