By Mohsen Pourahmadi

Methods for estimating sparse and massive covariance matrices

Covariance and correlation matrices play primary roles in each element of the research of multivariate facts gathered from numerous fields together with enterprise and economics, well-being care, engineering, and environmental and actual sciences. High-Dimensional Covariance Estimation provides available and entire assurance of the classical and sleek methods for estimating covariance matrices in addition to their purposes to the speedily constructing parts mendacity on the intersection of information and computer learning.

Recently, the classical pattern covariance methodologies were converted and superior upon to satisfy the desires of statisticians and researchers facing huge correlated datasets. High-Dimensional Covariance Estimation specializes in the methodologies in keeping with shrinkage, thresholding, and penalized probability with functions to Gaussian graphical versions, prediction, and mean-variance portfolio administration. The booklet is predicated seriously on regression-based rules and interpretations to attach and unify many current equipment and algorithms for the task.

High-Dimensional Covariance Estimation gains chapters on:

  • Data, Sparsity, and Regularization
  • Regularizing the Eigenstructure
  • Banding, Tapering, and Thresholding
  • Covariance Matrices
  • Sparse Gaussian Graphical Models
  • Multivariate Regression

The ebook is a perfect source for researchers in statistics, arithmetic, company and economics, desktop sciences, and engineering, in addition to an invaluable textual content or complement for graduate-level classes in multivariate research, covariance estimation, statistical studying, and high-dimensional information analysis.

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Extra resources for High-Dimensional Covariance Estimation: With High-Dimensional Data

Example text

Incoherence of the sensing matrix X which reflects the requirements that any k subset of its p columns are as orthogonal (uncorrelated) as they can be. The main goal of compressed sensing is to construct measurement matrices X, with the number of measurements n as small as possible relative to p, such that any k-sparse signal β ∈ R p , can be recovered exactly from the linear measurements Y = Xβ using a computationally efficient recovery algorithm. In compressed sensing, it is typical that p n.

P; t = 1, . . , T } is a multivariate or p-dimensional time series. When viewed as a data matrix its rows and columns are correlated. 1 displays the time series plots of the weekly log returns for three stocks from 2004 to 2010 with T = 365 observations. Even though the stocks are from three different industry groups, there are considerable similarities in their behavior over time. The temporal dependence is usually removed by fitting simple time series models to each row. The returns of an asset and the whole portfolio invariably depend on several economic and financial variables at the macroeconomics and company levels such as the growth rate of GDP, inflation rate, the industry type, company size, and the market value.

The key input to inferring such a network is either the p × p sample covariance matrix or a sparse precision matrix (see Chapter 5). In summary, there are numerous bioinformatics tasks that rely on the pairwise sample covariances as the basic ingredients in the analysis. A key question is how to obtain an accurate estimate of the population covariance (precision) matrix based on a dataset with p large and n small? In the next three examples both the rows and columns are dependent; such data matrices are called transposable matrices (Allen and Tibshirani, 2010).

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