Jose M. Peña
We extend path analysis by giving sufficient conditions for computing the partial covariance of two random variables from their covariance. This is specifically done by correcting the covariance with the product of some partial variance ratios. As a result, the partial covariance retains the covariance's salient feature of factorizing over the edges in the paths between the two variables of interest.
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Let $\boldsymbol{X}\in \mathbb{R}^p$ and $Y\in \mathbb{R}$. In this paper we propose an estimator of the conditional covariance matrix, $\mathrm{Cov}(\mathbb{E}[\boldsymbol{X}\vert Y])$, in an inverse regression setting. Based on the estimation of a quadratic functional, this methodology provides an efficient estimator from a semi parametric point of view. We consider a functional Taylor expansion of $\mathrm{Cov}(\mathbb{E}[\boldsymbol{X}\vert Y])$ under some mild conditions and the effect of using an estimate of the unknown joint distribution. The asymptotic properties of this estimator are …
Investigation of the reversibility of the directional hierarchy in the interdependency among the notions of conditional independence, conditional mean independence, and zero conditional covariance, for two random variables X and Y given a conditioning element Z which is not constrained by any topological restriction on its range, reveals that if the first moments of X, Y, and XY exist, then conditional independence implies conditional mean independence and conditional mean independence implies zero conditional covariance, but the direction of the hierarchy is not reversible in general. If the …
Gaussian dependence structure pairwise goodness-of-fit testing based on conditional covariance and the 20/60/20 rule
提出一种基于20/60/20规则和条件协方差的高斯依赖结构成对拟合优度检验框架，具有渐近枢轴分布，模拟和实证表明其优于流行基准。
Multiplier Bootstrap and Edge Phase Transitions of High-Dimensional Covariance Matrices
本文研究乘子自助法对高维样本协方差矩阵最大特征值渐近分布的影响，揭示其在不同乘子下的边缘相变，验证其用于高维谱推断的可行性与有效性。
In this short note we provide an analytical formula for the conditional covariance matrices of the elliptically distributed random vectors, when the conditioning is based on the values of any linear combination of the marginal random variables. We show that one could introduce the univariate invariant depending solely on the conditioning set, which greatly simplifies the calculations. As an application, we show that one could define uniquely defined quantile-based sets on which conditional covariance matrices must be equal to each other if only the vector is multivariate normal. The similar re…
The dynamic generalized covariance measure for conditional independence testing with nonstationary time series
本文提出首个适用于非平稳非线性过程单次实现的条件独立性检验框架，通过时变非线性回归等技术解决线性模型难以捕捉非线性动态的问题，为相关领域的随机过程关系识别提供了新工具。
We consider high-dimensional multivariate linear regression models, where the joint distribution of covariates and response variables is a multivariate normal distribution with a bandable covariance matrix. The main goal of this paper is to estimate the regression coefficient matrix, which is a function of the bandable covariance matrix. Although the tapering estimator of covariance has the minimax optimal convergence rate for the class of bandable covariances, we show that it has a sub-optimal convergence rate for the regression coefficient; that is, a minimax estimator for the class of banda…
Minimax estimation of the expected conditional covariance under bounds on the covariate density
本文针对协变量密度仅知上下界时，提出多分辨率估计器以最小最大意义估计期望条件协方差，证明最优速率优于先前猜想并解决长期猜想，实验显示其均方误差更小。
It is a common saying that testing for conditional independence, i.e., testing whether whether two random vectors $X$ and $Y$ are independent, given $Z$, is a hard statistical problem if $Z$ is a continuous random variable (or vector). In this paper, we prove that conditional independence is indeed a particularly difficult hypothesis to test for. Valid statistical tests are required to have a size that is smaller than a predefined significance level, and different tests usually have power against a different class of alternatives. We prove that a valid test for conditional independence does no…
We clarify relationships between conditional (CAR) and simultaneous (SAR) autoregressive models. We review the literature on this topic and find that it is mostly incomplete. Our main result is that a SAR model can be written as a unique CAR model, and while a CAR model can be written as a SAR model, it is not unique. In fact, we show how any multivariate Gaussian distribution on a finite set of points with a positive-definite covariance matrix can be written as either a CAR or a SAR model. We illustrate how to obtain any number of SAR covariance matrices from a single CAR covariance matrix by…
We present the path-sum formulation for exact statistical inference of marginals on Gaussian graphical models of arbitrary topology. The path-sum formulation gives the covariance between each pair of variables as a branched continued fraction of finite depth and breadth. Our method originates from the closed-form resummation of infinite families of terms of the walk-sum representation of the covariance matrix. We prove that the path-sum formulation always exists for models whose covariance matrix is positive definite: i.e.~it is valid for both walk-summable and non-walk-summable graphical mode…
We propose a new Conditional BEKK matrix-F (CBF) model for the time-varying realized covariance (RCOV) matrices. This CBF model is capable of capturing heavy-tailed RCOV, which is an important stylized fact but could not be handled adequately by the Wishart-based models. To further mimic the long memory feature of the RCOV, a special CBF model with the conditional heterogeneous autoregressive (HAR) structure is introduced. Moreover, we give a systematical study on the probabilistic properties and statistical inferences of the CBF model, including exploring its stationarity, establishing the as…
The precision matrix that encodes conditional linear dependency relations among a set of variables forms an important object of interest in multivariate analysis. Sparse estimation procedures for precision matrices such as the graphical lasso (Glasso) gained popularity as they facilitate interpretability, thereby separating pairs of variables that are conditionally dependent from those that are independent (given all other variables). Glasso lacks, however, robustness to outliers. To overcome this problem, one typically applies a robust plug-in procedure where the Glasso is computed from a rob…
We study pathwise invariances of centred random fields that can be controlled through the covariance. A result involving composition operators is obtained in second-order settings, and we show that various path properties including additivity boil down to invariances of the covariance kernel. These results are extended to a broader class of operators in the Gaussian case, via the Loève isometry. Several covariance-driven pathwise invariances are illustrated, including fields with symmetric paths, centred paths, harmonic paths, or sparse paths. The proposed approach delivers a number of promisi…
Phase transition for conditional covariance matrices estimated by importance sampling, and implications for cross-entropy schemes in high dimension
针对交叉熵算法中重要性采样估计协方差矩阵的问题，本文研究带依赖重尾项的随机矩阵模型的高维相变规律，揭示最小特征值与采样效果的关联，为高维稀有事件CE方案设计提供新视角。
Graphical model selection is a seemingly impossible task when many pairs of variables are never jointly observed; this requires inference of conditional dependencies with no observations of corresponding marginal dependencies. This under-explored statistical problem arises in neuroimaging, for example, when different partially overlapping subsets of neurons are recorded in non-simultaneous sessions. We call this statistical challenge the "Graph Quilting" problem. We study this problem in the context of sparse inverse covariance learning, and focus on Gaussian graphical models where we show tha…
Fundamental limits of distributed covariance matrix estimation via a conditional strong data processing inequality
本文提出 C-SDPI 系数刻画状态依赖信道的平均收缩，据此建立特征分割分布式协方差估计的近似紧 minimax 下界，并给出对数因子内最优及交互式通信协议。
A covariance graph is an undirected graph associated with a multivariate probability distribution of a given random vector where each vertex represents each of the different components of the random vector and where the absence of an edge between any pair of variables implies marginal independence between these two variables. Covariance graph models have recently received much attention in the literature and constitute a sub-family of graphical models. Though they are conceptually simple to understand, they are considerably more difficult to analyze. Under some suitable assumption on the proba…
The Projected Covariance Measure for assumption-lean variable significance testing
本文针对模型错误设定下变量显著性检验功效差的问题，提出一种基于投影协方差测度的无模型条件均值独立性检验框架，可结合非参数或机器学习方法，并证明样条回归版本达到极小极大最优速率。
A graphical model provides a compact and efficient representation of the association structure of a multivariate distribution by means of a graph. Relevant features of the distribution are represented by vertices, edges and other higher-order graphical structures, such as cliques or paths. Typically, paths play a central role in these models because they determine the independence relationships among variables. However, while a theory of path coefficients is available in models for directed graphs, little has been investigated about the strength of the association represented by a path in an u…
Binary Expansion Group Intersection Network
本文提出二进制展开群交网络（BEGIN），一种用于多变量二进制数据和位编码多项式变量的分布无关图表示。通过证明条件独立性与稀疏线性表示、块分解交互协方差矩阵和块对角化广义Schur补的等价性，拓展了高斯图模型的应用范围。
Sample Path Regularity of Gaussian Processes from the Covariance Kernel
本文研究了高斯过程样本路径的正则性，通过协方差核提出必要且充分条件，以Hölder正则性为核心，为机器学习中常用GPs的正则性提供精确刻画。
Conditioning of Banach Space Valued Gaussian Random Variables: An Approximation Approach Based on Martingales
本文通过鞅方法研究Banach空间值联合高斯随机变量的条件分布，证明其仍为高斯且可由有限维逼近确定，并应用于连续高斯过程的条件化，实现一致逼近，深化了对机器学习中高斯过程的理解。
Estimating sample paths of Gauss-Markov processes from noisy data
本文针对含噪观测下的高斯-马尔可夫过程，推导了任意样本路径点的条件均值与方差公式，并以布朗运动及布朗桥为例进行验证。
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