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Minimum Cost Data Aggregation with Localized Processing for Statistical Inference A. Anandkumar 1 L. Tong 1 A. Swami 2 A. Ephremides 3 1 ECE Dept., Cornell University, Ithaca, NY 14853 2 Army Research Laboratory, Adelphi MD 20783 3 EE Dept., University of Maryland College Park, MD 20742 IEEE INFOCOM 2008 . Supported by Army Research Laboratory CTA Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 1 / 21

Distributed Statistical Inference Sensor Network Applications Detection Estimation Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 2 / 21

Distributed Statistical Inference Sensor Network Applications Detection Estimation Classical Distributed Inference Fusion Center Sensors: take measurements Fusion Center: Final decision Statistical Model Y n = [ Y 1 , . . . , Y n ] Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 2 / 21

Routing for Inference Fusion Center Raw Data: Y n Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 3 / 21

Routing for Inference Fusion Center Raw Data: Y n Total cost Raw (SPT) Independent (MST) 0 20 40 60 80 100 120 140 160 180 Number of nodes n Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 3 / 21

Routing for Inference Fusion Center Raw Data: Y n Total cost Raw (SPT) Raw (SPT) Avg. cost Independent (MST) Independent (MST) 0 0 20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180 Number of nodes n Number of nodes n Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 3 / 21

Routing for Inference Fusion Center Raw Data: Y n Total cost Raw (SPT) Raw (SPT) Avg. cost Independent (MST) Independent (MST) 0 0 20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180 Number of nodes n Number of nodes n i.i.d. ∼ N ( θ, 1) Sufficient Statistics for Mean Estimation Y 1 , . . . , Y n � i Y i sufficient to estimate θ : no performance loss Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 3 / 21

Routing for Inference Fusion Center Fusion Center Raw Data: Y n Sum Function Total cost Raw (SPT) Raw (SPT) Avg. cost Independent (MST) Independent (MST) 0 0 20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180 Number of nodes n Number of nodes n i.i.d. ∼ N ( θ, 1) Sufficient Statistics for Mean Estimation Y 1 , . . . , Y n � i Y i sufficient to estimate θ : no performance loss Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 3 / 21

Routing for Inference Fusion Center Fusion Center Raw Data: Y n Sum Function Total cost Raw (SPT) Raw (SPT) Avg. cost Independent (MST) Independent (MST) 0 0 20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180 Number of nodes n Number of nodes n i.i.d. ∼ N ( θ, 1) Sufficient Statistics for Mean Estimation Y 1 , . . . , Y n � i Y i sufficient to estimate θ : no performance loss Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 3 / 21

Routing for Inference Fusion Center Fusion Center Raw Data: Y n Sum Function Total cost Raw (SPT) Raw (SPT) Avg. cost Independent (MST) Independent (MST) 0 0 20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180 Number of nodes n Number of nodes n i.i.d. ∼ N ( θ, 1) Sufficient Statistics for Mean Estimation Y 1 , . . . , Y n � i Y i sufficient to estimate θ : no performance loss Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 3 / 21

Routing for Inference Fusion Center Fusion Center Raw Data: Y n Sum Function Total cost Raw (SPT) Raw (SPT) Avg. cost Independent (MST) Independent (MST) 0 0 20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180 Number of nodes n Number of nodes n i.i.d. ∼ N ( θ, 1) Sufficient Statistics for Mean Estimation Y 1 , . . . , Y n � i Y i sufficient to estimate θ : no performance loss i.i.d. Binary Hypothesis Test: Y 1 , . . . , Y n ∼ f ( y ; H 0 ) or f ( y ; H 1 ) Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 3 / 21

Routing for Inference Fusion Center Fusion Center Raw Data: Y n Sum Function Total cost Raw (SPT) Raw (SPT) Avg. cost Independent (MST) Independent (MST) 0 0 20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180 Number of nodes n Number of nodes n i.i.d. ∼ N ( θ, 1) Sufficient Statistics for Mean Estimation Y 1 , . . . , Y n � i Y i sufficient to estimate θ : no performance loss i.i.d. Binary Hypothesis Test: Y 1 , . . . , Y n ∼ f ( y ; H 0 ) or f ( y ; H 1 ) i log f ( Y i ; H 0 ) , � i log f ( Y i ; H 1 )] sufficient to decide hypothesis [ � Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 3 / 21

Routing for Inference Fusion Center Fusion Center Raw Data: Y n Sum Function Total cost Raw (SPT) Raw (SPT) Avg. cost Independent (MST) Independent (MST) 0 0 20 40 60 80 100 120 140 160 180 20 40 60 80 100 120 140 160 180 Number of nodes n Number of nodes n i.i.d. ∼ N ( θ, 1) Sufficient Statistics for Mean Estimation Y 1 , . . . , Y n � i Y i sufficient to estimate θ : no performance loss i.i.d. Binary Hypothesis Test: Y 1 , . . . , Y n ∼ f ( y ; H 0 ) or f ( y ; H 1 ) i log f ( Y i ; H 0 ) , � i log f ( Y i ; H 1 )] sufficient to decide hypothesis [ � log f ( Y i ; H 1 ) LLR= � log f ( Y i ; H 1 ) minimally sufficient to decide hypothesis i Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 3 / 21

Minimum Cost In-Network Processing for Inference Minimal Sufficient Statistic for Binary Hypothesis Testing Log Likelihood Ratio: LLR ( Y V ) = log f ( Y V ; H 1 ) f ( Y V ; H 0 ) Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 4 / 21

Minimum Cost In-Network Processing for Inference Minimal Sufficient Statistic for Binary Hypothesis Testing Log Likelihood Ratio: LLR ( Y V ) = log f ( Y V ; H 1 ) f ( Y V ; H 0 ) Extent of Processing? Fusion Center ? LLR ( Y V ) = log f ( Y V ; H 1 ) log f ( Y V ; H 0 ) Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 4 / 21

Minimum Cost In-Network Processing for Inference Minimal Sufficient Statistic for Binary Hypothesis Testing Log Likelihood Ratio: LLR ( Y V ) = log f ( Y V ; H 1 ) f ( Y V ; H 0 ) Extent of Processing? Fusion Center ? Fusion Scheme? LLR ( Y V ) = log f ( Y V ; H 1 ) log f ( Y V ; H 0 ) Minimum Cost Data Fusion for Inference Min total costs s.t. LLR is delivered to fusion center Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 4 / 21

Minimum Cost In-Network Processing for Inference Minimal Sufficient Statistic for Binary Hypothesis Testing Log Likelihood Ratio: LLR ( Y V ) = log f ( Y V ; H 1 ) f ( Y V ; H 0 ) Extent of Processing? Fusion Center ? Fusion Scheme? LLR ( Y V ) = log f ( Y V ; H 1 ) log f ( Y V ; H 0 ) Minimum Cost Data Fusion for Inference Min total costs s.t. LLR is delivered to fusion center Spatial Correlation Model: Should Capture Full Correlation Range Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 4 / 21

Minimum Cost In-Network Processing for Inference Minimal Sufficient Statistic for Binary Hypothesis Testing Log Likelihood Ratio: LLR ( Y V ) = log f ( Y V ; H 1 ) f ( Y V ; H 0 ) Extent of Processing? Fusion Center ? Fusion Scheme? LLR ( Y V ) = log f ( Y V ; H 1 ) log f ( Y V ; H 0 ) Minimum Cost Data Fusion for Inference Min total costs s.t. LLR is delivered to fusion center Spatial Correlation Model: Should Capture Full Correlation Range Markov random field with dependency graph Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 4 / 21

Minimum Cost In-Network Processing for Inference Minimal Sufficient Statistic for Binary Hypothesis Testing Log Likelihood Ratio: LLR ( Y V ) = log f ( Y V ; H 1 ) f ( Y V ; H 0 ) Extent of Processing? Fusion Center ? Fusion Scheme? LLR ( Y V ) = log f ( Y V ; H 1 ) log f ( Y V ; H 0 ) Minimum Cost Data Fusion for Inference Min total costs s.t. LLR is delivered to fusion center Spatial Correlation Model: Should Capture Full Correlation Range Markov random field with dependency graph Structured LLR: sum over dependency graph cliques Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 4 / 21

Minimum Cost In-Network Processing for Inference Minimal Sufficient Statistic for Binary Hypothesis Testing Log Likelihood Ratio: LLR ( Y V ) = log f ( Y V ; H 1 ) f ( Y V ; H 0 ) Extent of Processing? Fusion Center ? Fusion Scheme? LLR ( Y V ) = log f ( Y V ; H 1 ) log f ( Y V ; H 0 ) Minimum Cost Data Fusion for Inference Min total costs s.t. LLR is delivered to fusion center Spatial Correlation Model: Should Capture Full Correlation Range Markov random field with dependency graph Structured LLR: sum over dependency graph cliques Local processing of clique data Anandkumar, Tong, Swami, Ephremides Minimum Cost Fusion for Inference IEEE INFOCOM ‘08 4 / 21