Quickest Detection and Its Allications

Zhu Han Department of Electrical and Computer Engineering University of Houston, Houston, TX, USA Quickest Detection and Its Allications

Outline • 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 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 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 paradigm

Receiver operating characteristic (ROC) curve Tradeoff between false alarm and detection probability

Basics 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. • Classification • Sequential detection: determine asap between two known distributions, starting from time 0. • Bayesian detection: at random time (known distribution), distribution changes between two known distribution. • CUSUM test: at random time (unknown distribution), distribution changes to known/unknown distribution. • Applications • Cognitive Radio: Primary user reappear • Multiuser Detection: Memory • Network Monitoring: • Medical Device: Fall or not

Markov 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

Sequential 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 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 sampling • s.d.r. to solve the optimization problem

Equivalent Rule

Optimal Detection Rule • We can rewrite the problem • Optimal stopping time • Optimal cost

S() 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 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

Example • 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.

Baysesian 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 for 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 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 Solution

Example • How to find optimal threshold • Detection vs. time example

Other 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. • an example is a delay penalty of polynomial type (T-t)p for fixed p>0 • The exponential penalty. (replace P(T<t) with P(T<t-ε) for fixed ε >0) • A alterative delay penalty

Bojdecki’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 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”.

Non-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 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 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 (Page, 1966) Hv: sequence has density f0 before v, and f1 after H0: sequence is stochastically homogeneous gn b This test minimizes the worst-average detection delay (in an asymptotic sense) Stopping time N

Example: Cognitive Radio Public Traffic Lane congested! Unlicensed Spectrum Or Secondary Users Treated as Harmful Interference Lane reserved for military Licensed Spectrum Or Primary Users

Spectrum 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 Sensing • Collaborative spectrum sensing 3- Fusion Center makes final decision: PU present or not 2- the SUs send their Local Sensing bits to a common fusion center 1- the SUs perform Local Sensing of PU signal

Collaborative 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 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

Power 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) z=Hx+e, for n power lines and m measurement, m<n H: Jacobean Matrix (n×n) x: State variable (n×1) z: Measurements (m×1), m<n e: 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 • 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 • 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 - NonBayesian Detection delay Detection time Actual time of active attack • 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.

Multi-thread CUSUM Algorithm How about the unknown? Declare the attacker is existing! Otherwise, continuous to the process. • 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:

Linear Solver for the Unknown The unknown is no long involved • 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:

Simulation: 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

Conclusion • 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? Thank you very much

Quickest Detection and Its Allications