We consider nonparametric estimation of a covariance function on the unit square, given a sample of discretely observed fragments of functional data. When each sample path is only observed on a subinterval of length $δ<1$, one has no statistical information on the unknown covariance outside a $δ$-band around the diagonal. The problem seems unidentifiable without parametric assumptions, but we show that nonparametric estimation is feasible under suitable smoothness and rank conditions on the unknown covariance. This remains true even when observation is discrete, and we give precise deterministic conditions on how fine the observation grid needs to be relative to the rank and fragment length for identifiability to hold true. We show that our conditions translate the estimation problem to a low-rank matrix completion problem, construct a nonparametric estimator in this vein, and study its asymptotic properties. We illustrate the numerical performance of our method on real and simulated data.
In this article, we consider the problem of recovering the underlying trajectory when the longitudinal data are sparsely and irregularly observed and noise-contaminated. Such data are popularly analyzed with functional principal component analysis via the Principal Analysis by Conditional Estimation (PACE) method. The PACE method may sometimes be numerically unstable because it involves the inverse of the covariance matrix. We propose a sparse orthonormal approximation (SOAP) method as an alternative. It estimates the optimal empirical basis functions in the best approximation framework rather…
Advanced Communications Research Institute (ACRI)（高级通信研究所（ACRI））；Sharif University of Technology（谢里夫理工大学）
Sparse Inverse Covariance Estimation (SICE) is useful in many practical data analyses. Recovering the connectivity, non-connectivity graph of covariates is classified amongst the most important data mining and learning problems. In this paper, we introduce a novel SICE approach using adaptive thresholding. Our method is based on updates in a transformed domain of the desired matrix and exponentially decaying adaptive thresholding in the main domain (Inverse Covariance matrix domain). In addition to the proposed algorithm, the convergence analysis is also provided. In the Numerical Experiments …
In this paper we consider the task of estimating the non-zero pattern of the sparse inverse covariance matrix of a zero-mean Gaussian random vector from a set of iid samples. Note that this is also equivalent to recovering the underlying graph structure of a sparse Gaussian Markov Random Field (GMRF). We present two novel greedy approaches to solving this problem. The first estimates the non-zero covariates of the overall inverse covariance matrix using a series of global forward and backward greedy steps. The second estimates the neighborhood of each node in the graph separately, again using …
Omics technologies are powerful tools for analyzing patterns in gene expression data for thousands of genes. Due to a number of systematic variations in experiments, the raw gene expression data is often obfuscated by undesirable technical noises. Various normalization techniques were designed in an attempt to remove these non-biological errors prior to any statistical analysis. One of the reasons for normalizing data is the need for recovering the covariance matrix used in gene network analysis. In this paper, we introduce a novel normalization technique, called the covariance shift (C-SHIFT)…
Evaluating the functional relationships between brain regions measured with neuroimaging provides insight into how the brain is sharing information at a macro scale. Many functional connectivity methods have been developed for dynamic imaging modalities such as functional MRI (fMRI), while less work has focused on models for static molecular imaging techniques such as FDG-PET and Tc-99m HMPAO SPECT across groups of individuals. In this work we provide a quantitative assessment of how well three functional connec- tivity models based on sparse inverse covariance estimation can accurately recove…
We consider the problem of estimating the covariance matrix of a random signal observed through unknown translations (modeled by cyclic shifts) and corrupted by noise. Solving this problem allows to discover low-rank structures masked by the existence of translations (which act as nuisance parameters), with direct application to Principal Components Analysis (PCA). We assume that the underlying signal is of length $L$ and follows a standard factor model with mean zero and $r$ normally-distributed factors. To recover the covariance matrix in this case, we propose to employ the second- and fourt…
Large-scale precision matrix estimation is of fundamental importance yet challenging in many contemporary applications for recovering Gaussian graphical models. In this paper, we suggest a new approach of innovated scalable efficient estimation (ISEE) for estimating large precision matrix. Motivated by the innovated transformation, we convert the original problem into that of large covariance matrix estimation. The suggested method combines the strengths of recent advances in high-dimensional sparse modeling and large covariance matrix estimation. Compared to existing approaches, our method is…
Recovering Latent Structures after Variational Bayesian Variable Selection: Fit Assessment and Factor-Number Selection in Partially Exploratory Factor Analysis
Faculty of Education, The University of Hong Kong（香港大学教育学院）；Department of Curriculum and Instruction, The Education University of Hong Kong（香港教育大学课程与教学系）
Physics-Informed Singular-Value Learning for Cross-Covariances Forecasting in Financial Markets
Université Paris-Saclay, CentraleSupélec, Mathématiques et Informatique pour la Complexité et les Systèmes（巴黎-萨克雷大学，CentraleSupélec，复杂性与系统数学与计算机科学）；Università degli Studi di Palermo, Dipartimento di Fisica e Chimica "Emilio Segrè"（巴勒莫大学，Emilio Segrè物理与化学系）
Many popular statistical models, such as factor and random effects models, give arise a certain type of covariance structures that is a summation of low rank and sparse matrices. This paper introduces a penalized approximation framework to recover such model structures from large covariance matrix estimation. We propose an estimator based on minimizing a non-likelihood loss with separable non-smooth penalty functions. This estimator is shown to recover exactly the rank and sparsity patterns of these two components, and thus partially recovers the model structures. Convergence rates under vario…
Consider the rank-1 spiked model: $\bf{X}=\sqrtνξ\bf{u}+ \bf{Z}$, where $ν$ is the spike intensity, $\bf{u}\in\mathbb{S}^{k-1}$ is an unknown direction and $ξ\sim \mathcal{N}(0,1),\bf{Z}\sim \mathcal{N}(\bf{0},\bf{I})$. Motivated by recent advances in analog-to-digital conversion, we study the problem of recovering $\bf{u}\in \mathbb{S}^{k-1}$ from $n$ i.i.d. modulo-reduced measurements $\bf{Y}=[\bf{X}]\mod Δ$, focusing on the high-dimensional regime ($k\gg 1$). We develop and analyze an algorithm that, for most directions $\bf{u}$ and $ν=\mathrm{poly}(k)$, estimates $\bf{u}$ to high accuracy …
Principal component analysis (PCA) is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix. However, PCA is well known to behave poorly in the ``large $p$, small $n$'' setting, in which the problem dimension $p$ is comparable to or larger than the sample size $n$. This paper studies PCA in this high-dimensional regime, but under the additional assumption that the maximal eigenvector is sparse, say, with at most $k$ nonzero components. We consider a spiked covariance model in which a base matrix is perturbed by adding a $k…
Principal components analysis (PCA) is a widely used dimension reduction technique with an extensive range of applications. In this paper, an online distributed algorithm is proposed for recovering the principal eigenspaces. We further establish its rate of convergence and show how it relates to the number of nodes employed in the distributed computation, the effective rank of the data matrix under consideration, and the gap in the spectrum of the underlying population covariance matrix. The proposed algorithm is illustrated on low-rank approximation and $\boldsymbol{k}$-means clustering tasks…
The present paper concerns large covariance matrix estimation via composite minimization under the assumption of low rank plus sparse structure. In this approach, the low rank plus sparse decomposition of the covariance matrix is recovered by least squares minimization under nuclear norm plus $l_1$ norm penalization. This paper proposes a new estimator of that family based on an additional least-squares re-optimization step aimed at un-shrinking the eigenvalues of the low rank component estimated at the first step. We prove that such un-shrinkage causes the final estimate to approach the targe…
Covariance-Based Structural Equation Modeling in Small-Sample Settings with $p>n$
Graduate School of Science and Technology, University of Tsukuba（筑波大学科学技术研究院）；Institute of Systems and Information Engineering, University of Tsukuba（筑波大学系统与信息工程研究所）；Tsukuba Institute for Advanced Research, University of Tsukuba（筑波大学先进研究所）；Center for Artificial Intelligence Research, University of Tsukuba（筑波大学人工智能研究中心）
The problem of estimating missing fragments of curves from a functional sample has been widely considered in the literature. However, a majority of the reconstruction methods rely on estimating the covariance matrix or the components of its eigendecomposition, a task that may be difficult. In particular, the accuracy of the estimation might be affected by the complexity of the covariance function and the poor availability of complete functional data. We introduce a non-parametric alternative based on a novel concept of depth for partially observed functional data. Our simulations point out tha…
We consider the problem of positive-semidefinite continuation: extending a partially specified covariance kernel from a subdomain $Ω$ of a rectangular domain $I\times I$ to a covariance kernel on the entire domain $I\times I$. For a broad class of domains $Ω$ called \emph{serrated domains}, we are able to present a complete theory. Namely, we demonstrate that a canonical completion always exists and can be explicitly constructed. We characterise all possible completions as suitable perturbations of the canonical completion, and determine necessary and sufficient conditions for a unique complet…
