Identification of ARX-models subject to missing data

In this paper we derive the maximum likelihood problem for missing data from a Gaussian model. We present in total eight different equivalent formulations of the resulting optimization problem, four out of which are nonlinear least squares formulations. Among these formulations are also formulations based on the expectation-maximization algorithm. Expressions for the derivatives needed in order to solve the optimization problems are presented. We also present numerical comparisons for two of the formulations for an ARMAX model.

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“…For certain problems there are closed form solutions, [Isa93]. In our case this is not true in general.…”

Section: Numerical Implementationmentioning

confidence: 97%

“…In [WR86] instead an innovation transformation is used to derive the log-likelihood function. In [Isa93] it is for ARX models suggested that the so-called Expectation Maximization (EM) algorithm could be used, see e.g. [DLR77,Wu83].…”

Section: Introductionmentioning

confidence: 99%

Maximum likelihood estimation of Gaussian models with missing data—Eight equivalent formulations

Hansson

Wallin

2012

Automatica

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“…A special system identification problems for the class of auto-regressive exogenous systems with missing data is considered in [10] and a method based on frequency domain techniques is proposed in [24]. These papers do not link the system identification problem to the block-Hankel low-rank approximation problem (SYSID), so that their approaches are different from ours.…”

Section: 22mentioning

confidence: 99%

Structured Low-Rank Approximation with Missing Data

Markovsky¹,

Usevich²

2013

SIAM J. Matrix Anal. & Appl.

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We consider low-rank approximation of affinely structured matrices with missing elements. The method proposed is based on reformulation of the problem as inner and outer optimization. The inner minimization is a singular linear least-norm problem and admits an analytic solution. The outer problem is a nonlinear least squares problem and is solved by local optimization methods: minimization subject to quadratic equality constraints and unconstrained minimization with regularized cost function. The method is generalized to weighted low-rank approximation with missing values and is illustrated on approximate low-rank matrix completion, system identification, and data-driven simulation problems. An extended version of the paper is a literate program, implementing the method and reproducing the presented results.

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“…The research literature on spectral analysis of gapped data is abundant ; see, e.g., Scargle (1997), Adorf (1995), Meisel (1979), Brown & Christensen-Dalsgaard (1990), Fahlman & Ulrych (1982), Swan (1982), Kuhn (1982), Roberts, Lehar, & Dreher (1987), Lomb (1976), Scargle (1982), Press & Rybicki (1989), Scargle (1989), and Rybicki & Press (1992) in astronomical journals and Isaksson (1993), Sacchi, Ulrych, & Walker (1998), and Bronez (1988) in signal processing journals. In the applications that motivated this literature, with astronomy as a notable example, the data come with gaps and hence require a di †erent set of analysis tools than those used to process full data sequences.…”

Section: Introductionmentioning

confidence: 99%

Adaptive Filter-bank Approach to Restoration and Spectral Analysis of Gapped Data

Stoica¹,

Larsson²,

Li³

2000

The Astronomical Journal

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The main topic of this paper is the nonparametric estimation of complex (both amplitude and phase) spectra from gapped data, as well as the restoration of such data. The focus is on the extension of the APES (amplitude and phase estimation) approach to data sequences with gaps. APES, which is one of the most successful existing nonparametric approaches to the spectral analysis of full data sequences, uses a bank of narrowband adaptive (both frequency and data dependent) Ðlters to estimate the spectrum. A recent interpretation of this approach showed that the Ðlterbank used by APES and the resulting spectrum minimize a least-squares (LS) Ðtting criterion between the Ðltered sequence and its spectral decomposition. The extended approach, which is called GAPES for somewhat obvious reasons, capitalizes on the aforementioned interpretation : it minimizes the APES-LS Ðtting criterion with respect to the missing data as well. This should be a sensible thing to do whenever the full data sequence is stationary, and hence the missing data have the same spectral content as the available data. We use both simulated and real data examples to show that GAPES estimated spectra and interpolated data sequences have excellent accuracy. We also show the performance gain achieved by GAPES over two of the most commonly used approaches for gapped-data spectral analysis, viz., the periodogram and the parametric CLEAN method.

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Identification of ARX-models subject to missing data

Cited by 121 publications

References 11 publications

Maximum likelihood estimation of Gaussian models with missing data—Eight equivalent formulations

Maximum likelihood estimation of Gaussian models with missing data—Eight equivalent formulations

Structured Low-Rank Approximation with Missing Data

Adaptive Filter-bank Approach to Restoration and Spectral Analysis of Gapped Data

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