Unstructured Sequential Change Detection in Sensor Networks Grigory Sokolov Department of Mathematics University of Southern California Los Angeles, California United States of America � gsokolov@usc.edu Joint work with Georgios Fellouris and Alexander Tartakovsky Fourth International Workshop in Sequential Methodologies University of Georgia, Athens, Georgia, USA July 20, 2013
Multi-sensor Change-point Detection • Given: K sensors , observing series { X k t } t ≥ 1 , k = 1 , 2 , · · · , K of independent data, collected sequentially, one at a time at each sensor. • Assumption: the change can occur in an unknown subset of sensors N : � g k ( x ) �≡ f k ( x ) , k ∈ N , i.i.d. i.i.d. X k X k ∼ f k ( x ) , t ≤ ν, ∼ t > ν, t t ∈ N , f k ( x ) , k / ∼ f ∼ f ∼ f ∼ g ∼ g X X X X X ... ... Sensor 1: ν ν + 1 ν + 2 ∼ f ∼ f ∼ f ∼ f ∼ f X X X X X ... ... Sensor 2: ν ν + 1 ν + 2 ∼ g K X K ∼ f K ∼ f K ∼ f K ∼ g K X K X K ... X K X K ... Sensor K : ν ν + 1 ν + 2 Change occurs time and the change-point, 0 ≤ ν ≤ ∞ , is unknown (deterministic). Fourth International Workshop in Sequential Methodologies 2 July 20, 2013
Multi-sensor Change-point Detection (Cont’d) • Let F k t = σ ( X k 1 , . . . , X k t ) for t � 0 , k = 1 , · · · , K with F k 0 = { ∅ , Ω } . • Notation: for t � 0 let P t ( · ) = P ( · | ν = t ) and E t [ · ] = E [ · | ν = t ] ; in particular, P ∞ ( · ) = P ( · | ν = ∞ ) and E ∞ [ · ] = E [ · | ν = ∞ ] . • Let T be a stopping time, adapted to {F k n } n � 1 . Define ARL( T ) = E ∞ [ T ] . • Use Lorden’s detection measure (Lorden 1971) ess sup E ν [( T − ν ) + |F ν ] . J ( T ) = sup ν ≥ 0 • Goal: To find stopping time T , such that J ( T ) is minimized within class { T : ARL( T ) � γ } for every γ � 1 . • Say that a stopping rule is first-order asymptotically optimal if J ( T ) inf T : ARL( T ) � γ J ( T ) → 1 , γ → ∞ . Fourth International Workshop in Sequential Methodologies 3 July 20, 2013
Multi-sensor Change-point Detection (Cont’d) • If the set of affected sensors, N , is known, the CUSUM stopping rule T CUSUM = inf { t ≥ 1 : W t ≥ a } , t log g k � � where W t = u t − min s ≤ t u s , u t = f k ( X i ) , i =1 k ∈N is optimal with respect to Lorden’s detection measure, when a is chosen so that ARL( T CUSUM ) = γ . • When N is unknown, Mei (2010’11) proposed the following procedure: � K � � W k t ≥ 1 : t ≥ b k } ≥ a T Mei = inf t 1 l { W k , k =1 t where log g k � u k W k t = u k s ≤ t u k t = f k ( X i ) , t − min s . i =1 This procedure is first-order asymptotically optimal. • A different approach was suggested by Xie and Siegmund (2013). Fourth International Workshop in Sequential Methodologies 4 July 20, 2013
Decentralized Change-point Detection Sensor Sensor K – 1 . . . . . . . . . . . . Sensor Sensor K . . . . . . . . . . . . . . . . . . . . . . Fusion Center • In a decentralized setting two types of constraints are usually considered: a) Sensors communicate with the fusion center at a given rate (e.g. Banerjee and Veeravalli 2012). b) Only a certain number of bits is permissible per transmission (e.g. Mei 2005, Fellouris and Moustakides 2013). • We will address both types of communication constraints. Fourth International Workshop in Sequential Methodologies 5 July 20, 2013
Proposed Communication Scheme • Transmit information to the fusion center at times τ k n . At any moment t , let τ k ( t ) be the last transmission time prior to t . • Random sampling (Fellouris and Moustakides 2013): � � k ) τ k t > τ k n − 1 : u k t − u k ∈ ( − ∆ k , ∆ n = inf n − 1 / , τ k k are design parameters. where ∆ k , ∆ k k ∆ ∆ Communication times ∆ k τ k τ k 0 k ∆ ∆ k ∆ k Fourth International Workshop in Sequential Methodologies 6 July 20, 2013
Proposed Communication Scheme (Cont’d) • Due to communication constraints, use u k τ k ( t ) instead of u k t . • Observe that u τ ( t ) can be written as u τ ( t ) = ( u τ 1 − u τ 0 ) + ( u τ 2 − u τ 1 ) + · · · + ( u τ mt − u τ mt − 1 ) · · · = ℓ 1 + ℓ 2 + + ℓ m t , where ℓ n = u τ n − u τ n − 1 is the accumulated log-likelihood ratio in the time-interval [ τ n , τ n − 1 ] , and m t is the number of the last transmission. • It suffices to transmit ℓ n to the fusion center. • Statistic is updated at communication times, and the stopping rule is K � � � V k V k t = u k u k t ≥ 1 : t ≥ a τ k ( t ) − min T = inf , m . τ k m ≤ m k k =1 t Fourth International Workshop in Sequential Methodologies 7 July 20, 2013
Proposed 1-bit Procedure • Due to quantization constraints, approximate u k u k t with ˜ τ k ( t ) from the last trans- mission time: u k τ ( t ) = ℓ k 1 + ℓ k 2 + · · · + ℓ k m t , ≈ ˜ 1 + ˜ 2 + · · · + ˜ ℓ k ℓ k ℓ k u k m t = ˜ τ ( t ) , ℓ k are approximations to ℓ k . where ˜ • Let each sensor transmit one-bit messages z n , the n -th message of whether the threshold was crossed up on down: k � if ℓ k n ≥ ∆ 1 , z k n = n − ≤ ∆ k if ℓ k − 1 , • ˜ ℓ is then defined following partial likelihood approach: n = log P 0 ( z k n =1 } + log P 0 ( z k n = − 1) n = 1) ˜ ℓ k n = 1) 1 l { z k n = − 1) 1 l { z k n = − 1 } . P ∞ ( z k P ∞ ( z k Fourth International Workshop in Sequential Methodologies 8 July 20, 2013
A Case Study: Gaussian Scenario • Let X k t ’s be standard Gaussian N (0 , 1) before the change, and N (0 . 5 , 1) for k ∈ N after the change. Put K = 5 , |N| = 2 . • Goal: Examine how the performance is affected by the choice of δ = E ∞ ( τ k 1 ) , the expected time to transmission. • We will examine several cases: a ) δ ≈ 1 . 5 ( frequent transmissions ), b ) δ ≈ 4 . 0 ( moderate transmission rate ), and c ) δ ≈ 10 . 0 ( infrequent transmis- sions ). Fourth International Workshop in Sequential Methodologies 9 July 20, 2013
Gaussian Scenario: 1-bit 65 55 0 (T) 45 E δ ≈ 1.5 35 δ ≈ 10.0 25 15 100 1000 10000 100000 ARL Figure 1: Dependency on average time between communications, δ . • One-bit procedure has two main disadvantages: a ) its performance drops for frequent transmissions, and b ) few levels of ARL are attainable for low transmis- sion rates. Fourth International Workshop in Sequential Methodologies 10 July 20, 2013
Gaussian Scenario: 1-bit 65 55 0 (T) 45 E δ ≈ 1.5 35 δ ≈ 10.0 Full 25 15 100 1000 10000 100000 ARL Figure 2: Dependency on average time between communications, δ . • One-bit procedure has two main disadvantages: a ) its performance drops for frequent transmissions, and b ) few levels of ARL are attainable for low transmis- sion rates. Fourth International Workshop in Sequential Methodologies 11 July 20, 2013
Proposed Multi-bit Procedure • Remedy: let each sensor transmit with alphabet of the form {− d, · · · , − 1 , 1 , · · · , d } , as opposed to 1 -bit messages. k k ∆ ∆ Overshoot ∆ k τ k τ k 0 k ∆ ∆ k Overshoot ∆ k Fourth International Workshop in Sequential Methodologies 12 July 20, 2013
Proposed Multi-bit Procedure (Cont’d) k k ∆ ∆ Overshoot percentiles ∆ k τ k τ k 0 k ∆ � 0 ∆ k Overshoot ∆ k Fourth International Workshop in Sequential Methodologies 13 July 20, 2013
Proposed Multi-bit Procedure (Cont’d) • Transmit messages z k n : k < ǫ k � if ǫ k j − 1 ≤ ℓ k j, n − ∆ j z k n = , 1 ≤ j ≤ d, n + ∆ k ≤ − ǫ k if − ǫ k j − 1 < ℓ k − j, j where the thresholds ǫ are the percentiles of ℓ n − ∆ . • As before, we approximate u k with ˜ τ k ( t ) = ˜ 1 + ˜ 2 + · · · + ˜ u k ℓ k ℓ k ℓ k t , where m k d log P 0 ( z k n = j } + log P 0 ( z k � n = − j ) � n = j ) ˜ � ℓ k n = n = j ) 1 l { z k n = − j ) 1 l { z k . n = − j } P ∞ ( z k P ∞ ( z k j =1 Fourth International Workshop in Sequential Methodologies 14 July 20, 2013
Gaussian Scenario, d = 2 65 55 0 (T) 45 E δ ≈ 1.5 35 δ ≈ 10.0 25 15 100 1000 10000 100000 ARL Figure 3: Dependency on average time between communications, δ . • For d = 2 procedure both effects are mitigated: a ) it performs well for frequent transmissions, and b ) for low transmission rates more levels of ARL are attainable. Fourth International Workshop in Sequential Methodologies 15 July 20, 2013
Gaussian Scenario, d = 2 65 55 0 (T) 45 E δ ≈ 1.5 35 δ ≈ 10.0 Full 25 15 100 1000 10000 100000 ARL Figure 4: Dependency on average time between communications, δ . • For d = 2 procedure both effects are mitigated: a ) it performs well for frequent transmissions, and b ) for low transmission rates more levels of ARL are attainable. Fourth International Workshop in Sequential Methodologies 16 July 20, 2013
Proposed procedures offer discrete set of run lengths 65 55 0 (T) 45 d = 1 E 35 2 d = d = 3 25 Full 15 100 1000 10000 100000 ARL Figure 5: Proposed procedures, δ = 10 . 0. • For practical purposes d = 3 procedure a ) performs well for frequent trans- missions, and b ) offers reasonably dense set of ARL levels even at very low transmission rates. Fourth International Workshop in Sequential Methodologies 17 July 20, 2013
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