An Empirical Bayes Approach for Distributed Estimation of Spatial Fields

In this paper we consider a network of spatially distributed sensors which collect measurement samples of a spatial field, and aim at estimating in a distributed way (without any central coordinator) the entire field by suitably fusing all network data. We propose a general probabilistic model that can handle both partial knowledge of the physics generating the spatial field as well as a purely data-driven inference. Specifically, we adopt an Empirical Bayes approach in which the spatial field is modeled as a Gaussian Process, whose mean function is described by means of parametrized equations. We characterize the Empirical Bayes estimator when nodes are heterogeneous, i.e., perform a different number of measurements. Moreover, by exploiting the sparsity of both the covariance and the (parametrized) mean function of the Gaussian Process, we are able to design a distributed spatial field estimator. We corroborate the theoretical results with two numerical simulations: a stationary temperature field estimation in which the field is described by a partial differential (heat) equation, and a data driven inference in which the mean is parametrized by a cubic spline

Distributed Learning from Interactions in Social Networks

We consider a network scenario in which agents can evaluate each other according to a score graph that models some interactions. The goal is to design a distributed protocol, run by the agents, that allows them to learn their unknown state among a finite set of possible values. We propose a Bayesian framework in which scores and states are associated to probabilistic events with unknown parameters and hyperparameters, respectively. We show that each agent can learn its state by means of a local Bayesian classifier and a (centralized) Maximum-Likelihood (ML) estimator of parameter-hyperparameter that combines plain ML and Empirical Bayes approaches. By using tools from graphical models, which allow us to gain insight on conditional dependencies of scores and states, we provide a relaxed probabilistic model that ultimately leads to a parameter-hyperparameter estimator amenable to distributed computation. To highlight the appropriateness of the proposed relaxation, we demonstrate the distributed estimators on a social interaction set-up for user profiling.Comment: This submission is a shorter work (for conference publication) of a more comprehensive paper, already submitted as arXiv:1706.04081 (under review for journal publication). In this short submission only one social set-up is considered and only one of the relaxed estimators is proposed. Moreover, the exhaustive analysis, carried out in the longer manuscript, is completely missing in this versio

Evaluation of Natural Robustness of Best Constant Weights to Random Communication Breakdowns

One of the most crucial aspects of an algorithmdesign for the wireless sensors networks is the failure tolerance.A high natural robustness and an effectively bounded executiontime are factors that can significantly optimize the overall energyconsumption and therefore, a great emphasis is laid on theseaspects in many applications from the area of the wireless sensornetworks. This paper addresses the robustness of the optimizedBest Constant weights of Average Consensus with a stoppingcriterion (i.e. the algorithm is executed in a finite time) and theirfive variations with a lower mixing parameter (i.e. slowervariants) to random communication breakdowns modeled as a stochastic event of a Bernoulli distribution. We choose threemetrics, namely the deviation of the least precise final estimatesfrom the average, the convergence rate expressed as the numberof the iterations for the consensus, and the deceleration of eachinitial setup, in order to evaluate the robustness of various initialsetups of Best Constant weights under a varying failureprobability and over 30 random geometric graphs of either astrong or a weak connectivity. Our contribution is to find themost robust initial setup of Best Constant weights according tonumerical experiments executed in Matlab. Finally, theexperimentally obtained results are discussed, compared to theresults from the error-free executions, and our conclusions arecompared with the conclusions from related papers

Comparative Study of Distributed Estimation Precision by Average Consensus Weight Models

Distributed algorithms for an aggregate function estimation are an important complement of many real-life applications based on wireless sensor networks. Achieving a high precision of an estimation in a shorter time can optimize the overall energy consumption. Therefore, the choice of a proper distributed algorithm is an important part of an application design. In this study, we focus our attention on the average consensus algorithm and evaluate six weight models appropriate for the implementation into real-life applications. Our aim is to find the most suitable model in terms of the estimation precision in various phases of the algorithm. We examine the deviation of the least precise estimate over iterations for a Gaussian, a Uniform and a Bernoulli distribution of the initial states in strongly and weakly connected networks with a randomly generated topology. We examine which model is the most and the least precise in various phases. Based on these findings, we determine the most suitable model for real-life applications

A Bayesian Framework for Distributed Estimation of Arrival Rates in Asynchronous Networks

In this paper, we consider a network of agents monitoring a spatially distributed arrival process. Each node measures the number of arrivals seen at its monitoring point in a given time interval with the objective of estimating the unknown local arrival rate. We propose an asynchronous distributed approach based on a Bayesian model with unknown hyperparameter, where each node computes the minimum mean-square error (MMSE) estimator of its local arrival rate in a distributed way. As a result, the estimation at each node “optimally” fuses the information from the whole network through a distributed optimization algorithm. Moreover, we propose an ad-hoc distributed estimator, based on a consensus algorithm for time-varying and directed graphs, which exhibits reduced complexity and exponential convergence. We analyze the performance of the proposed distributed estimators, showing that they i) are reliable even in presence of limited local data and ii) improve the estimation accuracy compared to the purely decentralized setup. Finally, we provide a statistical characterization of the proposed estimators. In particular, for the ad-hoc estimator, we show that as the number of nodes goes to infinity its mean square error converges to the optimal one. Numerical Monte Carlo simulations confirm the theoretical characterization and highlight the appealing performances of the estimators