Graphs, networks, and linear unbiased estimates Original Research Article

It is shown how a best linear unbiased estimate (blue) in the additive variety-block setting can be interpreted as a network flow, that is, a function on edges that obeys the Kirkhhoff laws, of minimum square norm. An explicit expression is then obtained for the coefficients of the blue in terms of invariants of the underlying network; specifically, the invariants are: the total number of spanning trees and the number of certain selective yet specific spanning forests with just two trees. The blue is also expressed as a linear combination of bases of paths in a constructive manner. It remains a conjecture as to whether there always exists a basis of paths in which the blue is a convex combination. Consequences to design optimality are explored.