Optimal Estimation from Relative Measurements: Error Scaling (Extended Abstract) Prabir Barooah Jo˜ ao P. Hespanha I. ESTIMATION FROM RELATIVE MEASUREMENTS We consider the problem of estimating a number of vector valued variables from a number of noisy “relative measurements”, i.e., measurement of the difference between certain pairs of these variables. This type of measurement model appears in several sensor network problems, such as sensor localization and time synchronization [1]. Consider n vector-valued variables x 1 ,x 2 ,...,x n ∈ R k , called node variables, one or more of which are known, and the rest are unknown. A number of noisy measurements of the difference x u − x v are available for certain pairs of nodes (u, v). We can associate the variables with the nodes V = {1, 2,...,n} of a directed graph G =(V, E) and the measurements with the edges E of it, consisting of ordered pairs (u, v) such that a noisy “relative” measurement between x u and x v is available: ζ uv = x u − x v + ǫ u,v ∈ R k , (u, v) ∈ E ⊂ V × V, (1) where the ǫ u,v ’s are uncorrelated zero-mean noise vectors with known covariance matrices P u,v = E[ǫ u,v ǫ T u,v ]. Just with relative measurements, determining the x u ’s is only possible up to an additive constant. To avoid this ambiguity, we assume that there is at least one reference node o ∈ V whose node variable x o is known. Distributed algorithms to compute the optimal estimate using only local information were reported in [1], where the Optimal estimate refers to the classical Best Linear Unbiased (BLU) Estimator, which achieves the minimum variance among all linear unbiased estimators. II. THE QUESTION OF ERROR SCALING One may wonder what are the fundamental limitations of estimation accuracy for truly large graphs. Reasons for concern arise from estimation problems such as the one associated with the simple graph shown in Figure 1. It is a chain of nodes with node 1 as the reference and with a single edge (u +1,u), 1 2 3 4 Fig. 1. A graph where xu’s optimal estimate has a error variance that grows linearly with distance from the reference. u ≥ 1 between consecutive nodes u and u +1. Without much difficulty, one can show that for such a graph the optimal estimate of x u is given by ˆ x u = ζ u,u−1 + ··· ζ 3,2 + ζ 2,1 + x 1 , and since each measurement introduces an additive error, the variance of the optimal estimation error ˆ x u − x u will increase linearly with u. This means that if u is very “far” from the reference node 1 then its estimate will necessarily be quite poor. Although the precise estimation error depends on the This material is based upon work supported by the Institute for Collaborative Biotechnologies through grant DAAD19-03-D-0004 from the U.S. Army Research Office and by the National Science Foundation under Grant No. CCR-0311084. Both authors are with the Dept. of Electrical and Computer Engineering and the Center for Control, Dynamical-Systems, and Computation at Univ. of California, Santa Barbara, CA 93106.