Error Probability Bounds for Balanced Binary Relay Trees (original) (raw)
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1Error Probability Bounds for Balanced Binary Relay Trees
2016
We study the detection error probability associated with a balanced binary relay tree, where the leaves of the tree correspond to N identical and independent detectors. The root of the tree represents a fusion center that makes the overall detection decision. Each of the other nodes in the tree are relay nodes that combine two binary messages to form a single output binary message. In this way, the information from the detectors is aggregated into the fusion center via the intermediate relay nodes. In this context, we describe the evolution of Type I and Type II error probabilities of the binary data as it propagates from the leaves towards the root. Tight upper and lower bounds for the total error probability at the fusion center as functions of N are derived. These characterize how fast the total error probability converges to 0 with respect to N, even if the individual sensors have error probabilities that converge to 1/2. Index Terms Binary relay tree, distributed detection, dec...
Error probability bounds for binary relay trees with unreliable communication links
Conference Record - Asilomar Conference on Signals, Systems and Computers, 2011
We study the detection error probability associated with balanced binary relay trees, in which sensor nodes fail with some probability. We consider N identical and independent crummy sensors, represented by leaf nodes of the tree. The root of the tree represents the fusion center, which makes the final decision between two hypotheses. Every other node is a relay node, which fuses at most two binary messages into one binary message and forwards the new message to its parent node. We derive tight upper and lower bounds for the total error probability at the fusion center as functions of N and characterize how fast the total error probability converges to 0 with respect to N . We show that the convergence of the total error probability is sub-linear, with the same decay exponent as that in a balanced binary relay tree without sensor failures. We also show that the total error probability converges to 0, even if the individual sensors have total error probabilities that converge to 1/2 and the failure probabilities that converge to 1, provided that the convergence rates are sufficiently slow.
Detection Performance in Balanced Binary Relay Trees With Node and Link Failures
IEEE Transactions on Signal Processing, 2000
We study the distributed detection problem in the context of a balanced binary relay tree, where the leaves of the tree correspond to N identical and independent sensors generating binary messages. The root of the tree is a fusion center making an overall decision. Every other node is a relay node that aggregates the messages received from its child nodes into a new message and sends it up toward the fusion center. We derive upper and lower bounds for the total error probability P N as explicit functions of N in the case where nodes and links fail with certain probabilities. These characterize the asymptotic decay rate of the total error probability as N goes to infinity. Naturally, this decay rate is not larger than that in the non-failure case, which is √ N. However, we derive an explicit necessary and sufficient condition on the decay rate of the local failure probabilities p k (combination of node and link failure probabilities at each level) such that the decay rate of the total error probability in the failure case is the same as that of the non-failure case. More precisely, we show that log P −1 N = Θ(√ N) if and only if log p −1 k = Ω(2 k/2).
Near-Optimal Distributed Detection in Balanced Binary Relay Trees
IEEE Transactions on Control of Network Systems, 2016
We study the distributed detection problem in a balanced binary relay tree, where the leaves of the tree are sensors generating binary messages. The root of the tree is a fusion center that makes an overall decision. Every other node in the tree is a relay node that fuses binary messages from its two child nodes into a new binary message and sends it to the parent node at the next level. We assume that the relay nodes at the same level use identical fusion rule. The goal is to find a string of fusion rules used at all the levels in the tree that maximizes the reduction in the total error probability between the leaf nodes and the fusion center. We formulate this problem as a deterministic dynamic program and express the optimal strategy in terms of Bellman's equation. Moreover, we use the notion of string-submodularity to show that the reduction in the total error probability is a string-submodular function. Consequentially, we show that the greedy strategy, which only maximizes the level-wise reduction in the total error probability, performs at least within a factor (1 − 1/e) of the optimal strategy in terms of reduction in the total error probability, even if the nodes and links in the trees are subject to random failures.
1Detection Performance in Balanced Binary Relay Trees with Node and Link Failures
2014
We study the distributed detection problem in the context of a balanced binary relay tree, where the leaves of the tree correspond to N identical and independent sensors generating binary messages. The root of the tree is a fusion center making an overall decision. Every other node is a relay node that aggregates the messages received from its child nodes into a new message and sends it up toward the fusion center. We derive upper and lower bounds for the total error probability PN as explicit functions of N in the case where nodes and links fail with certain probabilities. These characterize the asymptotic decay rate of the total error probability as N goes to infinity. Naturally, this decay rate is not larger than that in the non-failure case, which is N. However, we derive an explicit necessary and sufficient condition on the decay rate of the local failure probabilities pk (combination of node and link failure probabilities at each level) such that the decay rate of the total er...
Error probability bounds for M-ary relay trees
arXiv preprint arXiv:1202.1354, 2012
Abstract: We study the detection error probabilities associated with an M-ary relay tree, where the leaves of the tree correspond to identical and independent sensors. Only these leaves are sensors. The root of the tree represents a fusion center that makes the overall detection decision. Each of the other nodes in the tree is a relay node that combines M summarized messages from its immediate child nodes to form a single output message using the majority dominance rule. We derive tight upper and lower bounds for the Type I ...
Submodularity and optimality of fusion rules in balanced binary relay trees
Proceedings of the IEEE Conference on Decision and Control, 2012
We study the distributed detection problem in a balanced binary relay tree, where the leaves of the tree are sensors generating binary messages. The root of the tree is a fusion center that makes the overall decision. Every other node in the tree is a fusion node that fuses two binary messages from its child nodes into a new binary message and sends it to the parent node at the next level. We assume that the fusion nodes at the same level use the same fusion rule. We call a string of fusion rules used at different levels a fusion strategy. We consider the problem of finding a fusion strategy that maximizes the reduction in the total error probability between the sensors and the fusion center. We formulate this problem as a deterministic dynamic program and express the solution in terms of Bellman's equations. We introduce the notion of stringsubmodularity and show that the reduction in the total error probability is a string-submodular function. Consequentially, we show that the greedy strategy, which only maximizes the level-wise reduction in the total error probability, is within a factor (1 − e −1 ) of the optimal strategy in terms of reduction in the total error probability.
Detection performance of M-ary relay trees with non-binary message alphabets
2012 IEEE Statistical Signal Processing Workshop, SSP 2012, 2012
We study the detection performance of M -ary relay trees, where only the leaves of the tree represent sensors making measurements. The root of the tree represents the fusion center which makes an overall detection decision. Each of the other nodes is a relay node which aggregates M messages sent by its child nodes into a new compressed message and sends the message to its parent node. Building on previous work on the detection performance of M -ary relay trees with binary messages, in this paper we study the case of non-binary relay message alphabets. We characterize the exponent of the error probability with respect to the message alphabet size D, showing how the detection performance increases with D. Our method involves reducing a tree with non-binary relay messages into an equivalent higher-degree tree with only binary messages.
ICASSP, IEEE International Conference on Acoustics, Speech and Signal Processing - Proceedings, 2009
Wireless sensor networks are considered in which sensors convey binary decisions over fading channels to a common fusion center. The fusion center first takes each received signal and makes an estimate of the transmitted bit. The average of the estimated bits is compared to a threshold to make a global decision. Exponential error bounds are derived that allow one to trade off signal-to-noise ratio versus the number of sensors to achieve desired average error levels. An attractive feature of the bounds is that they do not require exact knowledge of the wireless channel statistics; approximations are sufficient.
Data fusion trees for detection: Does architecture matter?
Information Theory, IEEE …, 2008
We consider the problem of decentralized detection in a network consisting of a large number of nodes arranged as a tree of bounded height, under the assumption of conditionally independent, identically distributed observations. We characterize the optimal error exponent under a Neyman-Pearson formulation. We show that the Type II error probability decays exponentially fast with the number of nodes, and the optimal error exponent is often the same as that corresponding to a parallel configuration.