Estimation from relative measurements: error bounds from electrical analogy

We consider the problem of estimating vector-valued variables from noisy "relative" measurements. The measurement model can be expressed in terms of a graph, whose nodes correspond to the variables being estimated and the edges to noisy measurements of the difference between the two variables associated with the corresponding nodes (i.e., their relative values). This type of measurement model appears in several sensor networks problem. We take the value of one particular variable as a reference and consider the unbiased minimum variance (UMV) estimators for the differences between the remaining variables and the reference. We establish upper and lower bounds on the estimation error variance of a node's variable as a function of the Euclidean distance in a drawing of the graph between the node and the reference one. These bounds result in a classification of graphs: civilized and dense, based on how the variance grows with distance: at a rate greater than or less than linearly, logarithmically, or bounded. In deriving these results, we establish and exploit an analogy between the UMV estimator variance and the effective resistance in a generalized electrical network that is significant on its own.