Quickest Detection and Its Allications Zhu Han Department of Electrical and Computer Engineering...

Quickest Detection and Its Allications

Zhu Han

Department of Electrical and Computer Engineering

University of Houston, Houston, TX, USA

OutlineOutline Introduction– Basics– Markov stopping time

Quickest Detection– Sequential detection– Bayesian detection– CUSUM test

Applications– Cognitive radio network– Multiuser detection for memory– Medical applications– Smart grid

Conclusions

Classic Hypothesis TestClassic Hypothesis Test

Probability Space (Ω, F, P)– Ω is a set, a sample space

– F is a event

– P is the probability measure assign to the event

Detection: “Spot the Money”

Hypothesis TestingHypothesis Testing

Let the signal be y(t), model be h(t)

Hypothesis testing:

H0: y(t) = n(t) (no signal)

H1: y(t) = h(t) + n(t) (signal)

The optimal decision is given by the Likelihood ratio test (Nieman-Pearson Theorem), g is a threshold.

Select H1 if L(y) = log(P(y|H1)/P(y|H0)) > g;

otherwise select H0.

Signal detection paradigmSignal detection paradigm

Receiver operating characteristic (ROC) curveReceiver operating characteristic (ROC) curve

Tradeoff between false alarm and detection probability

Basics of Quickest DetectionBasics of Quickest Detection

A technique to detect distribution changes of a sequence of observations as quick as possible with the constraint of false alarm or detection probability.

Classification1. Sequential detection: determine asap between two

known distributions, starting from time 0.

2. Bayesian detection: at random time (known distribution), distribution changes between two known distribution.

3. CUSUM test: at random time (unknown distribution), distribution changes to known/unknown distribution.

Applications1. Cognitive Radio: Primary user reappear

2. Multiuser Detection: Memory

3. Network Monitoring:

4. Medical Device: Fall or not

Markov Stopping TimeMarkov Stopping Time For Markov process: memoriless property

likelihood of a given future state, at any given moment, depends only on its present state, and not on any past states

Random variable YT: a reward that can be claimed at time T Optimal stopping time that maximizes the reward

S is finite or infinite. For finite time S case

backward induction dynamic programming for Markov Case

For infinite time S case Define

Stopping time




Conclusions

Sequential DetectionSequential Detection How to reach a decision between two hypotheses after minimal

average trails? A real sample sequence, Zk;K=1,2… that obey one of

two hypotheses:

Stop the observation as soon as the decision is made Trade off between probability of error and decision time.

More accurate, more decision time. Quicker decision, less accurate

A sequential decision rule (s.d.r.) as the pair (T,δ), in which T declares the time to stop sampling and then δ takes the value 0 or 1 declaring which one of H1 , H0

Performance indices of interestPerformance indices of interest

Average cost of errors– False Alarm– Missing Probability– Average cost of errors, is the probability of event

– c0 and c1 are constants to balance the tradeoff

The cost of samplings.d.r. to solve the optimization problem

Equivalent RuleEquivalent Rule

Optimal Detection RuleOptimal Detection Rule

We can rewrite the problem

Optimal stopping time

Optimal cost

S(S() and thresholds) and thresholds

An illustration of s(π)

The thresholds are found from s(π)– One is for false alarm– The other is for missing prob.

Sequential probability ratio testSequential probability ratio test

Sequential probability ratio test (SPRT) with boundaries A and B : (SPRT(A, B))

– It exhibit minimal expected stopping time among all s.d.r.’s having given error probability.

– The stopping time T is equivalently be written as

ExampleExample

At the 1st exit of ∧k from (A,B), decides H1 if the exit is to the right of this interval and H0 if the exit is to the left.




Conclusions

Baysesian quickest detectionBaysesian quickest detection

The distribution changes with unknown time (but known distribution for the changing time). The objective of observer is to detect such a random change, if one occurs, as quickly as possible.

The difference from the sequential detection The design of quickest detection procedures involves the

optimization of a tradeoff between two types of performance indices: detection delay vs. false alarm.

For example, network from WIFI to Bluetooth Approaches

Shiryaev’s problem for Bayesian quickest detection Bojdecki’s quickest detection problem and other

constraints Ritov’s quickest detection problem: Game theory

approach

Shiryaev’s problem Shiryaev’s problem for Bayesian quickest detectionfor Bayesian quickest detection

Random sequence, Zk ; k=1,2,… suppose there is a change point, t, such that given Z1 , Z2…, Zt-1 with marginal distribution Q0 , and Zt , Zt+1…, ZT with marginal distribution Q1

Two performance indices– The expected detection delay:– The false alarm probability: The determination of optimal stopping time, T,

– It was a first posted by Shiryaev. It considers– C>0, is a constant controlling the balance between 2

indices.

Geometric distribution assumptionGeometric distribution assumption

To find the optimal stopping time, it need to assume a specific prior distribution for the change pint, t,

– π and ρ are the constant lying in the interval (0,1)– π, probability that a change already occurred when

the sequence observation start. – ρ, the conditional probability that the sequence will

transition to the post-change state at any time, given that it has not done so prior to that time

Optimal SolutionOptimal Solution

ExampleExample

How to find optimal threshold Detection vs. time example

Other penalty functionsOther penalty functions

The penalty parameters act like an optimal constraints (i.e. penalize combination of false alarms and detection delay) but the solutions ideally converge to the solution or the original one.

1. an example is a delay penalty of polynomial type (T-t)p for fixed p>0

2. The exponential penalty. (replace P(T<t) with P(T<t-ε) for fixed ε >0)

3. A alterative delay penalty

Bojdecki’s problemBojdecki’s problem

A different approach to detecting the change point t within Bayesian framework by maximizing the probability of selected the right estimator for t based on the observation.

B is an approx. measurable set and XT depends the observed Zk . If T* is existed, will be called optimal.

Let if maximizing the probability of stopping within m units of the change point t.

Omit other details

A game theoretic formulationA game theoretic formulation

An alternative approach: Ritov’s game-theoretic quickest detection problem

A game consists two player. – Player#1: “the statistician” is attempting to quickly

detect a random change point as in the preceding section

– Player#2: “nature” is attempting to choose the distribution of the change point and foil the Player#1.

– Given the probability of the change point

Is allowed to be a function of the past observation Z1~Zk-1, which is selected by “nature”.




Conclusions

Non-Bayesian quickest detectionNon-Bayesian quickest detection

Previously, Shiryaev’s problem for Bayesian quickest detection assumed the change point t, which is a random variable with given, prior distribution. – How to solve if the system has no pre-existing

statistical model for occurrence of event, like in surveillance or inspection system?

Lorden’s problem for non-Bayesian quickest detection– Problem definition– Page’s CUSUM test– Performance of Page’s test

Asymptotic results– Lorden’s approach

The false-alarm constraints

Lorden’s problemLorden’s problem

The detection delay is penalized by its worst case value :

– Where d(T) is the worst case delay, and dt (T) is the average delay under Pt

Constraint and Problem FormulationConstraint and Problem Formulation

The rate of false alarms can be quantified by the mean time between false alarms

The design criterion is then given by:

– is positive, finite constant, and is the stopping time for minimizing the worst-case delay within lower-bound constraint in the mean time between the false alarms.

Cusum test Cusum test (Page, 1966)(Page, 1966)

gn

b

Stopping time N

Hv: sequence has density f0 before v, and f1 after

H0: sequence is stochastically homogeneous

This test minimizes the worst-average detection delay (in an asymptotic sense)




Conclusions

Example: Cognitive RadioExample: Cognitive Radio

Lane reserved for militaryLicensed SpectrumOr Primary Users

Public TrafficLane congested!

Unlicensed SpectrumOr Secondary UsersTreated as Harmful Interference

Spectrum SensingSpectrum Sensing Secondary users must sense the spectrum to

– Detect the presence of the primary user for reducing interference on primary user

– Detect spectrum holes to be used for transmission

Spectrum sensing is to make a decision between two hypotheses– The primary user is present, hypothesis H0 – The primary user is absent, hypothesis H1

Quickest detection for spectrum sensing– A distribution change in frequency domain is detected in observations to

quit from or join into the licensed frequency band

– There exist unknown parameters after the primary radio emerges

Collaborative Spectrum SensingCollaborative Spectrum Sensing

Collaborative spectrum sensing

Common Secondary Fusion Center

Primary User (Licensed user)

Secondary User

Secondary User

Secondary UserSecondary User

1- the SUs perform Local Sensing of PU signal

2- the SUs send their Local Sensing bits to a common fusion center

3- Fusion Center makes final decision: PU present or not

Collaborative Quickest Spectrum SensingCollaborative Quickest Spectrum Sensing

The collaborative quickest spectrum sensing without communication coordination– An node made own broadcast decision

– The random time-slot selection

– The limited time slots for the secondary users to exchange information

The key issue is to determine

whether to broadcast based on the

current observation and the local

population of secondary user.– A threshold broadcast scheme

is proposed

Medical ApplicationsMedical Applications

Patient falling Quickest detection to detect as soon as possible to

prevent or report False alarm limitation

CUSUM test No prior information How to train the threshold Need real data

Computation Bluetooth between sensor and google phone Android Computation in Android using JAVA Communication through 3G or WIFI for reporting




Conclusions

Power System State Estimation ModelPower System State Estimation Model

Transmitted active power from bus i to bus j– High reactance over resistance ratio

– Linear approximation for small variance

– State vector , measure noise e with covariance Ʃe

– Actual power flow measurement for m active power-flow branches

– Define the Jacobian matrix

– We have the linear approximation

– H is known to the power system but not known to the attackers

State Estimation (SE)State Estimation (SE)

z=Hx+e, for n power lines and m measurement, m<nH: Jacobean Matrix (n×n)x: State variable (n×1)z: Measurements (m×1), m<ne: noise vector (n×1)

• Goal of system is to estimate x from z

• SE is a key function in building real-time models of electricity networks in Energy Management Centers (EMC)

• Real-time models of the network can be used by Independent System Operator (ISO) to make optimal decisions with respect to technical constraints (such as transmission line congestion, voltage and transient stability)

Bad Data Injection and Detection Bad Data Injection and Detection

Inject Bad data c: z=Hx+c+e

Bad data detection– Residual vector

– Without attacker

where

– Bad data detection (with threshold )

without attacker:

with attacker: otherwise

Stealth (unobservable) attack

– Hypothesis test would fail in detecting the attacker, since the control center believes that the true state is x + x.

QD System Model QD System Model

Assuming Bayesian framework:– the state variables are random with

The binary hypothesis test:

The distribution of measurement z under binary hyp: (differ only in mean)

We want a detector– False alarm and detection probabilities

Detection Model - NonBayesianDetection Model - NonBayesian Requiring a non-Bayesian approach due to unknown

prior probability, attacker statistic model

The unknown parameter exists in the post-change distribution and may changes over the detection process. – You do not know how attacker attacks.

Minimizing the worst-case effect via detection delay:

We want to detect the intruder as soon as possible while maintaining PD.

Actual time of active attack

Actual time of active attack

Detection time

Detection time

Detection delay

Detection delay

Multi-thread CUSUM AlgorithmMulti-thread CUSUM Algorithm

CUSUM Statistic:

where Likelihood ratio term of m measurements:

By recursion, CUSUM Statistic St at time t:

Average run length (ARL) for declaring the attack:

How about the unknown?

How about the unknown?

Declare the attacker is existing!

Otherwise, continuous to the process.

Linear Solver for the UnknownLinear Solver for the Unknown

Rao test – asymptotically equivalent model of GLRT:

The linear unknown solver for m measurements:– Omitting the necessity of [J-1] solo-parameter envir.

– Simplifying Quadratic form the unknown > 0

Recursive CUSUM Statistic w/ linear unknown parameter solve:The unknown is no long involved

The unknown is no long involved

Simulation: Adaptive CUSUM algorithmSimulation: Adaptive CUSUM algorithm

2 different detection tests: FAR: 1% and 0.1%

Active attack starts at time 6

Detection of attack at time 7 and 8, for different FARs

ConclusionConclusion

Different from the other detection techniques that minimize error, quickest detection minimizes the decision time.

Trade off between decision time and error probability (false alarm and error probabilities)

Depending on the different scenarios Sequential detection Bayesian detection Non-Bayesian detection

Applications Wireless network Medical applications Smart grid Other applications?

Questions?Questions?

Quickest Detection and Its Allications Zhu Han Department of Electrical and Computer Engineering...

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