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A new Bayesian approach to linear system identification
has been proposed in a series of recent papers. The
main idea is to frame linear system identification as predictor
estimation in an infinite dimensional space, with the aid of
regularization/Bayesian techniques. This approach guarantees
the identification of stable predictors based on the prediction
error minimization. Unluckily, the stability of the predictors
does not guarantee the stability of the impulse response of
the system. In this paper we propose and compare various
techniques to guarantee that the final model identified following
this Bayesian approach is stable. First, we consider the socalled
“LMI - constraint” approach and adapt it to constrain
the eigenvalues of the estimated model within the unit circle.
A second possibility which is being considered is to add to the
“classic” Stable-Spline algorithm a penalty term, depending on
the maximum absolute value of the eigenvalue of the system.
This last technique has the advantage of being integrated
directly inside the pre-existing optimization problem and not to
simply post-process the estimated model to guarantee stability.
Finally, we considered a Monte Carlo Markov Chain approach
sampling in both the space of hyper-parameters and of impulse
responses. Simulations results comparing these techniques will
be provided.

The knowledge of the size of a network, i.e.\ of the number of nodes composing it, is important for maintenance and organization purposes. In networks where the identity of the nodes or is not unique or cannot be disclosed for privacy reasons, the size-estimation problem is particularly challenging since the exchanged messages cannot be uniquely associated with a specific node. In this work, we propose a totally distributed anonymous strategy based on statistical inference concepts. In our approach, each node starts generating a vector of independent random numbers from a known distribution. Then nodes compute a common function via some distributed consensus algorithms, and finally they compute the Maximum Likelihood (ML) estimate of the network size exploiting opportune statistical inferences. In this work we study the performance that can be obtained following this computational scheme when the consensus strategy is either the maximum or the average. In the max-consensus scenario, when data come from absolutely continuous distributions, we provide a complete characterization of the ML estimator. In particular, we show that the squared estimation error decreases as $1/M$, where $M$ is the amount of random numbers locally generated by each node, independently of the chosen probability distribution. Differently, in the average-consensus scenario, we show that if the locally generated data are independent Bernoulli trials, then the probability for the ML estimator to return a wrong answer decreases exponentially in $M$. Finally, we provide a discussion as how the numerical errors may affect the estimators performance under different scenarios.

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.

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.

To maintain and organize distributed systems it is necessary to have a certain degree of knowledge of their status like the number of cooperating agents. The estimation of this number, usually referred as the network size, can pose challenging questions when agents' identification information cannot be disclosed, since the exchanged information cannot be associated to who originated it. In this paper we propose a totally distributed network size estimation strategy based on statistical inference concepts that can be applied under anonymity constraints. The scheme is based on the following paradigm: agents locally generate some Bernoulli trials, then distributedly compute averages of these generated data, finally locally compute the Maximum Likelihood estimate of the network size exploiting its probabilistic dependencies with the previously computed averages. In this work we study the statistical properties of this estimation strategy, and show how the probability of returning a wrong evaluation decreases exponentially in the number of locally generated trials. Finally, we discuss how practical implementation issues may affect the estimator, and show that there exists a neat phase transition between insensitivity to numerical errors and uselessness of the results.

Many modeling problems require to estimate ascalar output from one or more time series. Such problemsare usually tackled by extracting a fixed number of featuresfrom the time series (like their statistical moments), with aconsequent loss in information that leads to suboptimal predictivemodels. Moreover, feature extraction techniques usuallymake assumptions that are not met by real world settings (e.g.uniformly sampled time series of constant length), and failto deliver a thorough methodology to deal with noisy data.In this paper a methodology based on functional learningis proposed to overcome the aforementioned problems; theproposed Supervised Aggregative Feature Extraction (SAFE)approach allows to derive continuous, smooth estimates oftime series data (yielding aggregate local information), whilesimultaneously estimating a continuous shape function yieldingoptimal predictions. The SAFE paradigm enjoys severalproperties like closed form solution, incorporation of first andsecond order derivative information into the regressor matrix,interpretability of the generated functional predictor and thepossibility to exploit Reproducing Kernel Hilbert Spaces settingto yield nonlinear predictive models. Simulation studies areprovided to highlight the strengths of the new methodology withrespect to standard unsupervised feature selection approaches.

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.