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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.
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Decision ruleDecision rule::
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011
Bayesian formulationBayesian formulation
Likelihood ratio test:Likelihood ratio test:
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Neyman-Pearson formulationNeyman-Pearson formulation
For i.i.d.:For i.i.d.:
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011
Observations Observations xx11,...,,...,xxtt,...,... become available become available sequentiallysequentially
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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
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 6
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!
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 7
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)
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 8
Here there are Here there are twotwo thresholds thresholds AA < < 0 0 < < BB
Stopping rule:Stopping rule:
Decision rule:Decision rule:
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 9
uutt
tt
BB
AA
00TT
HH00
HH11
Infinite HorizonInfinite Horizon
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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
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 11
uutt
BB
AA
00
HH00
HH11
TT
tt KK
Finite HorizonFinite Horizon
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 12
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.
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Data from: Statistical Research Division of the US Bureau of the Census
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 14
Total number of record comparisons: 3703
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 15
Total number of record comparisons: 3703
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 16
In collaboration with V. VerykiosIn collaboration with V. Verykios
Sequential change detectionSequential change detection
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 17
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
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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 .....
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 19
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
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 20
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
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 21
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
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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.
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 23
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:
Optimum Optimum TT : :
subject tosubject to
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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.
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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:
Optimum Optimum TT : :
subject tosubject to
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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
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 27
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mt S
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GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 28
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.
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 29
Decentralized detectionDecentralized detection
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 30
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!
GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 GV MOUSTAKIDES: Sequential Detection, Overview & Open Problems, ISR, Dec. 2011 31
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.
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Thank you for your Thank you for your attentionattention
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