23 research outputs found
Maximum Entropy Vector Kernels for MIMO system identification
Recent contributions have framed linear system identification as a
nonparametric regularized inverse problem. Relying on -type
regularization which accounts for the stability and smoothness of the impulse
response to be estimated, these approaches have been shown to be competitive
w.r.t classical parametric methods. In this paper, adopting Maximum Entropy
arguments, we derive a new penalty deriving from a vector-valued
kernel; to do so we exploit the structure of the Hankel matrix, thus
controlling at the same time complexity, measured by the McMillan degree,
stability and smoothness of the identified models. As a special case we recover
the nuclear norm penalty on the squared block Hankel matrix. In contrast with
previous literature on reweighted nuclear norm penalties, our kernel is
described by a small number of hyper-parameters, which are iteratively updated
through marginal likelihood maximization; constraining the structure of the
kernel acts as a (hyper)regularizer which helps controlling the effective
degrees of freedom of our estimator. To optimize the marginal likelihood we
adapt a Scaled Gradient Projection (SGP) algorithm which is proved to be
significantly computationally cheaper than other first and second order
off-the-shelf optimization methods. The paper also contains an extensive
comparison with many state-of-the-art methods on several Monte-Carlo studies,
which confirms the effectiveness of our procedure
Bayesian and regularization approaches to multivariable linear system identification: the role of rank penalties
Recent developments in linear system identification have proposed the use of
non-parameteric methods, relying on regularization strategies, to handle the
so-called bias/variance trade-off. This paper introduces an impulse response
estimator which relies on an -type regularization including a
rank-penalty derived using the log-det heuristic as a smooth approximation to
the rank function. This allows to account for different properties of the
estimated impulse response (e.g. smoothness and stability) while also
penalizing high-complexity models. This also allows to account and enforce
coupling between different input-output channels in MIMO systems. According to
the Bayesian paradigm, the parameters defining the relative weight of the two
regularization terms as well as the structure of the rank penalty are estimated
optimizing the marginal likelihood. Once these hyperameters have been
estimated, the impulse response estimate is available in closed form.
Experiments show that the proposed method is superior to the estimator relying
on the "classic" -regularization alone as well as those based in atomic
and nuclear norm.Comment: to appear in IEEE Conference on Decision and Control, 201
Estimating effective connectivity in linear brain network models
Contemporary neuroscience has embraced network science to study the complex
and self-organized structure of the human brain; one of the main outstanding
issues is that of inferring from measure data, chiefly functional Magnetic
Resonance Imaging (fMRI), the so-called effective connectivity in brain
networks, that is the existing interactions among neuronal populations. This
inverse problem is complicated by the fact that the BOLD (Blood Oxygenation
Level Dependent) signal measured by fMRI represent a dynamic and nonlinear
transformation (the hemodynamic response) of neuronal activity. In this paper,
we consider resting state (rs) fMRI data; building upon a linear population
model of the BOLD signal and a stochastic linear DCM model, the model
parameters are estimated through an EM-type iterative procedure, which
alternately estimates the neuronal activity by means of the Rauch-Tung-Striebel
(RTS) smoother, updates the connections among neuronal states and refines the
parameters of the hemodynamic model; sparsity in the interconnection structure
is favoured using an iteratively reweighting scheme. Experimental results using
rs-fMRI data are shown demonstrating the effectiveness of our approach and
comparison with state of the art routines (SPM12 toolbox) is provided
Non-Parametric Bayesian Methods for Linear System Identification
Recent contributions have tackled the linear system identification problem by means of non-parametric Bayesian methods, which are built on largely adopted machine learning techniques, such as Gaussian Process regression and kernel-based regularized regression. Following the Bayesian paradigm, these procedures treat the impulse response of the system to be estimated as the realization of a Gaussian process. Typically, a Gaussian prior accounting for stability and smoothness of the impulse response is postulated, as a function of some parameters (called hyper-parameters in the Bayesian framework). These are generally estimated by maximizing the so-called marginal likelihood, i.e. the likelihood after the impulse response has been marginalized out. Once the hyper-parameters have been fixed in this way, the final estimator is computed as the conditional expected value of the impulse response w.r.t. the posterior distribution, which coincides with the minimum variance estimator. Assuming that the identification data are corrupted by Gaussian noise, the above-mentioned estimator coincides with the solution of a regularized estimation problem, in which the regularization term is the l2 norm of the impulse response, weighted by the inverse of the prior covariance function (a.k.a. kernel in the machine learning literature).
Recent works have shown how such Bayesian approaches are able to jointly perform estimation and model selection, thus overcoming one of the main issues affecting parametric identification procedures, that is complexity selection.
While keeping the classical system identification methods (e.g. Prediction Error Methods and subspace algorithms) as a benchmark for numerical comparison, this thesis extends and analyzes some key aspects of the above-mentioned Bayesian procedure. In particular, four main topics are considered.
1. PRIOR DESIGN. Adopting Maximum Entropy arguments, a new type of l2 regularization is derived: the aim is to penalize the rank of the block Hankel matrix built with Markov coefficients, thus controlling the complexity of the identified model, measured by its McMillan degree. By accounting for the coupling between different input-output channels, this new prior results particularly suited when dealing for the identification of MIMO systems
To speed up the computational requirements of the estimation algorithm, a tailored version of the Scaled Gradient Projection algorithm is designed to optimize the marginal likelihood.
2. CHARACTERIZATION OF UNCERTAINTY. The confidence sets returned by the non-parametric Bayesian identification algorithm are analyzed and compared with those returned by parametric Prediction Error Methods. The comparison is carried out in the impulse response space, by deriving “particle” versions (i.e. Monte-Carlo approximations) of the standard confidence sets.
3. ONLINE ESTIMATION. The application of the non-parametric Bayesian system identification techniques is extended to an online setting, in which new data become available as time goes. Specifically, two key modifications of the original “batch” procedure are proposed in order to meet the real-time requirements. In addition, the identification of time-varying systems is tackled by introducing a forgetting factor in the estimation criterion and by treating it as a hyper-parameter.
4. POST PROCESSING: MODEL REDUCTION. Non-parametric Bayesian identification procedures estimate the unknown system in terms of its impulse response coefficients, thus returning a model with high (possibly infinite) McMillan degree. A tailored procedure is proposed to reduce such model to a lower degree one, which appears more suitable for filtering and control applications. Different criteria for the selection of the order of the reduced model are evaluated and compared
The role of noise modeling in the estimation of resting-state brain effective connectivity
Causal relations among neuronal populations of the brain are studied through
the so-called effective connectivity (EC) network. The latter is estimated from
EEG or fMRI measurements, by inverting a generative model of the corresponding
data. It is clear that the goodness of the estimated network heavily depends on
the underlying modeling assumptions. In this present paper we consider the EC
estimation problem using fMRI data in resting-state condition. Specifically, we
investigate on how to model endogenous fluctuations driving the neuronal
activity
Sparse DCM for whole-brain effective connectivity from resting-state fMRI data
Contemporary neuroscience has embraced network science and dynamical systems to study the complex and self-organized structure of the human brain. Despite the developments in non-invasive neuroimaging techniques, a full understanding of the directed interactions in whole brain networks, referred to as effective connectivity, as well as their role in the emergent brain dynamics is still lacking. The main reason is that estimating brain connectivity requires solving a formidable large-scale inverse problem from indirect and noisy measurements. Building on the dynamic causal modelling framework, the present study offers a novel method for estimating whole-brain effective connectivity from resting-state functional magnetic resonance data. To this purpose sparse estimation methods are adapted to infer the parameters of our novel model, which is based on a linearized, region-specific haemodynamic response function. The resulting algorithm, referred to as sparse DCM, is shown to compare favorably with state-of-the art methods when tested on both synthetic and real data. We also provide a graph-theoretical analysis on the whole-brain effective connectivity estimated using data from a cohort of healthy individuals, which reveals properties such as asymmetry in the connectivity structure as well as the different roles of brain areas in favoring segregation or integration
Stabilizzazione in retroazione del moto di una bicicletta
La tesi consta di una prima parte dedicata alla definizione di modelli matematici descrittivi della bicicletta: vengono dapprima ricavati sistemi non lineari, dai quali in seguito alla linearizzazione, si perviene a funzioni di trasferimento e a modelli di stato. In un primo momento viene illustrato un modello classico che trascura alcune imnportanti caratteristiche geometriche del mezzo, per passare poi alla descrizione di un modello più completo che considera le proprietà geometriche della forcella anteriore, prevedendo come ingresso la coppia applicata dal ciclista sul manubrio e come uscita l'inclinazione laterale della bicicletta rispetto alla verticale. Definiti i modelli, si passa al progetto di compensatori e di matrici di retroazione dallo stato che permettano di stabilizzare tali sistemi e di migliorarne le prestazioni in termini di risposta al gradino, andando così a confrontare le due tecniche di controllo. La parte finale è invece dedicata alla valutazione dell'efficacia dei compensatori prima illustrati sui sistemi non lineari ricavati nella parte inizial
Online identification of time-varying systems: A Bayesian approach
We extend the recently introduced regularization/Bayesian System
Identification procedures to the estimation of time-varying systems.
Specifically, we consider an online setting, in which new data become available
at given time steps. The real-time estimation requirements imposed by this
setting are met by estimating the hyper-parameters through just one gradient
step in the marginal likelihood maximization and by exploiting the closed-form
availability of the impulse response estimate (when Gaussian prior and Gaussian
measurement noise are postulated). By relying on the use of a forgetting
factor, we propose two methods to tackle the tracking of time-varying systems.
In one of them, the forgetting factor is estimated by treating it as a
hyper-parameter of the Bayesian inference procedure