Consistency of the structured total least squares estimator in a multivariate errors-in-variables model

Kukush, Alexander; Markovsky, Ivan; Huffel, Sabine Van

doi:10.1016/j.jspi.2003.12.020

Cited by 47 publications

(41 citation statements)

References 17 publications

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“…In these socalled structured TLS problems, the data matrix [ X Y ] is structured, typically block Toeplitz or Hankel. In order to preserve maximum likelihood properties and consistency of the solution [16,17], the TLS problem formulation, given in Definition 1, must be extended with the additional constraint that any (affine) structure of X or [ X Y ] must be preserved in X or [ X Y ], where X and Y are chosen to minimize the error in the discrete L 1 , L 2 and L ∞ norm. For L 2 norm minimization, various computational algorithms have been presented, as surveyed in [4,5], and shown to reduce the computation time by exploiting the matrix structure in the computations.…”

Section: Total Least Squares: Problem Formulation Algorithms and Extmentioning

confidence: 99%

On the equivalence between total least squares and maximum likelihood PCA

Schuermans

Markovsky

Wentzell

et al. 2005

Analytica Chimica Acta

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The maximum likelihood PCA (MLPCA) method has been devised in chemometrics as a generalization of the well-known PCA method in order to derive consistent estimators in the presence of errors with known error distribution. For similar reasons, the total least squares (TLS) method has been generalized in the field of computational mathematics and engineering to maintain consistency of the parameter estimates in linear models with measurement errors of known distribution. The basic motivation for TLS is the following. Let a set of multidimensional data points (vectors) be given. How can one obtain a linear model that explains these data? The idea is to modify all data points in such a way that some norm of the modification is minimized subject to the constraint that the modified vectors satisfy a linear relation. Although the name "total least squares" appeared in the literature only 25 years ago, this method of fitting is certainly not new and has a long history in the statistical literature, where the method is known as "orthogonal regression", "errors-in-variables regression" or "measurement error modeling". The purpose of this paper is to explore the tight equivalences between MLPCA and element-wise weighted TLS (EW-TLS). Despite their seemingly different problem formulation, it is shown that both methods can be reduced to the same mathematical kernel problem, i.e. finding the closest (in a certain sense) weighted low rank matrix approximation where the weight is derived from the distribution of the errors in the data. Different solution approaches, as used in MLPCA and EW-TLS, are discussed. In particular, we will discuss the weighted low rank approximation (WLRA), the MLPCA, the EW-TLS and the generalized TLS (GTLS) problems. These four approaches tackle an equivalent weighted low rank approximation problem, but different algorithms are used to come up with the best approximation matrix. We will compare their computation times on chemical data and discuss their convergence behavior.

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Section: Total Least Squares: Problem Formulation Algorithms and Extmentioning

confidence: 99%

On the equivalence between total least squares and maximum likelihood PCA

Schuermans

Markovsky

Wentzell

et al. 2005

Analytica Chimica Acta

Self Cite

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“…SLRA solution can be interpreted as a maximum likelihood estimator of the true trajectory under assumption of Gaussian errors. Moreover, SLRA provides a consistent estimator of the system under weaker assumptions of zero-mean errors with covariance structure known up to a scalar factor [10].…”

Section: Introductionmentioning

confidence: 99%

Improved initial approximation for errors-in-variables system identification

Usevich

2012

2012 20th Mediterranean Conference on Control &Amp; Automation (MED)

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Abstract-Errors-in-variables system identification can be posed and solved as a Hankel structured low-rank approximation problem. In this paper different estimates based on suboptimal low-rank approximations are proposed. The estimates are shown to have almost the same efficiency and lead to the same minimum when supplied as an initial approximation for local optimization in the structured low-rank approximation problem. In this paper it is shown that increasing Hankel matrix window length improves initial approximation for autonomous systems and does not improve it in general for systems with inputs.

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“…The so-called structured total least squares estimator is studied in [8]. The construction of this estimator is based on an assumption that the true matrices as well as matrices of observations have a specific structure.…”

Section: Introductionmentioning

confidence: 99%

“…The construction of this estimator is based on an assumption that the true matrices as well as matrices of observations have a specific structure. For example, the construction in [8] is suitable for Toeplitz or Hankel matrices having the block structure.…”

Section: Introductionmentioning

confidence: 99%