Asynchronous random sampling for decentralized detection Georgios - PowerPoint PPT Presentation

Asynchronous random sampling for decentralized detection Georgios Fellouris , Columbia University, NY, USA George V. Moustakides , University of Patras, Greece

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 USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 2

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) USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 3

Bayes and Neyman-Pearson formulation Likelihood ratio test: Likelihood ratio test: For i.i.d i.i.d.: .: For WAIT until K samples become available, THEN WAIT THEN decide USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 4

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 make a 1 make a Can ξ Can ξ 1 Yes No 1 ξ 1 Stopping Rule Stopping Rule reliable decision? reliable decision? 2 ξ 1 , ξ 2 N ( ξ 1 ,..., ξ n )= { stop , continue } ... ... N ξ 1 ,..., ξ N We stop receiving Time N N is is a a Time observations stopping time stopping time Decision Rule Decision Rule D ( ξ 1 ,..., ξ N ) ∈ { 0,1 } USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 5

WHY sequential? 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 Changes with time time We define two thresholds A< 0 <B USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 6

Decision in favor of H 1 B u n N n 0 A Decision in favor of H 0 Stopping rule: Decision rule: USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 7

Remarkable optimality property of SPRT SPRT solves BOTH BOTH problems problems simultaneously simultaneously SPRT solves � Proved by Wald and Wolfowitz in 1948. USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 8

The Sequential change detection problem Also known as the Disorder problem or the Change- Point problem or the Quickest Detection problem. Change of Statistics Change of Statistics τ Time Detect as soon as possible Detect as soon as possible USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 9

Mathematical setup We are observing sequentially a process { ξ n } with the following statistics: for 0 < n 6 τ ~ f 0 ξ n ~ f 1 for τ < n Goal: Detect the change time Detect the change time τ τ “ “ as soon as possible as soon as possible ” ” Goal: � Change time τ : deterministic but 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), i.e. it is a stopping time. USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 10

Cumulative Sum (CUSUM) test We recall the running log-likelihood: The running minimum: m n = inf 0 6 s 6 n u s . Define the CUSUM process y n : y n = u n – m n The CUSUM stopping rule: S = inf n { n : y n > ν } We have a convenient recursion: USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 11

u n m n S USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 12

Lorden’s criterion (1971) The change time τ is deterministic and unknown. For any stopping time N define the criterion: J L ( N ) = sup τ essup E 1 [ ( N - τ ) + | F τ ] Optimization problem: inf N J L ( N ) ; subject to: E 0 [ N ] > γ . CUSUM solves the above optimization problem for the i.i.d. case (Moustakides 1986). USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 13

Decentralized detection and corresponding models ξ n ,1 ξ n ,2 Q 1 Q 2 Sensor 1 Sensor 2 ξ n ,1 z n ,2 z n ,1 ξ n ,2 Centralized Test (point of reference) Fusion Fusion High communication Center Center load z n , K Decentralized Test Quantization scheme ξ n , K Q K Sequential hypothesis Sensor K testing between f 0, i and ξ n , K f 1, i . USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 14

� 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 ) � Feedback with No Local Memory: z n , i = Q i ( ξ n , i ,[ z n -1,1 -1,1 ,..., ,..., z n -1, -1, K ]) � Feedback with Partial Local Memory: z n , i = Q i ( ξ n , i ,[ z n -1,1 -1,1 ,..., ,..., z n -1, -1, K ],...,[ ],...,[ z 1,1 1,1 ,..., ,..., z 1, 1, K ]) � Feedback with Full Local Memory: z n , i = Q i ( ξ n , i ,..., ξ 1, i ,[ z n -1,1 -1,1 ,..., ,..., z n -1, -1, K ],...,[ ],...,[ z 1,1 1,1 ,..., ,..., z 1, 1, K ]) USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 15

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: USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 16

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 . USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 17

An even better centralized scheme ! 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! USC (Applied Math seminar): Asynchronous random sampling for decentralized 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 . Instead of we propose the use of the following test statistic: Canonical sampling corresponds to: Stopping rule: Decision rule: USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 19

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 ? USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 20

� 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 . USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 21

Theorem: The detection delay of the proposed scheme differs from the centralized continuous-time optimum by a constant (order-2 asymptotic optimality). v t u t + C u t - C T u T l T USC (Applied Math seminar): Asynchronous random sampling for decentralized detection 22