Quickest Detection of a Dynamic Anomaly in a Heterogeneous Sensor Network Georgios Rovatsos, UIUC Venugopal V. Veeravalli, UIUC George V. Moustakides, Univ. of Patras June 21 – 26 G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 1 / 15
≈ ≈ ≈ Sensor Networks In modern applications, many engineering systems are monitored by set of sensors G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 2 / 15
Sensor Networks In modern applications, many engineering systems are monitored by set of sensors Data Concentrator 1 4 5 2 ≈ ≈ PMU 7 7 Station Station Control Control P 2 ,V 2 P 1 ,V 1 Control Center 9 8 PMU 9 PMU 8 Data Data Concentrator Concentrator 6 3 P 3 ,V 3 ≈ Station Control G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 2 / 15
Dynamic Anomaly in Sensor Networks Anomaly affects different node at different time instants Goal : Frame dynamic anomaly detection problem theoretically and design tractable and optimal algorithms G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 3 / 15
General Setting System monitored by a set of sensors denoted by [ L ] � { 1 , . . . , L } G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 4 / 15
General Setting System monitored by a set of sensors denoted by [ L ] � { 1 , . . . , L } Observations given to centralized decision maker sequentially G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 4 / 15
General Setting System monitored by a set of sensors denoted by [ L ] � { 1 , . . . , L } Observations given to centralized decision maker sequentially After unknown but deterministic time instant event leads system to enter abnormal state G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 4 / 15
General Setting System monitored by a set of sensors denoted by [ L ] � { 1 , . . . , L } Observations given to centralized decision maker sequentially After unknown but deterministic time instant event leads system to enter abnormal state Detect emergence of dynamic anomaly as quickly as possible subject to false alarm constraints G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 4 / 15
Observation Model Denote by ν ≥ 0 the unknown changepoint G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 5 / 15
Observation Model Denote by ν ≥ 0 the unknown changepoint Denote by g ℓ ( x ), f ℓ ( x ) the non-anomalous and anomalous distributions at sensor ℓ G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 5 / 15
Observation Model Denote by ν ≥ 0 the unknown changepoint Denote by g ℓ ( x ), f ℓ ( x ) the non-anomalous and anomalous distributions at sensor ℓ Anomalous node at time k : S [ k ] ∈ [ L ] G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 5 / 15
Observation Model Denote by ν ≥ 0 the unknown changepoint Denote by g ℓ ( x ), f ℓ ( x ) the non-anomalous and anomalous distributions at sensor ℓ Anomalous node at time k : S [ k ] ∈ [ L ] For fixed ν , S � { S [ k ] } ∞ k =1 , joint distribution of sensor observations characterized by non-anomalous and anomalous pdfs: L g ( x ) � � g ℓ ( x ℓ ) 1 ≤ k ≤ ν ℓ =1 X [ k ] ∼ � � p S [ k ] ( x ) � f S [ k ] ( x S [ k ] ) · � g ℓ ( x ℓ ) k > ν. ℓ � = S [ k ] G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 5 / 15
Problem Statement Delay metric: ess sup E S WADD ( τ ) = sup sup ν [ τ − ν | τ > ν, F ν ] S ν ≥ 0 G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 6 / 15
Problem Statement Delay metric: ess sup E S WADD ( τ ) = sup sup ν [ τ − ν | τ > ν, F ν ] S ν ≥ 0 Mean time to false alarm (MTFA): E ∞ [ τ ] G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 6 / 15
Problem Statement Delay metric: ess sup E S WADD ( τ ) = sup sup ν [ τ − ν | τ > ν, F ν ] S ν ≥ 0 Mean time to false alarm (MTFA): E ∞ [ τ ] Optimization problem: min WADD ( τ ) τ s.t. τ ∈ C γ where for γ > 0, C γ � { τ : E ∞ [ τ ] ≥ γ } G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 6 / 15
Mixture Statistical Model Plays important role in intuitive explanation and theoretical analysis of proposed algorithms G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 7 / 15
Mixture Statistical Model Plays important role in intuitive explanation and theoretical analysis of proposed algorithms Observation model that arises when at k , anomalous node chosen at random according to pmf α = { α ℓ : ℓ ∈ [ L ] } ∈ A L � g ( x ) � g ( x ℓ ) , 1 ≤ k ≤ ν ℓ =1 X [ k ] ∼ L � � p α ( x ) α ℓ p ℓ ( x ) , k > ν ℓ =1 G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 7 / 15
Mixture Statistical Model Plays important role in intuitive explanation and theoretical analysis of proposed algorithms Observation model that arises when at k , anomalous node chosen at random according to pmf α = { α ℓ : ℓ ∈ [ L ] } ∈ A L � g ( x ) � g ( x ℓ ) , 1 ≤ k ≤ ν ℓ =1 X [ k ] ∼ L � � p α ( x ) α ℓ p ℓ ( x ) , k > ν ℓ =1 Corresponding detection delay: α ν [ τ − ν | τ > ν, F ν ] WADD α ( τ ) = sup ess sup E ν ≥ 0 G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 7 / 15
Mixture Statistical Model Plays important role in intuitive explanation and theoretical analysis of proposed algorithms Observation model that arises when at k , anomalous node chosen at random according to pmf α = { α ℓ : ℓ ∈ [ L ] } ∈ A L � g ( x ) � g ( x ℓ ) , 1 ≤ k ≤ ν ℓ =1 X [ k ] ∼ L � � p α ( x ) α ℓ p ℓ ( x ) , k > ν ℓ =1 Corresponding detection delay: α ν [ τ − ν | τ > ν, F ν ] WADD α ( τ ) = sup ess sup E ν ≥ 0 Effective KL number: � log p α ( X [1]) � I α � E α . 0 g ( X [1]) G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 7 / 15
Mixture-CUSUM Test Proposed test is CUSUM test that detects transition from joint pre-change pdf to post-change model where anomalous node is picked randomly according to specific choice of α G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 8 / 15
Mixture-CUSUM Test Proposed test is CUSUM test that detects transition from joint pre-change pdf to post-change model where anomalous node is picked randomly according to specific choice of α Test statistic: W α [ k ] = ( W α [ k − 1]) + + log p α ( X [ k ]) g ( X [ k ]) where W α [0] � 0 G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 8 / 15
Mixture-CUSUM Test Proposed test is CUSUM test that detects transition from joint pre-change pdf to post-change model where anomalous node is picked randomly according to specific choice of α Test statistic: W α [ k ] = ( W α [ k − 1]) + + log p α ( X [ k ]) g ( X [ k ]) where W α [0] � 0 Stopping time: τ M ( b , α ) � inf { k ≥ 1 : W α [ k ] ≥ b } G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 8 / 15
Exact Optimality - Homogeneous Sensors 1 Theorem Fix γ > 0 . Assume that f ℓ = f ℓ ′ and g ℓ = g ℓ ′ for all ℓ, ℓ ′ ∈ [ L ] . The M-CUSUM test with uniform weights α ℓ = 1 L for all ℓ ∈ [ L ] and b chosen such that E ∞ [ τ M ] = γ is exactly optimal, i.e., WADD ( τ M ) = inf τ ∈C γ WADD ( τ ) 1 G. Rovatsos, G. V. Moustakides, and V. V. Veeravalli, “Quickest detection of a dynamic anomaly in a sensor network” G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 9 / 15
Asymptotic Optimality - Heterogeneous Sensors Theorem Choosing b = log γ implies that E ∞ [ τ M ] ≥ γ and that as γ → ∞ τ ∈C γ WADD ( τ ) ∼ WADD ( τ M ) ∼ log γ inf I α ∗ where α ∗ � arg min I α α ∈ A G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 10 / 15
KL-Number Minimizer Slope Property Lemma Let α ∗ � arg min I α α ∈ A We then have that α ∗ is an interior point of A and that � � p α ∗ ( X ) �� � � p α ∗ ( X ) �� E p ℓ log = E p ℓ ′ log g ( X ) g ( X ) for all ℓ, ℓ ′ ∈ [ L ] , where E p ℓ [ · ] denotes the expectation when ℓ is anomalous G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 11 / 15
Asymptotic Optimality Proof Sketch Universal lower bound: for any α ∈ A as γ → ∞ τ ∈C γ WADD α ( τ ) ∼ log γ τ ∈C γ WADD ( τ ) ≥ inf inf I α G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 12 / 15
Asymptotic Optimality Proof Sketch Universal lower bound: for any α ∈ A as γ → ∞ τ ∈C γ WADD α ( τ ) ∼ log γ τ ∈C γ WADD ( τ ) ≥ inf inf I α M-CUSUM upper bound: for any b > 0 as b → ∞ WADD ( τ M ( b , α ∗ )) ≤ b (1 + o (1)) I α ∗ G. Rovatsos - rovatso2@illinois.edu ISIT 2020 June 21 – 26 12 / 15
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