In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution. Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix. The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in Rp×p; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. In addition, if the random variable has normal distribution, the sample covariance matrix has Wishart distribution and a slightly differently scaled version of it is the maximum likelihood estimate. Cases involving missing data require deeper considerations. Another issue is the robustness to outliers: "Sample covariance matrices are extremely sensitive to outliers".
Statistical analyses of multivariate data often involve exploratory studies of the way in which the variables change in relation to one another and this may be followed up by explicit statistical models involving the covariance matrix of the variables. Thus the estimation of covariance matrices directly from observational data plays two roles:
- to provide initial estimates that can be used to study the inter-relationships;
- to provide sample estimates that can be used for model checking.
Estimates of covariance matrices are required at the initial stages of principal component analysis and factor analysis, and are also involved in versions of regression analysis that treat the dependent variables in a data-set, jointly with the independent variable as the outcome of a random sample.
Read more about Estimation Of Covariance Matrices: Estimation in A General Context, Maximum-likelihood Estimation For The Multivariate Normal Distribution, Shrinkage Estimation
Other articles related to "estimation of covariance matrices, covariance, estimation of, matrices, estimation":
... p is large, the above empirical estimators of covariance and correlation are very unstable ... Moreover, for n < p, the empirical estimate of the covariance matrix becomes singular, i.e ... As an alternative, many methods have been suggested to improve the estimation of the covariance matrix ...
... "Recollections from a 50-year random walk midst matrices, statistics and computing" ... "The infusion of matrices into statistics" ... "An overview of variance component estimation" ...
Famous quotes containing the word estimation:
“No man ever stood lower in my estimation for having a patch in his clothes; yet I am sure that there is greater anxiety, commonly, to have fashionable, or at least clean and unpatched clothes, than to have a sound conscience.”
—Henry David Thoreau (18171862)