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Graphical Diagnostics for Modeling Unstructured Covariance Matrices
Author(s) -
Pourahmadi Mohsen
Publication year - 2002
Publication title -
international statistical review
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.051
H-Index - 54
eISSN - 1751-5823
pISSN - 0306-7734
DOI - 10.1111/j.1751-5823.2002.tb00177.x
Subject(s) - cholesky decomposition , covariance , covariance matrix , mathematics , estimation of covariance matrices , statistics , matrix decomposition , covariance function , matrix (chemical analysis) , computer science , eigenvalues and eigenvectors , physics , materials science , quantum mechanics , composite material
Summary Prudent statistical analysis of correlated data requires accounting for the correlation among the measurements. Specifying a form for the covariance matrix of the data could reduce the high number of parameters of the covariance and increase efficiency of the inferences about the regression parameters. Motivated by the success of ordinary, partial and inverse correlograms in identifying parsimonious models for stationary time series, we introduce generalizations of these plots for nonstationary data. Their roles in detecting heterogeneity and correlation of the data and identifying parsimonious models for the covariance matrix are illuminated using a longitudinal dataset. Decomposition of a covariance matrix into “variance” and “dependence” components provides the necessary ingredients for the proposed graphs. This amounts to replacing a 3‐D correlation plot by a pair of 2‐D plots, providing complementary information about dependence and heterogeneity. Models identified and fitted using the variance‐correlation decomposition of a covariance matrix are not guaranteed to be positive definite, but those using the modified Cholesky decomposition are. Limitations of our graphical diagnostics for general multivariate data where the measurements are not (time‐) ordered are discussed.