We propose a new modeling paradigm for large dimensional aggregates of stochastic systems by Generalized Factor Analysis (GFA) models. These models describe the data as the sum of a flocking plus an uncorrelated idiosyncratic component. The flocking component describes a sort of collective orderly motion  which admits a much simpler   mathematical description than the whole ensemble while the idiosyncratic component describes weakly correlated noise. We first discuss static GFA representations and characterize in a rigorous way the properties of the two components. For wide-sense stationary sequences the character and existence of GFA models is completely clarified. The extraction of the flocking component of a random field is discussed for a simple class of separable random fields.

We discuss identifiability of dynamic SISO errors-in-variables (EIV) models with white measurement errors. Although this class of models turns out to be generically identifiable, it has been pointed out that in certain circumstances there may be two EIV models which are indistinguishable from external input–output experiments. This lack of (global) identifiability may be prejudicial to identification and needs better understanding. The identifiability conditions found in the literature guarantee uniqueness under certain coprimality assumptions on the (rational) transfer function of the ideal “true” system and the spectral density of the noiseless “true” input. Unfortunately these conditions are not testable since they concern precisely the unknowns of the problem which are not available to the experimenter. We provide new identifiability conditions which are instead expressible in terms of the external description of the observable signals, namely their joint power spectral densities.

In this paper, we study modeling and identification of stochastic systems by Generalized Factor Analysis models. Although this class of models was originally introduced for econometric purposes, we present some possible  applications of engineering interest. We show that there is a natural connection between Generalized Factor Analysis models and multi-agents systems. The common factor component of the model has an interpretation as a flocking component of the system behaviour.