ترغب بنشر مسار تعليمي؟ اضغط هنا

A unified framework for correlation mining in ultra-high dimension

114   0   0.0 ( 0 )
 نشر من قبل Yun Wei
 تاريخ النشر 2021
  مجال البحث الاحصاء الرياضي
والبحث باللغة English




اسأل ChatGPT حول البحث

An important problem in large scale inference is the identification of variables that have large correlations or partial correlations. Recent work has yielded breakthroughs in the ultra-high dimensional setting when the sample size $n$ is fixed and the dimension $p rightarrow infty$ ([Hero, Rajaratnam 2011, 2012]). Despite these advances, the correlation screening framework suffers from some serious practical, methodological and theoretical deficiencies. For instance, theoretical safeguards for partial correlation screening requires that the population covariance matrix be block diagonal. This block sparsity assumption is however highly restrictive in numerous practical applications. As a second example, results for correlation and partial correlation screening framework requires the estimation of dependence measures or functionals, which can be highly prohibitive computationally. In this paper, we propose a unifying approach to correlation and partial correlation mining which specifically goes beyond the block diagonal correlation structure, thus yielding a methodology that is suitable for modern applications. By making connections to random geometric graphs, the number of highly correlated or partial correlated variables are shown to have novel compound Poisson finite-sample characterizations, which hold for both the finite $p$ case and when $p rightarrow infty$. The unifying framework also demonstrates an important duality between correlation and partial correlation screening with important theoretical and practical consequences.



قيم البحث

اقرأ أيضاً

205 - Hidehiko Kamiya 2014
Two Bayesian models with different sampling densities are said to be marginally equivalent if the joint distribution of observables and the parameter of interest is the same for both models. We discuss marginal equivalence in the general framework of group invariance. We introduce a class of sampling models and establish marginal equivalence when the prior for the nuisance parameter is relatively invariant. We also obtain some robustness properties of invariant statistics under our sampling models. Besides the prototypical example of $v$-spherical distributions, we apply our general results to two examples---analysis of affine shapes and principal component analysis.
189 - Yehua Li , Tailen Hsing 2010
In this paper, we consider regression models with a Hilbert-space-valued predictor and a scalar response, where the response depends on the predictor only through a finite number of projections. The linear subspace spanned by these projections is cal led the effective dimension reduction (EDR) space. To determine the dimensionality of the EDR space, we focus on the leading principal component scores of the predictor, and propose two sequential $chi^2$ testing procedures under the assumption that the predictor has an elliptically contoured distribution. We further extend these procedures and introduce a test that simultaneously takes into account a large number of principal component scores. The proposed procedures are supported by theory, validated by simulation studies, and illustrated by a real-data example. Our methods and theory are applicable to functional data and high-dimensional multivariate data.
85 - Tiefeng Jiang , Ping Li 2021
Hotellings T-squared test is a classical tool to test if the normal mean of a multivariate normal distribution is a specified one or the means of two multivariate normal means are equal. When the population dimension is higher than the sample size, t he test is no longer applicable. Under this situation, in this paper we revisit the tests proposed by Srivastava and Du (2008), who revise the Hotellings statistics by replacing Wishart matrices with their diagonal matrices. They show the revised statistics are asymptotically normal. We use the random matrix theory to examine their statistics again and find that their discovery is just part of the big picture. In fact, we prove that their statistics, decided by the Euclidean norm of the population correlation matrix, can go to normal, mixing chi-squared distributions and a convolution of both. Examples are provided to show the phase transition phenomenon between the normal and mixing chi-squared distributions. The second contribution of ours is a rigorous derivation of an asymptotic ratio-unbiased-estimator of the squared Euclidean norm of the correlation matrix.
254 - Jeremie Kellner 2014
A new goodness-of-fit test for normality in high-dimension (and Reproducing Kernel Hilbert Space) is proposed. It shares common ideas with the Maximum Mean Discrepancy (MMD) it outperforms both in terms of computation time and applicability to a wide r range of data. Theoretical results are derived for the Type-I and Type-II errors. They guarantee the control of Type-I error at prescribed level and an exponentially fast decrease of the Type-II error. Synthetic and real data also illustrate the practical improvement allowed by our test compared with other leading approaches in high-dimensional settings.
We consider a $l_1$-penalization procedure in the non-parametric Gaussian regression model. In many concrete examples, the dimension $d$ of the input variable $X$ is very large (sometimes depending on the number of observations). Estimation of a $bet a$-regular regression function $f$ cannot be faster than the slow rate $n^{-2beta/(2beta+d)}$. Hopefully, in some situations, $f$ depends only on a few numbers of the coordinates of $X$. In this paper, we construct two procedures. The first one selects, with high probability, these coordinates. Then, using this subset selection method, we run a local polynomial estimator (on the set of interesting coordinates) to estimate the regression function at the rate $n^{-2beta/(2beta+d^*)}$, where $d^*$, the real dimension of the problem (exact number of variables whom $f$ depends on), has replaced the dimension $d$ of the design. To achieve this result, we used a $l_1$ penalization method in this non-parametric setup.
التعليقات
جاري جلب التعليقات جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
mircosoft-partner

هل ترغب بارسال اشعارات عن اخر التحديثات في شمرا-اكاديميا