Journal Article•
Maximum Likelihood Parameter and Rank Estimation in Reduced-Rank Linear Regression
About: This article is published in IEEE Signal Processing Magazine.The article was published on 1996-01-01 and is currently open access. It has received 11 citations till now. The article focuses on the topics: Rank (linear algebra) & Maximum likelihood sequence estimation.
Citations
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TL;DR: An estimator is derived that can, in an asymptotically optimal way, use, not only the structure implied by the Kronecker assumption, but also linear structure on the transmit- and receive covariance matrices.
36 citations
TL;DR: A new estimation procedure, based on instrumental variable principles, is derived and analyzed, designed to handle noise that is both spatially and temporally autocorrelated and outperforms previous methods when the noise is temporally correlated.
Abstract: This paper addresses parameter estimation in reduced rank linear regressions. This estimation problem has applications in several subject areas including system identification, sensor array processing, econometrics and statistics. A new estimation procedure, based on instrumental variable principles, is derived and analyzed. The proposed method is designed to handle noise that is both spatially and temporally autocorrelated. An asymptotical analysis shows that the proposed method outperforms previous methods when the noise is temporally correlated and that it is asymptotically efficient otherwise. A numerical study indicates that the performance is significantly improved also for finite sample set sizes. In addition, the Cramer-Rao lower bound (CRB) on unbiased estimator covariance for the data model is derived. A statistical test for rank determination is also developed. An important step in the new algorithm is the weighted low rank approximation (WLRA). As the WLRA lacks a closed form solution in its general form, two new, noniterative and approximate solutions are derived, both of them asymptotically optimal when part of the estimation procedure proposed here. These methods are also interesting in their own right since the WLRA has several applications.
21 citations
11 Jun 2001
TL;DR: This paper investigates the problem of channel estimation in the uplink of CDMA systems with base station antenna array and proposes to reduce the number of the unknowns in the channel estimation by constraining the space-time channel matrix of each user to be low rank.
Abstract: This paper investigates the problem of channel estimation in the uplink of CDMA systems with base station antenna array. The estimation is based on the transmission of training sequences with limited length. In order to improve multiuser receiver performance it is proposed to reduce the number of the unknowns in the channel estimation by constraining the space-time channel matrix of each user to be low rank. The rank-order is estimated according to the MDL criterion as this method provides the best trade-off between distortion (due to under-parametrization) and variance (due to the limited training length).
13 citations
17 May 2004
TL;DR: The weighted low-rank approximation (WLRA) problem is considered and non-iterative methods that are asymptotically optimal for the linear regression and related problems are developed.
Abstract: The weighted low-rank approximation (WLRA) problem is considered. The problem is that of approximating one matrix with another matrix of lower rank, such that the weighted norm of the difference is minimized. The problem is fundamental in a new method for reduced rank linear regression that is outlined, as well as in areas such as two-dimensional filter design and data mining. The WLRA problem has no known closed form solution in the general case, but iterative methods have previously been suggested. Non-iterative methods that are asymptotically optimal for the linear regression and related problems are developed. Computer simulations, where the new methods are compared to one step of the well-known alternating projections algorithm, show significantly improved performance.
9 citations
Cites methods from "Maximum Likelihood Parameter and Ra..."
...…≡ arg min a=vec(A) V (A, B) = arg min a ‖Q1/2ψ − Q1/2(B ⊗ Im)a‖22 = ( (BT ⊗ Im)Q(B ⊗ Im) )−1 (BT ⊗ Im)Qψ (9) Inserting (9) into (4) and defining B̄ ≡ Q1/2(B ⊗ Im) yield V (B) = ‖(Imn − B̄(B̄T B̄)−1B̄T )Q1/2ψ‖22 (10) Now, define N ∈ n×(n−r) to be a full column-rank matrix such that NT B = 0 (⇒ CN…...
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...The algorithm derived here differs from one iteration of the Newton algorithms presented in [4] in that the small residual assumption (2) is used to make simplifications that reduce the computational complexity drastically....
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TL;DR: In this paper, the authors incorporate the idea of reduced rank envelope to elliptical multivariate linear regres4 sion to improve the efficiency of estimation, which is more flexible and its estimator outperforms the estimator de10 rived for the normal case.
Abstract: 3 We incorporate the idea of reduced rank envelope [7] to elliptical multivariate linear regres4 sion to improve the efficiency of estimation. The reduced rank envelope model takes advantage 5 of both reduced rank regression and envelope model, and is an efficient estimation technique in 6 multivariate linear regression. However, it uses the normal log-likelihood as its objective func7 tion, and is most effective when the normality assumption holds. The proposed methodology 8 considers elliptically contoured distributions and it incorporates this distribution structure into 9 the modeling. Consequently, it is more flexible and its estimator outperforms the estimator de10 rived for the normal case. When the specific distribution is unknown, we present an estimator 11 that performs well as long as the elliptically contoured assumption holds. 12
8 citations