Detection of Gauss Markov Random Fields under Routing Energy - PowerPoint PPT Presentation

Detection of Gauss Markov Random Fields under Routing Energy Constraint A. Anandkumar 1 L. Tong 1 A. Swami 2 1 School of Electrical and Computer Engineering Cornell University, Ithaca, NY 14853 2 Army Research Laboratory, Adelphi MD 20783 Forty-fifth Annual Allerton Conference on Communication, Control, and Computing . Supported by the Army Research Laboratory CTA A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 1 / 26

Detection-Energy Tradeoff Distributed Detection Classical Routing Quantization rule @ sensors Generic Performance Metric Inference rule @ fusion center Layered architecture Shortcomings of Classical Detection For sensors in a large field, multi-hop routing is needed For energy-constrained networks, loss in detection performance Shortcomings of Classical Routing Need only likelihood ratio for inference, not raw data at fusion center Tradeoff between Routing and Detection in Wireless Sensor Networks A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 2 / 26

Tradeoff: Optimal Detection under Energy Constraint Optimal Detection of Binary Hypothesis Neyman Pearson: Min. miss detection subject to false alarm Large Networks: n → ∞ max − 1 n log P M subject to false alarm and avg. routing energy ¯ E Optimal Node Density λ ∗ ∆ subject to ¯ C ≤ ¯ = arg max λ ∗ λ> 0 D λ E D λ : Neyman-Pearson error exponent ¯ C: Average Routing Energy per node A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 3 / 26

Node Deployment Strategies for Optimal Tradeoff Node deployments Random Setup Random: Nodes drawn from uniform or Poisson distribution Constant density λ : n nodes in area n λ Factors Implications Signal & Energy Model λ ∗ → 0 or ∞ : Large/Small area Nature of Tradeoff λ ∗ ∈ (0 , ∞ ) : Careful Deployment A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 4 / 26

Node Deployment Strategies for Optimal Tradeoff Node deployments Random Setup Random: Nodes drawn from uniform or Poisson distribution Constant density λ : n nodes in area n λ Factors Implications Signal & Energy Model λ ∗ → 0 or ∞ : Large/Small area Nature of Tradeoff λ ∗ ∈ (0 , ∞ ) : Careful Deployment A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 4 / 26

Node Deployment Strategies for Optimal Tradeoff Node deployments Const. Density λ Setup Random: Nodes drawn from uniform or Poisson distribution Constant density λ : n nodes in area n λ Factors Implications Signal & Energy Model λ ∗ → 0 or ∞ : Large/Small area Nature of Tradeoff λ ∗ ∈ (0 , ∞ ) : Careful Deployment A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 4 / 26

Node Deployment Strategies for Optimal Tradeoff Node deployments Const. Density λ Setup Random: Nodes drawn from uniform or Poisson distribution Constant density λ : n nodes in area n λ Factors Implications Signal & Energy Model λ ∗ → 0 or ∞ : Large/Small area Nature of Tradeoff λ ∗ ∈ (0 , ∞ ) : Careful Deployment A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 4 / 26

Node Deployment Strategies for Optimal Tradeoff Node deployments λ ∗ → 0 or R → ∞ λ ∗ ∈ (0 , ∞ ) λ ∗ → ∞ or R → 0 Setup Random: Nodes drawn from uniform or Poisson distribution Constant density λ : n nodes in area n λ Factors Implications Signal & Energy Model λ ∗ → 0 or ∞ : Large/Small area Nature of Tradeoff λ ∗ ∈ (0 , ∞ ) : Careful Deployment A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 4 / 26

Example: Same Variances, No Energy Constraint Detection of Correlation H 1 : Correlated data vs. H 0 : Independent observations Assumptions Uniform signal field: same variance at every node, under H 0 and H 1 Correlation decays with distance under H 1 Only way to distinguish H 0 and H 1 : Correlation Intuition: to maximize correlation : Minimize inter-node distance In this case, λ ∗ → ∞ . What happens when variances are different? A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 5 / 26

Example: Same Variances, No Energy Constraint Detection of Correlation H 1 : Correlated data vs. H 0 : Independent observations Assumptions Uniform signal field: same variance at every node, under H 0 and H 1 Correlation decays with distance under H 1 Only way to distinguish H 0 and H 1 : Correlation Intuition: to maximize correlation : Minimize inter-node distance In this case, λ ∗ → ∞ . What happens when variances are different? A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 5 / 26

Example: Same Variances, No Energy Constraint Detection of Correlation H 1 : Correlated data vs. H 0 : Independent observations Assumptions Uniform signal field: same variance at every node, under H 0 and H 1 Correlation decays with distance under H 1 Only way to distinguish H 0 and H 1 : Correlation Intuition: to maximize correlation : Minimize inter-node distance In this case, λ ∗ → ∞ . What happens when variances are different? A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 5 / 26

Results on Optimal Node Density λ ∗ Variance Ratio K K is ratio of variances under alternative and null hypotheses Avg. Energy vs. λ Exponent vs. λ K < K t D ¯ C K > K ′ t Density λ Density λ No Energy constraint Feasible Energy A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 6 / 26

Results on Optimal Node Density λ ∗ Variance Ratio K K is ratio of variances under alternative and null hypotheses Avg. Energy vs. λ Exponent vs. λ K < K t D ¯ C K > K ′ t Density λ Density λ No Energy constraint Feasible Energy Optimal density λ ∗ Optimal density λ ∗ 0 0 Variance ratio K Variance ratio K K t K t A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 6 / 26

Results on Optimal Node Density λ ∗ Variance Ratio K K is ratio of variances under alternative and null hypotheses Avg. Energy vs. λ Exponent vs. λ K < K t D ¯ C K > K ′ t Density λ Density λ No Energy constraint Feasible Energy Optimal density λ ∗ Optimal density λ ∗ + ∞ + ∞ 0 0 Variance ratio K Variance ratio K K t K t A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 6 / 26

Results on Optimal Node Density λ ∗ Variance Ratio K K is ratio of variances under alternative and null hypotheses Avg. Energy vs. λ Exponent vs. λ K < K t D ¯ C K > K ′ t Density λ Density λ No Energy constraint Feasible Energy Optimal density λ ∗ Optimal density λ ∗ + ∞ + ∞ 0 0 Variance ratio K Variance ratio K K t K t A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 6 / 26

Results on Optimal Node Density λ ∗ Variance Ratio K K is ratio of variances under alternative and null hypotheses Avg. Energy vs. λ Exponent vs. λ K < K t D ¯ C K > K ′ t Density λ Density λ No Energy constraint Feasible Energy Optimal density λ ∗ Optimal density λ ∗ + ∞ + ∞ ? 0 0 K ′ Variance ratio K Variance ratio K K t K t t A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 6 / 26

Results on Optimal Node Density λ ∗ Variance Ratio K K is ratio of variances under alternative and null hypotheses Avg. Energy vs. λ Exponent vs. λ K < K t D ¯ C K > K ′ t Density λ Density λ No Energy constraint Feasible Energy Optimal density λ ∗ Optimal density λ ∗ + ∞ + ∞ λ E ? 0 0 K ′ Variance ratio K Variance ratio K K t K t t A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 6 / 26

Methodology Detection of Correlation H 1 : Correlated data vs. H 0 : Independent observations Modeling correlation Min. energy routing Gauss-Markov random field NP-hard (CISS 07) Correlation decays with dist. 2-approx. algo DFMRF Partial correlation at 0 Closed-form average energy Nearest-neighbor dependency Constraint: bound on λ Optimal node density Tractable performance metric Analyze error exponent Closed-form Neyman Pearson error exponent (ICASSP 07) A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 7 / 26

Methodology Detection of Correlation H 1 : Correlated data vs. H 0 : Independent observations Modeling correlation Min. energy routing Gauss-Markov random field NP-hard (CISS 07) Correlation decays with dist. 2-approx. algo DFMRF Partial correlation at 0 Closed-form average energy Nearest-neighbor dependency Constraint: bound on λ Optimal node density Tractable performance metric Analyze error exponent Closed-form Neyman Pearson error exponent (ICASSP 07) A. Anandkumar, L.Tong, A. Swami (Cornell) Detection Energy Tradeoff Allerton 2007 7 / 26