Non-Asymptotic Confidence Regions for Errors-In-Variables Systems

Parameter identification for nonlinear systems: Guaranteed confidence regions through LSCR

Asymptotic properties of SPS confidence regions

In this paper we consider the problem of estimating confidence regions for the parameters of ARMA models. Based on subsampling techniques and building on earlier exact finite sample results due to Hartigan, we compute the exact probability that the true parameters belong to certain regions in the parameter space. By intersecting these regions, a confidence region containing the true parameters with guaranteed probability is then obtained. All results hold true for a finite number of data points and no asymptotic theory is used. The usefulness of the approach is illustrated in a simulation example.

In this paper, the problem of constructing non-asymptotic confidence regions for Errors-In-Variables (EIV) systems is considered. In EIV systems both the input and the output measurements are noisy. The proposed method is based upon extending the Sign-Perturbed Sums (SPS) method with instrumental variables in [1] to EIV systems. The SPS method is a state-of-the-art identification method which provides non-asymptotic confidence regions containing the true parameter with a user-chosen probability under mild assumptions on noise. The existence of two noise sources in EIV problems makes the prediction error sequence dependent, and consequently the standard SPS method is not applicable. By adopting a new regression model and perturbing the input instead of the prediction error a new method for constructing a guaranteed non-asymptotic confidence region is proposed in this paper. The method is demonstrated through a numerical experiment.

In this paper we will investigate some non-asymptotic properties of the modified least squares estimates for the non-linear function f(λ*) by observations that nonlinearly depend on the parameter λ*. Non-asymptotic confidence regions with fixed sizes for the modified least squares estimate are used. The obtained confidence region is valid for a finite number of data points when the distributions of the observations are unknown. Asymptotically the suggested estimates represent usual estimates of the least squares. The paper presents the results of practical applications of the proposed method in C-OTDR monitoring systems.

A randomised subsampling method for change detectio

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.

Non-asymptotic confidence ellipsoids for the least-squares estimate

In this work, we consider identification of nonlinear errors-in-variables (EIV) system in the state-space form. As the direct identification of nonlinear structured model is difficult, the proposed strategy identifies nonlinear process as a linear parameter varying (LPV) EIV system. The method involves identification of various local linear models along an operating trajectory, and aggregating them using weights computed by means of local model identities. The entire problem is formulated as a maximum likelihood (ML) estimation problem, and is solved using the EM algorithm to obtain the LPV EIV model parameters. A pilot-scale experimental study is employed to demonstrate the effectiveness of the proposed approach.

Getting confidence regions for parameter estimates obtained from data collected by a wireless sensor network (WSN) is very important to assess the performance of the estimator. The sign perturbed sums (SPS) approach has been proposed recently to defined exact confidence regions in a centralized setting even if only few measurements are available. SPS may be distributed to get confidence regions at each node of a WSN. This paper investigates a data dissemination strategy called Tagged and Aggregated Sums (TAS), exploiting the particularities of SPS, to efficiently provide each node with the information necessary to evaluate locally the confidence region. TAS and flooding (FL) algorithms have been investigated through simulations and then implemented on commercial sensor nodes. The impact of collision avoidance mechanisms at the medium access control (MAC) layer is also experimentally assessed. Performance comparisons show that TAS outperforms FL in structured networks. <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup>

Dynamic errors-in-variables (EIV) systems, in which both the output and the input variables are corrupted by noises, are widely used in signal processing, communications, and control engineering. Although a number of different methods for identifying dynamic EIV systems have been proposed, the theoretical analysis of their properties has always been a difficult problem. This paper presents the convergence analysis of an adaptive EIV identification algorithm with no input restrictions. The methods of analysis currently available in literature often assume the input signals to be AR or ARMA process and persistent excitation. This restriction is removed for attenuating excitation in this paper. The convergence rate is derived and it is shown that the adaptive estimation can converge fast to the true system parameters values. Numerical simulations is conducted to demonstrate the theoretical analysis.

We propose confidence regions with asymptotically correct uniform coverage probability of parameters whose Fisher information matrix can be singular at important points of the parameter set. Our work is motivated by the need for reliable inference on scale parameters close or equal to zero in mixed models, which is obtained as a special case. The confidence regions are constructed by inverting a continuous extension of the score test statistic standardized by expected information, which we show exists at points of singular information under regularity conditions. Similar results have previously only been obtained for scalar parameters, under conditions stronger than ours, and applications to mixed models have not been considered. In simulations our confidence regions have near-nominal coverage with as few as $n = 20$ independent observations, regardless of how close to the boundary the true parameter is. It is a corollary of our main results that the proposed test statistic has an asymptotic chi-square distribution with degrees of freedom equal to the number of tested parameters, even if they are on the boundary of the parameter set.

LPV system identification under noise corrupted scheduling and output signal observations

Inspired by the framework of the Interacting Multiple Model (IMM), a method, called Interacting Multiple Gaussian Particle Filter (IMGPF), is proposed for solving the nonlinear Bayesian filtering problem with unknown continuous parameter. IMGPF regards the continuous parameter space as a union of disjoint subspaces, and each subspace is assigned to a model respectively. At each time step, for each model of IMGPF, under the assumption that the parameter belongs to the corresponding subspace, a Gaussian Particle Filter is applied to estimate the parameter and the state together. The parameter of each model of IMM is a fixed value, while the parameter of each model of IMGPF is a random variable need to be estimated. Thus IMGPF can achieve better estimation performance than IMM when the true parameter does not close to any element of the IMM model set. A simulation example of bearings only tracking problem is presented to demonstrate the effectiveness of IMGPF.

Pole-zero modeling of low-pass signals, such as an electromagnetic-scatterer response, is considered in this paper. It is shown by use of pencil-of-functions theorem that (a) the true parameters can be recovered in the ideal case (where the signal is the impulse response of a rational function H(z)), and (b) the parameters are optimal in the generalized least-squares sense when the observed data are corrupted by additive noise or by systematic error. Although the computations are more involved than in all-pole modeling, they are considerably less than those required in iterative schemes of pole-zero modeling. The advantages of the method are demonstrated by simulation example and through application to the electromagnetic response ofa scatterer. The paper also includes very recent and tantalizing results on a new approach to noise correction. In contradistinction with spectral subtraction techniques, where only amplitude information is emphasized (and phase is ignored), we propose a method that (a) estimates the sample variance for the particular data frame, and then performs the subtraction from the Gram matrix.