اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدEstimation of Large Covariance and Precision Matrices from Temporally Dependent Observations
160
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
We consider the estimation of large covariance and precision matrices from high-dimensional sub-Gaussian or heavier-tailed observations with slowly decaying temporal dependence. The temporal dependence is allowed to be long-range so with longer memory than those considered in the current literature. We show that several commonly used methods for independent observations can be applied to the temporally dependent data. In particular, the rates of convergence are obtained for the generalized thresholding estimation of covariance and correlation matrices, and for the constrained $ell_1$ minimization and the $ell_1$ penalized likelihood estimation of precision matrix. Properties of sparsistency and sign-consistency are also established. A gap-block cross-validation method is proposed for the tuning parameter selection, which performs well in simulations. As a motivating example, we study the brain functional connectivity using resting-state fMRI time series data with long-range temporal dependence.
قيم البحث
اقرأ أيضاً
Last decade witnesses significant methodological and theoretical advances in estimating large precision matrices. In particular, there are scientific applications such as longitudinal data, meteorology and spectroscopy in which the ordering of the va
riables can be interpreted through a bandable structure on the Cholesky factor of the precision matrix. However, the minimax theory has still been largely unknown, as opposed to the well established minimax results over the corresponding bandable covariance matrices. In this paper, we focus on two commonly used types of parameter spaces, and develop the optimal rates of convergence under both the operator norm and the Frobenius norm. A striking phenomenon is found: two types of parameter spaces are fundamentally different under the operator norm but enjoy the same rate optimality under the Frobenius norm, which is in sharp contrast to the equivalence of corresponding two types of bandable covariance matrices under both norms. This fundamental difference is established by carefully constructing the corresponding minimax lower bounds. Two new estimation procedures are developed: for the operator norm, our optimal procedure is based on a novel local cropping estimator targeting on all principle submatrices of the precision matrix while for the Frobenius norm, our optimal procedure relies on a delicate regression-based thresholding rule. Lepskis method is considered to achieve optimal adaptation. We further establish rate optimality in the nonparanormal model. Numerical studies are carried out to confirm our theoretical findings.
Consider the problem of estimating a low-rank matrix when its entries are perturbed by Gaussian noise. If the empirical distribution of the entries of the spikes is known, optimal estimators that exploit this knowledge can substantially outperform si
mple spectral approaches. Recent work characterizes the asymptotic accuracy of Bayes-optimal estimators in the high-dimensional limit. In this paper we present a practical algorithm that can achieve Bayes-optimal accuracy above the spectral threshold. A bold conjecture from statistical physics posits that no polynomial-time algorithm achieves optimal error below the same threshold (unless the best estimator is trivial). Our approach uses Approximate Message Passing (AMP) in conjunction with a spectral initialization. AMP algorithms have proved successful in a variety of statistical estimation tasks, and are amenable to exact asymptotic analysis via state evolution. Unfortunately, state evolution is uninformative when the algorithm is initialized near an unstable fixed point, as often happens in low-rank matrix estimation. We develop a new analysis of AMP that allows for spectral initializations. Our main theorem is general and applies beyond matrix estimation. However, we use it to derive detailed predictions for the problem of estimating a rank-one matrix in noise. Special cases of this problem are closely related---via universality arguments---to the network community detection problem for two asymmetric communities. For general rank-one models, we show that AMP can be used to construct confidence intervals and control false discovery rate. We provide illustrations of the general methodology by considering the cases of sparse low-rank matrices and of block-constant low-rank matrices with symmetric blocks (we refer to the latter as to the `Gaussian Block Model).
This paper focuses on exploring the sparsity of the inverse covariance matrix $bSigma^{-1}$, or the precision matrix. We form blocks of parameters based on each off-diagonal band of the Cholesky factor from its modified Cholesky decomposition, and pe
nalize each block of parameters using the $L_2$-norm instead of individual elements. We develop a one-step estimator, and prove an oracle property which consists of a notion of block sign-consistency and asymptotic normality. In particular, provided the initial estimator of the Cholesky factor is good enough and the true Cholesky has finite number of non-zero off-diagonal bands, oracle property holds for the one-step estimator even if $p_n gg n$, and can even be as large as $log p_n = o(n)$, where the data $y$ has mean zero and tail probability $P(|y_j| > x) leq Kexp(-Cx^d)$, $d > 0$, and $p_n$ is the number of variables. We also prove an operator norm convergence result, showing the cost of dimensionality is just $log p_n$. The advantage of this method over banding by Bickel and Levina (2008) or nested LASSO by Levina emph{et al.} (2007) is that it allows for elimination of weaker signals that precede stronger ones in the Cholesky factor. A method for obtaining an initial estimator for the Cholesky factor is discussed, and a gradient projection algorithm is developed for calculating the one-step estimate. Simulation results are in favor of the newly proposed method and a set of real data is analyzed using the new procedure and the banding method.
In this paper we study covariance estimation with missing data. We consider missing data mechanisms that can be independent of the data, or have a time varying dependency. Additionally, observed variables may have arbitrary (non uniform) and dependen
t observation probabilities. For each mechanism, we construct an unbiased estimator and obtain bounds for the expected value of their estimation error in operator norm. Our bounds are equivalent, up to constant and logarithmic factors, to state of the art bounds for complete and uniform missing observations. Furthermore, for the more general non uniform and dependent cases, the proposed bounds are new or improve upon previous results. Our error estimates depend on quantities we call scaled effective rank, which generalize the effective rank to account for missing observations. All the estimators studied in this work have the same asymptotic convergence rate (up to logarithmic factors).
Let $mathbf{X}_n=(x_{ij})$ be a $k times n$ data matrix with complex-valued, independent and standardized entries satisfying a Lindeberg-type moment condition. We consider simultaneously $R$ sample covariance matrices $mathbf{B}_{nr}=frac1n mathbf{Q}
_r mathbf{X}_n mathbf{X}_n^*mathbf{Q}_r^top,~1le rle R$, where the $mathbf{Q}_{r}$s are nonrandom real matrices with common dimensions $ptimes k~(kgeq p)$. Assuming that both the dimension $p$ and the sample size $n$ grow to infinity, the limiting distributions of the eigenvalues of the matrices ${mathbf{B}_{nr}}$ are identified, and as the main result of the paper, we establish a joint central limit theorem for linear spectral statistics of the $R$ matrices ${mathbf{B}_{nr}}$. Next, this new CLT is applied to the problem of testing a high dimensional white noise in time series modelling. In experiments the derived test has a controlled size and is significantly faster than the classical permutation test, though it does have lower power. This application highlights the necessity of such joint CLT in the presence of several dependent sample covariance matrices. In contrast, all the existing works on CLT for linear spectral statistics of large sample covariance matrices deal with a single sample covariance matrix ($R=1$).
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
