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We study the asymptotic distributions of the spiked eigenvalues and the largest nonspiked eigenvalue of the sample covariance matrix under a general covariance model with divergent spiked eigenvalues, while the other eigenvalues are bounded but otherwise arbitrary. The limiting normal distribution for the spiked sample eigenvalues is established. It has distinct features that the asymptotic mean relies on not only the population spikes but also the nonspikes and that the asymptotic variance in general depends on the population eigenvectors. In addition, the limiting Tracy–Widom law for the largest nonspiked sample eigenvalue is obtained. Estimation of the number of spikes and the convergence of the leading eigenvectors are also considered. The results hold even when the number of the spikes diverges. As a key technical tool, we develop a central limit theorem for a type of random quadratic forms where the random vectors and random matrices involved are dependent. This result can be of independent interest.

A common effect size metric used to quantify the outcome of experiments for ecological meta-analysis is the response ratio (RR): the log proportional change in the means of a treatment and control group. Estimates of the variance of RR are also important for meta-analysis because they serve as weights when effect sizes are averaged and compared. The variance of an effect size is typically a function of sampling error; however, it can also be influenced by study design. Here, I derive new variances and covariances for RR for several often-encountered experimental designs: when the treatment and control means are correlated; when multiple treatments have a common control; when means are based on repeated measures; and when the study has a correlated factorial design, or is multivariate. These developments are useful for improving the quality of data extracted from studies for meta-analysis and help address some of the common challenges meta-analysts face when quantifying a diversity of experimental designs with the response ratio.

Given a probability distribution in ℝn with general (nonwhite) covariance, a classical estimator of the covariance matrix is the sample covariance matrix obtained from a sample of N independent points. What is the optimal sample size N=N(n) that guarantees estimation with a fixed accuracy in the operator norm? Suppose that the distribution is supported in a centered Euclidean ball of radius \\(O(n)\\). We conjecture that the optimal sample size is N=O(n) for all distributions with finite fourth moment, and we prove this up to an iterated logarithmic factor. This problem is motivated by the optimal theorem of Rudelson (J. Funct. Anal. 164:60–72, 1999), which states that N=O(nlog n) for distributions with finite second moment, and a recent result of Adamczak et al. (J. Am. Math. Soc. 234:535–561, 2010), which guarantees that N=O(n) for subexponential distributions.

Sample covariance matrices are widely used in multivariate statistical analysis. The central limit theorems (CLTs) for linear spectral statistics of high-dimensional noncentralized sample covariance matrices have received considerable attention in random matrix theory and have been applied to many high-dimensional statistical problems. However, known population mean vectors are assumed for noncentralized sample covariance matrices, some of which even assume Gaussian-like moment conditions. In fact, there are still another two most frequently used sample covariance matrices: the ME (moment estimator, constructed by subtracting the sample mean vector from each sample vector) and the unbiased sample covariance matrix (by changing the denominator n as N = n — 1 in the ME) without depending on unknown population mean vectors. In this paper, we not only establish the new CLTs for noncentralized sample covariance matrices when the Gaussianlike moment conditions do not hold but also characterize the nonnegligible differences among the CLTs for the three classes of high-dimensional sample covariance matrices by establishing a substitution principle: by substituting the adjusted sample size N = n — 1 for the actual sample size n in the centering term of the new CLTs, we obtain the CLT of the unbiased sample covariance matrices. Moreover, it is found that the difference between the CLTs for the ME and unbiased sample covariance matrix is nonnegligible in the centering term although the only difference between two sample covariance matrices is a normalization by n and n — 1, respectively. The new results are applied to two testing problems for high-dimensional covariance matrices.

Meta‐analysis is a powerful approach to combine evidence from multiple studies to make inference about one or more parameters of interest, such as regression coefficients. The validity of the fixed effect model meta‐analysis depends on the underlying assumption that all studies in the meta‐analysis share the same effect size. In the presence of heterogeneity, the fixed effect model incorrectly ignores the between‐study variance and may yield false positive results. The random effect model takes into account both within‐study and between‐study variances. It is more conservative than the fixed effect model and should be favored in the presence of heterogeneity. In this paper, we develop a noniterative method of moments estimator for the between‐study covariance matrix in the random effect model multivariate meta‐analysis. To our knowledge, it is the first such method of moments estimator in the matrix form. We show that our estimator is a multivariate extension of DerSimonian and Laird’s univariate method of moments estimator, and it is invariant to linear transformations. In the simulation study, our method performs well when compared to existing random effect model multivariate meta‐analysis approaches. We also apply our method in the analysis of a real data example.

Fracture networks constitute essential conduits for fluid and heat transport in enhanced geothermal systems (EGSs), yet their characterization is challenging due to the inherent geological complexities. This study develops an integrated inversion framework for effective fracture network characterization. The framework consists of a novel fracture network parameterization method, the covariance matrix adaptation‐evolution strategy (CMA‐ES) for fracture parameter inversion, and the embedded discrete fracture model (EDFM) for robust forward simulation of flow and transport in fractured reservoirs. The proposed fracture network parameterization method uses a background fracture network with fixed geometries and non‐uniform fracture apertures to approximate real‐world fracture networks. CMA‐ES is employed to infer fracture parameters by matching both conservative and sorptive tracer measurements, and multiple parallel CMA‐ES runs are executed to obtain an ensemble of model realizations for uncertainty assessment. Three synthetic EGS case studies with varying complexities demonstrate the effectiveness of the inversion framework in capturing major flow and transport characteristics in fracture networks. The long‐term thermal performances of the EGS reservoirs are appropriately predicted with the inferred fracture network models. This integrated framework offers a feasible solution for fracture network characterization and thermal performance prediction in EGS and also has potential applications in unconventional gas/oil reservoir explorations. Key Points Dimension‐reduced parameterization eases complex fracture network characterization in enhanced geothermal reservoirs Covariance matrix adaptation‐evolution strategy fosters fracture parameter inversion and supports uncertainty assessment Well‐inferred fracture network models enable accurate predictions of the long‐term thermal performance of enhanced geothermal reservoirs

Global Navigation Satellite Systems (GNSS) integrated with Inertial Navigation Systems (INS) have been widely applied in many Intelligent Transport Systems. However, due to the influence of various factors, such as complex urban environments, etc., accurately describing the measurement noise statistics of GNSS receivers and inertial sensors is difficult. An inaccurate definition of the measurement noise covariance matrix will lead to the rapid divergence of the position error of the integrated navigation system. To overcome this problem, this paper proposed a Robust Adaptive Extended Kalman Filter (RAKF) method based on an improved measurement noise covariance matrix. By analyzing and considering the position accuracy factors, measurement factor, and position standard deviation in GNSS measurement results, this paper constructed the optimal measurement noise covariance matrix. Based on the Huber model, this paper constructed a two-stage robust adaptive factor expression and obtained the robust adaptive factors with and without abnormal disturbances. And robust adaptive filtering was carried out. To assess the performance of this method, the author conducted experiments on land vehicles by using a self-developed POS system (GNSS/INS combined navigation system). The classic Extended Kalman Filter algorithm (EKF), Adaptive Kalman Filter (AKF) algorithm, Robust Kalman Filter (RKF) algorithm, and the proposed method were compared through data processing. Experimental results show that compared with the classical EKF, AKF, and RKF, the positioning accuracies of the proposed method were improved by 72.43%, 2.54%, and 47.82%, respectively, in the vehicle land experiment. In order to further evaluate the performance of this method, the vehicle data were subjected to different times and degrees of disturbance experiments. Experimental results show that compared with EKF, AKF, and RKF, the heading angle accuracy had obvious advantages, and its accuracy was improved by 34.65%, 31.53%, and 18.36%, respectively. Therefore, this method can effectively monitor and isolate disturbance and improve the robustness, reliability, accuracy, and stability of GNSS/INS integrated navigation systems in complex urban environments.

By studying the family of p-dimensional scale mixtures, the paper shows for the first time a non-trivial example where the eigenvalue distribution of the corresponding sample covariance matrix does not converge to the celebrated Marčenko–Pastur law. A different and new limit is found and characterized. The reasons for failure of the Marčenko–Pastur limit in this situation are found to be a strong dependence between the p-co-ordinates of the mixture. Next, we address the problem of testing whether the mixture has a spherical covariance matrix. To analyse the traditional John's-type test we establish a novel and general central limit theorem for linear statistics of eigenvalues of the sample covariance matrix. It is shown that John's test and its recent high dimensional extensions both fail for high dimensional mixtures, precisely because of the different spectral limit above. As a remedy, a new test procedure is constructed afterwards for the sphericity hypothesis. This test is then applied to identify the covariance structure in model-based clustering. It is shown that the test has much higher power than the widely used integrated classification likelihood and Bayesian information criteria in detecting non-spherical component covariance matrices of a high dimensional mixture.

Semantic segmentation has been a fundamental task in interpreting remote sensing imagery (RSI) for various downstream applications. Due to the high intra-class variants and inter-class similarities, inflexibly transferring natural image-specific networks to RSI is inadvisable. To enhance the distinguishability of learnt representations, attention modules were developed and applied to RSI, resulting in satisfactory improvements. However, these designs capture contextual information by equally handling all the pixels regardless of whether they around edges. Therefore, blurry boundaries are generated, rising high uncertainties in classifying vast adjacent pixels. Hereby, we propose an edge distribution attention module (EDA) to highlight the edge distributions of leant feature maps in a self-attentive fashion. In this module, we first formulate and model column-wise and row-wise edge attention maps based on covariance matrix analysis. Furthermore, a hybrid attention module (HAM) that emphasizes the edge distributions and position-wise dependencies is devised combing with non-local block. Consequently, a conceptually end-to-end neural network, termed as EDENet, is proposed to integrate HAM hierarchically for the detailed strengthening of multi-level representations. EDENet implicitly learns representative and discriminative features, providing available and reasonable cues for dense prediction. The experimental results evaluated on ISPRS Vaihingen, Potsdam and DeepGlobe datasets show the efficacy and superiority to the state-of-the-art methods on overall accuracy (OA) and mean intersection over union (mIoU). In addition, the ablation study further validates the effects of EDA.

The paper deals with the estimation of a high dimensional covariance with a conditional sparsity structure and fast diverging eigenvalues. By assuming a sparse error covariance matrix in an approximate factor model, we allow for the presence of some cross-sectional correlation even after taking out common but unobservable factors. We introduce the principal orthogonal complement thresholding method 'POET' to explore such an approximate factor structure with sparsity. The POET-estimator includes the sample covariance matrix, the factor-based covariance matrix, the thresholding estimator and the adaptive thresholding estimator as specific examples. We provide mathematical insights when the factor analysis is approximately the same as the principal component analysis for high dimensional data. The rates of convergence of the sparse residual covariance matrix and the conditional sparse covariance matrix are studied under various norms. It is shown that the effect of estimating the unknown factors vanishes as the dimensionality increases. The uniform rates of convergence for the unobserved factors and their factor loadings are derived. The asymptotic results are also verified by extensive simulation studies. Finally, a real data application on portfolio allocation is presented.