Data Covariance Matrix Research Articles

Solving Geophysical Inversion Problems with Intractable Likelihoods: Linearized Gaussian Approximations Versus the Correlated Pseudo-marginal Method

A geophysical Bayesian inversion problem may target the posterior distribution of geological or hydrogeological parameters given geophysical data. To account for the scatter in the petrophysical relationship linking the target parameters to the geophysical properties, this study treats the intermediate geophysical properties as latent (unobservable) variables. To perform inversion in such a latent variable model, the intractable likelihood function of the (hydro)geological parameters given the geophysical data needs to be estimated. This can be achieved by approximation with a Gaussian probability density function based on local linearization of the geophysical forward operator, thereby, accounting for the noise in the petrophysical relationship by a corresponding addition to the data covariance matrix. The new approximate method is compared against the general correlated pseudo-marginal method, which estimates the likelihood by Monte Carlo averaging over samples of the latent variable. First, the performances of the two methods are tested on a synthetic test example, in which a multivariate Gaussian porosity field is inferred using crosshole ground-penetrating radar first-arrival travel times. For this example with rather small petrophysical uncertainty, the two methods provide near-identical estimates, while an inversion that ignores petrophysical uncertainty leads to biased estimates. The results of a sensitivity analysis are then used to suggest that the linearized Gaussian approach, while attractive due to its relative computational speed, suffers from a decreasing accuracy with increasing scatter in the petrophysical relationship. The computationally more expensive correlated pseudo-marginal method performs very well even for settings with high petrophysical uncertainty.

Indoor sound source localization under reverberation by extracting the features of sample covariance

Three dimensional indoor sound source localization is well known as a challenging task due to the complicated mechanics of reverberation. The conventional model-based methods generally require the prior knowledge of the microphone array geometry, the environment’s parameters, and the statistics of the signal and noise. On the other hand, the data-based methods could work in the absence of all/part of the aforementioned prior knowledge, while the localization performance depends on the established features of the data and the nature of neural network. In this work, a feature vector constructed from the sample covariance matrix of the microphone array data is fed to a designed deep neural network. Numerical simulation results show that the proposed method outperforms [21] in both ranging and direction finding, hence locating a human-speech source in a three-dimensional reverberant room space. An average relative localization error below 3% can be reached, which is robust to the room reverberation level of T60 from 0 to 600 ms when the signal-to-noise-ratio is not below 0 dB. Moreover, the robustness of the proposed method against different scenarios is verified: the increment of relative localization error is below 5% for various room dimensions, below 10% for different speech signals, and below 3% for microphone array offsets.

Gridless covariance matrix fitting methods for three dimensional acoustical source localization

In this article, we consider gridless source localization based on the spatial covariance matrix of acoustical data collected by an array of microphones. Covariance matrix fitting problems are formulated in infinite-dimensional settings, and solved by the Sliding Frank–Wolfe algorithm. The proposed method does not impose any constraint on the geometry of the array, the propagation model or the domain of interest, and does not necessitate a training phase. It is tested on simulated and experimental measurements for the localization of sources in a three-dimensional domain. Performances are compared to the state of the art, showing in particular a better resolution than MUSIC (MUltiple SIgnal Classification) at low SNR.

Efficient wave type fingerprinting and filtering by six-component polarization analysis

SUMMARYWe present a technique to automatically classify the wave type of seismic phases that are recorded on a single six-component recording station (measuring both three components of translational and rotational ground motion) at the Earth’s surface. We make use of the fact that each wave type leaves a unique ’fingerprint’ in the six-component motion of the sensor (i.e. the motion is unique for each wave type). This fingerprint can be extracted by performing an eigenanalysis of the data covariance matrix, similar to conventional three-component polarization analysis. To assign a wave type to the fingerprint extracted from the data, we compare it to analytically derived six-component polarization models that are valid for pure-state plane wave arrivals. For efficient classification, we make use of the supervised machine learning method of support vector machines that is trained using data-independent, analytically derived six-component polarization models. This enables the rapid classification of seismic phases in a fully automated fashion, even for large data volumes, such as encountered in land-seismic exploration or ambient noise seismology. Once the wave-type is known, additional wave parameters (velocity, directionality and ellipticity) can be directly extracted from the six-component polarization states without the need to resort to expensive optimization algorithms. We illustrate the benefits of our approach on various real and synthetic data examples for applications such as automated phase picking, aliased ground-roll suppression in land-seismic exploration and the rapid close-to real-time extraction of surface wave dispersion curves from single-station recordings of ambient noise. Additionally, we argue that an initial step of wave type classification is necessary in order to successfully apply the common technique of extracting phase velocities from combined measurements of rotational and translational motion.

Recognition method of basketball players' throwing action based on image segmentation

In order to solve the problems of low recognition accuracy and long recognition time in traditional basketball players' throwing action recognition methods, this paper proposes a new basketball players' throwing action recognition method based on image segmentation. The covariance matrix of noise data of basketball players' throwing action is constructed. The throwing action of basketball players is expressed by acceleration and angular velocity, and the acceleration vector of throwing action is obtained. The feature extraction of throwing action is completed by discrete Fourier transform algorithm. The image of basketball players' throwing action is segmented by threshold and edge, and the change features of throwing action are obtained by kernel function to complete the recognition of basketball players' throwing action. The experimental results show that the accuracy of the proposed method is about 98%, and the time cost is about 2.1 s.

Optimum “Eye Location” Problem for Spectral Clustering With Cosine Distance

It has recently been reported that Spectral Clustering gives state-of-the art clustering performance for many real-life benchmark datasets. When building the dissimilarity (distance) matrix for the Laplacian matrix, cosine distance is also reported to give the best performance among the other distance types for various real-life datasets. In this paper, we introduce an Optimum Sphere Location Problem for Spectral Clustering with cosine distance, and our investigations result in novel results, some of which are as follows: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</i> ) The Optimum “Eye Location” Problem (a.k.a. Optimum Sphere Location Problem) is a challenging problem due to the high complexity between the center of the unit sphere and the eigenvalues of the corresponding Laplacian matrix. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ii</i> ) As compared to the sphere at the origin, choosing the “eye location” at the data mean generally increases the eigenvalues of the Laplacian matrix. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iii</i> ) As compared to the sphere at the data mean, choosing the “eye location” at an appropriate point on the 2nd or 3rd dominant eigenvector of the data covariance matrix potentially gives better clustering performance for the same spectral clustering algorithm. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">iv</i> ) The proposed method includes the traditional methods, which are locating the unit sphere at the origin and at the data mean, as special cases. Computer simulations confirm the findings.

Enhanced BNC Approach for Non-Circular Signals Direction Finding with Sparse Array in the Scenario of Impulsive Noise

Sparse array is popular in the field of array signal processing. However, in direction of arrival (DOA) estimation, most research on sparse array assumes Gaussian noise, resulting in model mismatch in the practical scenarios of impulsive noise. We investigate the estimation performance of the bounded non-linear covariance (BNC) matrix based methods, which can be used to replace the original data covariance matrix of the received signal and achieve good robustness to the impulsive interference. We propose an enhanced BNC (EBNC) matrix with non-circular signal for nested array in the scenario of impulsive noise. The proposed EBNC matrix can fight against the impulsive noise outliers and its boundedness and convergence are shown. The proposed method's performance is assessed with simulations. Simulation results indicate that the proposed method provides better performance in DOA estimation than the classical BNC, correntropy-based covariance matrix and Gaussian covariance based methods, especially in highly impulsive noise scenario.

Multiple‐input multiple‐output sonar adaptive beamforming using transmission diversity smoothing and backward processing

AbstractBenefit from the transmission diversity smoothing (TDS) effect upon coherent targets decorrelation, the kind of adaptive beamformers can be directly applied for multiple‐input multiple‐output (MIMO) sonar applications. Increasing the number of transmitted signals helps improve the TDS‐based decorrelation performance. However, a large number of transmitters is not feasible for those sonar systems mounted on small platforms with limited space. To address the lack of TDS‐based degrees of freedom for smoothing, an improved TDS decorrelation method is proposed for MIMO sonar in this study. With the conjugated MIMO data covariance matrix being left‐multiplied and right‐multiplied, respectively, by the exchange matrix, the authors obtain a backward‐smoothed data covariance matrix. Afterwards, the authors average it with the original MIMO receiving data covariance matrix to reconstruct a new data covariance matrix. Compared with TDS, the improved TDS strengthens the decorrelation performance by increasing the number of covariance matrix for smoothing and thus enables the new rank restored signal covariance matrix to have better non‐singularity. Without increasing any aperture loss and system complexity, a lower correlation between echoes is obtained, and the performance of the minimum variance distortionless response beamformer is hence improved in the presence of coherent targets. The effectiveness of the proposed decorrelation method is verified by numerical simulations and real data from a water tank experiment.

Singular Spectrum Analysis Based EMG Artifact Removal from ECG Signal

Electromyogram (EMG) or muscle artifacts frequently affect electrocardiogram (ECG) readings. These artifacts make the required information in the ECG signal difficult to see. In this study, we introduced the singular spectrum analysis (SSA), a powerful subspace-based method for removing EMG artifacts from ECG data. In order to effectively extract the desired component from the tainted ECG data, we presented a new grouping approach and set a threshold. First, a process known as embedding converts a single channel signal into several channels of signals or data. The orthogonal eigenvectors are then calculated using singular value decomposition(SVD) from the multichannel data's covariance matrix. A threshold is selected to locate these eigenvectors, which are utilized to generate the required subspace. After locating the subspace, the multichannel data is simply projected into it, followed by a method called diagonal averaging which will create the original time series and extract the ECG signals. Keywords: Electrocardiogram, EMG artifact, Singular Spectrum Analysis, Embedding, SVD, Mobility.

A low-complexity two-dimensional DOA estimation algorithm based on an L-shaped sensor array

In this paper, a derivative-based MUSIC (multiple signal classification) algorithm for a mixture of circular and noncircular signals (DB-MUSIC-M) is proposed for two-dimensional (2D) direction of arrival (DOA) estimation employing an L-shaped uniform array. The DB-MUSIC-M transforms the 2D DOA estimation problem into a single one-dimensional (1D) estimation by finding the derivative of the objective function of the 2D improved MUSIC (I-MUSIC) algorithm, which greatly reduces its computational complexity. As it utilizes both the pseudo covariance matrix and the covariance matrix of the array data, the maximum number of signals that can be estimated is much higher than the total number of sensors. As a special case, the derivative-based MUSIC (DB-MUSIC) for circular signals is also proposed. There is no need for angle pairing for the proposed algorithms and they can handle the angle ambiguity problem effectively. As shown by simulation results, the proposed DB-MUSIC-M algorithm outperforms existing algorithms, with significantly reduced complexity compared to a direct 2D search method. Moreover, the proposed approach can be applied to some other array structures such as the uniform planar array.

Bayesian operational modal analysis of structures with tuned mass damper

Tuned mass damper (TMD) is a common strategy to reduce structural vibration in a passive manner without the need for active power. The basic parameters of a TMD include its mass ratio, natural frequency and damping ratio. While these parameters are factory-calibrated before installation, it would be desirable to assess the in-situ properties of the TMD and the ‘primary’ structure under operational state, e.g., to validate/assess performance and detect detuning over the service life. In this work, a Bayesian approach is developed for identifying the modal parameters of the TMD and primary structure using only the ambient vibration data measured on the primary structure, i.e., ‘operational modal analysis’. The likelihood function and theoretical PSD matrix of ambient data are formulated, accounting for primary-secondary structure dynamics with non-classical damping that is not treated in existing Bayesian formulations. An Expectation-Maximisation (EM) algorithm is developed for efficient computation of the most probable value of modal parameters. Analytical expressions are derived so that the ‘posterior’ (i.e., given data) covariance matrix can be determined accurately and efficiently. The proposed method is verified using synthetic data and applied to field data of a chimney with close modes response attenuated by a TMD.

Tracy–Widom at each edge of real covariance and MANOVA estimators

We study the sample covariance matrix for real-valued data with general population covariance, as well as MANOVA-type covariance estimators in variance components models under null hypotheses of global sphericity. In the limit as matrix dimensions increase proportionally, the asymptotic spectra of such estimators may have multiple disjoint intervals of support, possibly intersecting the negative half line. We show that the distribution of the extremal eigenvalue at each regular edge of the support has a GOE Tracy–Widom limit. Our proof extends a comparison argument of Ji Oon Lee and Kevin Schnelli, replacing a continuous Green function flow by a discrete Lindeberg swapping scheme.

Behavior of Some Hypothesis Tests for the Covariance Matrix of High Dimensional Data

The study of the structure of the covariance matrix when the dimension of the data is much greater than the sample size (high dimensional data) is a complicated problem, since we have many unknown parameters and few data. Several hypothesis tests for the covariance matrix, in the high dimensional context and in the classical case (where the dimension of the data is less than the sample size), can be found in the literature. It has been of interest the tests for the null hypothesis that the covariance matrix of Gaussian data is equal or proportional to the identity matrix, considering the classical case as well as the high dimensional context. Since it is important to have a wide comparison between these tests found in the literature, and for some of them it is difficult to have theoretical results about their powers, in this work we compare several tests by simulations, in terms of the size and power of the test. We also present some examples of application with real high dimensional data found in the literature.

Automatic detection of code-modulated visual evoked potentials using novel covariance estimators and short-time EEG signals

Background and Objective:Over the last years, code-modulated visual evoked potentials (cVEP)-based brain–computer interfaces (BCIs) have been developed as robust and non-invasive tools to construct high-speed communication systems. Recently, beamforming-based algorithms have extensively been used in cVEP-based BCI systems because of the need for short-time stimulation and less training data. One of the main drawbacks of the beamforming-based approaches is that their performance highly depends on estimating data covariance matrix and calculating activation patterns. Methods:In the present study, two novel covariance estimators (i.e., the modified convex combination (MCC) and the maximum likelihood (ML) techniques) are proposed to estimate a robust and more reliable covariance matrix. In the ML method, a new sparsity constraint is considered to express the specific eigendecomposition of the covariance matrix as a sparse matrix transform (SMT). Then, the SMT is calculated using the product of pairwise coordinate rotations. These rotations can be constructed by a cross-validation method. Two stimulation presentation rates of 60 and 120 Hz are used for the coding sequence. Results:Both of the suggested approaches (i.e., the MCC and SMT-based techniques) can efficiently improve the performance of the conventional spatiotemporal beamforming-based methods by providing a robust estimate of the covariance matrix in short stimulation times. Based on the experimental results, it can be concluded that the proposed SMT and MCC methods achieve the best results for the 60 and 120 Hz stimulus presentation rates, respectively. However, for both stimulus presentation rates, the proposed SMT and MCC-based methods remarkably outperform other state-of-the-art methods in cVEP-based BCI, such as conventional spatiotemporal beamforming and optimized support vector machines (SVM). Also, the results showed that the 120 Hz stimulus presentation rate provided faster communication. This procedure is performed by obtaining a maximal Information Transfer Rate (ITR) of 187.38 bits/minute. Conclusion:Finally, the present study suggested that the proposed MCC and SMT-based techniques could automatically detect the gazed targets. Also, these methods could be used as non-invasive alternatives over conventional methods.

Creating jackknife and bootstrap estimates of the covariance matrix for the two-point correlation function

ABSTRACT We present correction terms that allow delete-one Jackknife and Bootstrap methods to be used to recover unbiased estimates of the data covariance matrix of the two-point correlation function $\xi \left(\mathbf {r}\right)$. We demonstrate the accuracy and precision of this new method using a large set of 1000 QUIJOTE simulations that each cover a comoving volume of $1\rm {\left[h^{-1}Gpc\right]^3}$. The corrected resampling techniques recover the correct amplitude and structure of the data covariance matrix as represented by its principal components to within ∼10 per cent, the level of error achievable with the size of the sample of simulations used for the test. Our corrections for the internal resampling methods are shown to be robust against the intrinsic clustering of the cosmological tracers both in real- and redshift space using two snapshots at z = 0 and z = 1 that mimic two samples with significantly different clustering. We also analyse two different slicing of the simulation volume into $\, n_{\rm sv}\, =64$ or 125 sub-samples and show that the main impact of different $\, n_{\rm sv}\,$ is on the structure of the covariance matrix due to the limited number of independent internal realizations that can be made given a fixed $\, n_{\rm sv}\,$.

Approximate normality in testing an exchangeable covariance structure under large- and high-dimensional settings

In this paper the Rao score and likelihood ratio tests for hypothesis related to exchangeable structure of multivariate data covariance matrix are studied. Under the assumption of large-dimensionality the normal approximation of the Rao score test statistics distribution is proven as well as the exact and approximate distributions of the likelihood ratio test are derived. Simulation studies show the advantage of the Rao score test over the likelihood ratio test in both studied contexts: type I error and power. Moreover, the Rao score test is available in the case of high-dimensionality, and it is shown that the normal approximation matches well its distribution in this case. Thus, this latter approximation could be recommended for practical use.

Return of the templates: Revisiting the Galactic Center excess with multimessenger observations

The Galactic center excess (GCE) remains one of the most intriguing discoveries from the Fermi Large Area Telescope (LAT) observations. We revisit the characteristics of the GCE by first producing a new set of high-resolution galactic diffuse gamma-ray emission templates. This diffuse emission, which accounts for the bulk of the observed gamma rays, is ultimately due to cosmic-ray interactions with the interstellar medium. Using recent high-precision cosmic-ray observations, in addition to the continuing Fermi-LAT observations and observations from lower energy photons, we constrain the properties of the galactic diffuse emission. We describe a large set of diffuse gamma-ray emission templates which account for a very wide range of initial assumptions on the physical conditions in the inner galaxy. The broad properties of the GCE that we find in this work are qualitatively unchanged despite the introduction of this new set of templates, though its quantitative features appear mildly different than those obtained in previous analyses. In particular, we find a high-energy tail at higher significance than previously reported. This tail is very prominent in the northern hemisphere, and less so in the southern hemisphere. This strongly affects one prominent interpretation of the excess: known millisecond pulsars are incapable of producing this high-energy emission, even in the relatively softer southern hemisphere, and are therefore disfavored as the sole explanation of the GCE. The annihilation of dark matter particles of mass ${40}_{\ensuremath{-}7}^{+10}\text{ }\text{ }\mathrm{GeV}$ (95% CL) to $b$ quarks with a cross-section of $⟨{\ensuremath{\sigma}}_{A}v⟩=1.{4}_{\ensuremath{-}0.3}^{+0.6}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}26}\text{ }\text{ }{\mathrm{cm}}^{3}\text{ }{\mathrm{s}}^{\ensuremath{-}1}$ provides a good fit to the excess especially in the relatively cleaner southern sky. Dark matter of the same mass range annihilating to $b$ quarks or heavier dark matter particles annihilating to heavier Standard Model bosons can combine with millisecond pulsars to provide a good fit to the southern hemisphere emission as well, as can a broken power-law spectrum which would be related to recent cosmic-ray burst activity. As part of this paper, we make publicly available all of our templates and the data covariance matrix we have generated to account for systematic uncertainties.

Research on fault detection and identification methods of nonlinear dynamic process based on ICA

Abstract In the present study, nonlinear dynamic process data are mapped into the kernel state space by kernel gauge variable analysis method to obtain decorrelated state data. The time-lapse covariance matrix of the state data is weighted and summed to obtain the time-lapse structure matrix of the state data, and then supervised kernel independent component analysis (SKICA) is established, the independent component feature data is extracted from the status data and the monitoring statistics are constructed to detect the process faults. The data show that kernel independent component analysis (ICA) method (KICA) method can detect slow fault faster than the ICA method, except that the statistical detection ability of F3 and FS is reduced, and the KICA method can significantly improve the detection performance of other faults and statistics. By analyzing the detection results of SKICA method, it is obvious that in the detection process of all five kinds of slow faults, the fault detection capability of SKICA is better than that of ICA and KICA. The results of continuous stirred reactor simulation system show that, compared with the basic linear process, the slow fault detection has a good monitoring performance, it can detect the small deviation in the process sensitively and give alarm information to the slow fault in time, to improve the fault detection rate.

An Unsupervised Deep-Transfer-Learning-Based Motor Imagery EEG Classification Scheme for Brain-Computer Interface.

Brain–computer interface (BCI) research has attracted worldwide attention and has been rapidly developed. As one well-known non-invasive BCI technique, electroencephalography (EEG) records the brain’s electrical signals from the scalp surface area. However, due to the non-stationary nature of the EEG signal, the distribution of the data collected at different times or from different subjects may be different. These problems affect the performance of the BCI system and limit the scope of its practical application. In this study, an unsupervised deep-transfer-learning-based method was proposed to deal with the current limitations of BCI systems by applying the idea of transfer learning to the classification of motor imagery EEG signals. The Euclidean space data alignment (EA) approach was adopted to align the covariance matrix of source and target domain EEG data in Euclidean space. Then, the common spatial pattern (CSP) was used to extract features from the aligned data matrix, and the deep convolutional neural network (CNN) was applied for EEG classification. The effectiveness of the proposed method has been verified through the experiment results based on public EEG datasets by comparing with the other four methods.

Inverse Covariance Matrix Estimation for Low-Complexity Closed-Loop DPD Systems: Methods and Performance

In this article, we study closed-loop digital predistortion (DPD) systems and associated learning algorithms. Specifically, we propose various low-complexity approaches to estimate and manipulate the inverse of the input data covariance matrix (CM) and combine them with the so-called self-orthogonalized (SO) learning rule. The inherent simplicity of the SO algorithm, combined with the proposed solutions, allows for remarkably reduced complexity in the DPD system while maintaining similar linearization performance compared to other state-of-the-art methods. This is demonstrated with thorough over-the-air (OTA) mmW measurement results at 28 GHz, incorporating a state-of-the-art 64-element active antenna array, and very wide channel bandwidths up to 800 MHz. In addition, complexity analyses are carried out, which together with the measured linearization performance demonstrates favorable performance–complexity tradeoffs in linearizing mmW active array transmitters through the proposed solutions. The techniques can find application in systems where the power amplifier (PA) nonlinearities are time-varying and thus frequent or even constant updating of the DPD is required, good examples being mmW adaptive antenna arrays as well as terminal transmitters in 5G and beyond networks.