sequential detection overview & open problems george v. moustakides, university of patras,...

Sequential DetectionSequential DetectionOverview & Open ProblemsOverview & Open Problems

George V. Moustakides,George V. Moustakides,University of Patras, GREECEUniversity of Patras, GREECE

OutlineOutline Sequential hypothesis testingSequential hypothesis testing

SPRTSPRT Application from databasesApplication from databases Open problemsOpen problems

Sequential detection of changesSequential detection of changes The Shiryaev testThe Shiryaev test The Shiryaev-Roberts testThe Shiryaev-Roberts test The CUSUM testThe CUSUM test Open problemsOpen problems Decentralized detection (sensor networks)Decentralized detection (sensor networks)

GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 2

Sequential Hypothesis testingSequential Hypothesis testingConventional binary hypothesis testingConventional binary hypothesis testing::Fixed sample size observation vector Fixed sample size observation vector XX=[=[xx11,...,,...,xxKK]] XX satisfies the following two hypotheses: satisfies the following two hypotheses:

GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011

Given the data vector Given the data vector XX, , decidedecide between the two between the two hypotheses.hypotheses.

3

Decision ruleDecision rule::


Bayesian formulationBayesian formulation

Likelihood ratio test:Likelihood ratio test:

4

Neyman-Pearson formulationNeyman-Pearson formulation

For i.i.d.:For i.i.d.:


Observations Observations xx11,...,,...,xxtt,...,... become available become available sequentiallysequentially

5

Sequential binary hypothesis testingSequential binary hypothesis testing

Time Observations DecisionTime Observations Decision

Decide ReliablyDecide Reliably

Decide as soon Decide as soon as possible!as possible!

YeYess

NoNoNoNo


Time ObservationsTime Observations

We apply a We apply a two-rule two-rule procedureprocedure

11stst rule rule: : at each time instant at each time instant tt, evaluates whether , evaluates whether the observed data can lead to a reliable decisionthe observed data can lead to a reliable decision

Can Can xx11 lead to a lead to a reliable decision?reliable decision?Can Can xx11 lead to a lead to a

reliable decision?reliable decision?Can Can xx11,,xx22 lead to lead to

a reliable a reliable decision?decision?

Can Can xx11,,xx22 lead to lead to a reliable a reliable decision?decision?Can Can xx11,,xx22,...,,..., x xTT

lead to a reliable lead to a reliable decision?decision?

Can Can xx11,,xx22,...,,..., x xTT lead to a reliable lead to a reliable

decision?decision?STOPSTOP getting more data. getting more data.

22ndnd rule rule: : Familiar decision ruleFamiliar decision ruleTT is a is a stopping stopping rule rule Random!Random!TT is a is a stopping stopping rule rule Random!Random!


Why Sequential ?Why Sequential ?

In averageIn average, we need significantly less samples to , we need significantly less samples to reach a decision than the fixed sample size test, reach a decision than the fixed sample size test, for the same level of confidence (same error for the same level of confidence (same error probabilities)probabilities)

For the Gaussian case it is 4 - 5 times less samples.For the Gaussian case it is 4 - 5 times less samples.

SPRT SPRT (Wald 1945)(Wald 1945)


Here there are Here there are twotwo thresholds thresholds AA < < 0 0 < < BB

Stopping rule:Stopping rule:

Decision rule:Decision rule:


uutt

tt

BB

AA

00TT

HH00

HH11

Infinite HorizonInfinite Horizon


Amazing optimality property!!!Amazing optimality property!!!

SPRT solves both problemsSPRT solves both problems simultaneouslysimultaneously AA,,BB need to be selected to satisfy the two error need to be selected to satisfy the two error probability constraints with equalityprobability constraints with equality

I.i.d. observations (1948, Wald-Wolfowitz)I.i.d. observations (1948, Wald-Wolfowitz) Brownian motion (1967, Shiryaev)Brownian motion (1967, Shiryaev) Homogeneous Poisson (2000, Peskir-Shiryaev)Homogeneous Poisson (2000, Peskir-Shiryaev)

Open ProblemsOpen Problems: Dependency, Multiple Hypotheses: Dependency, Multiple Hypotheses


uutt

BB

AA

00

HH00

HH11

TT

tt KK

Finite HorizonFinite Horizon


attribute 1attribute 1 attribute 2attribute 2 attribute 3attribute 3 ... attribute attribute KK

...

...

...

......

...

Data Base (records)Data Base (records)

...00 00 11 11

Record linkageRecord linkage

xx11 xx22 xx33 xxKK

We assume known probabilities for We assume known probabilities for xxii =0 or 1 under match =0 or 1 under match (Hypothesis H(Hypothesis H11) or nonmatch (Hypothesis H) or nonmatch (Hypothesis H00) for each attribute.) for each attribute.


Data from: Statistical Research Division of the US Bureau of the Census


Total number of record comparisons: 3703


In collaboration with V. VerykiosIn collaboration with V. Verykios

Sequential change detectionSequential change detection


tt¿¿

Change in Change in statisticsstatistics

Change in Change in statisticsstatistics

Detect occurrence Detect occurrence as soon as as soon as possiblepossible

Detect occurrence Detect occurrence as soon as as soon as possiblepossible

xxtt


ApplicationsApplicationsMonitoring of quality of manufacturing process (1930’s)Biomedical EngineeringElectronic CommunicationsEconometricsSeismologySpeech & Image ProcessingVibration monitoringSecurity monitoring (fraud detection)Spectrum monitoringScene monitoringNetwork monitoring and diagnostics (router failures, intruder detection)Databases .....


Mathematical setupMathematical setup

We observe We observe sequentiallysequentially a process { a process {xxtt} that has the } that has the following statistical propertiesfollowing statistical properties

Changetime Changetime ¿¿ : either random with known prior or : either random with known prior or deterministic but unknown.deterministic but unknown. Both pdfs Both pdfs ff00, , ff11 are considered known. are considered known.

Detect occurrence of Detect occurrence of ¿¿ as soon as possible as soon as possible


We are interested in We are interested in sequentialsequential detection schemes. detection schemes.

WhateverWhatever sequential scheme one can think of, at sequential scheme one can think of, at every time instant every time instant tt it will have to make one of the it will have to make one of the following two decisions:following two decisions:

Either decide that a change didn’t take place Either decide that a change didn’t take place before before tt, therefore it needs to continue taking more , therefore it needs to continue taking more data.data. Or that a change took place before Or that a change took place before tt and therefore and therefore it should stop and issue an alarm.it should stop and issue an alarm.

Sequential Detector Sequential Detector Stopping rule Stopping rule


Shiryaev test Shiryaev test (Bayesian, Shiryaev 1963)(Bayesian, Shiryaev 1963)

Changetime Changetime ¿¿ is random with Geometric prior. is random with Geometric prior.

Optimum Optimum TT : :

If If TT is a stopping rule then we define the following is a stopping rule then we define the following cost functioncost function


Define the statistic:Define the statistic:

There exists such that the following rule is There exists such that the following rule is optimum. optimum.

In discrete time when {In discrete time when {xxtt} are i.i.d. before and after } are i.i.d. before and after the change.the change.In continuous time when {In continuous time when {xxtt} is a Brownian motion } is a Brownian motion with constant drift before and after the change.with constant drift before and after the change.In continuous time when {In continuous time when {xxtt} is Poisson with constant } is Poisson with constant rate before and after the change.rate before and after the change.


Shiryaev-Roberts test Shiryaev-Roberts test (Minmax, Pollak 1985)(Minmax, Pollak 1985)

Changetime Changetime ¿¿ is deterministic and unknown. is deterministic and unknown.

For any stopping rule For any stopping rule TT define the following define the following criterion:criterion:


subject tosubject to


In discrete time, when data are i.i.d. before and after In discrete time, when data are i.i.d. before and after the change with pdfs the change with pdfs ff00,,ff11..

Compute recursively the following statistic:Compute recursively the following statistic:

Yakir (AoS 1997) provided a proof of strict optimality.Yakir (AoS 1997) provided a proof of strict optimality.Mei (2006) showed that the proof was problematic.Mei (2006) showed that the proof was problematic.

Pollak (1985) Pollak (1985)

Tartakovsky (2011) found a counterexample.Tartakovsky (2011) found a counterexample.


CUSUM test CUSUM test (minmax, Lorden 1971)(minmax, Lorden 1971)

Changetime Changetime ¿¿ is deterministic and unknown. is deterministic and unknown.

For any stopping rule For any stopping rule TT define the following define the following criterion:criterion:


subject tosubject to


Compute the Cumulative Sum (CUSUM) statistic Compute the Cumulative Sum (CUSUM) statistic yytt

as follows:as follows:

Running LLRRunning LLR

Running minimumRunning minimum

For i.i.d.For i.i.d.

CUSUM testCUSUM test

CUSUM statisticCUSUM statistic


ut

mt S

º

º º


Moustakides (1986) strict optimality.Moustakides (1986) strict optimality.

Continuous timeContinuous timeShiryaev (1996), Beibel (1996) strict optimality for BMShiryaev (1996), Beibel (1996) strict optimality for BMMoustakides (2004) strict optimality for Ito processesMoustakides (2004) strict optimality for Ito processesMoustakides (>2012) strict optimality for Poisson Moustakides (>2012) strict optimality for Poisson processes.processes.

Open ProblemsOpen Problems: Dependent data, Non abrupt : Dependent data, Non abrupt changes, Transient changes,…changes, Transient changes,…

Discrete time:Discrete time: i.i.d. before and after the change i.i.d. before and after the changeLorden (1971) asymptotic optimality.Lorden (1971) asymptotic optimality.


Decentralized detectionDecentralized detection


tt

BB11

AA11

00

00

11

uutt11

tt11

tt22

uutt11 uutt

11

11-

If more than 1 bits,If more than 1 bits, quantize overshoot!quantize overshoot!


The Fusion center if at time The Fusion center if at time ttnn receives information receives information from sensor from sensor ii it updates an estimate of the global log- it updates an estimate of the global log-likelihood ratio:likelihood ratio:

and performs an SPRT (if hypothesis testing) or a and performs an SPRT (if hypothesis testing) or a CUSUM (if change detection) using the estimate of CUSUM (if change detection) using the estimate of the global log-likelihood ratio.the global log-likelihood ratio.


Thank you for your Thank you for your attentionattention

sequential detection overview & open problems george v. moustakides, university of patras,...

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