1 Optimal Fault-Tolerant Data Fusion in Sensor Networks Fundamental Limits and E cient

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Optimal Fault-Tolerant Data Fusion in Sensor

Networks: Fundamental Limits and Eﬃcient

Algorithms

Marian Temprana Alonso, Farhad Shirani , Sitharama S. Iyengar,

Florida International University,

Email: mtemp009@ﬁu.edu, fshirani@ﬁu.edu, iyengar@cis.ﬁu.edu

Abstract

Distributed estimation in the context of sensor networks is considered, where distributed agents

are given a set of sensor measurements, and are tasked with estimating a target variable. A subset of

sensors are assumed to be faulty. The objective is to minimize i) the mean square estimation error

at each node (accuracy objective), and ii) the mean square distance between the estimates at each

pair of nodes (consensus objective). It is shown that there is an inherent tradeoﬀbetween the former

and latter objectives. Assuming a general stochastic model, the sensor fusion algorithm optimizing this

tradeoﬀis characterized through a computable optimization problem, and a Cramer-Rao type lower bound

for the achievable accuracy-consensus loss is obtained. Finding the optimal sensor fusion algorithm is

computationally complex. To address this, a general class of low-complexity Brooks-Iyengar Algorithms

are introduced, and their performance, in terms of accuracy and consensus objectives, is compared to

that of optimal linear estimators through case study simulations of various scenarios.

I. Introduction

Distributed estimation arises naturally in various application scenarios including sensor networks [1]–

[4], robotics [5], navigation, tracking and radar networks [6], and monitoring and surveillance applications

[7]. The problem has been studied extensively in the literature under a range of assumptions and

formulations. In recent years, there has been signiﬁcant renewed interest in distributed estimation scenarios

involving sensor fusion due to advances in sensor and communication technologies, which has enabled

the application of massive non-homogeneous collections of sensors including radar, LiDAR, camera,

This work was supported in part by NSF grant CCF-2241057.

arXiv:2210.04156v3 [eess.SP] 22 Dec 2022

Agent 1

Agent 2

Agent 3

Agent 4

Sensor Network

𝐿!𝑈

𝐿"𝑈"𝐿#𝑈#

𝑋!

𝑋"#

𝑋#

𝑋$

Fig. 1: Each sensor measures the target variable Xand outputs a conﬁdence interval. Non-faulty sensors

are shown by solid black circles and faulty sensors by dashed red circles.

ultrasonic, GPS, IMU, and V2X in applications including tracking, navigation, and autonomous driving

[8].

This paper considers the distributed estimation scenario shown in Figure 1, where mdistributed agents

receive a set of nsensor measurements, in the form of conﬁdence intervals [Li,j,Ui,j],i∈[n],j∈[m], and

are tasked with ﬁnding the best estimate of a target variable Xwith respect to ﬁdelity criteria on both in-

dividual accuracy and collective consensus among sensors. In general, the sensor measurements are noisy,

and additionally a subset of sensors may be faulty, where a faulty sensor provides measurements which are

independent of the target variable. Furthermore, faulty sensors transmit independent measurements to each

of the distributed agents. Faulty sensors model both sensor malfunction as well as adversarial interference

in the sensor measurement and transmission phases [9], [10]. The distributed estimation system has two

objectives: i) Accuracy objective: to minimize the mean square error of the estimate of the target variable

by each agent, and ii) Consensus objective: to minimize the square distance between pairs of estimates of

the target variable by each of the distributed agents. The former objective is a local performance objective

focusing on the individual accuracy of each sensor, whereas the latter objective is a global performance

objective focusing on the collective agreement among distributed sensors. The consensus objective is

of interest in applications such as clock synchronization, robot convergence and gathering, autonomous

driving, blockchain technologies, and distributed voting, among others. The consensus objective has been

studied extensively in the context of the Byzantine Agreement Problem [11]–[14] as well as the study

of consensus-based ﬁlters [15], [16].

The study of the distributed estimation scenario considered in this paper was initiated in [17], [18].

Marzullo proposed a sensor fusion algorithm in [19] and derived the corresponding worst case perfor-

mance guarantees. An improved algorithm was proposed by Brooks and Iyengar (BI) in [20], and worst-

case theoretical guarantees for both accuracy and consensus objectives were derived in [21]. It was shown

that the BI algorithm improves upon Marzullo’s algorithm in terms of the worst-case performance in the

consensus objective. On the theory side, prior works have considered various other formulations of the

distributed estimation problem under Bayesian, mean square error, and Dempster–Shafer theory paradigms

[15], [16], [22], [23]. In this paper, we ﬁrst formulate a general distributed estimation problem, and

provide an analytical framework to evaluate the performance limits in terms of the accuracy and consensus

objectives. We show that there is a fundamental tradeoﬀbetween these two objectives, and characterize the

fusion operation optimizing the tradeoﬀin the form of a convex optimization problem. Furthermore, we

provide a lower bound on the achievable accuracy-consensus loss. The bound is quantiﬁed by the average

Fisher information of the sensor output variables and is obtained via a Cramer-Rao type inequality.

Analytical characterization of the fusion operation optimizing the aforementioned tradeoﬀrequires prior

knowledge of the underlying statistics, which is not possible in many real-world applications. Furthermore,

even in presence of accurate estimates of the underlying statistics, deriving the optimal decision function

has high computational complexity. In the second part of the paper, we propose a generalized class

of practical low-complexity Brooks-Iyengar fusion algorithms and evaluate their performance through

various simulations of practical scenarios. The main contributions of this work are summarized below:

•To characterize a tradeoﬀin the accuracy and consensus objectives and to provide a computable

optimization problem for ﬁnding the fusion algorithm optimizing this tradeoﬀ.

•To provide a Cramer-Rao type bound on the performance of the optimal algorithm in terms of accuracy

and consensus.

•To introduce a generalized class of BI algorithms (GBI).

•To prove the optimality of the GBI algorithms, in terms of the accuracy objective, under speciﬁc

statistical assumptions.

•To provide case study simulations of the performance of the proposed algorithms.

Notation: The set {1,2,··· ,n},n∈Nis represented by [n]. An n-length vector is written as [xi]i∈[n]

and an n×mmatrix is written as [hi,j]i,j∈[n]×[m]. We write xand hinstead of [xi]i∈[n]and [hi,j]i,j∈[n]×[m],

respectively, when the dimension is clear from context. The vector emis the m-length all-ones vector. Sets

are denoted by calligraphic letters such as X.Φrepresent the empty set. Bdenotes the Borel σ-ﬁeld. For

the event E, the variable (E) denotes the indicator of the event. For random variable Xwith probability

space (X,FX,PX), the Hilbert space of functions of Xwith ﬁnite variance is denoted by L2(X).

II. Problem Formulation

We consider a distributed estimation problem involving magents and nsensors. In the most general

formulation, each agent is tasked with estimating the target variable Xdeﬁned on the probability space

(R,B,PX), where Bdenotes the Borel σ-algebra. Each sensor measures the target variable X separately,

and the measurement output is assumed to be in the form of an interval [L,U], where L<U. The

ith sensor transmits the pair (Li,j,Ui,j) to the jth agent, where i∈[n],j∈[m] and Li,j≤Ui,j. This is

formalized below.

Deﬁnition 1 (Distributed Estimation Setup).A distributed estimation setup is characterized by the

tuple (n,m,PX,(Li,j,Ui,j)i∈[n],j∈[m]), where n,m∈Nand PX,(Li,j,Ui,j)i∈[n],j∈[m]is a probability measure deﬁned on

Rnm+1such that P(Li,j≤Ui,j,i∈[n],j∈[m]) =0.

The fusion algorithm is formally deﬁned below.

Deﬁnition 2 (Fusion Algorithm).For a distributed estimation setup characterized by the tuple (n,m,

PX,(Li,j,Ui,j)i∈[n],j∈[m]), a fusion algorithm fconsists of a collection of functions fj:R2n→R,j∈[m]. The

estimate of X at the jth agent is b

Xj,fj(Li,j,Ui,j,i∈[n]),j∈[m].

The fusion algorithm is evaluated with respect to an accuracy objective and a consensus objective as

formalized below.

Deﬁnition 3 (Fusion Objectives).For a given a distributed estimation setup (n,m,PX,(Li,j,Ui,j)i∈[n],j∈[m])and

fusion algorithm f=(fj(·),j∈[m]), the accuracy is parametrized by the vector (mse(fj),j∈[m]), where:

mse(fj),EX−b

Xj2,j∈[m].

The consensus is parametrized by the vector (cns(fj,fj0),j,j0∈[m]), where:

cns(fj,fj0),Eb

Xj−b

Xj02,j,j0∈[m].

Given parameter λ∈Rsuch that 0≤λ≤1, the fusion algorithm f∗

λis called λ-optimal if it is the

solution to the following optimization problem:

f∗

λ=arg min

f=(fj)j∈[m]

λX

j∈[m]

mse(fj)+λ

m−1X

j,j0∈[m]

j,j0

cns(fj,fj0),(1)

where the minimum is taken over all f=(fj)j∈[m]such that fj∈ L2(Qi∈[n]Li,j×Ui,j),j∈[m]and λ,1−λ.

In Section III, we evaluate the accuracy-consensus tradeoﬀunder the general distributed estimation

model described above. We characterize the λ-optimal sensor fusion algorithm in the form of a computable

optimization problem. The optimization requires knowledge of the underlying distribution and has high

computational complexity. In Section IV, we restrict our study to a speciﬁc subset of distributed esti-

mation scenarios involving faulty sensor measurements, and provide practical fault-tolerant sensor fusion

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1OptimalFault-TolerantDataFusioninSensorNetworks:FundamentalLimitsandEcientAlgorithmsMarianTempranaAlonso,FarhadShirani,SitharamaS.Iyengar,FloridaInternationalUniversity,Email:mtemp009@u.edu,fshirani@u.edu,iyengar@cis.u.eduAbstractDistributedestimationinthecontextofsensornetworksisconsidered,whe...

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