Estimation of structured covariance matrices

Burg, John P.; Luenberger, David G.; Wenger, D.L.

doi:10.1109/proc.1982.12427

Cited by 312 publications

(188 citation statements)

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“…Hence, in these assumptions, one can prove that the function is bounded and admits a computable maximum (Burg et al, 1982). A proper (R, B) tuple for the inversion system is necessarily a maximum of the function (Dee, 1995).…”

Section: Maximum Of Likelihoodmentioning

confidence: 99%

Towards better error statistics for atmospheric inversions of methane surface fluxes

et al. 2013

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We adapt general statistical methods to estimate the optimal error covariance matrices in a regional inversion system inferring methane surface emissions from atmospheric concentrations. Using a minimal set of physical hypotheses on the patterns of errors, we compute a guess of the error statistics that is optimal in regard to objective statistical criteria for the specific inversion system. With this very general approach applied to a real-data case, we recover sources of errors in the observations and in the prior state of the system that are consistent with expert knowledge while inferred from objective criteria and with affordable computation costs. By not assuming any specific error patterns, our results depict the variability and the inter-dependency of errors induced by complex factors such as the misrepresentation of the observations in the transport model or the inability of the model to reproduce well the situations of steep gradients of concentrations. Situations with probable significant biases (e.g., during the night when vertical mixing is ill-represented by the transport model) can also be diagnosed by our methods in order to point at necessary improvement in a model. By additionally analysing the sensitivity of the inversion to each observation, guidelines to enhance data selection in regional inversions are also proposed. We applied our method to a recent significant accidental methane release from an offshore platform in the North Sea and found methane fluxes of the same magnitude than what was officially declared

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Section: Maximum Of Likelihoodmentioning

confidence: 99%

Towards better error statistics for atmospheric inversions of methane surface fluxes

et al. 2013

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“…Significant improvement is observed in terms of noise variance estimation, an important parameter for detection threshold design. In practice, the complexity of the iteration described in [6], which is applied at the parameter estimation stage of the proposed method, can become too costly. Research is underway in order to devise alternative methods with lower computational cost.…”

Section: Discussionmentioning

confidence: 99%

“…This problem amounts to one of structured covariance matrix estimation from Gaussian observations [6], where the structure ofR(σ) is defined by (4). This estimation problem has unfortunately no closed-form solution, though a fixed point iteration exists [6] that converges to the ML estimateσ ML (S). Substituting the ML estimateR S .…”

Section: Estimation From Compressed Datamentioning

confidence: 99%

Wideband spectral estimation from compressed measurements exploiting spectral a priori information in Cognitive Radio systems

Vazquez-Vilar

López-Valcarce

Mosquera

et al. 2010

2010 IEEE International Conference on Acoustics, Speech and Signal Processing

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In Cognitive Radio scenarios channelization information from primary network may be available to the spectral monitor. Under this assumption we propose a spectral estimation algorithm from compressed measurements of a multichannel wideband signal. The analysis of the Cramer-Rao Lower Bound (CRLB) for this estimation problem shows the importance of detecting the underlaying sparsity pattern of the signal. To this end we describe a Bayesian based iterative algorithm that discovers the set of active signals conforming the band and simultaneously reconstructs the spectrum. This iterative spectral estimator is shown to perform close to a GenieAided CRLB that includes full knowledge about the sparsity pattern of the channels.

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“…Consider the problem of estimating the covariance matrix Σ of a random vector x with components x[n] when it is known that Σ is a linear combination of the matrices in the set S. This problem has a long history and a wide range of applications [1], [2]. Now consider a modification of the same problem where only a subset of the samples x[n] is available, i.e., when the observations are y[m] = x[n m ] for some I = {n 0 , n 1 , .…”

Section: Problem Statementmentioning

confidence: 99%

“…Note that, in view of the covariance sampling criterion, we can confine our attention to linearly independent sets S. Thus, a universal covariance sampler can be defined as a sampler that preserves linear independence 1 . The subsequent sections aim to formalize this notion and to provide simpler conditions that may assist in the design of universal covariance samplers.…”

Section: Universal Covariance Samplersmentioning

confidence: 99%

Compressive covariance sampling

Romero

Leus

2013

2013 Information Theory and Applications Workshop (ITA)

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Abstract-Most research efforts in the field of compressed sensing have been pointed towards analyzing sampling and reconstruction techniques for sparse signals, where sampling rates below the Nyquist rate can be reached. When only second-order statistics or, equivalently, covariance information is of interest, perfect signal reconstruction is not required and rate reductions can be achieved even for non-sparse signals. This is what we will refer to as compressive covariance sampling. In this paper, we will study minimum-rate compressive covariance sampling designs within the class of non-uniform samplers. Necessary and sufficient conditions for perfect covariance reconstruction will be provided and connections to the well-known sparse ruler problem will be highlighted. I. PROBLEM STATEMENTConsider the problem of estimating the covariance matrix Σ of a random vector x with components x[n] when it is known that Σ is a linear combination of the matrices in the set S. This problem has a long history and a wide range of applications [1], [2]. Now consider a modification of the same problem where only a subset of the samples x[n] is available, i.e., when the observations are y[m] = x[n m ] for some I = {n 0 , n 1 , . . .} ⊂ Z. Intuitively, if the grid defined by I is dense enough, then this estimation problem, which we label as compressive covariance sampling, can still be solved.More specifically, the above sample selection produces a transformation of the problem: the vector y collecting the compressed observations y[m], has a covariance matrixΣ, which is a linear combination of the matrices inS. The coefficients in the linear combination are in both problems the same so that solving the latter amounts to solving the former. In this paper we address the optimal design of the index set I so that the coefficients can be estimated, i.e., we look for sets of indices with a minimum number of elements guaranteeing that these parameters remain statistically identifiable. A. Relation to Compressive SamplingRecent interest in sampling signals below their Nyquist rate owes to compressive sampling (or compressed sensing) [3], which has motivated a great deal of research efforts in the last few years. In compressive sampling we are interested in reducing the number of samples by focusing on a special family of signals referred to as sparse. These signals are intrinsically redundant, so that this reduction does not entail any information loss if properly done.Non-uniform sampling [4] is a particular case of compressive sampling. Mathematically, this process can be described as first acquiring a continuous-index signal x(t) at rate 1/T , resulting in the sequence x[n] = x(nT ), and then downsampling x[n] to obtain y[m] according to a set of indices I, in a periodic or non-periodic fashion (periodic non-uniform sampling is also known as multi-coset sampling).Either in the context of compressive sampling (the nonuniform sampling version) or in the context of what we call compressive covariance sampling, the design of the set I has ...

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Estimation of structured covariance matrices

Cited by 312 publications

References 1 publication

Towards better error statistics for atmospheric inversions of methane surface fluxes

Towards better error statistics for atmospheric inversions of methane surface fluxes

Wideband spectral estimation from compressed measurements exploiting spectral a priori information in Cognitive Radio systems

Compressive covariance sampling

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