By Mohsen Pourahmadi

Methods for estimating sparse and massive covariance matrices

Covariance and correlation matrices play primary roles in each point of the research of multivariate facts gathered from quite a few fields together with enterprise and economics, wellbeing and fitness care, engineering, and environmental and actual sciences. High-Dimensional Covariance Estimation provides obtainable and complete assurance of the classical and glossy methods for estimating covariance matrices in addition to their purposes to the swiftly constructing components mendacity on the intersection of facts and computing device learning.

Recently, the classical pattern covariance methodologies were converted and greater upon to fulfill the desires of statisticians and researchers facing huge correlated datasets. High-Dimensional Covariance Estimation specializes in the methodologies in line 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 latest tools 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 booklet is a perfect source for researchers in facts, 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 info for High-Dimensional Covariance Estimation: With High-Dimensional Data

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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. Techniques from time series and multivariate analysis are central to the study of multiple assets and portfolio returns, and often they involve high-dimensional statistical models with challenging computational issues.

The magical interaction between high-dimensionality and sparsity is explained next using Stein’s shrinkage idea when estimating the mean of a homoscedastic normal random vector. Example 7 (Stein’s Phenomenon, Sparsity, and High-Dimensionality) Consider a p-dimensional random vector Y ∼ N p (μ, I ) where μ = 0 and the goal is to estimate ˜ the mean vector using the single (n = 1) data vector Y . The risk of any estimator μ is measured using the squared error loss ˜ = E||μ ˜ − μ||2 . R(μ) Note that the risk of the maximum likelihood estimator (MLE) μ = Y , computed using the fact that var(Yi ) = 1 is p R(μ) = E(Yi − μi )2 = pvar(Y1 ) = p.

The two common loss functions with the corresponding risk functions for an arbitrary estimator = (S) are L 1 ( , ) = tr L 2 ( , ) = tr −1 −1 − log | 2 −I , −1 | − p, and Ri ( , ) = E L i ( , ), i = 1, 2. An estimator is considered better than the sample covariance matrix S, if its risk function is smaller than that of S. The loss function L 1 was advocated by Stein (1956) and is usually called the entropy loss or the Kullback–Liebler divergence of two multivariate normal densities corresponding to the two covariance matrices.

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