Gianluigi Pillonetto

A new kernel-based approach for linear system identification

This paper describes a new kernel-based approach for linear system identification of stable systems. We model the impulse response as the realization of a Gaussian process whose statistics, differently from previously adopted priors, include information not only on smoothness but also on BIBO-stability. The associated autocovariance defines what we call a stable spline kernel. The corresponding minimum variance estimate belongs to a reproducing kernel Hilbert space which is spectrally characterized. Compared to parametric identification techniques, the impulse response of the system is searched for within an infinite-dimensional space, dense in the space of continuous functions. Overparametrization is avoided by tuning few hyperparameters via marginal likelihood maximization. The proposed approach may prove particularly useful in the context of robust identification in order to obtain reduced order models by exploiting a two-step procedure that projects the nonparametric estimate onto the space of nominal models. The continuous-time derivation immediately extends to the discrete-time case. On several continuous- and discrete-time benchmarks taken from the literature the proposed approach compares very favorably with the existing parametric and nonparametric techniques.

Survey Kernel methods in system identification, machine learning and function estimation: A survey

Most of the currently used techniques for linear system identification are based on classical estimation paradigms coming from mathematical statistics. In particular, maximum likelihood and prediction error methods represent the mainstream approaches to identification of linear dynamic systems, with a long history of theoretical and algorithmic contributions. Parallel to this, in the machine learning community alternative techniques have been developed. Until recently, there has been little contact between these two worlds. The first aim of this survey is to make accessible to the control community the key mathematical tools and concepts as well as the computational aspects underpinning these learning techniques. In particular, we focus on kernel-based regularization and its connections with reproducing kernel Hilbert spaces and Bayesian estimation of Gaussian processes. The second aim is to demonstrate that learning techniques tailored to the specific features of dynamic systems may outperform conventional parametric approaches for identification of stable linear systems.

Prediction error identification of linear systems: A nonparametric Gaussian regression approach

A novel Bayesian paradigm for the identification of output error models has recently been proposed in which, in place of postulating finite-dimensional models of the system transfer function, the system impulse response is searched for within an infinite-dimensional space. In this paper, such a nonparametric approach is applied to the design of optimal predictors and discrete-time models based on prediction error minimization by interpreting the predictor impulse responses as realizations of Gaussian processes. The proposed scheme describes the predictor impulse responses as the convolution of an infinite-dimensional response with a low-dimensional parametric response that captures possible high-frequency dynamics. Overparameterization is avoided because the model involves only a few hyperparameters that are tuned via marginal likelihood maximization. Numerical experiments, with data generated by ARMAX and infinite-dimensional models, show the definite advantages of the new approach over standard parametric prediction error techniques and subspace methods both in terms of predictive capability on new data and accuracy in reconstruction of system impulse responses.

Sensing, Compression, and Recovery for WSNs: Sparse Signal Modeling and Monitoring Framework

We address the problem of compressing large and distributed signals monitored by a Wireless Sensor Network (WSN) and recovering them through the collection of a small number of samples. We propose a sparsity model that allows the use of Compressive Sensing (CS) for the online recovery of large data sets in real WSN scenarios, exploiting Principal Component Analysis (PCA) to capture the spatial and temporal characteristics of real signals. Bayesian analysis is utilized to approximate the statistical distribution of the principal components and to show that the Laplacian distribution provides an accurate representation of the statistics of real data. This combined CS and PCA technique is subsequently integrated into a novel framework, namely, SCoRe1: Sensing, Compression and Recovery through ON-line Estimation for WSNs. SCoRe1 is able to effectively self-adapt to unpredictable changes in the signal statistics thanks to a feedback control loop that estimates, in real time, the signal reconstruction error. We also propose an extensive validation of the framework used in conjunction with CS as well as with standard interpolation techniques, testing its performance for real world signals. The results in this paper have the merit of shedding new light on the performance limits of CS when used as a recovery tool in WSNs.

Numerical non-identifiability regions of the minimal model of glucose kinetics : superiority of Bayesian estimation

The so-called minimal model (MM) of glucose kinetics is widely employed to estimate insulin sensitivity (S(I)) both in clinical and epidemiological studies. Usually, MM is numerically identified by resorting to Fisherian parameter estimation techniques, such as maximum likelihood (ML). However, unsatisfactory parameter estimates are sometimes obtained, e.g. S(I) estimates virtually zero or unrealistically high and affected by very large uncertainty, making the practical use of MM difficult. The first result of this paper concerns the mathematical demonstration that these estimation difficulties are inherent to MM structure which can expose S(I) estimation to the risk of numerical non-identifiability. The second result is based on simulation studies and shows that Bayesian parameter estimation techniques are less sensitive, in terms of both accuracy and precision, than the Fisherian ones with respect to these difficulties. In conclusion, Bayesian parameter estimation can successfully deal with difficulties of MM identification inherently due to its structure.

System Identification Via Sparse Multiple Kernel-Based Regularization Using Sequential Convex Optimization Techniques

Model estimation and structure detection with short data records are two issues that receive increasing interests in System Identification. In this paper, a multiple kernel-based regularization method is proposed to handle those issues. Multiple kernels are conic combinations of fixed kernels suitable for impulse response estimation, and equip the kernel-based regularization method with three features. First, multiple kernels can better capture complicated dynamics than single kernels. Second, the estimation of their weights by maximizing the marginal likelihood favors sparse optimal weights, which enables this method to tackle various structure detection problems, e.g., the sparse dynamic network identification and the segmentation of linear systems. Third, the marginal likelihood maximization problem is a difference of convex programming problem. It is thus possible to find a locally optimal solution efficiently by using a majorization minimization algorithm and an interior point method where the cost of a single interior-point iteration grows linearly in the number of fixed kernels. Monte Carlo simulations show that the locally optimal solutions lead to good performance for randomly generated starting points.

A New Kernel-Based Approach for NonlinearSystem Identification

We present a novel nonparametric approach for identification of nonlinear systems. Exploiting the framework of Gaussian regression, the unknown nonlinear system is seen as a realization from a Gaussian random field. Its covariance encodes the idea of “fading” memory in the predictor and consists of a mixture of Gaussian kernels parametrized by few hyperparameters describing the interactions among past inputs and outputs. The kernel structure and the unknown hyperparameters are estimated maximizing their marginal likelihood so that the user is not required to define any part of the algorithmic architecture, e.g., the regressors and the model order. Once the kernel is estimated, the nonlinear model is obtained solving a Tikhonov-type variational problem. The Hilbert space the estimator belongs to is characterized. Benchmarks problems taken from the literature show the effectiveness of the new approach, also comparing its performance with a recently proposed algorithm based on direct weight optimization and with parametric approaches with model order estimated by AIC or BIC.

Motion planning using adaptive random walks

We propose a novel single-shot motion-planning algorithm based on adaptive random walks. The proposed algorithm turns out to be simple to implement, and the solution it produces can be easily and efficiently optimized. Furthermore, the algorithm can incorporate adaptive components, so the developer is not required to specify all the parameters of the random distributions involved, and the algorithm itself can adapt to the environment it is moving in. Proofs of the theoretical soundness of the algorithm are provided, as well as implementation details. Numerical comparisons with well-known algorithms illustrate its effectiveness.

An inequality constrained nonlinear Kalman-Bucy smoother by interior point likelihood maximization

Kalman-Bucy smoothers are often used to estimate the state variables as a function of time in a system with stochastic dynamics and measurement noise. This is accomplished using an algorithm for which the number of numerical operations grows linearly with the number of time points. All of the randomness in the model is assumed to be Gaussian. Including other available information, for example a bound on one of the state variables, is non trivial because it does not fit into the standard Kalman-Bucy smoother algorithm. In this paper we present an interior point method that maximizes the likelihood with respect to the sequence of state vectors satisfying inequality constraints. The method obtains the same decomposition that is normally obtained for the unconstrained Kalman-Bucy smoother, hence the resulting number of operations grows linearly with the number of time points. We present two algorithms, the first is for the affine case and the second is for the nonlinear case. Neither algorithm requires the optimization to start at a feasible sequence of state vector values. Both the unconstrained affine and unconstrained nonlinear Kalman-Bucy smoother are special cases of the class of problems that can be handled by these algorithms.

An

Robustness is a major problem in Kalman filtering and smoothing that can be solved using heavy tailed distributions; e.g., ℓ1-Laplace. This paper describes an algorithm for finding the maximum a posteriori (MAP) estimate of the Kalman smoother for a nonlinear model with Gaussian process noise and ℓ1 -Laplace observation noise. The algorithm uses the convex composite extension of the Gauss-Newton method. This yields convex programming subproblems to which an interior point path-following method is applied. The number of arithmetic operations required by the algorithm grows linearly with the number of time points because the algorithm preserves the underlying block tridiagonal structure of the Kalman smoother problem. Excellent fits are obtained with and without outliers, even though the outliers are simulated from distributions that are not ℓ1 -Laplace. It is also tested on actual data with a nonlinear measurement model for an underwater tracking experiment. The ℓ1-Laplace smoother is able to construct a smoothed fit, without data removal, from data with very large outliers.