Statistical efficiency of structured CPD estimation applied to Wiener-Hammerstein modeling

Goulart, José Henrique de Morais; Boizard, Maxime; Boyer, Rémy; Favier, Gérard; Comon, Pierre

doi:10.1109/eusipco.2015.7362523

Cited by 2 publications

(12 citation statements)

References 15 publications

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“…This leads to the following result. 2 In this case, to preserve the generality of the model, only two factors should have the scaling of their columns fixed.…”

Section: Propositionmentioning

confidence: 99%

“…Without loss of generality, we assume θ (2) = θ (3) . The parameter vector is thus written as η = [θ (1) ,θ (2) , λ] T . The Jacobian is then given by…”

Section: Propositionmentioning

confidence: 99%

“…Let X 3 = UΣV H denote the SVD of X 3 . Then there exists an R × R invertible matrix M such that UM = A (3) , and M −1 ΣV H = Diag(λ)(A (2) A (1) ) T . The idea is to solve the former for M and θ (3) , which is possible if the number of unknowns, U 3 −1+R 2 , is smaller than the number of equations, I 3 R. This can be done by just solving the linear system:…”

Section: B Subspace-based Closed Form Solutionsmentioning

confidence: 99%

“…Hence we have matrix Ω = A (2) A (1) up to scaling. If ω r denotes the rth column of matrix Ω, the rth column of matrices A (1) and A (2) can eventually be obtained by computing the best rank-one approximation of the I 1 × I 2 matrix Unvec(ω r ). The appropriate normalization of columns of A (1) and A (2) finally gives λ.…”

Section: B Subspace-based Closed Form Solutionsmentioning

confidence: 99%

“…Assume that A (1) and A (2) are structured and independently follow model (7) with parameter vectors θ (1) and θ (2) , respectively. We do not assume any structure for matrix A (3) .…”

Section: B Subspace-based Closed Form Solutionsmentioning

confidence: 99%

See 4 more Smart Citations

Tensor CP Decomposition With Structured Factor Matrices: Algorithms and Performance

Goulart

Boizard

Boyer

et al. 2016

IEEE J. Sel. Top. Signal Process.

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The canonical polyadic decomposition (CPD) of high-order tensors, also known as Candecomp/Parafac, is very useful for representing and analyzing multidimensional data. This paper considers a CPD model having structured matrix factors, as e.g. Toeplitz, Hankel or circulant matrices, and studies its associated estimation problem. This model arises in signal processing applications such as Wiener-Hammerstein system identification and cumulant-based wireless communication channel estimation. After introducing a general formulation of the considered structured CPD (SCPD), we derive closed-form expressions for the Cramér-Rao bound (CRB) of its parameters under the presence of additive white Gaussian noise. Formulas for special cases of interest, as when the CPD contains identical factors, are also provided. Aiming at a more relevant statistical evaluation from a practical standpoint, we discuss the application of our formulas in a Bayesian context, where prior distributions are assigned to the model parameters. Three existing algorithms for computing SCPDs are then described: a constrained alternating least squares (CALS) algorithm, a subspace-based solution and an algebraic solution for SCPDs with circulant factors. Subsequently, we present three numerical simulation scenarios, in which several specialized estimators based on these algorithms are proposed for concrete examples of SCPD involving circulant factors. In particular, the third scenario concerns the identification of a Wiener-Hammerstein system via the SCPD of an associated Volterra kernel. The statistical performance of the proposed estimators is assessed via Monte Carlo simulations, by comparing their Bayesian mean-square error with the expected CRB.

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“…This leads to the following result. 2 In this case, to preserve the generality of the model, only two factors should have the scaling of their columns fixed.…”

Section: Propositionmentioning

confidence: 99%

“…Without loss of generality, we assume θ (2) = θ (3) . The parameter vector is thus written as η = [θ (1) ,θ (2) , λ] T . The Jacobian is then given by…”

Section: Propositionmentioning

confidence: 99%

Section: B Subspace-based Closed Form Solutionsmentioning

confidence: 99%

Section: B Subspace-based Closed Form Solutionsmentioning

confidence: 99%

“…Assume that A (1) and A (2) are structured and independently follow model (7) with parameter vectors θ (1) and θ (2) , respectively. We do not assume any structure for matrix A (3) .…”

Section: B Subspace-based Closed Form Solutionsmentioning

confidence: 99%

See 3 more Smart Citations

Tensor CP Decomposition With Structured Factor Matrices: Algorithms and Performance

Goulart

Boizard

Boyer

et al. 2016

IEEE J. Sel. Top. Signal Process.

Self Cite

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Identification of Fifth-order Wiener and Hammerstein Channels based on the Estimation of an Associated Volterra Kernel

Ahmed¹,

Favier²,

Derbel³

2015

IJCA

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In this paper, we consider the problem of identification of fifthorder Wiener and Hammerstein nonlinear communication channels using the estimation of an associated Volterra kernel. We exploit the special form of the fifth-order associated Volterra kernel for deriving two algorithms that allow to estimate the parameters of the linear part of these channels. In the case of a Wiener channel, the associated Volterra kernel is a tensor satisfying a rank-one PA-RAFAC decomposition whose the parameters can be estimated by means of an alternating least squares (ALS) algorithm. In the case of a Hammerstein channel, its associated Volterra kernel is a diagonal tensor, which leads to a closed-form solution for estimating the parameters of the linear block. The coefficients of the nonlinear block modeled as a fifth degree polynomial are then estimated by means of the standard non recursive least squares (LS) algorithm. The performance of the proposed identification methods is illustrated by means of Monte Carlo simulation results.

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Statistical efficiency of structured CPD estimation applied to Wiener-Hammerstein modeling

Cited by 2 publications

References 15 publications

Tensor CP Decomposition With Structured Factor Matrices: Algorithms and Performance

Tensor CP Decomposition With Structured Factor Matrices: Algorithms and Performance

Identification of Fifth-order Wiener and Hammerstein Channels based on the Estimation of an Associated Volterra Kernel

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