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In many manufacturing or service systems, the product quality and/or process outcome is described in terms of a large number of quality characteristics. Since the use of large sample sizes in such systems imposes significant costs on the manufacturer, the sample size is usually considered smaller than the process dimension. In this condition, which is referred to high-dimensionality, the sample covariance matrix is not a positive semi-definite matrix. Moreover, another issue to establish variability control charts under high-dimensional setting is their speed to detect the process disturbances under both diagonal and off-diagonal shift patterns. To solve these drawbacks, in this study, a double sampling-based adaptive thresholding LASSO control chart called DSATL is developed to monitor the variability matrix of high-dimensional processes under sparsity assumption. The run length distribution of the DSATL chart under seven out-of-control patterns is evaluated taking a different number of quality characteristics into account. The simulation results show that the detectability of the DSATL chart outperforms the classical adaptive thresholding LASSO one under various changes in the process covariance matrix. The results also confirm that the sensitivity of the DSATL chart for real-time detection of covariance matrix anomalies improves as the process dimension increases.

The performance of a brain–computer interface (BCI) will generally improve by increasing the volume of training data on which it is trained. However, a classifier’s generalization ability is often negatively affected when highly non-stationary data are collected across both sessions and subjects. The aim of this work is to reduce the long calibration time in BCI systems by proposing a transfer learning model which can be used for evaluating unseen single trials for a subject without the need for training session data. A method is proposed which combines a generalization of the previously proposed subject-specific “multivariate empirical-mode decomposition” preprocessing technique by taking a fixed band of 8–30Hz for all four motor imagery tasks and a novel classification model which exploits the structure of tangent space features drawn from the Riemannian geometry framework, that is shared among the training data of multiple sessions and subjects. Results demonstrate comparable performance improvement across multiple subjects without subject-specific calibration, when compared with other state-of-the-art techniques.

The paper uses empirical data, including 42 globally main stock indices in the period 1996–2014, to systematically study the evolution of fluctuation modes and inner structures of global stock markets. The data are large in scale considering both time and space. A covariance matrix-based principle fluctuation mode analysis (PFMA) is used to explore the properties of the global stock markets. It has been ignored by previous studies that covariance matrix is more suitable than the correlation matrix to be the basis of PFMA. It is found that the principle fluctuation modes of global stock markets are in the same directions, and global stock markets are divided into three clusters, which are found to be closely related to the countries’ locations with exceptions of China, Russia and Czech Republic. A time-stable correlation network constructing method is proposed to solve the problem of high-level statistical uncertainty when the estimated periods are very short, and the complex dynamic network (CDN) is constructed to investigate the evolution of inner structures. The results show when the clusters emerge and how long the clusters exist. When the 2008 financial crisis broke out, the indices form one cluster. After these crises, only the European cluster still exists. These findings complement the previous studies, and can help investors and regulators to understand the global stock markets.

Coherence arises from the superposition principle, where it plays a central role in quantum mechanics. In Phys. Rev. Lett.114, 210401 (2015), it has been shown that the freezing phenomenon of quantum correlations beyond entanglement is intimately related to the freezing of quantum coherence (QC). In this paper, we compare the behavior of entanglement and quantum discord with quantum coherence in two different subsystems (optical and mechanical). We use respectively the entanglement of formation (EoF) and the Gaussian quantum discord (GQD) to quantify entanglement and quantum discord. Under thermal noise and optomechanical coupling effects, we show that EoF, GQD and QC behave in the same way. Remarkably, when entanglement vanishes, GQD and QC remain almost unaffected by thermal noise, keeping nonzero values even for high-temperature, which is in concordance with Phys. Rev. Lett.114, 210401 (2015). Also, we find that the coherence associated with the optical subsystem is more robust — against thermal noise — than those of the mechanical subsystem. Our results confirm that optomechanical cavities constitute a powerful resource of QC.

Intrusion detection is an important part of assuring the reliability of computer systems. Different intrusion detection approaches vary with different patterns used and different intrusions addressed. However, what patterns are effective in constructing a detection system is still a challenge. This paper attempts to apply the traditional covariance matrix concept to the detection of multiple known and unknown network anomalies. With respect to the initiation of typical flood-based network intrusions, the proposed approach takes the measure of covariance matrix to reflect the changes of sequential correlativity of the network traffic when flood-based attacks happen. The differences among covariance matrices of network samples collected in temporal sequences of fixed and equal length are directly evaluated to detect multiple network anomalies. Extensive experiments on the subset of KDDCUP 1999 dataset show that the covariance matrix, as a new pattern, can be directly utilized to construct an effective detection system for flood-based attacks. It also points out that utilizing the covariance matrix in the detection of flood-based attacks can achieve higher performance over traditional approaches.

Intracellular metabolic pathways are highly interlinked networks. Such a network inevitably involves stochasticity due to the finite number of molecules of individual chemical species in biochemical reactions, which in turn would affect the development and function of living cells. Here, we investigate noise in the metabolic pools using linear noise approximation. By analyzing several typical motifs, we show non-propagation of noise in a clearer way in contrast to previous studies. Numerical simulation based on molecule-level Monte Carlo simulations further verifies the theoretic prediction. Our results would be helpful for understanding intracellular processes.

It is well-known that the crucial ingredient for a vector Gaussian random function is its covariance matrix, where a diagonal entry termed a direct covariance is simply the covariance function of a component but it seems no simple interpretation for an off-diagonal entry termed a cross covariance, which often make it hard to specify. In this paper we employ three approaches to derive vector random functions in space and/or time, which are not homogeneous (stationary) in general but contain the stationary case as a special case, and have long-range or short-range dependence.

We show that, in spite of rather common opinion, correlations of observables on subsystems of a composite quantum system can be represented as correlations of classical Gaussian variables. We restrict our model to the finite dimensional case (which is important, e.g., in quantum information theory). Here quantum correlations are represented by integrals with Gaussian densities which can be directly calculated. In particular, the EPR–Bell correlations for polarizations of entangled photons can be represented as correlations of Gaussian random variables. The tricky point of our construction is the necessity to introduce the Gaussian noise ("vacuum fluctuations") to obtain positively defined covariance matrices for "prequantum random variables." Quantum mechanics can be considered as a method to proceed by subtracting the influence of this Gaussian background noise.

Estimations and optimal experimental designs for two-dimensional Haar-wavelet regression models are discussed. It is shown that the eigenvalues of the covariance matrix of the best linear unbiased estimator of the unknown parameters in a two-dimensional linear Haar-wavelet model can be represented in closed form. Some common discrete optimal designs for the model are constructed analytically from the eigenvalues. Some equivalences among these optimal designs are also given, and an example is demonstrated.

This paper considers the minimax estimation of the high-dimensional sparse covariance matrices in the presence of missing observations. Based on random missing data, the upper bounds of convergence rate about the data-driven thresholding estimator are constructed over a large class of sparse covariance matrices under the $l_1$ norm and the Frobenius norm. In addition, we use Le Cam’s lemma and the relation between the total variation affinity and the Kullback–Leibler divergence to establish the lower bounds which illustrate the desired upper bounds cannot be improved. It is worth mentioning that the approach we adopt to get the lower bounds is simpler than the existing ones.

Quantum steering refers to the apparent possibility of exploiting nonseparable quantum correlations to remotely influence the quantum state of an observer via local measurements. Different from entanglement and Bell nonlocality, quantum steering exhibits an inherent asymmetric property, which makes it relevant for many asymmetric quantum information processing tasks. Here, we study Gaussian quantum steering between two mechanical modes in an optomechanical ring cavity. Using experimentally feasible parameters, we show that the state of the two considered modes can exhibit two-way steering and even one-way steering. Instead of using unbalanced losses or noises, we propose a simple practical way to control the direction of one-way steering. A comparative study between the steering and entanglement of the studied modes shows that both steering and entanglement undergo a sudden death-like behavior. In particular, steering is found more fragile against thermal noise remaining constantly upper-bounded by entanglement. The proposed scheme may be meaningful for one-sided device-independent quantum key distribution, where the security of such protocol depends fundamentally on the direction of steering.

The aim of this paper is to propose numerical models and methods for solving (n×n) linear equations, AX=b where parameters in the matrix A and the vector b are random and correlated. In case of a probabilistic b and a deterministic A, the amount of second order co-variances computation is equivalent to solving the (n×n) problem sequentially n times with additional efforts of performing a Cholesky factorization plus a symmetric matrix multiplication of size n×n while the complexity for kth order correlation can be proven to be O(n^k+1). For randomness in both A and b with an assumption of multidimensional Gaussian distribution, two systems of linear equations with size n²(n+1) and n(n+1)/2 are derived and can be solved sequentially to obtain the desired statistical parameters up to the second order of X. Both approaches are coded as Matlab M-files, complied and run to test their efficiency. For higher order correlation with non-Gaussian distribution, an approximation scheme is proposed. Two applications in optimization are discussed.

We study the dynamics and redistribution of entanglement and coherence in three time-dependent coupled harmonic oscillators. We resolve the Schrödinger equation by using time-dependent Euler rotation together with a linear quench model to obtain the state of vacuum solution. Such state can be translated to the phase space picture to determine the Wigner distribution. We show that its Gaussian matrix $𝔾(t)$ can be used to directly cast the covariance matrix $\sigma (t)$. To quantify the mixedness and entanglement of the state, one uses respectively linear and von Neumann entropies for three cases: fully symmetric, bi-symmetric and fully nonsymmetric. Then we determine the coherence, tripartite entanglement and local uncertainties and derive their dynamics. We show that the dynamics of all quantum information quantities are driven by the Ermakov modes. Finally, we use an homodyne detection to redistribute both resources of entanglement and coherence.

A time domain method for identifying random dynamic loads is proposed based on spectral decomposition and regularization, which to some extent makes up for the deficiency of frequency domain methods. The random dynamic loads are descripted with their time domain mean functions and covariance matrix, which can intuitively reflect the statistical characteristics of the loads. Therein the random dynamic load identification is transformed into the load mean function identification and covariance matrix reconstruction. The forward identification models are mainly established based on Green’s kernel function method, and then spectral decomposition is conducted to transform the identification of load covariance matrix into a series of identifications of eigenvectors. To overcome the ill-posedness in the inverse process, the least-square QR iterative regularization is adopted. Two numerical examples and an application on the reconstruction of road excitations acting on a vehicle are studied to verify the effectiveness of the proposed method.

In this paper, we will propose two new estimators for sparse covariance matrix. Our starting point is to make the estimator of each element of covariance matrix more robust. More precisely, we will trim the observations for each pairwise product of components of population as a first step. Then we form the sample covariance matrices based on the trimmed data. Finally, we apply the thresholding to the derived sample covariance matrices. These two new estimators will be shown to achieve the optimal convergence rate.

Let $x_1,\dots ,x_n$ be independent and identically distributed (i.i.d.) real-valued random vectors from distribution $N_p(\mu ,\mathrm{\Sigma })$, where the sample size $n$ and the vector dimension $p$ satisfy $n-1>p\to \infty$. We are interested in the exponential convergence rate of the likelihood ratio test (LRT) statistics for testing $\mathrm{\Sigma }$ equal to a given matrix and $(\mu ,\mathrm{\Sigma })$ equal to a given pair. In traditional statistical theory, the LRT statistics have been studied under the null hypothesis and finite-dimensional conditions. In this paper, we prove the moderate deviation principle (MDP) under the high-dimensional conditions for the two LRT statistics. We show that our results hold under the null hypothesis and the alternative hypothesis as well. Some numerical simulations indicate that our conclusions have good performance.

Multi-label classification is a supervised learning problem that predicts multiple labels simultaneously. One of the key challenges in such tasks is modelling the correlations between multiple labels. In this paper we use a ‘decision tree’ algorithm, which models label dependency locally. The proposed algorithm establishes a hybrid multi-label decision tree (HMLDT) that interpolates between two baseline methods: binary relevance, which assumes all labels independent; and label powerset, which learns the joint label distribution. The key idea is a splitting criterion based on the label covariance matrix. Empirical results on 12 datasets show strong performance of the HMLDT, particularly on datasets with hundreds of labels.

The new extended mathematical model for evaluation uncertainties of the multivariable indirect measurements, which upgrades the method of GUM-Supplement 2 is presented. In this model the uncertainties and correlations of the processing function parameters are also taken into account. It can be used for multivariable measurements and also to describe the accuracy of instruments and systems that perform such measurements. An example is included on estimated uncertainties of indirectly-measured output voltage and current of a twoport circuit, by considering uncertainties and correlations of its elements.

Linear regression models are frequently used for modelling measurements with larger amount of output quantities. Very often the least squares method (LSM) is used for determination of estimation of unknown parameter values. When value estimation is done by LSM then it is possible to make use of the input quantity uncertainties. When unknown parameter estimation is obtained by the LSM according to uncertainty propagation law then the uncertainties and covariances amongst them are determined. In wide range of publications only a few cases had considered the type B evaluations of uncertainties (e.g. caused by measuring gauge error) in this matter. A question arises if there are cases where the type B uncertainties should be omitted while unknown model parameters are estimated. In this paper the method of regression model parameters estimation is presented for the case when type B evaluation of uncertainties does not affected the estimations.

In high resolution critical dimensional metrology, when modeling measurements, a library of curves is usually assembled through the simulation of a multi-dimensional parameter space. A nonlinear regression routine described in this paper is then used to identify an optimum set of parameters that yields the closest experiment-to-theory agreement and generates the model-based measurements for the desired parameters. To improve the model-based measurements, other measurement techniques can also be used to provide a priori information. In this paper, a Bayesian statistical approach is proposed to allow the combination of different measurement techniques that are based on different physical measurements. The effect of this hybrid metrology approach is shown to reduce the uncertainties of the parameter estimators, i.e., the model-based measurements.