Cleaning large-dimensional covariance matrices for correlated samples

关于控制相关正态分布均值对双侧备择进行多重检验中的错误发现率

On Controlling the False Discovery Rate in Multiple Testing of the Means of Correlated Normals Against Two-Sided Alternatives

本文针对相关正态均值双侧多重检验中 BH 方法能否无条件控制 FDR 的开放猜想，在协方差矩阵已知或为已知矩阵未知标量倍数的两种情形下，证明了基于加权 z 或 t 统计量的 BH 型逐步上升方法能够控制 FDR。

在高维因子模型中通过自助法样本协方差矩阵检验公共因子数目

Testing the number of common factors by bootstrapped sample covariance matrix in high-dimensional factor models

本文研究了高维因子模型中自助法对样本协方差矩阵特征值分布的影响，利用尖刺与非尖刺特征值的不同渐近行为，提出了检验公共因子数目的创新方法，并证明了其在弱因子和截面相关误差下的可靠性。

Hypothesis Testing for the Covariance Matrix in High-Dimensional Transposable Data with Kronecker Product Dependence Structure

The matrix-variate normal distribution is a popular model for high-dimensional transposable data because it decomposes the dependence structure of the random matrix into the Kronecker product of two covariance matrices: one for each of the row and column variables. We develop tests for assessing the form of the row (column) covariance matrix in high-dimensional settings while treating the column (row) dependence structure as a nuisance. Our tests are robust to normality departures provided that the Kronecker product dependence structure holds. In simulations, we observe that the proposed tests…

A Supremum-Norm Based Test for the Equality of Several Covariance Functions

In this paper, we propose a new test for the equality of several covariance functions for functional data. Its test statistic is taken as the supremum value of the sum of the squared differences between the estimated individual covariance functions and the pooled sample covariance function, hoping to obtain a more powerful test than some existing tests for the same testing problem. The asymptotic random expression of this test statistic under the null hypothesis is obtained. To approximate the null distribution of the proposed test statistic, we describe a parametric bootstrap method and a non…

Determining the Dimension and Structure of the Subspace Correlated Across Multiple Data Sets

Detecting the components common or correlated across multiple data sets is challenging due to a large number of possible correlation structures among the components. Even more challenging is to determine the precise structure of these correlations. Traditional work has focused on determining only the model order, i.e., the dimension of the correlated subspace, a number that depends on how the model-order problem is defined. Moreover, identifying the model order is often not enough to understand the relationship among the components in different data sets. We aim at solving the complete modelse…

某些相关复样本协方差矩阵的特征值分布

Eigenvalue distributions for some correlated complex sample covariance matrices

矩阵乘积$Z^\dagger Z$的最小和最大特征值的分布表示为$m \times m$行列式，其中$Z$是沿行和向下列都具有相关性的$n \times m$复高斯矩阵。在沿行的相关性的情况下，这些表达式的计算效率比那些涉及分区和舒尔多项式最近报道的相同的分布。

相关Wishart矩阵的谱矩

Spectral Moments of Correlated Wishart Matrices

我们提出了一种分析方法来确定相关Wishart随机矩阵的协方差矩阵的谱特性。该方法在大矩阵的极限下，给出了协方差矩阵的谱矩与特征值密度及其估计量之间的精确解析关系。其结果可用于在实践中提取信息的真实相关性从给定的实验实现的随机矩阵。

Decompositions of log-correlated fields with applications

In this article we establish novel decompositions of Gaussian fields taking values in suitable spaces of generalized functions, and then use these decompositions to prove results about Gaussian multiplicative chaos. We prove two decomposition theorems. The first one is a global one and says that if the difference between the covariance kernels of two Gaussian fields, taking values in some Sobolev space, has suitable Sobolev regularity, then these fields differ by a Hölder continuous Gaussian process. Our second decomposition theorem is more specialized and is in the setting of Gaussian fields …

Matrix Moments in a Real, Doubly Correlated Algebraic Generalization of the Wishart Model

The Wishart model of random covariance or correlation matrices continues to find ever more applications as the wealth of data on complex systems of all types grows. The heavy tails often encountered prompt generalizations of the Wishart model, involving algebraic distributions instead of a Gaussian. The mathematical properties pose new challenges, particularly for the doubly correlated versions. Here we investigate such a doubly correlated algebraic model for real covariance or correlation matrices. We focus on the matrix moments and explicitly calculate the first and the second one, the compu…

幂律尾数据经验协方差矩阵的谱性质

Spectral properties of empirical covariance matrices for data with power-law tails

提出了一种计算相关数据经验协方差矩阵谱密度的解析方法。在这种方法中，数据被表示为矩形随机矩阵，其列对应于系统的采样状态。该方法适用于一类随机矩阵的径向措施，包括那些沉重的（幂律）尾部的概率分布。作为一个例子，我们将其应用到多元学生分布。

广义Whittle-Mat $\acute{\text{E}}$ rn随机场作为关联涨落模型

Generalized Whittle-Mat$\acute{\text{E}}$rn random field as a model of correlated fluctuations

本文考虑了协方差函数为Whittle-Mat $\acute{\text{e}}$ rn族的高斯随机场的推广.这样一个随机场可以作为具有两个分数阶的分数阶随机微分方程的解而得到。研究了广义Whittle-Mat $\acute{\text{e}}$ rn族中协方差函数的渐近性质，并利用这些协方差函数导出了随机场的样本路径性质. Whittle-Mat $\acute{\text{e}}$ rn场已被广泛用于模拟地质统计数据，如海束数据、风速、场温度和土壤数据。在这篇文章中，我们表明，广义Whittle-Mat $\acute{\text{e}}$ rn场提供了一个更灵活的模型风速数据。

相关元随机矩阵的高斯涨落

Gaussian fluctuations for random matrices with correlated entries

对于随机矩阵合奏与非高斯矩阵元素，可能会表现出一些相关性，它示出，在矩阵中的多项式的中心轨迹收敛分布的高斯过程，其协方差矩阵是对角的切比雪夫多项式的基础上。证明是组合的，并适应维格纳的论点，显示收敛的状态密度的循环法。

凸性破缺与热力学极限

Convex Replica Simmetry Breaking From Positivity and Thermodynamic Limit

考虑一个相关的高斯随机能量模型，通过连续地将一个粒子（自旋）添加到系统中并施加相关协方差矩阵的正性来构建。我们表明，最近孤立的条件，确保热力学极限的存在的有效性，迫使协方差矩阵表现出的Parisi复本对称性破缺计划的凸性条件的矩阵元素。