Table 5 - uploaded by Ashwini Maurya
Content may be subject to copyright.
Source publication
We develop a method for estimating well-conditioned and sparse covariance and inverse co-variance matrices from a sample of vectors drawn from a sub-gaussian distribution in high dimensional setting. The proposed estimators are obtained by minimizing the quadratic loss function and joint penalty of 1 norm and variance of its eigenvalues. In contras...
Similar publications
The complementary exponential-geometric (CEG) distribution is a useful model for analyzing lifetime data. For this distribution, some recurrence relations satisfied by marginal and joint moment generating functions of k-th lower record values were established. They enable the computation of the means, variances, and covariances of k-th lower record...
The complementary exponential-geometric distribution is a useful model for analyzing lifetime data. For this distribution, we established some recurrence relations satisfied by marginal and joint moment generating functions of kth lower record values. These recurrence relations enable the computation of the means, variances and covariances of kth l...
In this paper, we establish some new recurrence relations for the single and product
moments of progressively Type-II right censored order statistics from the Erlangtruncated
exponential distribution. These relations generalize those established by
Aggarwala and Balakrishnan (1996) for standard exponential distribution. These
recurrence relations e...
In this paper, we consider the estimation problem for the semiparametric regression model with censored data in which the number of explanatory variables p in the linear part is much larger than sample size n, often denoted as p ≫ n. The purpose of this paper is to study the effects of covariates on a response variable censored on the right by a ra...
This article presents SPSTS, an automated sequential procedure for computing point and Confidence-Interval (CI) estimators for the steady-state mean of a simulation-generated process subject to user-specified requirements for the CI coverage probability and relative half-length. SPSTS is the first sequential method based on Standardized Time Series...
Citations
... Numerous empirical studies, including those [10], [11], [12], [13], [14], [15], and [16], have highlighted the inadequacy of the sample covariance matrix S S S as an estimator of the true population covariance when the dataset's dimensions exceed the sample's observations. Furthermore, these studies emphasized that the eigenvalues of the sample covariance matrix S S S exhibit over-dispersion, particularly due to (p-n+1) eigenvalues being exactly equal to zero, as depicted in Fig. 5. Thus, it is crucial to consider alternative methods for covariance estimation in high-dimensional datasets where the number of dimensions surpasses the number of observations to ensure accurate and reliable results in statistical analyses. ...
... The corresponding JPEN covariance matrix estimator [16] is; ...
Principal Component Analysis (PCA) aims to reduce the dimensions of datasets by transforming them into uncorrelated Principal Components (PCs), retaining most of the data’s variation with fewer components. However, standard PCA struggles in high-dimensional settings where there are more variables than observations due to limitations in covariance estimation. PCA relies on covariance matrices to measure variable relationships, using eigenvectors to determine data distribution directions and assessing eigenvalues’ significance. This article examines the pros and cons of estimating high-dimensional covariance matrices and emphasizes the importance of well-conditioned covariance estimation for accurate finite sample PCA. Various methods are available for estimating population covariance, among which Ledoit-Wolf estimation is deemed optimal in scenarios where the number of observations is smaller than the number of variables. However, it tends to excessively shrink the sample covariates matrix, resulting in an underestimation of the true eigen spectrum and a dearth of sparsity. Therefore, there’s a need for sparse and well-conditioned covariance matrix estimation to enhance PC estimation accuracy.
... In estimating covariance matrix, often quadratic loss instead of the normal likelihood is used together with the l 1 -penalty (Rothman, 2012;Xue et al., 2012;Maurya, 2016). Lam and Fan (2009) derived the convergence rates under the Frobenius norm and sparsistency of the covariance and precision matrix estimators when the l 1 -penalty is applied to the Cholesky factor, the precision matrix, or the covariance matrix. ...
In this paper, we propose a scalable Bayesian method for sparse covariance matrix estimation by incorporating a continuous shrinkage prior with a screening procedure. In the first step of the procedure, the off-diagonal elements with small correlations are screened based on their sample correlations. In the second step, the posterior of the covariance with the screened elements fixed at $0$ is computed with the beta-mixture prior. The screened elements of the covariance significantly increase the efficiency of the posterior computation. The simulation studies and real data applications show that the proposed method can be used for the high-dimensional problem with the `large p, small n'. In some examples in this paper, the proposed method can be computed in a reasonable amount of time, while no other existing Bayesian methods work for the same problems. The proposed method has also sound theoretical properties. The screening procedure has the sure screening property and the selection consistency, and the posterior has the optimal minimax or nearly minimax convergence rate under the Frobeninus norm.
... par Partial Least Squares dans (Bouhlel et al. 2016) ou maximum de vraisemblance dans (Tripathy, Bilionis, and Gonzalez 2016). Le grand nombre de points de données N est typiquement abordé par le remplacement des données par des points résumés (inducing points) (Hensman, Fusi, and Lawrence 2013), des fonctions de covariance à support restreints (Maurya 2016) ou des combinaisons linéaires de processus gaussiens bâtis sur une partie des données (Rullière et al. 2018). La plupart de ces avancées sont compatibles entre elles. ...
L'objectif de la thèse est d’améliorer des algorithmes actuellement utilisés en assistance aux expertises de sûreté pour analyser la réponse d’un code de calcul simulant un phénomène physique. Les simulations étant coûteuses, les algorithmes utilisés construisent des plans d’expériences de manière itérative, grâce à des métamodèles capables de prédire les résultats des simulations. On s’intéresse en particulier aux simulations produites par des chaînes de calcul (enchaînement séquentiel de plusieurs codes de calculs distincts). Les informations produites par les premiers codes permettent une analyse dite « introspective » de la grandeur finale.
Dans un premier temps, nous travaillons sur différents métamodèles introspectifs admissibles. Nous étudions ainsi les différentes formes de cokrigeage et proposons une amélioration au problème de l’optimisation de ses paramètres. Nous développons également un autre métamodèle introspectif, l’hyperkrigeage. Nous faisons ensuite une comparaison des performances prédictives de ces métamodèles. Dans un second temps, nous travaillons sur les algorithmes. Un premier algorithme d’optimisation, appelé « Step or Stop Optimization », prenant en compte les spécificités du cas introspectif a été développé. Il permet de ne pas poursuivre des calculs si les premières étapes n’y encouragent pas. Des comparaisons avec l’algorithme actuel tendent à confirmer une économie importante des ressources de calcul grâce à la meilleure prise en compte de la physique intermédiaire de la simulation. La stratégie utilisée par cet algorithme est généralisée, et étendue à des problématiques autres que l’optimisation.
... par Partial Least Squares dans (Bouhlel et al. 2016) ou maximum de vraisemblance dans (Tripathy, Bilionis, and Gonzalez 2016). Le grand nombre de points de données N est typiquement abordé par le remplacement des données par des points résumés (inducing points) (Hensman, Fusi, and Lawrence 2013), des fonctions de covariance à support restreints (Maurya 2016) ou des combinaisons linéaires de processus gaussiens bâtis sur une partie des données (Rullière et al. 2018). La plupart de ces avancées sont compatibles entre elles. ...
[Thesis in French]
In this thesis, we aim to improve the algorithms used by safety experts to analyze the outputs of their computer simulations of physical phenomena. Since simulations are expensive, the related algorithms build the design of experiments sequentially, using metamodels to predict the outcome of the simulations. We pay special attention to chain-produced simulations (where outputs of one code are used as inputs in other numerical simulators). The information given by the first codes allows a so-called "introspective" analysis of the final quantity.
Firstly, we review different possible introspective metamodels, which treat the results as a multi-outputs function. We thus study the different forms of co-kriging and propose an improvement for the optimization of its parameters. We also develop another introspective metamodel, the hyper-kriging. We then make a comparison between the predictive performance of these metamodels. Secondly, we work on the algorithms. An optimization algorithm, called "Step or Stop Optimization", is developed, taking into account the specificities of the introspective case. It allows to stop the computation if the first steps are disengaging. Comparisons with currently used algorithm tend to confirm a significant saving in computing resources thanks to a better consideration of the intermediate physics of the simulation. The strategy used in this algorithm can be generalized, and extended to be used beyond the context of optimization problems.
... Only in scenario (2i) under the Frobenius loss is the null target preferred over the diagonal targets. Perhaps this is not surprising: For the Frobenius norm the slowest rate of convergence of the estimator comes from the diagonal entries (Rothman, 2012;Maurya, 2016). From the losses as defined above we get that, in a sense, the Frobenius norm emphasizes proportionality, while the quadratic norm emphasizes the diagonal. ...
We consider the problem of jointly estimating multiple inverse covariance matrices from high-dimensional data consisting of distinct classes. An L2-penalized maximum likelihood approach is employed. The suggested approach is flexible and generic, incorporating several other L2-penalized estimators as special cases. In addition, the approach allows specification of target matrices through which prior knowledge may be incorporated and which can stabilize the estimation procedure in high-dimensional settings. The result is a targeted fused ridge estimator that is of use when the precision matrices of the constituent classes are believed to chiefly share the same structure while potentially differing in a number of locations of interest. It has many applications in (multi)factorial study designs. We focus on the graphical interpretation of precision matrices with the proposed estimator then serving as a basis for integrative or meta-analytic Gaussian graphical modeling. Situations are considered in which the classes are defined by data sets and subtypes of diseases. The performance of the proposed estimator in the graphical modeling setting is assessed through extensive simulation experiments. Its practical usability is illustrated by the differential network modeling of 12 large-scale gene expression data sets of diffuse large B-cell lymphoma subtypes. The estimator and its related procedures are incorporated into the R-package rags2ridges.
... Both methods show limitations when the scale dependence is respectively short and large. For more details and references, see Stein (2014), Maurya (2016) and Bachoc et al. (2017). Another drawback-which to the best of our knowledge is little discussed in the literature-is the difficulty to use these methods when the dimension of the input space is large (say larger than 10, which is frequent in computer experiments or machine learning). ...
This work falls within the context of predicting the value of a real function f at some input locations given a limited number of observations of this function. Kriging interpolation technique (or Gaussian process regression) is often considered to tackle such problem but the method suffers from its computational burden when the number of observation points n is large. We introduce in this article nested Kriging estimators which are constructed by aggregating sub-models based on subsets of observation points. This approach is proven to have better theoretical properties than other aggregation methods that can be found in the literature. In particular, contrary to some other methods which are shown inconsistent, we prove the consistency of our proposed aggregation method. Finally, the practical interest of the proposed method is illustrated on simulated datasets and on an industrial test case with 10 4 observations in a 6-dimensional space.
... with s(a jl , κ) = sign(a jl ) max(|a jl | − κ, 0) 1 {j =l} , see e.g. [6,11,12,17]. Then the Γ-Step has a closed form given by ...
The precision matrix is the inverse of the covariance matrix. Estimating large sparse precision matrices is an interesting and a challenging problem in many fields of sciences, engineering, humanities and machine learning problems in general. Recent applications often encounter high dimensionality with a limited number of data points leading to a number of covariance parameters that greatly exceeds the number of observations, and hence the singularity of the covariance matrix. Several methods have been proposed to deal with this challenging problem, but there is no guarantee that the obtained estimator is positive definite. Furthermore, in many cases, one needs to capture some additional information on the setting of the problem. In this paper, we introduce a criterion that ensures the positive definiteness of the precision matrix and we propose the inner-outer alternating direction method of multipliers as an efficient method for estimating it. We show that the convergence of the algorithm is ensured with a sufficiently relaxed stopping criterion in the inner iteration. We also show that the proposed method converges, is robust, accurate and scalable as it lends itself to an efficient implementation on parallel computers.
Although Phase I analysis of multivariate processes has been extensively discussed, the discussion on techniques for Phase I monitoring of high‐dimensional processes is still limited. In high‐dimensional applications, it is common to observe that a large number of components but only a limited number of them change at the same time. The shifted components are often sparse and unknown a priori in practice. Motivated by this, this article studies Phase I monitoring of high‐dimensional process mean vectors under an unknown sparsity level of shifts. The basic idea of the proposed monitoring scheme is to first employ the false discovery rate procedure to estimate the sparsity level of mean shifts, and then to monitor the mean changes based on the maximum of the directional likelihood ratio statistics over all the possible shift directions. The comparison results based on extensive simulations favor the proposed monitoring scheme. A real example is presented to illustrate the implementation of the new monitoring scheme.
