Umberto Soverini

The Frisch scheme in dynamic system identification

Abstract The use of the Frisch scheme in the identification of linear dynamic systems is investigated in order to describe the whole family of models that can explain given input-output noisy sequences. Unlike the algebraic case, it is shown that, in general, only a single model is compatible with the data. These results are first proposed for single-input single-output systems and then generalized to the multivariable case.

Perspectives on errors-in-variables estimation for dynamic systems

The paper gives all overview of various methods for identifying dynamic errors-in-variables systems. Several approaches are classified by how the original information in time-series data of the noisy input and output measurements is condensed before further processing. For some methods, such as instrumental variable estimators, the information is condensed into a nonsymmetric covariance matrix as a first step before further processing. In a second class of methods, where a symmetric covariance matrix is used instead, the Frisch scheme and other bias-compensation approaches appear. When dealing with the estimation problem in the frequency domain, a milder data reduction typically takes place by first computing spectral estimators of the noisy input-output data. Finally, it is also possible to apply maximum likelihood and prediction error approaches using the original time-domain data in a direct fashion. This alternative will often require quite high computational complexity but yield good statistical efficiency. The paper is also presenting various properties of parameter estimators for the errors-in-variables problem, and a few conjectures are included, as well as some perspectives and experiences by the authors.

Speech Enhancement Combining Optimal Smoothing and Errors-In-Variables Identification of Noisy AR Processes

In the framework of speech enhancement, several parametric approaches based on an a priori model for a speech signal have been proposed. When using an autoregressive (AR) model, three issues must be addressed. (1) How to deal with AR parameter estimation? Indeed, due to additive noise, the standard least squares criterion leads to biased estimates of AR parameters. (2) Can an estimation of the variance of the additive noise for each speech frame be obtained? A voice activity detector is often used for its estimation. (3) Which estimation rules and techniques (filtering, smoothing, etc.) can be considered to retrieve the speech signal? Our contribution in this paper is threefold. First, we propose to view the identification of the noisy AR process as an errors-in-variables problem. This blind method has the advantage of providing accurate estimations of both the AR parameters and the variance of the additive noise. Second, we propose an alternative algorithm to standard Kalman smoothing, based on a constrained minimum variance estimation procedure with a lower computational cost. Third, the combination of these two steps is investigated. It provides better results than some existing speech enhancement approaches in terms of signal-to-noise-ratio (SNR), segmental SNR, and informal subjective tests.

A New Criterion in EIV Identification and Filtering Applications

Abstract One of the advantages of errors-in-variables (EIV) models consists in the symmetrical description of all variables. These models, on the other hand, are characterized by more complex identification schemes that require, when applied to real data, the definition of suitable criteria. This paper introduces a new efficient and robust criterion based on covariance-matching properties and tests its performance by means of a Monte Carlo simulation concerning also the application of the identified models in EIV filtering.

Brief paper: Maximum likelihood identification of noisy input-output models

This work deals with the identification of errors-in-variables models corrupted by white and uncorrelated Gaussian noises. By introducing an auxiliary process, it is possible to obtain a maximum likelihood solution of this identification problem, by means of a two-step iterative algorithm. This approach allows also to estimate, as a byproduct, the noise-free input and output sequences. Moreover, an analytic expression of the finite Cramer-Rao lower bound is derived. The method does not require any particular assumption on the input process, however, the ratio of the noise variances is assumed as known. The effectiveness of the proposed algorithm has been verified by means of Monte Carlo simulations.

Identification of dynamic errors-in-variables models

This paper deals with the identifiability of scalar dynamic errors-in-variables models characterized by rational spectra. The hypothesis of causality for the underlying dynamic system is taken into account. By making use of stochastic realization theory and of the structural properties of state-space representations, it is shown that, under mild assumptions, the model is uniquely identified.

Brief Optimal errors-in-variables filtering

This paper deals with optimal (minimal variance) filtering in an errors-in-variables framework. Differently from many other contexts, errors-in-variables models treat all variables in a symmetric way (no partition of the variables into inputs and outputs is required) and assume additive noise on all the variables. The filtering technique described in this paper can be easily implemented in a recursive way and does not require the use of a Riccati equation at every update. The results of Monte Carlo simulations have shown the effectiveness and consistency of the approach.

Brief Identification of dynamic errors-in-variables models: Approaches based on two-dimensional ARMA modeling of the data

In this paper we propose a parametric and a non-parametric identification algorithm for dynamic errors-in-variables model. We show that the two-dimensional process composed of the input-output data admits a finite order ARMA representation. The non-parametric method uses the ARMA structure to compute a consistent estimate of the joint spectrum of the input and the output. A Frisch scheme is then employed to extract an estimate of the joint spectrum of the noise free input-output data, which in turn is used to estimate the transfer function of the system. The parametric method exploits the ARMA structure to give estimates of the system parameters. The performances of the algorithms are illustrated using the results obtained from a numerical simulation study.

Identification of ARX and ARARX Models in the Presence of Input and Output Noises

ARX (AutoRegressive models with eXogenous variables) are the simplest models within the equation error family but are endowed with many practical advantages concerning both their estimation and their predictive use since their optimal predictors are always stable. Similar considerations can be repeated for ARARX models where the equation error is described by an AR process instead of a white noise. The ARX and ARARX schemes can be enhanced by introducing the assumption of the presence of additive white noise on the input and output observations. These schemes, that will be denoted as “ARX + noise” and “ARARX + noise”, can be seen as errors-in-variables models where both measurement errors and process disturbances are taken into account. This paper analyzes the problem of identifying ARX + noise and ARARX + noise models. The proposed identification algorithms are derived on the basis of the procedures developed for the solution of the dynamic Frisch scheme. The paper reports also Monte Carlo simulations that confirm the effectiveness of the proposed procedures.

Kalman filtering in extended noise environments

This note introduces an extended environment for Kalman filtering that considers also the presence of additive noise on input observations in order to solve the problem of optimal (minimal variance) estimation of noise-corrupted input and output sequences. This environment includes as subcases both errors-in-variables filtering (optimal estimate of inputs and outputs from noisy observations) and traditional Kalman filtering (optimal estimate of state and output in presence of state and output noise). A Monte Carlo simulation shows that the performance of this extended filtering technique leads to the expected minimal variance estimates.