Guaranteed characterization of exact non-asymptotic confidence regions as defined by LSCR and SPS

International audienceIn parameter estimation, it is often desirable to supplement the estimates with an assessment of their quality. A new family of methods proposed by Campi et al. for this purpose is particularly attractive, as it makes it possible to obtain exact, non-asymptotic confidence regions under mild assumptions on the noise distribution. A bottleneck of this approach, however, is the numerical characterization of these confidence regions. So far, it has been carried out by gridding, which provides no guarantee as to its results and is only applicable to low dimensional spaces. This paper shows how interval analysis can contribute to removing this bottleneck

Guaranteed characterization of exact non-asymptotic confidence regions

accepté à AutomaticaIn parameter estimation, it is often desirable to supplement the estimates with an assessment of their quality. A new family of methods proposed by Campi et al. for this purpose is particularly attractive, as it makes it possible to obtain exact, non-asymptotic con dence regions under mild assumptions on the noise distribution. A bottleneck of this approach, however, is the numerical characterization of these con dence regions. So far, it has been carried out by gridding, which provides no guarantee as to its results and is only applicable to low dimensional spaces. This paper shows how interval analysis can contribute to removing this bottleneck

Finite-sample system identification: An overview and a new correlation method

Finite-sample system identification algorithms can be used to build guaranteed confidence regions for unknown model parameters under mild statistical assumptions. It has been shown that in many circumstances these rigorously built regions are comparable in size and shape to those that could be built by resorting to the asymptotic theory. The latter sets are, however, not guaranteed for finite samples and can sometimes lead to misleading results. The general principles behind finite-sample methods make them virtually applicable to a large variety of even nonlinear systems. While these principles are simple enough, a rigorous treatment of the attendant technical issues makes the corresponding theory complex and not easy to access. This is believed to be one of the reasons why these methods have not yet received widespread acceptance by the identification community and this letter is meant to provide an easy access point to finite-sample system identification by presenting the fundamental ideas underlying these methods in a simplified manner. We then review three (classes of) methods that have been proposed so far-1) Leave-out Sign-dominant Correlation Regions (LSCR); 2) Sign-Perturbed Sums (SPS); 3) Perturbed Dataset Methods (PDMs). By identifying some difficulties inherent in these methods, we also propose in this letter a new sign-perturbation method based on correlation which overcome some of these difficulties

Perturbed Datasets Methods for Hypothesis Testing and Structure of Corresponding Confidence Sets

Hypothesis testing methods that do not rely on exact distribution assumptions have been emerging lately. The method of sign-perturbed sums (SPS) is capable of characterizing confidence regions with exact confidence levels for linear regression and linear dynamical systems parameter estimation problems if the noise distribution is symmetric. This paper describes a general family of hypothesis testing methods that have an exact user chosen confidence level based on finite sample count and without relying on an assumed noise distribution. It is shown that the SPS method belongs to this family and we provide another hypothesis test for the case where the symmetry assumption is replaced with exchangeability. In the case of linear regression problems it is shown that the confidence regions are connected, bounded and possibly non-convex sets in both cases. To highlight the importance of understanding the structure of confidence regions corresponding to such hypothesis tests it is shown that confidence sets for linear dynamical systems parameter estimates generated using the SPS method can have non-connected parts, which have far reaching consequences

Sign-perturbed sums: A new system identification approach for constructing exact non-asymptotic confidence regions in linear regression models

We propose a new system identification method, called Sign - Perturbed Sums (SPS), for constructing nonasymptotic confidence regions under mild statistical assumptions. SPS is introduced for linear regression models, including but not limited to FIR systems, and we show that the SPS confidence regions have exact confidence probabilities, i.e., they contain the true parameter with a user-chosen exact probability for any finite data set. Moreover, we also prove that the SPS regions are star convex with the Least-Squares (LS) estimate as a star center. The main assumptions of SPS are that the noise terms are independent and symmetrically distributed about zero, but they can be nonstationary, and their distributions need not be known. The paper also proposes a computationally efficient ellipsoidal outer approximation algorithm for SPS. Finally, SPS is demonstrated through a number of simulation experiments

Guaranteed characterization of exact confidence regions for FIR models under mild assumptions on the noise via interval analysis

International audienceSPS is one of the two methods proposed recently by Campi et al. to obtain exact, non-asymptotic confidence regions for parameter estimates under mild assumptions on the noise distribution. It does not require the measurement noise to be Gaussian (or to have any other known distribution for that matter). The numerical characterization of the resulting confidence regions is far from trivial, however, and has only be carried out so far on very low-dimensional problems via methods that could not guarantee their results and could not be extended to large-scale problems because of their intrinsic complexity. The aim of the present paper is to show how interval analysis can contribute to a guaranteed characterization of exact confidence regions in large-scale problems. The application considered is the estimation of the parameters of finite-impulse response (FIR) models. The structure of the problem makes it possible to define a very efficient specific contractor, allowing the treatement of models with a large number of parameters, as is the rule for FIR models, and thus escaping the curse of dimensionality that often plagues interval methods

Distributed SPS Algorithms for Non-Asymptotic Confidence Region Evaluation

In this paper, the distributed computation of confidence regions for parameter estimation is considered. Some information diffusion strategies are proposed and compared in terms of the required number of data exchanges to get the corresponding region. The effects of algorithms truncation is also addressed. As support for the theoretical part, numerical results are presented

Towards D-Optimal Input Design for Finite-Sample System Identification

Finite-sample system identification methods provide statistical inference, typically in the form of confidence regions, with rigorous non-asymptotic guarantees under minimal distributional assumptions. Data Perturbation (DP) methods constitute an important class of such algorithms, which includes, for example, Sign-Perturbed Sums (SPS) as a special case. Here we study a natural input design problem for DP methods in linear regression models, where we want to select the regressors in a way that the expected volume of the resulting confidence regions are minimized. We suggest a general approach to this problem and analyze it for the fundamental building blocks of all DP confidence regions, namely, for ellipsoids having confidence probability exactly 1/2. We also present experiments supporting that minimizing the expected volumes of such ellipsoids significantly reduces the average sizes of the constructed DP confidence regions

Energy-efficient Communication and Estimation in Wireless Sensor Networks

This thesis focuses on the energy efficiency in wireless networks under the transmission and information diffusion points of view. In particular, on one hand, the communication efficiency is investigated, attempting to reduce the consumption during transmissions, while on the other hand the energy efficiency of the procedures required to distribute the information among wireless nodes in complex networks is taken into account. For what concerns energy efficient communications, an innovative transmission scheme reusing source of opportunity signals is introduced. This kind of signals has never been previously studied in literature for communication purposes. The scope is to provide a way for transmitting information with energy consumption close to zero. On the theoretical side, starting from a general communication channel model subject to a limited input amplitude, the theme of low power transmission signals is tackled under the perspective of stating sufficient conditions for the capacity achieving input distribution to be discrete. Finally, the focus is shifted towards the design of energy efficient algorithms for the diffusion of information. In particular, the endeavours are aimed at solving an estimation problem distributed over a wireless sensor network. The proposed solutions are deeply analyzed both to ensure their energy efficiency and to guarantee their robustness against losses during the diffusion of information (against information diffusion truncation more in general)