Local Covariance Matrices Research Articles

Robust second-order stationary spatial blind source separation using generalized sign matrices

Consider a spatial blind source separation model in which the observed multivariate spatial data are assumed to be a linear mixture of latent stationary spatially uncorrelated random fields. The objective is to recover an unknown mixing procedure as well as the latent random fields. Recently, spatial blind source separation methods that are based on the simultaneous diagonalization of two or more scatter matrices were proposed. In cases involving uncontaminated data, such methods can solve the blind source separation problem, however, in the presence of outlying observations, these methods perform poorly. We propose a robust blind source separation method that employs robust global and local covariance matrices based on generalized spatial signs in simultaneous diagonalization. Simulation studies are employed to illustrate the robustness and efficiency of the proposed methods in various scenarios.

Inference on the maximal rank of time-varying covariance matrices using high-frequency data

We study the rank of the instantaneous or spot covariance matrix ΣX(t) of a multidimensional process X(t). Given high-frequency observations X(i/n), i=0,…,n, we test the null hypothesis rank(ΣX(t))≤r for all t against local alternatives where the average (r+1)st eigenvalue is larger than some signal detection rate vn. A major problem is that the inherent averaging in local covariance statistics produces a bias that distorts the rank statistics. We show that the bias depends on the regularity and spectral gap of ΣX(t). We establish explicit matrix perturbation and concentration results that provide nonasymptotic uniform critical values and optimal signal detection rates vn. This leads to a rank estimation method via sequential testing. For a class of stochastic volatility models, we determine data-driven critical values via normed p-variations of estimated local covariance matrices. The methods are illustrated by simulations and an application to high-frequency data of U.S. government bonds.

Distributed State and Input Estimation Over Sensor Networks Under Deception Attacks

This paper is concerned with the joint state and input estimation problem for linear discrete time-varying systems over peer-to-peer sensor networks, in which deception attacks are taken into account. By resorting to singular value decomposition, a local estimator structure is proposed to jointly estimate the system states and unknown inputs. Then, a distributed state estimator is constructed by fusing local estimates and covariance matrices. In addition, a sufficient condition is provided to ensure the uniformly bounded estimation error (in mean square) in each node. Finally, a numerical example is provided to show the effectiveness of the proposed estimation algorithm.

Portfolio allocation under asymmetric dependence in asset returns using local Gaussian correlations

It is well known that there are asymmetric dependence structures between financial returns. This paper describes a portfolio selection method rooted in the classical mean–variance framework that incorporates such asymmetric dependency structures using a nonparametric measure of local dependence, the local Gaussian correlation (LGC). It is shown that the portfolio optimization process for financial returns with asymmetric dependence structures is straightforward using local covariance matrices. The new method is shown to outperform the equally weighted (“1/N”) portfolio and the classical Markowitz portfolio when applied to historical data on six assets.

A stability with optimality analysis of consensus-based distributed filters for discrete-time linear systems

In this paper we investigate how stability and optimality of consensus-based distributed filters depend on the number of consensus steps in a discrete-time setting for both directed and undirected graphs. By introducing two new algorithms, a simpler one based on dynamic averaging of the estimates and a more complex version where local error covariance matrices are exchanged as well, we are able to derive a complete theoretical analysis. In particular we show that dynamic averaging alone suffices to approximate the optimal centralized estimate if the number of consensus steps is large enough and that the number of consensus steps needed for stability can be computed in a distributed way. These results shed light on the advantages as well as the fundamental limitations shared by all the existing proposals for this class of algorithms in the basic case of linear time-invariant systems, that are relevant for the analysis of more complex situations.

The Optimal Distributed Kalman Filtering Fusion With Linear Equality Constraint

In this paper, the optimal distributed Kalman filtering fusion with linear equality constraint (LEC) is proposed. When the Kalman filter subject to LEC is applied in state estimation of distributed linear dynamic system, all local error covariance matrices are singular, which is a challenge for most of the existing distributed fusion algorithms. However, the new fusion algorithm is rigorously derived with matrix analysis technique on the Moore-Penrose inverse. Updating state estimate and error covariance matrix, the new fusion algorithm only uses local estimates and their corresponding error covariances. We have proved theoretically that the distributed Kalman filtering with LEC (DKF-LEC) just using all local measurements is equivalent to the centralized Kalman filtering with LEC (CKF-LEC), which means it is optimal. Furthermore, we also prove the optimality of the proposed fusion formula for Kalman filtering with LEC and feedback. In addition, the simulation results show that the proposed fusion formula for Kalman filtering with LEC have the same performance of the centralized Kalman filtering with LEC whether considering feedback or not. It is consistent with the theoretical result that the DKF-LEC is equivalent to CKF-LEC as the estimate of CKF-LEC is disassembled into local estimates in distributed system. And the performance of the new fusion formula for Kalman filtering with LEC and feedback is obviously better than those without feedback.

Unbiased least-squares modification of Stokes\u2019 formula

As the KTH method for geoid determination by combining Stokes integration of gravity data in a spherical cap around the computation point and a series of spherical harmonics suffers from a bias due to truncation of the data sets, this method is based on minimizing the global mean square error (MSE) of the estimator. However, if the harmonic series is increased to a sufficiently high degree, the truncation error can be considered as negligible, and the optimization based on the local variance of the geoid estimator makes fair sense. Such unbiased types of estimators, derived in this article, have the advantage to the MSE solutions not to rely on the imperfectly known gravity signal degree variances, but only the local error covariance matrices of the observables come to play. Obviously, the geoid solution defined by the local least variance is generally superior to the solution based on the global MSE. It is also shown, at least theoretically, that the unbiased geoid solutions based on the KTH method and remove–compute–restore technique with modification of Stokes formula are the same.

GarNet++: Improving Fast and Accurate Static 3D Cloth Draping by Curvature Loss.

In this paper, we tackle the problem of static 3D cloth draping on virtual human bodies. We introduce a two-stream deep network model that produces a visually plausible draping of a template cloth on virtual 3D bodies by extracting features from both the body and garment shapes. Our network learns to mimic a physics-based simulation (PBS) method while requiring two orders of magnitude less computation time. To train the network, we introduce loss terms inspired by PBS to produce plausible results and make the model collision-aware. To increase the details of the draped garment, we introduce two loss functions that penalize the difference between the curvature of the predicted cloth and PBS. Particularly, we study the impact of mean curvature normal and a novel detail-preserving loss both qualitatively and quantitatively. Our new curvature loss computes the local covariance matrices of the 3D points, and compares the Rayleigh quotients of the prediction and PBS. This leads to more details while performing favorably or comparably against the loss that considers mean curvature normal vectors in the 3D triangulated meshes. We validate our framework on four garment types for various body shapes and poses. Finally, we achieve superior performance against a recently proposed data-driven method.

Local temporal common spatial patterns modulated with phase locking value

In the field of electroencephalogram (EEG)-based brain–computer interfaces (BCI), the method of common spatial patterns (CSP) is formulated as a problem of eigen-decomposition of covariance matrices. By using temporally local samples to construct the covariance matrices, the approach of local temporal common spatial patterns (LTCSP) are developed, which performs manifold modeling. It is useful to consider the intrinsic structure of samples in defining the temporally local covariance matrices. From the perspective of neurophysiological knowledge, phase synchronization indicates information communication. In this paper, we apply phase locking value (PLV) to quantify the phase relationship between samples, which are then adopted as weights to define the temporally local covariance matrices. As a result, we obtain the PLV-modulated LTCSP. More discriminative features are discovered with the approach proposed. Experiments of EEG classification on three EEG data sets (i.e., the data sets IIIa and IVa of BCI competition III and the data set IIa of BCI competition IV) demonstrate the effectiveness of the proposed technique. The average classification accuracies of the proposed method on the three data sets are 90.56%, 83.25%, and 83.26%. In the case of noise introduced, the average classification accuracy of the proposed method exceeds the conventional CSP and LTCSP by nearly 10% and 6%, respectively. The hypothesis test indicates the superiority of the proposed method in terms of statistical significance.

Two-sample statistics based on anisotropic kernels.

The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between [Formula: see text] data points and a set of [Formula: see text] reference points, where [Formula: see text] can be drastically smaller than [Formula: see text]. While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as [Formula: see text], and a finite-sample lower bound of the testing power is obtained. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.

A novel anomaly detection method based on adaptive Mahalanobis-squared distance and one-class kNN rule for structural health monitoring under environmental effects

Anomaly detection by Mahalanobis-squared distance (MSD) is a popular unsupervised learning approach to structural health monitoring (SHM). Despite the popularity and high applicability of the MSD-based anomaly detection method, some major challenging issues and limitations such as environmental variability, determination of an inappropriate threshold limit, estimation of an inaccurate covariance matrix, and non-Gaussianity of training data may lead to false alarms and erroneous results of damage detection. The main objective of this article is to propose a novel anomaly detection method based on adaptive Mahalanobis-squared distance and one-class kNN rule called AMSD-kNN for SHM under varying environmental conditions. The central idea behind the proposed method is to find sufficient nearest neighbors of training and testing datasets in a two-stage procedure for removing the environmental variability conditions and estimate local covariance matrices. An effective approach based on a multivariate normality hypothesis test is proposed to find sufficient nearest neighbors that guarantee the estimate of well-conditioned local covariance matrices. The great novelty of the proposed AMSD-kNN method is to create a novel unsupervised learning strategy for SHM by a new multivariate distance measure and one-class kNN rule. Generalized extreme value distribution modeling by the block maxima (BM) method is presented to determine an accurate threshold limit. Due to the importance of choosing adequate blocks in the BM method, a goodness-of-fit measure via the Kolmogorov-Smirnov hypothesis test is applied to select an optimal block number. The performance and effectiveness of the proposed methods are verified by two well-known benchmark structures. Several comparative studies are also conducted to demonstrate the superiority of the proposed methods over some state-of-the-art techniques. Results show that the proposed AMSD-kNN and BM methods highly succeed in detecting damage under environmental variability conditions.

Under-Determined Convolutive Blind Source Separation Combining Density-Based Clustering and Sparse Reconstruction in Time-Frequency Domain

Blind source separation (BSS) in time-frequency (TF) domain is a versatile framework to recover sources from the recorded mixture signals in a reverberant environment. In general, a two-stage strategy is one of the popular BSS frameworks for the underdetermined BSS case (the number of mixtures is less than the number of sources), which is a tough problem due to the mixing matrix is not invertible. In this paper, we propose a new two-stage scheme combining density-based clustering and sparse reconstruction to estimate mixing matrix and sources, respectively. At the first stage, we transform the mixing matrix estimation as an eigenvector clustering problem based on a particular local dominant assumption. The eigenvectors are first exploited from the rank-one structure of local covariance matrices of mixture TF vectors. These eigenvectors are then clustered and adjusted to give estimated mixing matrix by cooperating density-based clustering and weight clustering. At the second stage, we transform the source reconstruction as a $\ell _{p}$ norm $(0 minimization by an iterative Lagrange multiplier method. With a proper initialization, the obtained solution is a global minimum for any $p$ in (0, 1] with convergence guarantee. The proposed approach is demonstrated to be superior to the state-of-the-art baseline methods in various underdetermined experiments.

Estimating Number of Speakers via Density-Based Clustering and Classification Decision

It is crucial to robustly estimate the number of speakers (NoS) from the recorded audio mixtures in a reverberant environment. Some popular time-frequency (TF) methods approach this NoS estimation problem by assuming that only one of the speech components is active at each TF slot. However, this condition is violated in many scenarios where the speeches are convolved with long length of room impulse response coefficients, which causes degenerated performance of NoS estimation. To tackle this problem, a density-based clustering strategy is proposed to estimate NoS based on a local dominance assumption of speeches. Our method consists of several steps from clustering to classification of speakers with the consideration of robustness. First, the leading eigenvectors are extracted from the local covariance matrices of mixture TF components and ranked by the combination of local density and minimum distance to other leading eigenvectors with higher density. Second, a gap-based method is employed to determine the cluster centers from the ranked leading eigenvectors at each frequency bin. Third, a criterion based on averaged volume of cluster centers is proposed to select reliable clustering results at some frequency bins for the classification decision of NoS. The experiment results demonstrate that the proposed algorithm is superior to the existing methods in various reverberation cases with noise-free condition or noise condition.

Fusion estimation in clustering sensor networks under stochastic deception attacks

ABSTRACTIn this paper, a fusion estimation problem in clustering sensor networks under stochastic deception attacks is investigated. The sensors are divided into several clusters, and a local estimator is embedded in each cluster. During the network transmissions, deception attacks are considered to randomly occur and the statuses of the attacks are described by a set of Bernoulli variables. By using the clustering sampled information, a local minimum variance filter is derived and the filter gains are recursively determined. By using the local estimates and the local estimation error covariance matrices, a covariance intersection fusion algorithm is presented for the fusion centre to generate the final estimate. Moreover, the relationship between the trace of the local estimation error covariance matrix and the uniform successful attack rate is analysed. An illustrative example is given to shown the effectiveness of the proposed theoretical results.

Integration of spectral and spatial information via local covariance matrices for segmentation and classification of hyperspectral images

In this work, a novel approach is presented for the feature extraction step in hyperspectral image processing to form more discriminative features between different pixel regions. The proposed method combines both spatial and spectral information, which is very important for segmentation and classification of hyperspectral images. For comparison, five different feature sets are formed using eigen decomposition of local covariance matrices of subcubes located around a pixel of interest in the scene. Subcubes of neighbor pixels are obtained by a windowed structure to expose pattern similarities. As a novel approach, local covariance matrices are computed in eigenspace and proposed feature sets are created after this stage. Before the formation of feature sets in eigenspace, the original input space is transferred to eigenspace by linear and nonlinear manner by principal component analysis (PCA) and its kernelized version (KPCA) and they are used in the experiments comparatively. In the simulations, one hyperspectral scene with ground-truth and one without ground-truth are used for the segmentation and classification tasks. Results of experiments are evaluated with four different unsupervised learning algorithms for data without ground-truth and three different supervised learning algorithms for data with ground-truth comparatively.

Subpixel hyperspectral target detection using local spectral and spatial information

We present two methods to improve three hyperspectral stochastic algorithms for target detection; the algorithms are the constrained energy minimization, the generalized likelihood ratio test, and the adaptive coherence estimator. The original algorithms rely solely on spectral information and do not use spatial information; this usage is normally justified in subpixel target detection, since the target size is smaller than the size of a pixel. However, we found that since the background (and the false alarms) may be spatially correlated and the point spread function can distribute the energy of a point target between several neighboring pixels, the implementation of spatial filtering algorithms considerably improved target detection. Our first improvement used the local spatial mean and covariance matrices, which take into account the spatial local mean instead of the global mean. While this concept has been found in the literature, the effect of its implementation in both the estimated mean and the covariance matrix is examined quantitatively here. The second was based on the fact that the effect of a target of physical subpixel size will extend to a cluster of pixels. We tested our algorithms by using the data set and scoring methodology of the Rochester Institute of Technology Target Detection Blind Test project. The results showed that both spatial methods independently improved the basic spectral algorithms mentioned above, and when the two methods were used together, the results were even better.

Physical properties of the Schur complement of local covariance matrices

General properties of global covariance matrices representing bipartite Gaussian states can be decomposed into properties of local covariance matrices and their Schur complements. We demonstrate that given a bipartite Gaussian state ρ12 described by a 4 × 4 covariance matrix V, the Schur complement of a local covariance submatrix V1 of it can be interpreted as a new covariance matrix representing a Gaussian operator of party 1 conditioned to local parity measurements on party 2. The connection with a partial parity measurement over a bipartite quantum state and the determination of the reduced Wigner function is given and an operational process of parity measurement is developed. Generalization of this procedure to an n-partite Gaussian state is given, and it is demonstrated that the n − 1 system state conditioned to a partial parity projection is given by a covariance matrix such that its 2 × 2 block elements are Schur complements of special local matrices.

Erratum: Minimal Set of Local Measurements and Classical Communication for Two-Mode Gaussian State Entanglement Quantification [Phys. Rev. Lett.98, 150501 (2007)

We develop the minimal requirements for the complete entanglement quantification of an arbitrary two-mode bipartite Gaussian state via local measurements and a classical communication channel. The minimal set of measurements is presented as a reconstruction protocol of local covariance matrices and no previous knowledge of the state is required but its Gaussian character. The protocol becomes very simple mostly when dealing with Gaussian states transformed to its standard form, since photocounting/intensity measurements define the whole set of entangled states. In addition, conditioned on some prior information, the protocol is also useful for a complete global state reconstruction.

Minimal Set of Local Measurements and Classical Communication for Two-Mode Gaussian State Entanglement Quantification

We develop the minimal requirements for the complete entanglement quantification of an arbitrary two-mode bipartite Gaussian state via local measurements and a classical communication channel. The minimal set of measurements is presented as a reconstruction protocol of local covariance matrices and no previous knowledge of the state is required but its Gaussian character. The protocol becomes very simple mostly when dealing with Gaussian states transformed to its standard form, since photocounting or intensity measurements define the whole set of entangled states. In addition, conditional on some prior information, the protocol is also useful for a complete global state reconstruction.

KWMeshVisu: A Mesh Visualization Tool for Shape Analysis

Many statistical shape analysis methods produce various types of data about the analyzed surfaces, such as p-value maps, distance maps, 3D difference vectors and local covariance matrices. This data is often too large and thus difficult to be properly evaluated on a qualitative basis. A visual representation of this data strongly simplifies qualitative evaluation by humans and thus greatly enhances the value of the statistical results. In this paper we present a new tool for visualizing various datasets on surfaces represented as triangle meshes. Our tool, KWMeshVisu, is implemented using the Insight Toolkit ITK, www.itk.org, the Visualization Toolkit VTK, www.vtk.org, and the KWWidgets user interface toolkit, www.kwwidgets.org. The source code for KWMeshVisu, as well as input data used to generate the images in this paper, is provided with this document.