[Paper Review] Recursive Distributed Detection for Composite Hypothesis Testing: Algorithms and Asymptotics

Anit Kumar Sahu, Soummya Kar|arXiv (Cornell University)|Jan 19, 2016

Distributed Sensor Networks and Detection Algorithms37 references3 citations

TL;DR

This paper proposes two distributed recursive generalized likelihood ratio tests, CILRT and CIGLRT, for composite hypothesis testing in sparsely connected networks of agents. By combining local sensing with consensus-based information sharing, the algorithms enable agents to jointly detect signals corrupted by noise, achieving asymptotically decaying error probabilities under global observability and large deviations performance guarantees in linear models.

ABSTRACT

This paper studies recursive composite hypothesis testing in a network of sparsely connected agents. The network objective is to test a simple null hypothesis against a composite alternative concerning the state of the field, modeled as a vector of (continuous) unknown parameters determining the parametric family of probability measures induced on the agents' observation spaces under the hypotheses. Specifically, under the alternative hypothesis, each agent sequentially observes an independent and identically distributed time-series consisting of a (nonlinear) function of the true but unknown parameter corrupted by Gaussian noise, whereas, under the null, they obtain noise only. Two distributed recursive generalized likelihood ratio test type algorithms of the \emph{consensus+innovations} form are proposed, namely $\mathcal{CILRT}$ and $\mathcal{CIGLRT}$, in which the agents estimate the underlying parameter and in parallel also update their test decision statistics by simultaneously processing the latest local sensed information and information obtained from neighboring agents. For $\mathcal{CIGLRT}$, for a broad class of nonlinear observation models and under a global observability condition, algorithm parameters which ensure asymptotically decaying probabilities of errors~(probability of miss and probability of false detection) are characterized. For $\mathcal{CILRT}$, a linear observation model is considered and large deviations decay exponents for the error probabilities are obtained.

Motivation & Objective

• To develop distributed, recursive algorithms for composite hypothesis testing in networks of sparsely connected agents.
• To enable agents to jointly detect a signal in noise when the signal's parametric form is unknown but lies in a continuous family.
• To characterize algorithm parameters that ensure asymptotically decaying probabilities of miss and false detection under global observability.
• To derive large deviations decay exponents for error probabilities in the linear observation model case.

Proposed method

• The algorithms use a consensus+innovations architecture, where agents update their local estimates and test statistics using both new observations and information from neighbors.
• CIGLRT is designed for nonlinear observation models and employs recursive parameter estimation alongside likelihood ratio testing.
• CILRT is tailored for linear models and uses a linearized likelihood ratio test with recursive parameter updates.
• Agents maintain local sufficient statistics and iteratively refine their estimates and decision rules using a distributed, recursive update rule.
• The algorithms ensure asymptotic consistency of parameter estimates and convergence of test statistics to the true hypothesis.
• Global observability is assumed as a condition to guarantee that the network as a whole can identify the true parameter and distinguish hypotheses.

Experimental results

Research questions

• RQ1 Can distributed recursive algorithms achieve asymptotically decaying error probabilities in composite hypothesis testing under nonlinear observation models?
• RQ2 What parameter choices in the CIGLRT algorithm ensure that both false alarm and miss probabilities decay to zero?
• RQ3 How do the large deviations decay exponents of error probabilities behave in the linear model case under the CILRT algorithm?
• RQ4 How does the consensus+innovations structure enable reliable detection in sparse, decentralized networks?

Key findings

• For CIGLRT, under a global observability condition, algorithm parameters can be chosen such that both the probability of miss and the probability of false detection decay asymptotically to zero.
• For CILRT in the linear model, the paper derives explicit large deviations decay exponents for the error probabilities, quantifying the rate of decay.
• The proposed algorithms achieve consistent parameter estimation and reliable hypothesis testing through distributed, recursive updates.
• The consensus+innovations framework enables agents to combine local sensing with neighbor information effectively, improving detection performance.