The spatial sign covariance matrix with unknown location

The well-known spatial sign covariance matrix (SSCM) carries out a radial transform which moves all data points to a sphere, followed by computing the classical covariance matrix of the transformed data. Its popularity stems from its robustness to outliers, fast computation, and applications to correlation and principal component analysis. In this paper we study more general radial functions. It is shown that the eigenvectors of the generalized SSCM are still consistent and the ranks of the eigenvalues are preserved. The influence function of the resulting scatter matrix is derived, and it is shown that its asymptotic breakdown value is as high as that of the original SSCM. A simulation study indicates that the best results are obtained when the inner half of the data points are not transformed and points lying far away are moved to the center.

The one-sample multivariate shape problem is considered. The shape matrix is a scatter matrix rescaled or normalized in a certain way. The procedures here are based on multivariate spatial signs and spatial ranks. Tests and estimates based on the spatial sign covariance matrix and the spatial rank covariance matrices of different types are considered. The asymptotic efficiencies show a good performance of these methods, particularly for heavy-tailed distributions.

Spatial sign correlation

This paper proposes a practical way to solve the Uplink-Downlink Covariance Conversion (UDCC) problem in a frequency Division Duplex (FDD) communication system. The UDCC problem consists in the estimation of the Downlink (DL) spatial covariance matrix from the prior knowledge of the Uplink (UL) spatial covariance matrix without the need of a feedback transmission from the User Equipment (UE) to the Base Station (BS). Estimating the DL sample spatial covariance matrix is unfeasible in current massive Multiple-Input Multiple-Output (MIMO) deployments in frequency selective or fast fading channels due to the required large training overhead. Our method is based on the application of sparse filtering ideas to the estimation of a quantized version of the so-called Angular Power Spectrum (APS), being the common factor between the UL and DL spatial channel covariance matrices.

A new representation of spatio-temporal random processes is proposed in this work. In practical applications, such processes are used to model velocity fields, temperature distributions, response of vibrating systems, to name a few. Finding an efficient representation for any random process leads to encapsulation of information which makes it more convenient for a practical implementations, for instance, in a computational mechanics problem. For a single-parameter process such as spatial or temporal process, the eigenvalue decomposition of the covariance matrix leads to the well-known Karhunen-Loève (KL) decomposition. However, for multiparameter processes such as a spatio-temporal process, the covariance function itself can be defined in multiple ways. Here the process is assumed to be measured at a finite set of spatial locations and a finite number of time instants. Then the spatial covariance matrix at different time instants are considered to define the covariance of the process. This set of square, symmetric, positive semi-definite matrices is then represented as a third-order tensor. A suitable decomposition of this tensor can identify the dominant components of the process, and these components are then used to define a closed-form representation of the process. The procedure is analogous to the KL decomposition for a single-parameter process, however, the decompositions and interpretations vary significantly. The tensor decompositions are successfully applied on (i) a heat conduction problem, (ii) a vibration problem, and (iii) a covariance function taken from the literature that was fitted to model a measured wind velocity data. It is observed that the proposed representation provides an efficient approximation to some processes. Furthermore, a comparison with KL decomposition showed that the proposed method is computationally cheaper than the KL, both in terms of computer memory and execution time.

In motor imagery (MI) tasks for brain computer interfaces (BCIs), the spatial covariance matrix (SCM) of electroencephalogram (EEG) signals plays a critical role in accurate classification. Given that SCMs are symmetric positive definite (SPD), Riemannian geometry is widely utilized to extract classification features. However, calculating distances between SCMs is computationally intensive due to operations like eigenvalue decomposition, and classical optimization techniques, such as gradient descent, cannot be directly applied to Riemannian manifolds, making the computation of the Riemannian mean more complex and reliant on iterative methods or approximations. In this paper, we propose a novel multiclass classification framework that integrates Riemannian geometry and neural networks to mitigate these challenges. The framework comprises two modules: a Riemannian module with multiple branches and a classification module. During training, a fusion loss function is introduced to update the branch corresponding to the true label, while other branches are updated using different loss functions along with the classification module. Comprehensive experiments on four sets of MI EEG data demonstrate the efficiency and effectiveness of the proposed model.

This paper proposes a novel method for simultaneously solving the problems of underdetermined blind source separation (BSS), source activity detection, dereverberation and direction-of-arrival (DOA) estimation by introducing an extension of the "multichannel factorial hidden Markov model (MFH-MM)." The MFHMM is an extension of the multichannel non-negative matrix factorization (NMF) modeL in which the basis spectra are allowed to vary over time according to the transitions of the hidden states. This model has allowed us to perform source separation, source activity detection and dereverberation in a unified manner. In our previous model, the spatial covariance of each source has been treated as a model parameter. This has led the entire generative model to have an unnecessarily high degree of freedom, and thus the parameter inference has been prone to getting trapped into undesired local optima. To reasonably restrict the solution space of the spatial covariance matrix of each source, we propose to describe it as a weighted sum of the fixed spatial covariance matrix corresponding to the discrete set of DOAs. Through the parameter inference, the proposed model allows us to simultaneously solve the problems of underdetermined BSS, source activity detection, dereverberation and DOA estimation. Experimental results revealed that the proposed method was superior to a previous method in terms of the signal-to-distortion ratios of separated signals.

Spatial channel covariance information can replace full channel state information for designing analog precoders in millimeter wave (mmWave) hybrid MIMO systems. Hybrid MIMO architectures, however, make it challenging to estimate the spatial channel covariance matrix because the estimator in baseband can only see the low-dimensional projections of the original channel. In this paper, we propose two key ideas for developing the covariance estimation techniques based on compressive sensing techniques. One is to use the Hermitian property of the covariance matrix, and the other is to use a time-varying analog combining matrix to effectively extend the measurement size.

The coordinatewise median of a multivariate data set is a highly robust location estimator, but it depends on the choice of coordinates. A popular alternative which avoids this drawback is the spatial median, defined as the value that minimizes the sum of distances to the individual data points. In this paper we introduce and discuss another orthogonal equivariant version of the multivariate median, obtained by averaging the coordinatewise median over all orthogonal transformations. We investigate the asymptotic behavior of this estimator and compare it to the spatial median.

This paper investigates limiting spectral properties of a high-dimensional sample spatial-sign covariance matrix when both the dimension of the observations and the sample size grow to infinity. The underlying population is general enough to include the popular independent components model and the family of elliptical distributions. The first result of the paper shows that the empirical spectral distribution of a high dimensional sample spatial-sign covariance matrix converges to a generalized Marčenko-Pastur distribution. Secondly, a new central limit theorem for a class of related linear spectral statistics is established.

In this article, we propose a new blind speech extraction (BSE) method that robustly extracts a directional speech from background diffuse noise by combining independent low-rank matrix analysis (ILRMA) and efficient rank-constrained spatial covariance matrix (SCM) estimation. To achieve more accurate BSE than ILRMA, which assumes each source to be a point source (rank-1 spatial model), the proposed method restores the lost spatial basis for the full-rank SCM of diffuse noise. We adopt the multivariate complex generalized Gaussian distribution (GGD) as the statistical generative model to express various types of observed signal. To estimate the model parameters for an arbitrary shape parameter of the multivariate GGD, we derive a new inequality for rank-constrained SCMs. Also, we propose new acceleration methods to accomplish much faster extraction than conventional blind source separation methods. In BSE experiments using simulated and real recorded data, we confirm that the proposed method achieves more accurate and faster speech extraction than conventional methods.

To mitigate the asynchronous ICI (Inter-Cell Interference), the CFR (Channel Frequency Response) and SCM (Spatial Covariance Matrix) should be accurately estimated. Therefore, the enhanced asynchronous ICI mitigation scheme which can effectively reduce the estimation error of both CFR and SCM, and improve performance for MIMO (Multiple-Input Multiple-Output)-OFDMA (Orthogonal Frequency Division Multiple Access) system is derived. In particular, the lower-bound performance is approached to within 1 dB, by using the proposed method for both MMSE (Minimum Mean Square Error) and ML (Maximum Likelihood) receiver in the presence of asynchronous ICI.

The affine equivariant sign covariance matrix: asymptotic behavior and efficiencies

In this thesis, we discuss several dependence related problems in statistics, which covers three topics I worked on during my PhD study. In the first part, we introduce a new concept of robust-equitability for (nonlinear) dependence measures and identify a robust-equitable copula dependence measure (RCD). We also apply RCD in the application of feature ranking and selection. In the second part, we focus on the estimation of high dimensional spatial covariance matrix. Under the block bandable assumption which arises naturally from spatially correlated data, we propose a rate optimal double tapering estimator for estimating the spatial covariance matrix and demonstrate its advantages over the sample covariance matrix. The third part generalizes the former one in the way that it considers the dependence from both nearby and remote distance grid points, and propose a far-near covariance model which could potentially be used to model teleconnection effect in climate research.

In this study, we propose a robust estimator of the multivariate location parameter by means of the spatial median based on data trimming which extending trimmed mean in the univariate setup. The trimming quantity of this estimator is determined by the bootstrap method, and its covariance matrix is estimated by using the double bootstrap method. This extends the work of Jhun et al. (1993) to the multivariate case. Monte Carlo study shows that the proposed trimmed spatial median estimator yields better efficiency than a spatial median, while its covariance matrix based on double bootstrap overcomes the under-estimating problem occurred on single bootstrap method.