We address covariance estimation in the sense of minimum mean-squared error (MMSE) when the samples are Gaussian distributed. Specifically, we consider shrinkage methods which are suitable for high dimensional problems with a small number of samples (large p small n). First, we improve on the Ledoit-Wolf (LW) method by conditioning on a sufficient statistic. By the Rao-Blackwell theorem, this yields a new estimator called RBLW, whose mean-squared error dominates that of LW for Gaussian variables. Second, to further reduce the estimation error, we propose an iterative approach which approximates the clairvoyant shrinkage estimator. Convergence of this iterative method is established and a closed form expression for the limit is determined, which is referred to as the oracle approximating shrinkage (OAS) estimator. Both RBLW and OAS estimators have simple expressions and are easily implemented. Although the two methods are developed from different perspectives, their structure is identical up to specified constants. The RBLW estimator provably dominates the LW method for Gaussian samples. Numerical simulations demonstrate that the OAS approach can perform even better than RBLW, especially when n is much less than p . We also demonstrate the performance of these techniques in the context of adaptive beamforming.

Shrinkage Algorithms for MMSE Covariance Estimation

Markowitz (1952) portfolio selection requires estimates of (i) the vector of expected returns and (ii) the covariance matrix of returns. Many proposals to address the first question exist already. This paper addresses the second question. We promote a new nonlinear shrinkage estimator of the covariance matrix that is more flexible than previous linear shrinkage estimators and has ‘just the right number’ of free parameters (that is, the Goldilocks principle). In a stylized setting, the nonlinear shrinkage estimator is asymptotically optimal for portfolio selection. In addition to theoretical analysis, we establish superior real-life performance of our new estimator using backtest exercises.

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Nonlinear Shrinkage of the Covariance Matrix for Portfolio Selection: Markowitz Meets Goldilocks

Recently, a new robust adaptive beamforming (RAB) technique was proposed to remove the signal of interest (SOI) component from the sample covariance matrix based on interference-plus-noise covariance matrix reconstruction, which utilizes the Capon spectrum estimator integrated over a region separated from the direction of the SOI. However, the extreme condition of the reconstruction-based technique, that the precise information about the array structure is known in advance, is almost impossible in practice. In this paper, a novel method to reconstruct the interference-plus-noise covariance matrix is proposed. Considering the imprecise prior information about the array structure, which means that the array may be uncalibrated, we use an annulus uncertainty set to constrain the steering vectors of the interferences. Then we integrate the Capon spectrum over the surface of the annulus, by which we can obtain the reconstructed interference matrix without containing the SOI. Since the integral interval is a high-dimensional domain, which is very difficult to solve, we use a discrete sum method to calculate the integral approximately. With the reconstructed interference-plus-noise matrix, the nominal steering vector can be corrected via maximizing the beamformer output power by solving a quadratically constrained quadratic programming (QCQP) problem. The previous reconstruction method can be seen as a special case of the proposed one. The main advantage is that the proposed algorithm is robust against unknown arbitrary-type mismatches. Theoretical analysis and simulation results demonstrate the effectiveness and robustness of the proposed algorithm.

Robust Adaptive Beamforming With a Novel Interference-Plus-Noise Covariance Matrix Reconstruction Method

Robust adaptive beamforming based on interference covariance matrix sparse reconstruction

In the practical radar with multiple antennas, the antenna imperfections degrade the system performance. In this paper, the problem of estimating the direction of arrival (DOA) in a multiple-input and multiple-output (MIMO) radar system with unknown mutual coupling effect between antennas is investigated. To exploit the target sparsity in the spatial domain, the compressed sensing based methods have been proposed by discretizing the detection area and formulating the dictionary matrix, so an off-grid gap is caused by the discretization processes. In this paper, different from the present DOA estimation methods, both the off-grid gap due to the sparse sampling and the unknown mutual coupling effect between antennas are considered at the same time, and a novel sparse system model for DOA estimation is formulated. Then, a novel sparse Bayesian learning (SBL)-based method named sparse Bayesian learning with the mutual coupling (SBLMC) is proposed, where an expectation-maximum-based method is established to estimate all the unknown parameters including the noise variance, the mutual coupling vectors, the off-grid vector, and the variance vector of scattering coefficients. Additionally, the prior distributions for all the unknown parameters are theoretically derived. With regard to the DOA estimation performance, the proposed SBLMC method can outperform state-of-the-art methods in the MIMO radar with unknown mutual coupling effect, while keeping the acceptable computational complexity.

Off-Grid DOA Estimation Using Sparse Bayesian Learning in MIMO Radar With Unknown Mutual Coupling

The main drawback of the conventional diagonal loading (DL) approaches is that there is no clear guideline on how to choose the DL level reliably or how to select user parameters appropriately. An algorithm that can be used to compute the DL level completely automatically from the given data without the need of specifying any user parameter is considered. In this algorithm an enhanced covariance matrix estimate obtained via a shrinkage method, instead of the sample covariance matrix, is used in the standard Capon beamforming formulation. The performance of the resulting beamformer is illustrated via numerical examples, and it is compared with several other adaptive beamformers.

Fully Automatic Computation of Diagonal Loading Levels for Robust Adaptive Beamforming

One of the most well-known robust adaptive beamforming approaches is diagonal loading. However, there are usually no clear guidelines on how to choose the diagonal loading level reliably. In this paper, we present algorithms that can compute the diagonal loading level fully automatically from the given data without the need of specifying any user parameters. The proposed diagonal loading algorithms use shrinkage-based covariance matrix estimates, instead of the conventional sample covariance matrix, in the standard Capon beamforming formulation. The performance of the resulting beamformers is illustrated via numerical examples and compared with other adaptive beamforming techniques.

Fully automatic computation of diagonal loading levels for robust adaptive beamforming

Review of user parameter-free robust adaptive beamforming algorithms

This paper provides a comprehensive review of user parameter-free robust adaptive beamforming algorithms, including ridge regression Capon beamformers (RRCBs), the mid-way (MW) algorithm, the shrinkage based approaches, and iterative beamforming algorithms, namely the iterative adaptive approach (IAA), maximum likelihood based IAA (IAA-ML) and M-SBL (multi-snapshot sparse Bayesian learning). The purpose of these algorithms is to mitigate the negative effects of model errors on the standard Capon beamformer (SCB). We provide a thorough evaluation of these methods under various scenarios and give insights into which algorithm is the best choice under which circumstances.

Doppler spectrogram analysis of the human gait is a useful tool for discriminating various microDoppler tracks due to the movements of different body parts. A data-dependent algorithm, namely the short-time iterative adaptive approach, is used to obtain a more accurate spectrogram than the one provided by the conventional short-time Fourier transform-based approaches. The performance of the approaches is demonstrated and contrasted using both simulated and measured human gait data.

Lin Du

Papers

Fully Automatic Computation of Diagonal Loading Levels for Robust Adaptive Beamforming

Fully automatic computation of diagonal loading levels for robust adaptive beamforming

Review of user parameter-free robust adaptive beamforming algorithms

Review of user parameter-free robust adaptive beamforming algorithms

Doppler spectrogram analysis of human gait via iterative adaptive approach