Bibliography on higher-order statistics

Swami, Ananthram; Giannakis, Georgios B.; Zhou, G.T.

doi:10.1016/s0165-1684(97)00065-0

Cited by 86 publications

(40 citation statements)

References 662 publications

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“…This secondorder response (j¼2) is well a particular case of (2) i i i i 1 2 3 4 ω ω ω related to the higher order cross-cumulants of the structural response find applications in various fields of science and engineering [47]; they are also expressed in terms of the bispectrum and trispectrum of the loading [16]. In an MDOF context to be general, and under stationary conditions, the third-and fourth-order spectra of the response read Again, they may be viewed as particular cases of (2).…”

Section: Spectral Analysismentioning

confidence: 99%

Multiple timescale spectral analysis

Denoël

2015

Probabilistic Engineering Mechanics

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a b s t r a c tSpectral analysis is a classical tool for the structural analysis of structures subjected to random excitations. The most common application of spectral analysis is the determination of the steady-state second order cumulant of a linear oscillator, under the action of a stationary loading prescribed by means of its power spectral density. There exists however a broad variety of such similar problems, extending the concept to multi degree-of-freedom systems, non Gaussian excitation, slightly nonlinear oscillators or even transient excitations. In this wide class of problems, the cumulants of the response are obtained as the result of the integral of corresponding spectra over the frequency space, which is possibly multidimensional. Application of standard numerical integration techniques may be prohibitive, a reason why the spectral approach is often left aside. Besides, many engineering problems involve a clear timescale separation, usually of those pertaining to the loading and to the mechanical behavior of the system. In these problems, a proper consideration of the timescale separation results in dropping the order of integration by one, at least. This offers the possibility to derive analytical solutions, whenever the order of integration drops to zero, or to make numerical integration competitive. The paper presents this general method, together with some applications in wind and marine engineering.

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Section: Spectral Analysismentioning

confidence: 99%

Multiple timescale spectral analysis

Denoël

2015

Probabilistic Engineering Mechanics

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“…Higher-order information can also be used to obtain improved estimates, and, of course, to estimate those parameters which are not covered by second-order information. We again refer to Swami, Giannakis and Zhou (1997), where many further areas of application of higher-order statistics are indicated. Some recent references and examples can also be found in Anh, Leonenko and Sakhno (2002b).…”

Section: Introductionmentioning

confidence: 99%

Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence

Anh¹,

Leonenko²,

Sakhno³

2004

Journal of Applied Probability

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Abstract. This paper provides a quasilikelihood/minimum contrast-type method for parameter estimation of random &elds in the frequency domain based on higher-order information. The estimation technique uses the spectral density of the general k-th order and allows for possible long-range dependence in the random &elds. To avoid bias due to edge effects, data tapering is incorporated in the method. The suggested minimum contrast functional is linear with respect to the periodogram of k-th order, hence kernel estimation for the spectral densities is not needed. Furthermore, discretisation is not required in the estimation of continuously observed random &elds. The consistency and asymptotic normality of the resulting estimators are established. Illustrative application of the method to some problems in mathematical &nance and signal detection will be indicated.

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“…Introduction. Higher-order tensors (three-way arrays) have been used as a tool in higher-order statistics [39,36,47,38] and independent component analysis (ICA) [13,14,19,9] for several decades. Other application areas include chemometrics, scientific computing, biomedical signal processing, image processing, and telecommunications.…”

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confidence: 99%

Jacobi Algorithm for the Best Low Multilinear Rank Approximation of Symmetric Tensors

Ishteva

Absil

Dooren

2013

SIAM J. Matrix Anal. & Appl.

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Abstract. The problem discussed in this paper is the symmetric best low multilinear rank approximation of third-order symmetric tensors. We propose an algorithm based on Jacobi rotations, for which symmetry is preserved at each iteration. Two numerical examples are provided indicating the need for such algorithms. An important part of the paper consists of proving that our algorithm converges to stationary points of the objective function. This can be considered an advantage of the proposed algorithm over existing symmetry-preserving algorithms in the literature.Key words. multilinear algebra, higher-order tensor, rank reduction, singular value decomposition, Jacobi rotation AMS subject classifications. 15A69, 65F99 DOI. 10.1137/11085743X1. Introduction. Higher-order tensors (three-way arrays) have been used as a tool in higher-order statistics [39,36,47,38] and independent component analysis (ICA) [13,14,19,9] for several decades. Other application areas include chemometrics, scientific computing, biomedical signal processing, image processing, and telecommunications. For an exhaustive list and references we refer to [46,34,32,8,12].Let us first consider the general low multilinear rank approximation of thirdorder tensors. The problem consists of finding the best approximation of a given tensor A ∈ R I1×I2×I3 , subject to a constraint on the multilinear rank of the approximation. The concept of multilinear rank was first introduced in [25,26] and is simply a generalization of the row and column rank of matrices to higher-order tensors. We define the problem more precisely in the next section.A closed-form solution of the best low multilinear rank approximation problem is not known. A generalization of the singular value decomposition (SVD) [23, sect. 2.5] called higher-order SVD (HOSVD) has been studied in [15]. A variation of this decomposition is know as the Tucker decomposition [49,50]. In general, truncation of the HOSVD leads to a good but not necessarily to the best low multilinear rank approximation. Recent iterative algorithms solving the problem include geometric Newton [21,29]

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Bibliography on higher-order statistics

Cited by 86 publications

References 662 publications

Multiple timescale spectral analysis

Multiple timescale spectral analysis

Quasi-likelihood-based higher-order spectral estimation of random fields with possible long-range dependence

Jacobi Algorithm for the Best Low Multilinear Rank Approximation of Symmetric Tensors

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