A Structure Theory for Linear Dynamic Errors-in-Variables Models

Abstract

We deal with problems connected with the identification of linear dynamic systems in situations when inputs and outputs may be contaminated by noise. The case of uncorrelated noise components and the bounded noise case is considered. If also the inputs may be contaminated by noise, a number of additional complications in identification arise, in particular the underlying system is not uniquely determined from the population second moments of the observations. A description of classes of observationally equivalent systems is given, continuity properties of mappings relating classes of observationally equivalent systems to the spectral densities of the observations are derived and the classes of spectral densities corresponding to a given maximum number of outputs are studied.

Statistical Analysis of a Third-Order Cumulants Based Algorithm for Discrete-Time Errors-in-Variables Identification

“…The problem of identifying this errors-in-variables system is concerned with 1 This research was partially supported by The Swedish Research Council, contract 621-2005-42071. consistently estimating the parameter vector θ from the measured noisy data {u(t k ), y(t k )} N k=1 . It is well known Anderson and Deistler [1984], Scherrer and Deistler [1998] that if only the second-order statistics are exploited and without introducing more explicit assumptions, a unique solution to identification of the errorsin-variables systems does not exist. It is thus natural to study alternative methods based on higher-order statistics (HOS).…”

Generalized linear dynamic factor models - a structure theory

“…The idea is to generalize and combine linear dynamic factor models with strictly idiosyncratic noise (in the sense that the noise components are uncorrelated at any lead and lag) as analyzed in [8] and [9] and generalized linear static factor models. Factor models in a time series setting may be used to compress information contained in the data in both the cross-sectional dimension N and in the time dimension T .…”

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“…As has been shown in [18], for dynamic factor models with idiosyncratic noise the set of all spectral densities f x (λ) described by (3) for given r is a "thin" subset of the set of all spectral densities f x (λ), if r < n − √ n holds.…”

“…The assumptions imposed so far do not determine a reasonable model class, in the sense that for given f x , or Σ x respectively, too many models would be possible, see [18]. Thus, in order to obtain reasonable model classes, further assumptions have to be imposed.…”

“…For dynamic factor models with idiosyncratic noise, a rather complete structure theory has been developed in [18]. The answer to the dynamic analogon of the first identifiability question above is, that, for given spectral density f x , the spectra f χ = Λ(e −iλ )f ξ (λ)Λ(e −iλ ) * and f u are generically unique for r ≤ n − √ n.…”