  We address the problem of identifying linear relations among variables based
on noisy measurements. This is, of course, a central question in problems
involving "Big Data." Often a key assumption is that measurement errors in each
variable are independent. This precise formulation has its roots in the work of
Charles Spearman in 1904 and of Ragnar Frisch in the 1930's. Various topics
such as errors-in-variables, factor analysis, and instrumental variables, all
refer to alternative formulations of the problem of how to account for the
anticipated way that noise enters in the data. In the present paper we begin by
describing the basic theory and provide alternative modern proofs to some key
results. We then go on to consider certain generalizations of the theory as
well applying certain novel numerical techniques to the problem. A central role
is played by the Frisch-Kalman dictum which aims at a noise contribution that
allows a maximal set of simultaneous linear relations among the noise-free
variables --a rank minimization problem. In the years since Frisch's original
formulation, there have been several insights including trace minimization as a
convenient heuristic to replace rank minimization. We discuss convex
relaxations and certificates guaranteeing global optimality. A complementary
point of view to the Frisch-Kalman dictum is introduced in which models lead to
a min-max quadratic estimation error for the error-free variables. Points of
contact between the two formalisms are discussed and various alternative
regularization schemes are indicated.
