Decentralized Sequential Hypothesis Testing. Change Detection

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1 Decentralized Sequential Hypothesis Testing & Change Detection Giorgos Fellouris, Columbia University, NY, USA George V. Moustakides, University of Patras, Greece

2 Outline Sequential hypothesis testing and SPRT Sequential change detection and CUSUM Decentralized detection and corresponding models Centralized schemes (points of reference) Decentralized detection using asynchronous random sampling Simulation comparisons UTT: Decentralized Hypothesis testing & Change detection 2

3 Sequential hypothesis testing and SPRT Conventional binary hypothesis testing (fixed sample size): Collection of observations ξ 1,...,ξ K H 0 : ξ 1,...,ξ K ~ f 0 (ξ 1,...,ξ K ); H 1 : ξ 1,...,ξ K ~ f 1 (ξ 1,...,ξ K ); Decision rule D(ξ 1,...,ξ K ) {0,1} P(D=1 H 1 ) (Correct decision) P(D=1 H 0 ) (Type I error) P(D=0 H 1 ) (Type II error) P(D=0 H 0 ) (Correct decision) UTT: Decentralized Hypothesis testing & Change detection 3

4 Bayes and Neyman-Pearson formulation Likelihood ratio test: For i.i.d.:.: WAIT until K samples become available, THEN decide UTT: Decentralized Hypothesis testing & Change detection 4

5 Sequential binary hypothesis testing Observations ξ 1,...,ξ n,... are supplied sequentially. H 0 : ξ 1,...,ξ n,... ~ f 0 (ξ n ) H 1 : ξ 1,...,ξ n,... ~ f 1 (ξ n ) Time Observations 1 ξ 1 2 ξ 1,ξ N ξ 1,...,ξ N Time N is RANDOM Decision Rule D(ξ 1,...,ξ N ) {0,1} Can ξ 1 Yes No make a reliable decision? Stopping Rule N(ξ 1,...,ξ n )= {stop,continue} We stop receiving observations UTT: Decentralized Hypothesis testing & Change detection 5

6 WHY sequential? For the same level of confidence with a sequential test we need, in the average, (significantly) less samples than a fixed sample size test, to reach a decision. The Sequential Probability Ratio Test (SPRT) (Wald 1947) Changes with time We define two thresholds A< 0 <B UTT: Decentralized Hypothesis testing & Change detection 6

7 B Decision in favor of H 1 u n 0 n N A Stopping rule: Decision rule: Decision in favor of H 0 UTT: Decentralized Hypothesis testing & Change detection 7

8 Remarkable optimality property of SPRT SPRT solves BOTH problems simultaneously Proved by Wald and Wolfowitz in UTT: Decentralized Hypothesis testing & Change detection 8

9 The Sequential change detection problem Also known as the Disorder problem or the Change- Point problem or the Quickest Detection problem. Change of Statistics Detect as soon as possible τ Time UTT: Decentralized Hypothesis testing & Change detection 9

10 Applications Monitoring of quality of manufacturing process (1930 s) Biomedical Engineering Electronic Communications Econometrics Seismology Speech & Image Processing Vibration monitoring Security monitoring (fraud detection) Spectrum monitoring Scene monitoring Network monitoring and diagnostics (router failures, intruder detection) Databases... UTT: Decentralized Hypothesis testing & Change detection 10

11 Mathematical setup We are observing sequentially a process {ξ n } with the following statistics: ξ n ~ f 0 for 0 < n6 τ ~ f 1 for τ <n Goal: Detect the change time τ as soon as possible Change time τ : unknown Densities f 0, f 1 : known At every time instant n we perform a test and decide whether there was a change or not. In the former case we stop in the latter we continue sampling. The test at time n must be based on the available information up to time n (and not on any future information). UTT: Decentralized Hypothesis testing & Change detection 11

12 Cumulative Sum (CUSUM) test We recall the running log-likelihood: The running minimum: m n = inf 06s 6n u s. Define the CUSUM process y n : The CUSUM stopping rule: N = inf n { n: y n > ν } We have a convenient recursion: y n = u n m n UTT: Decentralized Hypothesis testing & Change detection 12

13 u n m n N UTT: Decentralized Hypothesis testing & Change detection 13

14 Decentralized detection and corresponding models ξ n,1 ξ n,2 Sensor 2 ξ n,k Q 2 Q K z n,2 ξ n,2 ξ n,1 z n,k ξ n,k Sensor K Fusion Center z n,1 Sensor 1 Sequential hypothesis testing between f 0,i and f 1,i. UTT: Decentralized Hypothesis testing & Change detection 14 Q 1 Centralized Test (point of reference) High communication load Decentralized Test Quantization scheme

15 No Local Memory: z n,i =Q i (ξ n,i ) Full Local Memory: z n,i =Q i (ξ n,i,ξ n-1,i,...,ξ 1,i ) Mei (2008) Feedback with No Local Memory: z n,i =Q i (ξ n,i,[z n-1,1,...,z n-1,k ]) Feedback with Partial Local Memory: z n,i =Q i (ξ n,i,[z n-1,1,...,z n-1,k ],...,[z 1,1,...,z 1,K ]) Feedback with Full Local Memory: z n,i =Q i (ξ n,i,...,ξ 1,i,[z n-1,1,...,z n-1,k ],...,[z 1,1,...,z 1,K ]) UTT: Decentralized Hypothesis testing & Change detection 15

At the Fusion center we form the running log-likelihood ratio and

16 Centralized tests We recall that in this case the sensors send the observations ξ n,i to the Fusion center. At the Fusion center we form the running log-likelihood ratio and apply an SPRT: Stopping rule: Decision rule: UTT: Decentralized Hypothesis testing & Change detection 16

17 Remark 1: In ALL previous detection structures it is assumed the existence of a GLOBAL CLOCK. Synchronization of distant sensors with the fusion center is practically difficult (especially in sensor networks). Remark 2: In most practical applications the observation samples ξ n,i come from canonical sampling of a continuous time process ξ t,i where ξ n,i = ξ nt,i i.e. we sample ξ t,i at the time instances t n =nt. UTT: Decentralized Hypothesis testing & Change detection 17

Stopping rule: Decision rule: The continuous time SPRT is better than the discrete time SPRT due

18 An even better centralized scheme The fusion center instead of receiving the samples ξ n,i it can receive the CONTINUOUS TIME PROCESSES ξ t,i to form an SPRT. Stopping rule: Decision rule: The continuous time SPRT is better than the discrete time SPRT due to infinite time resolution. It constitutes the ultimate point of reference! UTT: Decentralized Hypothesis testing & Change detection 18

Asynchronous random sampling Let be increasing sequence of sampling times NOT necessarily canonical. At these times we sample the local log-likelihood u t,i in the form.

19 Asynchronous random sampling Let be increasing sequence of sampling times NOT necessarily canonical. At these times we sample the local log-likelihood u t,i in the form. Instead of we propose the use of the following expression: Canonical sampling corresponds to: Stopping rule: Decision rule: UTT: Decentralized Hypothesis testing & Change detection 19

20 How do we transmit the local log-likelihoods from the sensors to the Fusion center? We observe To form the local log-likelihood at the fusion center, Sensor i needs to transmit the differences We select so that the difference takes the value A i or B i which are specified before hand. What is the sampling strategy at Sensor i? UTT: Decentralized Hypothesis testing & Change detection 20

21 Repeated SPRTs at each sensor. Communication with the Fusion center If the difference is equal to B i send 1 if it is equal to A i send 0. The Fusion center knows that 1 corresponds to B i and 0 to A i and can therefore update the partial log-likelihood then compute v t and perform the test by comparing v t to A and B. UTT: Decentralized Hypothesis testing & Change detection 21

22 Every time new information arrives at the Fusion center (even from one sensor) the Fusion center updates and performs the test. Communication is Asynchronous and Random!!! How do we select the local thresholds A i, B i? We can specify a communication rate between sensors and Fusion center. If the sensors must communicate, in the average, every T time units, then this condition specifies completely the thresholds. We must select the thresholds so that the average detection delay of the local SPRTs is equal to T. UTT: Decentralized Hypothesis testing & Change detection 22

23 Simulations ξ t,2 Sensor 2 ξ t,1 Sensor 1 Fusion Center ξ t,i : both standard Brownian Motions H 0 : drift = 0; H 1 : drift = 1. Average communication period T=3. Canonical sampling generates: H 0 : Gaussian(0,3); H 1 : Gaussian(3,3) α = β UTT: Decentralized Hypothesis testing & Change detection 23

24 Average Detection Delay Centralized Cont. Time Centralized Discr. Time Mei s Detection scheme Proposed α = β UTT: Decentralized Hypothesis testing & Change detection 24

25 UTT: Decentralized Hypothesis testing & Change detection 25

26 FIN Merci de votre attention UTT: Decentralized Hypothesis testing & Change detection 26

Sequential Detection. Changes: an overview. George V. Moustakides

Sequential Detection. Changes: an overview. George V. Moustakides Sequential Detection of Changes: an overview George V. Moustakides Outline Sequential hypothesis testing and Sequential detection of changes The Sequential Probability Ratio Test (SPRT) for optimum hypothesis