Covariance Selection Task
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A Covariance Selection Task is an estimation task of a concentration matrix.
References
2013
- (Di Stefano, 2013) ⇒ Di Stefano, L. (2013). The Geometry of Covariance Selection.
- The particular subject of this project is Gaussian graphical models, i.e. undirected graphical models whose variables are jointly normally distributed. These were introduced by Dempster [1972] as covariance selection models, without the associated graphical representation. Gaussian graphical models possess an elegant and appealing theory, particularly with respect to maximum likelihood estimation. Edges in the graph correspond to elements of the inverse covariance matrix (also called the concentration or precision matrix), and represent conditional dependencies among the component random variables, which are represented by vertices.
Covariance selection models must be distinguished from what are sometimes called covariance graph models, in which edges correspond to entries of the covariance matrix and represent marginal, rather than conditional, dependencies (i.e. overall correlation rather than partial correlation).
- The particular subject of this project is Gaussian graphical models, i.e. undirected graphical models whose variables are jointly normally distributed. These were introduced by Dempster [1972] as covariance selection models, without the associated graphical representation. Gaussian graphical models possess an elegant and appealing theory, particularly with respect to maximum likelihood estimation. Edges in the graph correspond to elements of the inverse covariance matrix (also called the concentration or precision matrix), and represent conditional dependencies among the component random variables, which are represented by vertices.
2007
- (Yuan & Lin, 2007) ⇒ Ming Yuan, and Yi Lin. (2007). “Model Election and Estimation in the Gaussian Graphical Model.” In: Biometrica, 90. doi:10.1093/biomet/asm018.
- Abstract - We propose penalized likelihood methods for estimating the concentration matrix in the Gaussian graphical model. The methods lead to a sparse and shrinkage estimator of the concentration matrix that is positive definite, and thus conduct model selection and estimation simultaneously. The implementation of the methods is nontrivial because of the positive definite constraint on the concentration matrix, but we show that the computation can be done effectively by taking advantage of the efficient maxdet algorithm developed in convex optimization. We propose a BIC-type criterion for the selection of the tuning parameter in the penalized likelihood methods. The connection between our methods and existing methods is illustrated. Simulations and real examples demonstrate the competitive performance of the new methods.
1972
- (Dempster, 1972) ⇒ Arthur P. Dempster. (1972). “Covariance Selection.” In: Biometrics, 28.
- Abstract - The covariance structure of a multivariate normal population can be simplified by setting elements of the inverse of the covariance matrix to zero. Reasons for adopting such a model and a rule for estimating its parameters are given in section 2. It is also proposed to select the zeros in the inverse from sample data. A numerical illustration of the proposed technique is given in section 3. Appendix A sketches the general theory of exponential families which underlies the special results of section 2, and Appendix B describes two approaches to computation of the proposed estimator.